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A Two-network thermomechanical model of a shape memory polymer
International Journal of Engineering Science 49 (2011) 823–838
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A Two-network thermomechanical model of a shape memory polymer Pritha Ghosh, A.R. Srinivasa ⇑ Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, United States
a r t i c l e
i n f o
Article history: Received 25 January 2010 Received in revised form 7 February 2011 Accepted 19 April 2011 Available online 12 May 2011 Keywords: Shape memory polymers Yield stress Hysteresis Thermal expansion Multinetwork model Helmhotz potential Shape fixity Shape recovery
a b s t r a c t The aim of this work is to demonstrate a Helmholtz potential based approach for the development of the constitutive equations for a shape memory polymer undergoing a thermomechanical cycle. The model is able to simulate the response of the material during heating and cooling cycles and the sensitive dependence of the response on thermal expansion. We notice that the yield-stress of the material controls the gross features of the response of the model, and suggests that the material yields differently depending on not just the current value of the temperature but also on whether the temperature of the material dropped or increased from the previous time-step somewhat similar to the Bauschinger effect in plasticity, except that here the controlling parameter is the rate of temperature change rather than rate of plastic strain. The results of the simulation are in qualitative and quantitative agreement with experiments performed on two different shape memory polymer samples: polyurethane and epoxy resin. We find that modeling the hysteresis of the yield stress of the material during temperature changes is the key to the results. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Shape memory polymers (SMPs) belong to a class of smart materials, that are capable of holding a temporary shape that was caused by the application of forces, and can subsequently regain or ‘‘remember’’ their original shape ‘‘on demand’’ by the application of non-mechanical stimuli. Temperature-responsive shape memory polymers use changes in temperature as this stimulus, and undergo structural changes at a certain temperature called the transition temperature. This transition temperature could correspond to glass-transition (for amorphous polymers) or the melting temperature (for polymers with crystalline phases), which gives rise to different kinds of shape-memory behavior (Kelch & Lendlein, 2002). They can be easily deformed to give a temporary shape at low-working temperatures and upon heating above their transformation temperature, return to their original structure. Examples of shape memory polymers are polyurethanes, polyurethanes with ionic or mesogenic components, block copolymers consisting of polyethyleneterephthalate and polyethyleneoxide, block copolymers containing polystyrene and polybutadiene, polyesterurethanes with methylenebis and butanediol, filler modified epoxy, thermoset epoxy resins, epoxy composites (Kelch & Lendlein, 2002). This feature which may be termed ‘‘shape retention and recovery on demand’’ can be exploited to design components that are used to enhance product performance and quality under changeable environments. Shape memory polymers have found numerous applications on account of their unique thermomechanical properties, and are currently being investigated for a wide range of applications where their shape fixity and shape recovery behavior are useful (Hu, 2007, 2008; Kelch & Lendlein, 2002; Lendlein & Langer, 2002; Liu, Qin, & Mather, 2007; Ni, Zhang, Fu, Dai, & Kimura, 2007). The key aspect to the use of shape memory polymers is the large recoverable change in properties across a narrow temperature range, which can be used to monitor and actuate various mechanisms. ⇑ Corresponding author. E-mail address: [email protected] (A.R. Srinivasa). 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.04.003
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Glassy shape memory polymers allow large reversible change in properties across the glass transition temperature (hg). Specifically, the material can change from a glassy state to a rubbery state across hg, accompanied by a change in its modulus. An increase in temperature allows the material to become more flexible, therefore easily deformable, and a decrease in temperature hardens the material into a glassy polymer, sustaining the new shape. It is important to note that the glass transition temperature is a kinetic parameter, and thus depends on the cooling rate. Thus the slower the cooling rate, the lower hg. Establishing appropriate models to simulate the shape memory processes and therefore to predict the shape memory properties should be helpful for the development and application of SMPs. In a recent article, Liu, Gall, Dunn, Greenberg, and Diani (2006) have observed that relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. They emphasize that constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. The shape memory behavior is dependent on both the structure and the thermomechanical conditions. Theories of the shape fixity and recovery behavior have been developed by Tobushi, Hara, Yamada, and Hayashi (1996), Tobushi, Hashimoto, Hayashi, and Yamada (1997), Tobushi, Hashimoto, Ito, Hayashi, and Yamada (1998), Tobushi, Hayashi, Hoshio, and Ejiri (2008), Abrahamson, Lake, Munshi, and Gall (2003), Takahashi, Hayashi, and Hayashi (1996), Lee, Kim, and Kim (2004), Lin and Chen (1999), Bhattacharyya and Tobushi (2000), Hong, Yu, and Youk (2007), Liu et al. (2006) and Diani, Liu, and Gall (2006). The literature on the shape-memory effect in polymers mostly deal with the development of theoretical concepts of the mechanisms of thermally stimulated strain recovery (Tobushi et al., 2008); new approaches to the description of the deformation and relaxation behavior of polymers (Takahashi et al., 1996); new information on the shape recovery features of deformed samples; and results of investigations of synthetic polymeric materials possessing unusual properties (Beloshenko, Varyukhin, & Voznyak, 2005). For example, the model proposed by Lin and Chen (1999) employs two Maxwell elements connected in parallel to describe the shape memory properties of shape memory polyurethane. Their modeling results and experimental data show some deviation, which the authors ascribe to the polydispersed glass transition temperature of the studied samples. The model can qualitatively explain the occurrence of shape memory behaviors. However, since the dampers in the model are both viscous, there is no truly irrecoverable strain at the end of the process. Furthermore, the model was not developed using thermodynamical principles. Abrahamson et al. (2003) utilize a friction element in their model to account for the irrecoverable strain at the end of the cycle, that progresses from fully stuck to fully free over a finite range of strain. It was found that the stress–strain curve predicted by the model and that obtained by experiment agreed well. It will become apparent that the material response is sensitive to the thermal expansion of the material. However, Abrahamson’s model does not take the thermal expansion of the material into consideration during the change of temperature. The model developed by Tobushi et al. (1997) takes irreversible deformation and thermal strains of shape memory polyurethanes into account. Tobushi and coworkers have added a friction element into the standard linear viscoelastic model to simulate the behaviors of shape memory polyurethanes. They have carried out a series of creep tests of shape memory polyurethanes at different temperatures. They propose that if the strain exceeds particular threshold strain, irreversible deformation occurs. Also, the temperature change that would cause thermal expansion is accounted for by simply adding a coefficient of thermal expansion to terms involving the rate of strain, without any thermodynamic considerations as seen in Eq. (4) in Tobushi et al. (1997). As a result, their model can be shown to be theormodynamically inconsistent even for pure thermoelasticity. The current work deals with the development of a two-network thermodynamically consistent model in a state-space form and shows that most of the gross features of a SMP depend on the yield-stress of the material. We demonstrate that a systematic application of fundamental thermodynamic principles coupled with a suitable choice for the Helmholtz potential and suitable kinetic equations for the inelastic variables will allow the development of a complete simulation of the behavior of a class of shape memory polymeric materials. Although the model can be generalized to 3-dimensions, there is a paucity of experimental data. For this reason, we will restrict ourselves to a one dimensional model. A number of significant features of the response can be deduced by a careful observation of the available experimental data. (1) There is a hysteresis in the yield-stress function of the material during heating and cooling. (2) The yield stress of the material evolves differently depending on the extent to which the material has been strained. (3) The thermal expansion behavior of the material significantly enhances the stress rise in the material during cooling due to contraction, but opposes the strain recovery by heating due to expansion. The contrasting behavior of the thermal expansion during cooling and heating makes the material response very sensitive to the coefficient of thermal expansion of the material. The model that we propose takes into account these three phenomena. Given the lack of experimental data on the actual polymer temperature versus that of the ambient atmosphere, and the extremely slow deformation rates, we will assume that the temperature of the polymer is the same as that of the surrounding atmosphere. Thus, we shall not be dealing with the heat equation in this work, rather we shall take temperature changes to the material as a defined input. The solution of the system equations for a simple thermomechanical cycle shall be simulated in MATLAB and its results shall be shown to be in qualitative and quantitative agreement with experiments performed on polyurethane and epoxy resin.
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Barot and Rao (2006) and later Barot, Rao, and Rajagopal (2008) have developed models for SMPs undergoing shape setting through partial crystallization using a thermodynamical approach (developed by Rajagopal & Srinivasa (2000)) based on maximum rate of dissipation. Our approach, while similar in the use of thermodynamical ideas, differs from these models in the following ways: (1) We are focussed on shape setting through glass transition and not through a real phase change. This makes our modeling task much simpler. (2) Their approach is a fully three dimensional history dependent approach as seen from Eq. (34) in Barot et al. (2008). This leads to considerable complications in solving boundary value problems. Our approach is based on a much simpler rate type constitutive equations using a plasticity like approach, leading to simple ODEs for homogenous deformations and coupled PDEs and ODEs for general boundary value problems. A few comments are in order with regard to our focus on a one dimensional model as opposed to full three dimensional constitutive theory. Although there are many applications of SMP that require full 3D models, there are several applications in which a 1D model will be more than sufficient, such as widely used SMP sutures with complex loading, SMP orthodontic wires for dental realignments, SMP thrombectomy devices, etc. Also there is a paucity in experimental data in the 3D response of these materials, as we have mentioned above, especially in view of the fact that there are indications of considerable anisoptropy in the response. Also, in view of the uncertainties associated in 3D models, and to make sure the paper is accessible to a wider audience, especially with regard to our central point, i.e. the temperature hysteresis of the yielding phenomena, we have chosen to focus on an entirely 1D model. 2. Behavior of shape memory polymers The experiment that we shall be considering in this work is a thermomechanical cycle on a shape memory polymer from the work carried out on a polyurethane sample by Tobushi et al. (1997) and on an epoxy resin sample by Liu et al. (2006). The process involved here are (refer to Fig. 1): 1. Initial conditions: at (a): The material is considered at a stress free state at a temperature above the glass transition temperature hg = hg + 20°C. All strains are to be measured from this state. This current shape is thus recognized as the ‘‘permanent shape’’ of the material. 2. Process A: from (a) to (b): High temperature stretching: The temperature is held fixed at hmax and the strain is increased steadily at a constant prescribed rate g1 to give the temporary shape to the material, and then the strain is held constant for a time 0 < t < t1. 3. Process B: from (b) to (c): Cooling and fixing the temporary shape: The strain is fixed at g1 and the temperature is lowered to hmin = hg 20°C at a predetermined rate ‘f1’. 4. Process C: from (c) to (d): Relaxing the stress: Now the temperature is fixed at hmin and the stress is gradually relaxed to zero at a predetermined rate ‘ g2’. During this process, material is observed to still be in its temporary shape. 5. Process D: from (d) to (a): Recovering the original shape: Now the body is heated at rate ‘f1’ in a stress-free state back to the original temperature to recover the original shape. The strain slowly relaxes and the material is back to its original shape. The typical material response for maximum strain of 10% for the thermomechanical tests carried by Tobushi et al. (1997) is shown in Fig. 1-right. The thermomechanical cycle also involves strain and stress-relaxing rate, and cooling and heating rates, which is not evident in these graphs. In the following work, we have extracted information from the above experimental data and plotted them against time.
Fig. 1. (a) The processes taking place in a thermomechanical cycle. (b) Typical (non-dimensionalized) experimental data for a polyurethane sample. The curve a-b indicates the high temperature deformations. Curve b–c is the shape fixing process at constant strain, under cooling. Curve c–d is the relaxation of the stress at constant low temperature. Curve d–e is the strain recovery process under no load condition through heating.
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3. Development of the proposed model for the SMP behavior To model this phenomenon, we borrow ideas from multi-network theory dating back to Tobolsky and Andrews (1945) and Rajagopal and Wineman (1992), and introduce just two networks; one of which is a permanent-network responsible for permanent shape retention and recovery while the other network is a temporary-network which can be made to persist over long periods of time by suitably lowering the temperature (Srinivasa & Ghosh, 2008). This latter network is responsible for the shape fixity properties of the polymer at low temperatures. Furthermore the interactions between the two networks are related to the shape fixity and shape recovery parameters. Change in morphology: (Refer to Fig. 2). 1. State (a) to (b): As the polymer is stretched, the permanent networks, i.e. polymer chains with permanent nodes (shown as black circles) in the figure, deforms due to the partial uncoiling of the polymeric chains between the cross-links. When two chains come close enough they stick together momentarily and form temporary nodes (shown as grey circles) before breaking off (shown as white jagged circles) continuously at high temperature, because of the electrostatic attraction between the individual chains. The connections between these temporary nodes form the temporary network. 2. State (b) to (c): material is cooled, mobility of the chains decreases and the temporary nodes that are formed hold parts of the chains immobile exhibiting ‘‘stickiness’’ and makes the polymer glassy. 3. State (c) to (d): on unloading of the material, the permanent network will have a tendency to coil back to its initial configuration. However, the temporary nodes ‘‘lock-in’’ the deformed state of the permanent network here as they are extant at low temperatures. 4. State (d) to (a): subsequent heating increases the mobility of the chains, and the rate of breaking and reforming of temporary nodes starts rising once again. This ‘‘unlocks’’ the permanent networks which now take over and recoil the polymer back to its original state. It is clear from this discussion that any model that we develop for this kind of response should have: 1. Two elastic moduli: A rubbery one which is apparent at high temperatures for the permanent network and a glassy one which is apparent at high temperatures for the temporary network, 2. ‘‘Yield’’ behavior which is temperature sensitive so that it allows strains to be ‘‘locked in’’ at low temperatures and ‘‘unlocked’’ at high temperatures, 3. Viscous behavior that will accommodate the sticky slippage of the temporary network chains over time. The model shown in Fig. 3 represents a mechanical visualization of a model that has all the necessary features to model for the SMP response in a thermomechanical cycle: In Fig. 3, E1 and E2 denote the modulus of the rubbery and glassy springs respectively, l is the viscosity of the viscous dashpot, and k is the yield stress of the friction dashpot of the model. The axial stress is r. The rubbery spring in the model represents the non-linear rubbery response of the material, at temperatures beyond the glass transition. The glassy spring
Fig. 2. Change in morphology in a two-network model of the material during a thermomechanical cycle. Network 1: permanent network consisting of polymer chains with permanent nodes, and network 2: temporary network consisting of connections between the temporary nodes
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Fig. 3. The proposed mechanical model for the SMP. The spring E1 together with the viscous dashpot l represents the rubbery response of the material, the spring E2 represents that glassy response and the frictional element k represents the locking behavior.
represents the glassy response at temperatures below the glass transition (when the rubbery spring is ‘‘locked’’) and the stress is relaxed. To elaborate, the dashpot is a frictional dashpot that is temperature sensitive. Below the glass transition, the frictional dashpot locks, and prevents the rubbery network from deforming further, leading to a ‘‘locked in’’ strain. On the other hand, at high temperature, the dashpot ceases to operate i.e., its resistance vanishes, and the material behaves like an elastic rubber. In this paper, we will use the finite strain measure = (k 1), where k is the stretch ratio, and all our stress measures will be with respect to the reference configuration. In other words, we are using a referential description for the 1D response in this paper. Also note that Fig. 3 should not be taken literally but is only meant as a conceptualization of the different aspects of the overall response. Accordingly we will assign the stress r1, which is a first Piola Kirchoff stress, as the non-linear rubbery response and, stress r2 as the dissipative response. p is the strain of the dampers in parallel, representing the locked in shape of the polymer, while e is the strain of the glassy spring. To the mechanical strain 1 shown in the figure we shall have to add the thermal strain of the material so that we now have the total strain of the material. Thus, taking a to be the thermal expansion of the material, the elastic strain is given as:
e ¼ p aðh hhigh Þ
ð1Þ
In the above equation a is not a constant, but is actually a function of temperature, and h is the absolute temperature of the SMP sample (assumed to be uniform), and the form a(h hhigh) is written for convenience. We will begin the modeling of this material by considering the flow of mechanical power into the system. When such a material system is supplied with mechanical power, only a part of it recoverable; the remaining portion of the power that is supplied is dissipated i.e. irrecoverably lost. The proportion that is recoverable and that which is dissipated is what gives the material its characteristic behavior or response. Consider one-dimensional continuum, i.e., a wire, made of shape memory polymer, lying along the X-axis of a co-ordinate system and is heated above its glass-transition temperature so that it is in its rubbery state. A preliminary list of state variables is given by S = (r, , p, h). The foundation of the thermodynamical approach presented here is that, the non-dissipative properties of the material are derivable from a single potential, namely the Helmholtz potential for the continuum, while the dissipative properties are explored through the manifestation of the second law of thermodynamics, using the role of the rate of mechanical dissipation as a mechanism for entropy generation (Rajagopal & Srinivasa, 2000). Following Callen (1985), we can write the equation of state in terms of the internal energy as:
^ ð; p ; gÞ u¼u
ð2Þ
where u is the internal energy per unit refenence length of the system, g is the entropy of the 1-D continuum per unit reference length. In terms of Eq. (2), the absolute temperature h is given by h ¼ @@ug. Then the energy conservation equation in the absence of body forces can be written as:
qr
du d dq ¼r þr dt dt dX
where, q is the axial heat flux and r is the lateral rate of heat transfer per unit reference length.
ð3Þ
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The Helmholtz potential being the legendre transform of the internal energy with respect to the entropy, it is related to the energy through
w ¼ u hg
ð4Þ
The entropy g now becomes a function of Helmholtz free energy as below:
g¼
@w @h
ð5Þ
Now, we can take the time derivative of the internal energy function in Eq. (4):
du d ^ @w dh d @w ¼ wð; p ; hÞ h dt dt @h dt dt @h
ð6Þ
We reduce the energy conservation equation in terms of the Helmholtz potential by substituting Eqs. (4)–(6) into Eq. (3) and arrive at the heat equation:
hg_ ¼
r_
@w @w dq _ þ _p þ r @ @ p dX
ð7Þ
The terms in the equation above are:
Rate of heating ¼ fRate of heating due to mechanical effectsg þ axial heat flow þ latent heat The terms within the curly brackets in Eq. (7) is representative of the thermomechanical coupling. The first term inside the curly brackets represents the mechanical power (or deformation power) supplied to the wire, while the second two terms represent the rate of decrease in recoverable mechanical work. The sum of these terms represents the net mechanical power dissipated by the system. We hence introduce the rate of dissipation n through
r_
@w @w _ þ _p @ @ p
ð8Þ
:¼ n
where n P 0 as effected by the second law of thermodynamics. The terms in the equation above are:
Supply of mechanical power Rate of recov erable mechanical work ¼ Net mechanical power dissipated The above identity is referred to as the dissipation relation or the reduced energy relation. This identity can also be equated in the following manner, which represents the reduced energy-rate of dissipation relation for the system at a particular temperature:
_ r_ wj h fixed ¼ n
ð9Þ
We now have to characterize the material response by relating the energy storage elements in the proposed model to the Helmholtz potential and the dissipated elements in the model to the rate of dissipation function in the equation above. For this development, we shall be dealing with the approximate temperature dependence of the thermal expansion of shape memory polyurethanes (Gunes, Cao, & Jana, 2008), refer to Section (6.2), and the thermal strains directly in the case of epoxy resin (Liu et al., 2006), refer to Section (7.2). The Helmholtz potential w function depends on the state variables (, p, h) and is assumed to be of the form:
2 1 w ¼ a2p b4p þ E2 p aðh hhigh Þ 2
ð10Þ
The above form is not expected to be valid for arbitrary strains but only for strains up to 10%–15%. The rate of dissipation will also be a function of the state variables (, p, h), and will depend on how the viscous damper and the friction damper dissipate energy as indicated below:
n ¼ l_ 2p þ kj_p j
ð11Þ
The first term represents the rate of dissipation through the viscous damper modeled as a viscous quadratic Raleigh element, while the second term accounts for the rate of dissipation through the friction damper modeled as dry friction that is homogenous first degree in _p . Substituting Eq. (10) in Eq. (8) results in the following:
r _ E2 ð p aðh hhigh ÞÞ 2ap 4b3p E2 ð p aðh hhigh ÞÞ _ p ¼ ^nð_p Þ We will assume that the total stress r for the total strain
r¼
@w ¼ E2 p aðh hhigh Þ @
In the light of Eq. (13), Eq. (12) reduces to
ð12Þ
is given by: ð13Þ
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^nð_p Þ ¼ 2ap 4b3 E2 ð p aðh hhigh ÞÞ _ p p
ð14Þ
Using the stress in Eq. (13), the above equation can be reduced further as below, and this is used as the constraint for the system
^nð_p Þ ¼
r ð2ap 4b3p Þ _ p
ð15Þ
We are thus led to the development of a kinetic equation for p which will satisfy the above equation. Now we shall use the maximum rate of dissipation criterion (Srinivasa & Srinivasan, 2009), which states that the system will evolve such that the actual value of _p is that which maximizes n subject to the constraint Eq. (15). Hence by using the standard method of calculus of constrained maximization as explained in Segel and Handelman (1987), we extremize n subject to the constraint Eq. (15) using the method of Lagrange multipliers: Thus the value of _p that extremizes n is:
_ p ¼
1
l
r ð2ap 4b3p Þ þ kðsgnð_p ÞÞ
ð16Þ
The dissipative response of the rubbery network is described by means of this kinetic equation. As can be seen from the model figure, the change in amount of stretch of the rubbery spring, is related to the rate of change of p and this in turn is determined by the nature of the frictional dashpot. Notice that the above expression holds only when _p – 0, which occurs only ^ when r2 > kðhÞ, i.e. only when the stress in the friction damper is greater than the yield-stress, the damper will have a veloc^ ity. Otherwise if r2 6 kðhÞ, then _p will have to remain zero, because the stress in the friction damper has not yet exceeded its yield-stress, hence the damper remains locked. The second law of thermodynamics requires that the dissipation of the system considered always be non-negative, which is automatically satisfied by the above consideration. The cases that arise from this lead us to three threshold conditions which are stated as the kinetic equation of the model:
^
r ð2ap 4b3p Þ 6 kðhÞ ^ ^ _p ¼ l1 r ð2ap 4b3p Þ kðhÞ ; r ð2ap 4b3p Þ > kðhÞ > > > > : 1 r ð2a 4b3 Þ þ kðhÞ ^ ^ ; r ð2ap 4b3p Þ < kðhÞ p p l 8 > 0; > > > <
ð17Þ
The above set of cases can be written in a compact form as
_p ¼
1 nD
l
E
D
^ ^ r ð2ap 4b3p Þ kðhÞ r þ ð2ap 4b3p Þ kðhÞ
Eo
ð18Þ
where hxi ¼ 12 ðx þ kxkÞ. The state of the material is represented by the variables S = (r, , p, h). Eqs. (13) and (17) represent two constitutive equations for the response of the material. The remaining two equations needed to solve for this four-variable system can be specified in the control equation of stress/strain input and the input function of the temperature h. 4. Normalization of system equations 4.1. Normalizing variables (1) (2) (3) (4)
The The The The
glass transition temperature hg. maximum strain applied 0 from experimental results. typical stress response at high temperature and constant applied strain, r0, from experimental results. non-dimensionalization of the time, since this is connected with the kinetic response.
¼ ah0 . Note that a is still a function of The parameters used to normalize the equations here are C ¼ r00 and a h (ref: Sec0 tion (6.2)). Thereafter, the non-dimensional quantities are tabulated as in Table 1. Besides the above non-dimensional quantities, we take the values for the parameters enlisted in Table 2, which are decided based on normalizing the experimental
Table 1 Dimensional quantities and corresponding non-dimensional quantities. Dim Nondim
r
r ¼ rr0
¼
Dim Nondim
a ¼ 2a a C
b ¼ 4b20 b
0
C
t t ¼ t t0
h h ¼ hhg
E2
l
E2 ¼ EC2
l ¼ t0lC
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data from Tobushi et al.’s (1997) and Liu et al.’s (2006) experimental data so that comparison of model and experimental data is sensible. 4.2. Non-dimensional set of system equations We view the SMP as a dynamical system i.e., one whose state changes are given by a suitable differential equation. We have obtained the system of model equations for the material in state evolution form. Force or Strain Control:
_ þ BðtÞ_ ¼ gðtÞ AðtÞr
ð19Þ
State Equation: Eq. (13) can be rewritten in rate form as follows:
r_ E2 _ þ E2 _p þ E2
da h hhigh Þ þ a h_ ¼ 0 ð hg dh
ð20Þ
Kinetic Equation:
p 2 Þp 6 k r þb ða 2 2 1 _p ¼ l r ða þ bp Þp k ; r ða þ bp Þp > k > > : 1 r 2 Þ þ k ; r 2 Þ < k þb þb ða ða p p p p l 8 0; > > <
ð21Þ
Temperature Specification:
h_ ¼ f ðtÞ
ð22Þ
5. Simulation of the SMP response 5.1. Implementation of the system equations The state evolution form derived in the previous section will translate as follows to be fed into a suitable manner into MATLAB:
S_ ¼ f ðS; tÞ
ð23Þ
where S represents the variables that model the current state of the system and f is a function of the state variables and time. We will include the yield-stress as another variable for convenience in solving these equations in a numerical solver, details of this are discussed in Section (6.1). Thus, we can proceed to feed the Eq. (19) to (22) in the form Aðx;h;tÞ x_ ¼ pðx;h;tÞ þ Q ðx;h;tÞ x, which is solved by the ODE45 solver in MATLAB.
2
AðtÞ
6 0 6 6 6 1 6 6 4 0 0
BðtÞ 0 E2
E2 ðddah ðh
0
0
1
0 þ aÞ E2
hhigh Þ hg
0
0
1
0
ddhk
^
0
3 2 32 _ 3 2 r 0 gðtÞ 7 6 6 7 6 0 _ 7 6 07 76 7 6 f ðtÞ 7 6 7 6 76 _ 7 6 7 6 07 h 7¼6 6 0 7þ6 0 76 7 6 7 61 2 76 _ 7 6 0 54 p 5 4 0 5 4 l a _ 0 1 0 k 0
0 0 0 0
0 0
0 0
0
2 Þa2 þb 0 0 l1 ða p 0 0 0
0 0
32
3
r 76 7 76 7
76 7 76 h 7 76 7 1 3 76 7 l a 54 p 5 k 0 0
ð24Þ
Table 2 Parameter values for normalization and their significance in the samples. Parameter
r0 Mpa
0 hg K t0 s
max for tests on Polyurethane
max for tests on Epoxy resin
2.4%
4%
10%
9.1%
9.1%
0%
0.7968 0.0230 328 100
1.2513 0.04017
2.06997 0.0962
0.85 0.091 343 100
0.8 0.091
0 0
Parameter
Significance
r0 Mpa
Experimental value of stress at constant hhigh and applied constant Experimental applied constant max Glass-transition temperature of sample Suitable fraction of rate experimental processes
0 hg K t0 s
max
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P. Ghosh, A.R. Srinivasa / International Journal of Engineering Science 49 (2011) 823–838 Table 3 Process inputs w.r.t time. Process
Stress/ Strain rate input ðgtÞ
Stress Control: AðtÞ
Strain Control: BðtÞ
Temperature rate i/p i.e. ðf tÞ
a–b: 0 < t < t1 b–c: t1 < t < t2 c–d: t1 < t < t2 d–e: t3 < t < t4
g 1 ðtÞ 0 g 2 ðtÞ 0
0 0 1 1
1 1 0 0
0 f1 ðtÞ 0 f1 ðtÞ
Table 4 Specifications of the processes obtained from experimental data of Polyurethane and Epoxy resin. Process specification
For polyurethane
For Epoxy resin
Strain-rate Loading and unloading hold-time Heating and cooling rate Total rise or drop in temperature
5%–50% per minute 120 min ±4 K/min 40 K
0.03%–3% per minute 127 min ±1 K/min 85 K
where,
8 > > < 0; a ¼ 1; > > : 1;
p 2 Þp 6 k r þb ða 2 r ða þ bp Þp > k r ða þ bp 2 Þp < k
ð25Þ
5.2. Simulation specifications of the thermomechanical cycle Since we are dealing with rate equations to describe the system, we have extracted the experimental data from Tobushi’s and Liu’s work with respect to time, which are then used in the governing equations as shown below.
; Initial conditions: In terms of the state variables as shown in the section above, r ; h; p ; k ¼ f0; 0; hhigh ; 0; 0g. For this the specifications of the processes are obtained from experimental data of Polyurethane (Tobushi et al., 1997) and Epoxy resin (Liu et al., 2006) as tabulated in Tables 3 and 4. 6. Temperature dependence form of material parameters 6.1. The yield stress function The key to the response of the SMP material is the variation of the yield stress. The yield-stress of the material has a sensitive dependence on temperature, and the material yields differently depending on not just the current value of the temperature but also on whether the temperature of the material dropped or increased from the previous time-step. Thus there is a hysteresis of the yield-stress from the cooling to the heating cycle, which gives the different trends of the stress-rise during cooling and the strain-recovery during heating. The unrecovered strain at the end of the cycle is also explained by this hysteresis, because of the plastic yielding of the material in the thermomechanical cycle. These considerations suggest that the yield-stress rate has the following functional form, and thus the yield-stress varies as shown in Fig. 4.
dh _ ¼ ^f h; ; sgn dh k dt dt
ð26Þ
The factors that become important in this setup are as follows: (a) hhigh and hlow : the value of the yield stress is kept at a low minimum kmin, almost zero, so that in the initial range of high temperature stretching, the instantaneous rise in stress in the dry-friction damper overcomes its yield-stress and hence the plastic strain starts rising in the dampers. Once the stress reaches a constant value, there is no more sliding of the dry-friction damper, and hence the plastic strain gets ‘‘locked-in’’ after the initial rise. On the other hand, at low temperature towards the end of the cooling range, the yield stress of the dashpots reaches its maximum limit for that particular temperature. The stress in the dashpot network, even though in compression in the stress-relaxation range, its magnitude never exceeds the yield-stress and hence the friction dashpot helps ‘‘lock’’ the plastic strain for this range as well. (b) h: the yield-stress of the material is temperature dependent and changes during heating or cooling of the material. During cooling, the stress of the material rises. This is effected mainly by allowing stress in the dashpots to the rise by varying the yield stress of the material. The stress in the dashpots overcomes the yield-stress and hence the friction
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1.6
1.2
1.4
1 strain & yieldstress
stress & yieldstress
832
1.2 1 0.8 0.6 0.4
0
0.4
0 1 1.05 temp total stress yieldstress cooling yieldstress heating
1 0.8 0.6 0.4 0.2 0.95
1 temp
1.05
0.95
yield stress rate for heating
0.95
yield stress rate for cooling
0.6
0.2
0.2
0
0.8
1 1.05 temp total strain yieldstress cooling yieldstress heating
−0.1 −0.15 −0.2 −0.25 0.95
1 temp
1.05
Fig. 4. Variation of (a) yield-stress with temperature in a thermomechanical cycle and (b) yield-stress rate with temperature during cooling and heating processes, for experimental data of polyurethane with maximum strain of 4%.
_ here is prescribed to be zero till almost above glassdashpot slips further. The rate of yield-stress of the material k transition, and starts rising rapidly only at lower temperatures below glass-transition (Richeton, Ahzi, Daridon, & Rémond, 2005). Because of this, the rise in stress can be controlled carefully, and kept constant till temperature drops to hg, and then the stress is made to rise in desired fashion at lower temperatures. Thus a sigmoidal function ^f 1 is used which asymptotes between kmin and kmax. (c) ddht : the yield stress controls the way the plastic strain evolves. Having the same function of rate of yield-stress during heating, as specified for cooling f1 in the above point, leads to unsatisfactory strain recovery from the model. Also, complete recovery of the yield-stress implies complete recovery of the plastic strain at the end of heating, which will not mimic the residual strain recovery at the end of the cycle. For finer control on the response curves of stress and
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1 and k 2 (see strain during cooling and heating, respectively, it becomes obvious that different trends of yieldstress k Fig. 4(a)) are needed depending on whether the material is cooled or heated, i.e. whether sign ddht < or > 0. Thus Eq. (26) will depend on three cases as follows:
8 0; if ddht ¼ 0 > > < _ ¼ ^f ðh; Þ dh ; if sgn dh ¼ v e k 1 dt dt > > : ^ dh f 2 ðh; Þ dt ; if sgn ddht ¼ þv e
ð27Þ
This hypothesis suggests that the material yields differently depending on not just the current value of the temperature but also on whether the temperature of the material dropped or increased from the previous time-step. ^f 2 is adjusted so that desired shape recovery is obtained and is reduced to a value that will give ideal residual shape recovery response. This adjustment was done by extracting a Gaussian curve-fit to get an approximate idea of the functional form that will fit data for all three strain levels for validation. The residual yield-stress at the end of heating range will affect the next cycle of loading, and will account for the thermal hardening of the material under cyclic loading. Thus the stress and strain response during cooling and heating, respectively, can be controlled by adjusting ^f 1 and ^f 2 . This can be witnessed in Fig. 4(b). Details of this effect max high encan be seen in Fig. 6. For example, the initial constant strain response during heating is maintained by keeping k ough at the end of the cooling cycle, so that the yield stress takes a while before it becomes comparable to the stress in the dampers. Only then the plastic strain gets ‘‘unlocked’’, and strain recovery begins. (d) : Although results of the previous hypothesis give considerably improved results, it becomes evident that the rate of yield stress of the material being dependent on h and sgn ddht , it has the same trend regardless of the strain level. However the strain recovery of polyurethane during heating for the three different strain levels we are working with for validation, have different trends (refer Section 7). It is also evident in the compression and tension experiments of epoxy resin, that the strain level implies different yield tendencies of the material. Although the maximum stress rise during cooling and the thermal strain opposing factor during heating (refer to Section (6.2)) contribute in the strain recovery characteristics at different strain levels, the changes accounted for are not satisfactory. Thus, the yield stress rate needs to take into account the strain level the material is at, to affect the strain recovery differently at different strain levels. Note that the rate yield stress is only a linear function of the strain, and this dependence will disappear for the yield-stress, and hence not cause any inconsistencies with the assumptions for the rate of dissipation function. _ ¼ f^1 ð _ ¼ f^2 ð Þ ddht for cooling and k Þ ddht for heating. Thus we have the rate of yield stress as k h; h; Taking all these factors into consideration, the final equations of rate of yield stress have the following form:
f1 ¼ a þ d1 bðsinhðcpðh h1 ÞÞÞ f2 ¼ d d2 e 1 ðf tanhðmh þ nÞÞ2 where, m = 2/(hmax h2), n = 1 mhmaxand, a, b, c, d, e, f are constants depending on material type and applied max, h1 and h2 are limiting values of h until which there is no rise/fall in the stress/strain during the cooling/heating cycle. This is determined from the experimental data, for example, refer to the experimental data for maximum applied strain of 4% in Fig. 7, where these values will be h1 = 0.98 (value of h until which there is no stress rise in the cooling process) and h2 = 0.95 (value of h until which there is no strain recovery in the heating process), and
d1 ¼ d2 ¼
1; 8h 6 h1 0;
8h > h 1
0;
8h < h 2
1; 8h P h2
ð28Þ
ð29Þ
The rate of yield-stress specified by the above equations for the experimental data corresponding to Fig. 7 is as shown in Fig. 4(b). 6.2. The thermal expansion The thermal expansion of the material is dependent on temperature. Let us consider the typical shape memory polyurethane thermal response as shown in Fig. 5. The average thermal expansion coefficient in the rubbery state (a1 = 21.6X105K1) is about two times of that in the glassy state (a2 = 11.8X105K1) (Gunes et al., 2008). All thermal strains being measured with reference to the high temperature, the thermal strains at lower temperatures, are the largest. The thermal behavior of the material although aids in the cooling process, it counteracts the recovery behavior during the heating process (refer to Fig. 6). This counteraction is what causes the total strain to rise at the initial period of heating, and this can be controlled by the thermal strain completely. The material response is very sensitive to the thermal expansion coefficient, and the final rise of stress at the end of cooling range and the rise of strain during the start of heating range are affected severely by it. Therefore, a more accurate specification of the thermal expansion gives closer results to the actual
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Fig. 5. Variation of thermal strain with temperature. Take note of (a) Thermal strain for constant high temp value a1, axis on LHS and associated unaccounted thermal strain, (b) thermal strain for constant low temp value a2, axis on RHS and associated unaccounted thermal strain, (c) thermal strain for varying values of a, axis on LHS.
total strain elastic strain thermal strain strain in dampers
1 0.8
Strains
0.6 0.4 0.2 0 −0.2 0.94 0.96 0.98 1 1.02 1.04 1.06 temperature Fig. 6. Plots of damper, thermal and elastic strain and their combination to produce the total strain in the complete cycle. Notice that the thermal strain is opposite to the damper strain during recovery, and the effect is pronounced at the beginning of recovery.
experimental data. Note however that the thermal strains of the material is dependent on the temperature alone, and this role of thermal expansion in the strain recovery process reduces with increasing strain levels, because the total amount of strain to be recovered is rising while the amount gained back by the thermal expansion remains the same at all strain levels. Hence, for the polyurethane experiments, at low strain levels, the thermal strain accounts for almost 20% of the total strain as shown in the contributing factor C. F. values in Table 5, but at high strain levels its effect is only 4%. This implies that at increasing strain levels, 96% of the recovery will have to be dictated by the yielding of the material and not the thermal expansion. Also, as discussed previously, the strain recovery process and the thermal expansion process during heating counteract each other, as seen in Fig. 6. This implies that a relatively low value of thermal expansion in comparison to recovered
Table 5 Comparison of the strain responses and thermal strains, of shape memory polyurethane at various applied strain (max) levels.
max (%)
Shape fixity F (%)
Shape recovery R (%)
th ð%Þ C:F ¼ max
O:F ¼ C:F R
2.4 4 10
16.66 12.5 17
12.5 17.5 6
19.35 11.61 4.64
1.54 0.66 0.77
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strain is desired for improved shape memory effect. This is measured in terms of opposing factor O.F, as shown in Table 5, lower O.Fs being desirable. 7. Results 7.1. For shape memory polyurethane The glass transition of the material was 328 K, the thickness of the specimen was about 70 lm. The width was 5 mm, gauge length 25 mm and total length of 75 mm (Tobushi et al., 1997). The yield-stress function was chosen such that the model gave reasonable results for all three strain levels.
stress and strain vs temp
stress vs temp
1
stress
stress
1.5 1 0.5
0 0.5 strain
0
1 0.95 temp
1.05
0 0.95
1 1.05 temp stress vs strain
strain vs temp 1 stress
strain
1.5 0.5
1 0.5 0
0 0.95
1 temp
1.05
0
0.5 strain
1
Fig. 7. Comparison of the experimental data of PolyUrethane by Tobushi et al. (1997) with the prediction of the model developed here for
stress and strain vs temp
stress vs temp 2 stress
2 stress
max = 4%.
1 0 1
1.5 1 0.5
0.5 strain
0
1 0.95 temp
1.05
0 0.95
1 1.05 temp stress vs strain
strain vs temp 1 stress
strain
2 0.5
1.5 1 0.5
0
0 0.95
1 temp
1.05
0
0.5 strain
1
Fig. 8. Comparison of the experimental data of PolyUrethane by Tobushi et al. (1997) with the prediction of the model developed here for maximum applied strain of 2.4%.
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stress and strain vs temp
stress vs temp
stress
stress
1.5 1 0.5 0 0.5 strain
0
1 0.95 temp
1.05
1 0.5 0 0.95
1 1.05 temp stress vs strain
strain vs temp 1 stress
strain
1.5 0.5
1 0.5
0
0 0.95
1 temp
1.05
0
0.5 strain
1
Fig. 9. Comparison of the experimental data of PolyUrethane by Tobushi et al. (1997) with the prediction of the model developed here for maximum applied strain of 10%.
The simulation results shown in Fig. 7 compare the results of the model and experimental data for max = 4%. The parameters a, b and l are selected to control the stress response of the model during the initial high-temperature stretching (process a–b). The stress vs. strain graph will show how the non-linearity of the response comes into play. For higher strains this can be seen in Section 7, Fig. 9 for 10 % strain. The yield-stress functions are chosen to control the stress-rise during cooling (process b–c) and the strain recovery during heating (process d–e) as discussed in Section 6.1. The parameter E2 is selected to control the strain response i.e. the shape fixity, during low temperature stress relaxation (process c–d). The yield-stress rate function chosen for this material response is dependent on strain levels, and hence with change in the maximum strain applied in the thermomechanical cycle, the yield-stress changes slightly to accommodate the changing responses in the strain levels as can be seen in Figs. 7, 8 and 9 for 4%, 2.4% and 10% strain respectively. The non-linear response becomes obvious at this higher strain level as can be seen in the stress vs. strain plot in Fig. 9. The parameters a and b were chosen to exhibit non-linearity in the model. The stress rise during cooling, for the chose yieldstress function, is not satisfactory and this consequently affects the strain recovery during heating.
7.2. For shape memory epoxy resin thermal ¼ ^f ð A thermal strain function hÞ is fitted from experimental data, and used in the model in the rate form directly as previously. below, rather than in terms of the conductivity a
r_ E2 _ þ E2 _p þ E2 _ thermal ¼ 0
ð30Þ
The validation results for shape memory epoxy resin undergoing tension, compression and no load strain conditions are provided in Figs. 10–12 respectively.
stress vs temp
strain vs temp strain
stress
4 3 2 1 0.8
0.85
0.9 0.95 temp
1
0.8 0.6 0.4 0.2 0.85
0.9 0.95 temp
1
Fig. 10. Comparison of the experimental data of epoxy by Liu et al. (2006) with the prediction of the model developed here under tension of 9.1%.
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stress vs temp 3 strain
stress
2 1 0 −1 0.8
0.85
0.9 0.95 temp
strain vs temp
0 −0.2 −0.4 −0.6 −0.8
1
0.8
0.85
0.9 0.95 temp
1
Fig. 11. Comparison of the experimental data of epoxy by Liu et al. (2006) with the prediction of the model developed here under compression of 9.1%.
stress vs temp
2
−0.5
1 0 0.8
strain vs temp
0 strain
stress
3
0.85
0.9 0.95 temp
1
−1 0.8
0.85
0.9 0.95 temp
1
Fig. 12. Comparison of the experimental data of epoxy by Liu et al. (2006) with the prediction of the model developed here in the undeformed case.
8. Conclusions The yield-stress of the friction dashpot plays the lead role in controlling the response of the model during heating and cooling. Different functions of the yield-stress evolution during heating and cooling processes give improved results in strain recovery. This suggests that there is a hysteresis in the yield stress of the material when it undergoes a heating–cooling cycle. The yield stress rate, when made to be a function of the strain of the material, affects the recovery behavior during heating, which helps simulate the significantly different strain recoveries at the different strain levels. This suggests that the yielding of the material is influenced by the extent to which it is loaded. The variation of the thermal strain of the material during the heating and cooling processes assists in the stress rise during cooling, but opposes the strain recovery during heating. It was concluded that low temperature thermal expansion coefficient has strong influence on the stress response during cooling and strain recovery during heating. However, the significance of the contribution of the thermal strains on the material response reduces with increasing strain levels of the experiments. The model exhibits non-linearity of the response at high strain levels for the polyurethane experiments, which is accounted by the non-linear response of the rubbery spring. However, even though the non-linear behavior of the material is evident at higher applied strain levels during loading, it does not affect the recovery of the material significantly. This may suggest that the recovery responses at high strain levels are governed mainly by the yielding of the material, and since the recovery patterns are not monotonic with increasing applied strain levels, as can be seen in the shape recovery factors R% in Table 5, there may be some other mode of yielding or failure that has to be taken into consideration. Acknowledgement The authors would like to acknowledge the support of the National Science Foundation CMMI Grant 1000790 in carrying out this research. References Abrahamson, E., Lake, M., Munshi, N., & Gall, K. (2003). Shape memory mechanics of an elastic memory composite resin. Journal of Intelligent Material Systems and Structures, 14(10), 623–632. Barot, G., & Rao, I. (2006). Constitutive modeling of the mechanics associated with crystallizable shape memory polymers. Zeitschrift f ur¨ Angewandte Mathematik und Physik (ZAMP), 57(4), 652–681. Barot, G., Rao, I., & Rajagopal, K. (2008). A thermodynamic framework for the modeling of crystallizable shape memory polymers. International Journal of Engineering Science, 46(4), 325–351. Beloshenko, V., Varyukhin, V., & Voznyak, Y. (2005). The shape memory effect in polymers. Russian Chemical Reviews, 74(3), 265–283. Bhattacharyya, A., & Tobushi, H. (2000). Analysis of the isothermal mechanical response of a shape memory polymer rheological model. Polymer Engineering and Science, 40(12), 2498–2510. Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics. New York: John Wiley & Sons.
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Diani, J., Liu, Y., & Gall, K. (2006). Finite strain 3D thermoviscoelastic constitutive model for shape memory polymers. Polymer Engineering and Science, 46(4), 486. Gunes, I., Cao, F., & Jana, S. (2008.) Effect of thermal expansion on shape memory behavior of polyurethane and its nanocomposites. Journal of Polymer Science Part B: Polymer Physics, 46(14). Hong, S., Yu, W., & Youk, J. (2007). Thermomechanical deformation analysis of shape memory polymers using viscoelasticity. In: AIP conference proceedings, AIP (Vol. 907, pp. 853–858). Hu, J. (2007). Shape memory polymers and textiles. Woodhead Publishing Ltd.. Hu, J. (2008). New developments in elastic fibers. Polymer Reviews, 48(2), 275–301. Kelch, S., & Lendlein, A. (2002). Shape memory polymers. Angew Chemte – International Edition in English, 41, 2034–2057. Lee, S., Kim, J., & Kim, B. (2004). Shape memory polyurethanes having crosslinks in soft and hard segments. Smart Materials and Structures, 13(6), 1345–1350. Lendlein, A., & Langer, R. (2002). Biodegradable, elastic shape-memory polymers for potential biomedical applications. Science, 296(5573), 1673–1676. Lin, J., & Chen, L. (1999). Shape-memorized crosslinked ester-type polyurethane and its mechanical viscoelastic model. Journal of Applied Polymer Science, 73(7), 1305–1319. Liu, Y., Gall, K., Dunn, M., Greenberg, A., & Diani, J. (2006). Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling. International Journal of Plasticity, 22(2), 279–313. Liu, C., Qin, H., & Mather, P. (2007). Review of progress in shape-memory polymers. Journal of Materials Chemistry, 17(16), 1543–1558. Ni, Q., Zhang, C., Fu, Y., Dai, G., & Kimura, T. (2007). Shape memory effect and mechanical properties of carbon nanotube/shape memory polymer nanocomposites. Composite Structures, 81(2), 176–184. Rajagopal, K., & Srinivasa, A. (2000). A thermodynamic frame work for rate type fluid models. Journal of Non-Newtonian Fluid Mechanics, 88(3), 207–227. Rajagopal, K., & Wineman, A. (1992). A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes. International Journal of Plasticity, 8(4), 385–395. Richeton, J., Ahzi, S., Daridon, L., & Rémond, Y. (2005). A formulation of the cooperative model for the yield stress of amorphous polymers for a wide range of strain rates and temperatures. Polymer, 46(16), 6035–6043. Segel, L., & Handelman, G. (1987). Mathematics applied to continuum mechanics. New York: Dover Publications. Srinivasa, A., & Ghosh, P. (2008). A simple, Gibbs potential based multinetwork model for shape memory polymers. In AIP conference proceedings (Vol. 1029, p. 58). Srinivasa, A. R., & Srinivasan, S. M. (2009). Inelasticity of materials: An engineering approach and practical guide. World Scientific Pub. Co. Inc. Takahashi, T., Hayashi, N., & Hayashi, S. (1996). Structure and properties of shape-memory polyurethane block copolymers. Journal of Applied Polymer Science, 60(7), 1061–1069. Tobolsky, A., & Andrews, R. (1945). Systems manifesting superposed elastic and viscous behavior. The Journal of Chemical Physics, 13, 3–27. Tobushi, H., Hara, H., Yamada, E., & Hayashi, S. (1996). Thermomechanical properties in a thin film of shape memory polymer of polyurethane series. Smart Materials & Structures, 5(4), 483–491. Tobushi, H., Hashimoto, T., Hayashi, S., & Yamada, E. (1997). Thermomechanical constitutive modeling in shape memory polymer of polyurethane series. Journal of Intelligent Material Systems and Structures, 8(8), 711–718. Tobushi, H., Hashimoto, T., Ito, N., Hayashi, S., & Yamada, E. (1998). Shape fixity and shape recovery in a film of shape memory polymer of polyurethane series. Journal of Intelligent Material Systems and Structures, 9(2), 127–136. Tobushi, H., Hayashi, S., Hoshio, K., & Ejiri, Y. (2008). Shape recovery and irrecoverable strain control in polyurethane shape-memory polymer. Science and Technology of Advanced Materials, 9(1), 015009 (7).
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
A Two-network thermomechanical model of a shape memory polymer Pritha Ghosh, A.R. Srinivasa ⇑ Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, United States
a r t i c l e
i n f o
Article history: Received 25 January 2010 Received in revised form 7 February 2011 Accepted 19 April 2011 Available online 12 May 2011 Keywords: Shape memory polymers Yield stress Hysteresis Thermal expansion Multinetwork model Helmhotz potential Shape fixity Shape recovery
a b s t r a c t The aim of this work is to demonstrate a Helmholtz potential based approach for the development of the constitutive equations for a shape memory polymer undergoing a thermomechanical cycle. The model is able to simulate the response of the material during heating and cooling cycles and the sensitive dependence of the response on thermal expansion. We notice that the yield-stress of the material controls the gross features of the response of the model, and suggests that the material yields differently depending on not just the current value of the temperature but also on whether the temperature of the material dropped or increased from the previous time-step somewhat similar to the Bauschinger effect in plasticity, except that here the controlling parameter is the rate of temperature change rather than rate of plastic strain. The results of the simulation are in qualitative and quantitative agreement with experiments performed on two different shape memory polymer samples: polyurethane and epoxy resin. We find that modeling the hysteresis of the yield stress of the material during temperature changes is the key to the results. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Shape memory polymers (SMPs) belong to a class of smart materials, that are capable of holding a temporary shape that was caused by the application of forces, and can subsequently regain or ‘‘remember’’ their original shape ‘‘on demand’’ by the application of non-mechanical stimuli. Temperature-responsive shape memory polymers use changes in temperature as this stimulus, and undergo structural changes at a certain temperature called the transition temperature. This transition temperature could correspond to glass-transition (for amorphous polymers) or the melting temperature (for polymers with crystalline phases), which gives rise to different kinds of shape-memory behavior (Kelch & Lendlein, 2002). They can be easily deformed to give a temporary shape at low-working temperatures and upon heating above their transformation temperature, return to their original structure. Examples of shape memory polymers are polyurethanes, polyurethanes with ionic or mesogenic components, block copolymers consisting of polyethyleneterephthalate and polyethyleneoxide, block copolymers containing polystyrene and polybutadiene, polyesterurethanes with methylenebis and butanediol, filler modified epoxy, thermoset epoxy resins, epoxy composites (Kelch & Lendlein, 2002). This feature which may be termed ‘‘shape retention and recovery on demand’’ can be exploited to design components that are used to enhance product performance and quality under changeable environments. Shape memory polymers have found numerous applications on account of their unique thermomechanical properties, and are currently being investigated for a wide range of applications where their shape fixity and shape recovery behavior are useful (Hu, 2007, 2008; Kelch & Lendlein, 2002; Lendlein & Langer, 2002; Liu, Qin, & Mather, 2007; Ni, Zhang, Fu, Dai, & Kimura, 2007). The key aspect to the use of shape memory polymers is the large recoverable change in properties across a narrow temperature range, which can be used to monitor and actuate various mechanisms. ⇑ Corresponding author. E-mail address: [email protected] (A.R. Srinivasa). 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.04.003
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Glassy shape memory polymers allow large reversible change in properties across the glass transition temperature (hg). Specifically, the material can change from a glassy state to a rubbery state across hg, accompanied by a change in its modulus. An increase in temperature allows the material to become more flexible, therefore easily deformable, and a decrease in temperature hardens the material into a glassy polymer, sustaining the new shape. It is important to note that the glass transition temperature is a kinetic parameter, and thus depends on the cooling rate. Thus the slower the cooling rate, the lower hg. Establishing appropriate models to simulate the shape memory processes and therefore to predict the shape memory properties should be helpful for the development and application of SMPs. In a recent article, Liu, Gall, Dunn, Greenberg, and Diani (2006) have observed that relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. They emphasize that constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. The shape memory behavior is dependent on both the structure and the thermomechanical conditions. Theories of the shape fixity and recovery behavior have been developed by Tobushi, Hara, Yamada, and Hayashi (1996), Tobushi, Hashimoto, Hayashi, and Yamada (1997), Tobushi, Hashimoto, Ito, Hayashi, and Yamada (1998), Tobushi, Hayashi, Hoshio, and Ejiri (2008), Abrahamson, Lake, Munshi, and Gall (2003), Takahashi, Hayashi, and Hayashi (1996), Lee, Kim, and Kim (2004), Lin and Chen (1999), Bhattacharyya and Tobushi (2000), Hong, Yu, and Youk (2007), Liu et al. (2006) and Diani, Liu, and Gall (2006). The literature on the shape-memory effect in polymers mostly deal with the development of theoretical concepts of the mechanisms of thermally stimulated strain recovery (Tobushi et al., 2008); new approaches to the description of the deformation and relaxation behavior of polymers (Takahashi et al., 1996); new information on the shape recovery features of deformed samples; and results of investigations of synthetic polymeric materials possessing unusual properties (Beloshenko, Varyukhin, & Voznyak, 2005). For example, the model proposed by Lin and Chen (1999) employs two Maxwell elements connected in parallel to describe the shape memory properties of shape memory polyurethane. Their modeling results and experimental data show some deviation, which the authors ascribe to the polydispersed glass transition temperature of the studied samples. The model can qualitatively explain the occurrence of shape memory behaviors. However, since the dampers in the model are both viscous, there is no truly irrecoverable strain at the end of the process. Furthermore, the model was not developed using thermodynamical principles. Abrahamson et al. (2003) utilize a friction element in their model to account for the irrecoverable strain at the end of the cycle, that progresses from fully stuck to fully free over a finite range of strain. It was found that the stress–strain curve predicted by the model and that obtained by experiment agreed well. It will become apparent that the material response is sensitive to the thermal expansion of the material. However, Abrahamson’s model does not take the thermal expansion of the material into consideration during the change of temperature. The model developed by Tobushi et al. (1997) takes irreversible deformation and thermal strains of shape memory polyurethanes into account. Tobushi and coworkers have added a friction element into the standard linear viscoelastic model to simulate the behaviors of shape memory polyurethanes. They have carried out a series of creep tests of shape memory polyurethanes at different temperatures. They propose that if the strain exceeds particular threshold strain, irreversible deformation occurs. Also, the temperature change that would cause thermal expansion is accounted for by simply adding a coefficient of thermal expansion to terms involving the rate of strain, without any thermodynamic considerations as seen in Eq. (4) in Tobushi et al. (1997). As a result, their model can be shown to be theormodynamically inconsistent even for pure thermoelasticity. The current work deals with the development of a two-network thermodynamically consistent model in a state-space form and shows that most of the gross features of a SMP depend on the yield-stress of the material. We demonstrate that a systematic application of fundamental thermodynamic principles coupled with a suitable choice for the Helmholtz potential and suitable kinetic equations for the inelastic variables will allow the development of a complete simulation of the behavior of a class of shape memory polymeric materials. Although the model can be generalized to 3-dimensions, there is a paucity of experimental data. For this reason, we will restrict ourselves to a one dimensional model. A number of significant features of the response can be deduced by a careful observation of the available experimental data. (1) There is a hysteresis in the yield-stress function of the material during heating and cooling. (2) The yield stress of the material evolves differently depending on the extent to which the material has been strained. (3) The thermal expansion behavior of the material significantly enhances the stress rise in the material during cooling due to contraction, but opposes the strain recovery by heating due to expansion. The contrasting behavior of the thermal expansion during cooling and heating makes the material response very sensitive to the coefficient of thermal expansion of the material. The model that we propose takes into account these three phenomena. Given the lack of experimental data on the actual polymer temperature versus that of the ambient atmosphere, and the extremely slow deformation rates, we will assume that the temperature of the polymer is the same as that of the surrounding atmosphere. Thus, we shall not be dealing with the heat equation in this work, rather we shall take temperature changes to the material as a defined input. The solution of the system equations for a simple thermomechanical cycle shall be simulated in MATLAB and its results shall be shown to be in qualitative and quantitative agreement with experiments performed on polyurethane and epoxy resin.
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Barot and Rao (2006) and later Barot, Rao, and Rajagopal (2008) have developed models for SMPs undergoing shape setting through partial crystallization using a thermodynamical approach (developed by Rajagopal & Srinivasa (2000)) based on maximum rate of dissipation. Our approach, while similar in the use of thermodynamical ideas, differs from these models in the following ways: (1) We are focussed on shape setting through glass transition and not through a real phase change. This makes our modeling task much simpler. (2) Their approach is a fully three dimensional history dependent approach as seen from Eq. (34) in Barot et al. (2008). This leads to considerable complications in solving boundary value problems. Our approach is based on a much simpler rate type constitutive equations using a plasticity like approach, leading to simple ODEs for homogenous deformations and coupled PDEs and ODEs for general boundary value problems. A few comments are in order with regard to our focus on a one dimensional model as opposed to full three dimensional constitutive theory. Although there are many applications of SMP that require full 3D models, there are several applications in which a 1D model will be more than sufficient, such as widely used SMP sutures with complex loading, SMP orthodontic wires for dental realignments, SMP thrombectomy devices, etc. Also there is a paucity in experimental data in the 3D response of these materials, as we have mentioned above, especially in view of the fact that there are indications of considerable anisoptropy in the response. Also, in view of the uncertainties associated in 3D models, and to make sure the paper is accessible to a wider audience, especially with regard to our central point, i.e. the temperature hysteresis of the yielding phenomena, we have chosen to focus on an entirely 1D model. 2. Behavior of shape memory polymers The experiment that we shall be considering in this work is a thermomechanical cycle on a shape memory polymer from the work carried out on a polyurethane sample by Tobushi et al. (1997) and on an epoxy resin sample by Liu et al. (2006). The process involved here are (refer to Fig. 1): 1. Initial conditions: at (a): The material is considered at a stress free state at a temperature above the glass transition temperature hg = hg + 20°C. All strains are to be measured from this state. This current shape is thus recognized as the ‘‘permanent shape’’ of the material. 2. Process A: from (a) to (b): High temperature stretching: The temperature is held fixed at hmax and the strain is increased steadily at a constant prescribed rate g1 to give the temporary shape to the material, and then the strain is held constant for a time 0 < t < t1. 3. Process B: from (b) to (c): Cooling and fixing the temporary shape: The strain is fixed at g1 and the temperature is lowered to hmin = hg 20°C at a predetermined rate ‘f1’. 4. Process C: from (c) to (d): Relaxing the stress: Now the temperature is fixed at hmin and the stress is gradually relaxed to zero at a predetermined rate ‘ g2’. During this process, material is observed to still be in its temporary shape. 5. Process D: from (d) to (a): Recovering the original shape: Now the body is heated at rate ‘f1’ in a stress-free state back to the original temperature to recover the original shape. The strain slowly relaxes and the material is back to its original shape. The typical material response for maximum strain of 10% for the thermomechanical tests carried by Tobushi et al. (1997) is shown in Fig. 1-right. The thermomechanical cycle also involves strain and stress-relaxing rate, and cooling and heating rates, which is not evident in these graphs. In the following work, we have extracted information from the above experimental data and plotted them against time.
Fig. 1. (a) The processes taking place in a thermomechanical cycle. (b) Typical (non-dimensionalized) experimental data for a polyurethane sample. The curve a-b indicates the high temperature deformations. Curve b–c is the shape fixing process at constant strain, under cooling. Curve c–d is the relaxation of the stress at constant low temperature. Curve d–e is the strain recovery process under no load condition through heating.
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3. Development of the proposed model for the SMP behavior To model this phenomenon, we borrow ideas from multi-network theory dating back to Tobolsky and Andrews (1945) and Rajagopal and Wineman (1992), and introduce just two networks; one of which is a permanent-network responsible for permanent shape retention and recovery while the other network is a temporary-network which can be made to persist over long periods of time by suitably lowering the temperature (Srinivasa & Ghosh, 2008). This latter network is responsible for the shape fixity properties of the polymer at low temperatures. Furthermore the interactions between the two networks are related to the shape fixity and shape recovery parameters. Change in morphology: (Refer to Fig. 2). 1. State (a) to (b): As the polymer is stretched, the permanent networks, i.e. polymer chains with permanent nodes (shown as black circles) in the figure, deforms due to the partial uncoiling of the polymeric chains between the cross-links. When two chains come close enough they stick together momentarily and form temporary nodes (shown as grey circles) before breaking off (shown as white jagged circles) continuously at high temperature, because of the electrostatic attraction between the individual chains. The connections between these temporary nodes form the temporary network. 2. State (b) to (c): material is cooled, mobility of the chains decreases and the temporary nodes that are formed hold parts of the chains immobile exhibiting ‘‘stickiness’’ and makes the polymer glassy. 3. State (c) to (d): on unloading of the material, the permanent network will have a tendency to coil back to its initial configuration. However, the temporary nodes ‘‘lock-in’’ the deformed state of the permanent network here as they are extant at low temperatures. 4. State (d) to (a): subsequent heating increases the mobility of the chains, and the rate of breaking and reforming of temporary nodes starts rising once again. This ‘‘unlocks’’ the permanent networks which now take over and recoil the polymer back to its original state. It is clear from this discussion that any model that we develop for this kind of response should have: 1. Two elastic moduli: A rubbery one which is apparent at high temperatures for the permanent network and a glassy one which is apparent at high temperatures for the temporary network, 2. ‘‘Yield’’ behavior which is temperature sensitive so that it allows strains to be ‘‘locked in’’ at low temperatures and ‘‘unlocked’’ at high temperatures, 3. Viscous behavior that will accommodate the sticky slippage of the temporary network chains over time. The model shown in Fig. 3 represents a mechanical visualization of a model that has all the necessary features to model for the SMP response in a thermomechanical cycle: In Fig. 3, E1 and E2 denote the modulus of the rubbery and glassy springs respectively, l is the viscosity of the viscous dashpot, and k is the yield stress of the friction dashpot of the model. The axial stress is r. The rubbery spring in the model represents the non-linear rubbery response of the material, at temperatures beyond the glass transition. The glassy spring
Fig. 2. Change in morphology in a two-network model of the material during a thermomechanical cycle. Network 1: permanent network consisting of polymer chains with permanent nodes, and network 2: temporary network consisting of connections between the temporary nodes
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Fig. 3. The proposed mechanical model for the SMP. The spring E1 together with the viscous dashpot l represents the rubbery response of the material, the spring E2 represents that glassy response and the frictional element k represents the locking behavior.
represents the glassy response at temperatures below the glass transition (when the rubbery spring is ‘‘locked’’) and the stress is relaxed. To elaborate, the dashpot is a frictional dashpot that is temperature sensitive. Below the glass transition, the frictional dashpot locks, and prevents the rubbery network from deforming further, leading to a ‘‘locked in’’ strain. On the other hand, at high temperature, the dashpot ceases to operate i.e., its resistance vanishes, and the material behaves like an elastic rubber. In this paper, we will use the finite strain measure = (k 1), where k is the stretch ratio, and all our stress measures will be with respect to the reference configuration. In other words, we are using a referential description for the 1D response in this paper. Also note that Fig. 3 should not be taken literally but is only meant as a conceptualization of the different aspects of the overall response. Accordingly we will assign the stress r1, which is a first Piola Kirchoff stress, as the non-linear rubbery response and, stress r2 as the dissipative response. p is the strain of the dampers in parallel, representing the locked in shape of the polymer, while e is the strain of the glassy spring. To the mechanical strain 1 shown in the figure we shall have to add the thermal strain of the material so that we now have the total strain of the material. Thus, taking a to be the thermal expansion of the material, the elastic strain is given as:
e ¼ p aðh hhigh Þ
ð1Þ
In the above equation a is not a constant, but is actually a function of temperature, and h is the absolute temperature of the SMP sample (assumed to be uniform), and the form a(h hhigh) is written for convenience. We will begin the modeling of this material by considering the flow of mechanical power into the system. When such a material system is supplied with mechanical power, only a part of it recoverable; the remaining portion of the power that is supplied is dissipated i.e. irrecoverably lost. The proportion that is recoverable and that which is dissipated is what gives the material its characteristic behavior or response. Consider one-dimensional continuum, i.e., a wire, made of shape memory polymer, lying along the X-axis of a co-ordinate system and is heated above its glass-transition temperature so that it is in its rubbery state. A preliminary list of state variables is given by S = (r, , p, h). The foundation of the thermodynamical approach presented here is that, the non-dissipative properties of the material are derivable from a single potential, namely the Helmholtz potential for the continuum, while the dissipative properties are explored through the manifestation of the second law of thermodynamics, using the role of the rate of mechanical dissipation as a mechanism for entropy generation (Rajagopal & Srinivasa, 2000). Following Callen (1985), we can write the equation of state in terms of the internal energy as:
^ ð; p ; gÞ u¼u
ð2Þ
where u is the internal energy per unit refenence length of the system, g is the entropy of the 1-D continuum per unit reference length. In terms of Eq. (2), the absolute temperature h is given by h ¼ @@ug. Then the energy conservation equation in the absence of body forces can be written as:
qr
du d dq ¼r þr dt dt dX
where, q is the axial heat flux and r is the lateral rate of heat transfer per unit reference length.
ð3Þ
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The Helmholtz potential being the legendre transform of the internal energy with respect to the entropy, it is related to the energy through
w ¼ u hg
ð4Þ
The entropy g now becomes a function of Helmholtz free energy as below:
g¼
@w @h
ð5Þ
Now, we can take the time derivative of the internal energy function in Eq. (4):
du d ^ @w dh d @w ¼ wð; p ; hÞ h dt dt @h dt dt @h
ð6Þ
We reduce the energy conservation equation in terms of the Helmholtz potential by substituting Eqs. (4)–(6) into Eq. (3) and arrive at the heat equation:
hg_ ¼
r_
@w @w dq _ þ _p þ r @ @ p dX
ð7Þ
The terms in the equation above are:
Rate of heating ¼ fRate of heating due to mechanical effectsg þ axial heat flow þ latent heat The terms within the curly brackets in Eq. (7) is representative of the thermomechanical coupling. The first term inside the curly brackets represents the mechanical power (or deformation power) supplied to the wire, while the second two terms represent the rate of decrease in recoverable mechanical work. The sum of these terms represents the net mechanical power dissipated by the system. We hence introduce the rate of dissipation n through
r_
@w @w _ þ _p @ @ p
ð8Þ
:¼ n
where n P 0 as effected by the second law of thermodynamics. The terms in the equation above are:
Supply of mechanical power Rate of recov erable mechanical work ¼ Net mechanical power dissipated The above identity is referred to as the dissipation relation or the reduced energy relation. This identity can also be equated in the following manner, which represents the reduced energy-rate of dissipation relation for the system at a particular temperature:
_ r_ wj h fixed ¼ n
ð9Þ
We now have to characterize the material response by relating the energy storage elements in the proposed model to the Helmholtz potential and the dissipated elements in the model to the rate of dissipation function in the equation above. For this development, we shall be dealing with the approximate temperature dependence of the thermal expansion of shape memory polyurethanes (Gunes, Cao, & Jana, 2008), refer to Section (6.2), and the thermal strains directly in the case of epoxy resin (Liu et al., 2006), refer to Section (7.2). The Helmholtz potential w function depends on the state variables (, p, h) and is assumed to be of the form:
2 1 w ¼ a2p b4p þ E2 p aðh hhigh Þ 2
ð10Þ
The above form is not expected to be valid for arbitrary strains but only for strains up to 10%–15%. The rate of dissipation will also be a function of the state variables (, p, h), and will depend on how the viscous damper and the friction damper dissipate energy as indicated below:
n ¼ l_ 2p þ kj_p j
ð11Þ
The first term represents the rate of dissipation through the viscous damper modeled as a viscous quadratic Raleigh element, while the second term accounts for the rate of dissipation through the friction damper modeled as dry friction that is homogenous first degree in _p . Substituting Eq. (10) in Eq. (8) results in the following:
r _ E2 ð p aðh hhigh ÞÞ 2ap 4b3p E2 ð p aðh hhigh ÞÞ _ p ¼ ^nð_p Þ We will assume that the total stress r for the total strain
r¼
@w ¼ E2 p aðh hhigh Þ @
In the light of Eq. (13), Eq. (12) reduces to
ð12Þ
is given by: ð13Þ
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^nð_p Þ ¼ 2ap 4b3 E2 ð p aðh hhigh ÞÞ _ p p
ð14Þ
Using the stress in Eq. (13), the above equation can be reduced further as below, and this is used as the constraint for the system
^nð_p Þ ¼
r ð2ap 4b3p Þ _ p
ð15Þ
We are thus led to the development of a kinetic equation for p which will satisfy the above equation. Now we shall use the maximum rate of dissipation criterion (Srinivasa & Srinivasan, 2009), which states that the system will evolve such that the actual value of _p is that which maximizes n subject to the constraint Eq. (15). Hence by using the standard method of calculus of constrained maximization as explained in Segel and Handelman (1987), we extremize n subject to the constraint Eq. (15) using the method of Lagrange multipliers: Thus the value of _p that extremizes n is:
_ p ¼
1
l
r ð2ap 4b3p Þ þ kðsgnð_p ÞÞ
ð16Þ
The dissipative response of the rubbery network is described by means of this kinetic equation. As can be seen from the model figure, the change in amount of stretch of the rubbery spring, is related to the rate of change of p and this in turn is determined by the nature of the frictional dashpot. Notice that the above expression holds only when _p – 0, which occurs only ^ when r2 > kðhÞ, i.e. only when the stress in the friction damper is greater than the yield-stress, the damper will have a veloc^ ity. Otherwise if r2 6 kðhÞ, then _p will have to remain zero, because the stress in the friction damper has not yet exceeded its yield-stress, hence the damper remains locked. The second law of thermodynamics requires that the dissipation of the system considered always be non-negative, which is automatically satisfied by the above consideration. The cases that arise from this lead us to three threshold conditions which are stated as the kinetic equation of the model:
^
r ð2ap 4b3p Þ 6 kðhÞ ^ ^ _p ¼ l1 r ð2ap 4b3p Þ kðhÞ ; r ð2ap 4b3p Þ > kðhÞ > > > > : 1 r ð2a 4b3 Þ þ kðhÞ ^ ^ ; r ð2ap 4b3p Þ < kðhÞ p p l 8 > 0; > > > <
ð17Þ
The above set of cases can be written in a compact form as
_p ¼
1 nD
l
E
D
^ ^ r ð2ap 4b3p Þ kðhÞ r þ ð2ap 4b3p Þ kðhÞ
Eo
ð18Þ
where hxi ¼ 12 ðx þ kxkÞ. The state of the material is represented by the variables S = (r, , p, h). Eqs. (13) and (17) represent two constitutive equations for the response of the material. The remaining two equations needed to solve for this four-variable system can be specified in the control equation of stress/strain input and the input function of the temperature h. 4. Normalization of system equations 4.1. Normalizing variables (1) (2) (3) (4)
The The The The
glass transition temperature hg. maximum strain applied 0 from experimental results. typical stress response at high temperature and constant applied strain, r0, from experimental results. non-dimensionalization of the time, since this is connected with the kinetic response.
¼ ah0 . Note that a is still a function of The parameters used to normalize the equations here are C ¼ r00 and a h (ref: Sec0 tion (6.2)). Thereafter, the non-dimensional quantities are tabulated as in Table 1. Besides the above non-dimensional quantities, we take the values for the parameters enlisted in Table 2, which are decided based on normalizing the experimental
Table 1 Dimensional quantities and corresponding non-dimensional quantities. Dim Nondim
r
r ¼ rr0
¼
Dim Nondim
a ¼ 2a a C
b ¼ 4b20 b
0
C
t t ¼ t t0
h h ¼ hhg
E2
l
E2 ¼ EC2
l ¼ t0lC
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data from Tobushi et al.’s (1997) and Liu et al.’s (2006) experimental data so that comparison of model and experimental data is sensible. 4.2. Non-dimensional set of system equations We view the SMP as a dynamical system i.e., one whose state changes are given by a suitable differential equation. We have obtained the system of model equations for the material in state evolution form. Force or Strain Control:
_ þ BðtÞ_ ¼ gðtÞ AðtÞr
ð19Þ
State Equation: Eq. (13) can be rewritten in rate form as follows:
r_ E2 _ þ E2 _p þ E2
da h hhigh Þ þ a h_ ¼ 0 ð hg dh
ð20Þ
Kinetic Equation:
p 2 Þp 6 k r þb ða 2 2 1 _p ¼ l r ða þ bp Þp k ; r ða þ bp Þp > k > > : 1 r 2 Þ þ k ; r 2 Þ < k þb þb ða ða p p p p l 8 0; > > <
ð21Þ
Temperature Specification:
h_ ¼ f ðtÞ
ð22Þ
5. Simulation of the SMP response 5.1. Implementation of the system equations The state evolution form derived in the previous section will translate as follows to be fed into a suitable manner into MATLAB:
S_ ¼ f ðS; tÞ
ð23Þ
where S represents the variables that model the current state of the system and f is a function of the state variables and time. We will include the yield-stress as another variable for convenience in solving these equations in a numerical solver, details of this are discussed in Section (6.1). Thus, we can proceed to feed the Eq. (19) to (22) in the form Aðx;h;tÞ x_ ¼ pðx;h;tÞ þ Q ðx;h;tÞ x, which is solved by the ODE45 solver in MATLAB.
2
AðtÞ
6 0 6 6 6 1 6 6 4 0 0
BðtÞ 0 E2
E2 ðddah ðh
0
0
1
0 þ aÞ E2
hhigh Þ hg
0
0
1
0
ddhk
^
0
3 2 32 _ 3 2 r 0 gðtÞ 7 6 6 7 6 0 _ 7 6 07 76 7 6 f ðtÞ 7 6 7 6 76 _ 7 6 7 6 07 h 7¼6 6 0 7þ6 0 76 7 6 7 61 2 76 _ 7 6 0 54 p 5 4 0 5 4 l a _ 0 1 0 k 0
0 0 0 0
0 0
0 0
0
2 Þa2 þb 0 0 l1 ða p 0 0 0
0 0
32
3
r 76 7 76 7
76 7 76 h 7 76 7 1 3 76 7 l a 54 p 5 k 0 0
ð24Þ
Table 2 Parameter values for normalization and their significance in the samples. Parameter
r0 Mpa
0 hg K t0 s
max for tests on Polyurethane
max for tests on Epoxy resin
2.4%
4%
10%
9.1%
9.1%
0%
0.7968 0.0230 328 100
1.2513 0.04017
2.06997 0.0962
0.85 0.091 343 100
0.8 0.091
0 0
Parameter
Significance
r0 Mpa
Experimental value of stress at constant hhigh and applied constant Experimental applied constant max Glass-transition temperature of sample Suitable fraction of rate experimental processes
0 hg K t0 s
max
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P. Ghosh, A.R. Srinivasa / International Journal of Engineering Science 49 (2011) 823–838 Table 3 Process inputs w.r.t time. Process
Stress/ Strain rate input ðgtÞ
Stress Control: AðtÞ
Strain Control: BðtÞ
Temperature rate i/p i.e. ðf tÞ
a–b: 0 < t < t1 b–c: t1 < t < t2 c–d: t1 < t < t2 d–e: t3 < t < t4
g 1 ðtÞ 0 g 2 ðtÞ 0
0 0 1 1
1 1 0 0
0 f1 ðtÞ 0 f1 ðtÞ
Table 4 Specifications of the processes obtained from experimental data of Polyurethane and Epoxy resin. Process specification
For polyurethane
For Epoxy resin
Strain-rate Loading and unloading hold-time Heating and cooling rate Total rise or drop in temperature
5%–50% per minute 120 min ±4 K/min 40 K
0.03%–3% per minute 127 min ±1 K/min 85 K
where,
8 > > < 0; a ¼ 1; > > : 1;
p 2 Þp 6 k r þb ða 2 r ða þ bp Þp > k r ða þ bp 2 Þp < k
ð25Þ
5.2. Simulation specifications of the thermomechanical cycle Since we are dealing with rate equations to describe the system, we have extracted the experimental data from Tobushi’s and Liu’s work with respect to time, which are then used in the governing equations as shown below.
; Initial conditions: In terms of the state variables as shown in the section above, r ; h; p ; k ¼ f0; 0; hhigh ; 0; 0g. For this the specifications of the processes are obtained from experimental data of Polyurethane (Tobushi et al., 1997) and Epoxy resin (Liu et al., 2006) as tabulated in Tables 3 and 4. 6. Temperature dependence form of material parameters 6.1. The yield stress function The key to the response of the SMP material is the variation of the yield stress. The yield-stress of the material has a sensitive dependence on temperature, and the material yields differently depending on not just the current value of the temperature but also on whether the temperature of the material dropped or increased from the previous time-step. Thus there is a hysteresis of the yield-stress from the cooling to the heating cycle, which gives the different trends of the stress-rise during cooling and the strain-recovery during heating. The unrecovered strain at the end of the cycle is also explained by this hysteresis, because of the plastic yielding of the material in the thermomechanical cycle. These considerations suggest that the yield-stress rate has the following functional form, and thus the yield-stress varies as shown in Fig. 4.
dh _ ¼ ^f h; ; sgn dh k dt dt
ð26Þ
The factors that become important in this setup are as follows: (a) hhigh and hlow : the value of the yield stress is kept at a low minimum kmin, almost zero, so that in the initial range of high temperature stretching, the instantaneous rise in stress in the dry-friction damper overcomes its yield-stress and hence the plastic strain starts rising in the dampers. Once the stress reaches a constant value, there is no more sliding of the dry-friction damper, and hence the plastic strain gets ‘‘locked-in’’ after the initial rise. On the other hand, at low temperature towards the end of the cooling range, the yield stress of the dashpots reaches its maximum limit for that particular temperature. The stress in the dashpot network, even though in compression in the stress-relaxation range, its magnitude never exceeds the yield-stress and hence the friction dashpot helps ‘‘lock’’ the plastic strain for this range as well. (b) h: the yield-stress of the material is temperature dependent and changes during heating or cooling of the material. During cooling, the stress of the material rises. This is effected mainly by allowing stress in the dashpots to the rise by varying the yield stress of the material. The stress in the dashpots overcomes the yield-stress and hence the friction
P. Ghosh, A.R. Srinivasa / International Journal of Engineering Science 49 (2011) 823–838
1.6
1.2
1.4
1 strain & yieldstress
stress & yieldstress
832
1.2 1 0.8 0.6 0.4
0
0.4
0 1 1.05 temp total stress yieldstress cooling yieldstress heating
1 0.8 0.6 0.4 0.2 0.95
1 temp
1.05
0.95
yield stress rate for heating
0.95
yield stress rate for cooling
0.6
0.2
0.2
0
0.8
1 1.05 temp total strain yieldstress cooling yieldstress heating
−0.1 −0.15 −0.2 −0.25 0.95
1 temp
1.05
Fig. 4. Variation of (a) yield-stress with temperature in a thermomechanical cycle and (b) yield-stress rate with temperature during cooling and heating processes, for experimental data of polyurethane with maximum strain of 4%.
_ here is prescribed to be zero till almost above glassdashpot slips further. The rate of yield-stress of the material k transition, and starts rising rapidly only at lower temperatures below glass-transition (Richeton, Ahzi, Daridon, & Rémond, 2005). Because of this, the rise in stress can be controlled carefully, and kept constant till temperature drops to hg, and then the stress is made to rise in desired fashion at lower temperatures. Thus a sigmoidal function ^f 1 is used which asymptotes between kmin and kmax. (c) ddht : the yield stress controls the way the plastic strain evolves. Having the same function of rate of yield-stress during heating, as specified for cooling f1 in the above point, leads to unsatisfactory strain recovery from the model. Also, complete recovery of the yield-stress implies complete recovery of the plastic strain at the end of heating, which will not mimic the residual strain recovery at the end of the cycle. For finer control on the response curves of stress and
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833
1 and k 2 (see strain during cooling and heating, respectively, it becomes obvious that different trends of yieldstress k Fig. 4(a)) are needed depending on whether the material is cooled or heated, i.e. whether sign ddht < or > 0. Thus Eq. (26) will depend on three cases as follows:
8 0; if ddht ¼ 0 > > < _ ¼ ^f ðh; Þ dh ; if sgn dh ¼ v e k 1 dt dt > > : ^ dh f 2 ðh; Þ dt ; if sgn ddht ¼ þv e
ð27Þ
This hypothesis suggests that the material yields differently depending on not just the current value of the temperature but also on whether the temperature of the material dropped or increased from the previous time-step. ^f 2 is adjusted so that desired shape recovery is obtained and is reduced to a value that will give ideal residual shape recovery response. This adjustment was done by extracting a Gaussian curve-fit to get an approximate idea of the functional form that will fit data for all three strain levels for validation. The residual yield-stress at the end of heating range will affect the next cycle of loading, and will account for the thermal hardening of the material under cyclic loading. Thus the stress and strain response during cooling and heating, respectively, can be controlled by adjusting ^f 1 and ^f 2 . This can be witnessed in Fig. 4(b). Details of this effect max high encan be seen in Fig. 6. For example, the initial constant strain response during heating is maintained by keeping k ough at the end of the cooling cycle, so that the yield stress takes a while before it becomes comparable to the stress in the dampers. Only then the plastic strain gets ‘‘unlocked’’, and strain recovery begins. (d) : Although results of the previous hypothesis give considerably improved results, it becomes evident that the rate of yield stress of the material being dependent on h and sgn ddht , it has the same trend regardless of the strain level. However the strain recovery of polyurethane during heating for the three different strain levels we are working with for validation, have different trends (refer Section 7). It is also evident in the compression and tension experiments of epoxy resin, that the strain level implies different yield tendencies of the material. Although the maximum stress rise during cooling and the thermal strain opposing factor during heating (refer to Section (6.2)) contribute in the strain recovery characteristics at different strain levels, the changes accounted for are not satisfactory. Thus, the yield stress rate needs to take into account the strain level the material is at, to affect the strain recovery differently at different strain levels. Note that the rate yield stress is only a linear function of the strain, and this dependence will disappear for the yield-stress, and hence not cause any inconsistencies with the assumptions for the rate of dissipation function. _ ¼ f^1 ð _ ¼ f^2 ð Þ ddht for cooling and k Þ ddht for heating. Thus we have the rate of yield stress as k h; h; Taking all these factors into consideration, the final equations of rate of yield stress have the following form:
f1 ¼ a þ d1 bðsinhðcpðh h1 ÞÞÞ f2 ¼ d d2 e 1 ðf tanhðmh þ nÞÞ2 where, m = 2/(hmax h2), n = 1 mhmaxand, a, b, c, d, e, f are constants depending on material type and applied max, h1 and h2 are limiting values of h until which there is no rise/fall in the stress/strain during the cooling/heating cycle. This is determined from the experimental data, for example, refer to the experimental data for maximum applied strain of 4% in Fig. 7, where these values will be h1 = 0.98 (value of h until which there is no stress rise in the cooling process) and h2 = 0.95 (value of h until which there is no strain recovery in the heating process), and
d1 ¼ d2 ¼
1; 8h 6 h1 0;
8h > h 1
0;
8h < h 2
1; 8h P h2
ð28Þ
ð29Þ
The rate of yield-stress specified by the above equations for the experimental data corresponding to Fig. 7 is as shown in Fig. 4(b). 6.2. The thermal expansion The thermal expansion of the material is dependent on temperature. Let us consider the typical shape memory polyurethane thermal response as shown in Fig. 5. The average thermal expansion coefficient in the rubbery state (a1 = 21.6X105K1) is about two times of that in the glassy state (a2 = 11.8X105K1) (Gunes et al., 2008). All thermal strains being measured with reference to the high temperature, the thermal strains at lower temperatures, are the largest. The thermal behavior of the material although aids in the cooling process, it counteracts the recovery behavior during the heating process (refer to Fig. 6). This counteraction is what causes the total strain to rise at the initial period of heating, and this can be controlled by the thermal strain completely. The material response is very sensitive to the thermal expansion coefficient, and the final rise of stress at the end of cooling range and the rise of strain during the start of heating range are affected severely by it. Therefore, a more accurate specification of the thermal expansion gives closer results to the actual
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Fig. 5. Variation of thermal strain with temperature. Take note of (a) Thermal strain for constant high temp value a1, axis on LHS and associated unaccounted thermal strain, (b) thermal strain for constant low temp value a2, axis on RHS and associated unaccounted thermal strain, (c) thermal strain for varying values of a, axis on LHS.
total strain elastic strain thermal strain strain in dampers
1 0.8
Strains
0.6 0.4 0.2 0 −0.2 0.94 0.96 0.98 1 1.02 1.04 1.06 temperature Fig. 6. Plots of damper, thermal and elastic strain and their combination to produce the total strain in the complete cycle. Notice that the thermal strain is opposite to the damper strain during recovery, and the effect is pronounced at the beginning of recovery.
experimental data. Note however that the thermal strains of the material is dependent on the temperature alone, and this role of thermal expansion in the strain recovery process reduces with increasing strain levels, because the total amount of strain to be recovered is rising while the amount gained back by the thermal expansion remains the same at all strain levels. Hence, for the polyurethane experiments, at low strain levels, the thermal strain accounts for almost 20% of the total strain as shown in the contributing factor C. F. values in Table 5, but at high strain levels its effect is only 4%. This implies that at increasing strain levels, 96% of the recovery will have to be dictated by the yielding of the material and not the thermal expansion. Also, as discussed previously, the strain recovery process and the thermal expansion process during heating counteract each other, as seen in Fig. 6. This implies that a relatively low value of thermal expansion in comparison to recovered
Table 5 Comparison of the strain responses and thermal strains, of shape memory polyurethane at various applied strain (max) levels.
max (%)
Shape fixity F (%)
Shape recovery R (%)
th ð%Þ C:F ¼ max
O:F ¼ C:F R
2.4 4 10
16.66 12.5 17
12.5 17.5 6
19.35 11.61 4.64
1.54 0.66 0.77
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strain is desired for improved shape memory effect. This is measured in terms of opposing factor O.F, as shown in Table 5, lower O.Fs being desirable. 7. Results 7.1. For shape memory polyurethane The glass transition of the material was 328 K, the thickness of the specimen was about 70 lm. The width was 5 mm, gauge length 25 mm and total length of 75 mm (Tobushi et al., 1997). The yield-stress function was chosen such that the model gave reasonable results for all three strain levels.
stress and strain vs temp
stress vs temp
1
stress
stress
1.5 1 0.5
0 0.5 strain
0
1 0.95 temp
1.05
0 0.95
1 1.05 temp stress vs strain
strain vs temp 1 stress
strain
1.5 0.5
1 0.5 0
0 0.95
1 temp
1.05
0
0.5 strain
1
Fig. 7. Comparison of the experimental data of PolyUrethane by Tobushi et al. (1997) with the prediction of the model developed here for
stress and strain vs temp
stress vs temp 2 stress
2 stress
max = 4%.
1 0 1
1.5 1 0.5
0.5 strain
0
1 0.95 temp
1.05
0 0.95
1 1.05 temp stress vs strain
strain vs temp 1 stress
strain
2 0.5
1.5 1 0.5
0
0 0.95
1 temp
1.05
0
0.5 strain
1
Fig. 8. Comparison of the experimental data of PolyUrethane by Tobushi et al. (1997) with the prediction of the model developed here for maximum applied strain of 2.4%.
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stress and strain vs temp
stress vs temp
stress
stress
1.5 1 0.5 0 0.5 strain
0
1 0.95 temp
1.05
1 0.5 0 0.95
1 1.05 temp stress vs strain
strain vs temp 1 stress
strain
1.5 0.5
1 0.5
0
0 0.95
1 temp
1.05
0
0.5 strain
1
Fig. 9. Comparison of the experimental data of PolyUrethane by Tobushi et al. (1997) with the prediction of the model developed here for maximum applied strain of 10%.
The simulation results shown in Fig. 7 compare the results of the model and experimental data for max = 4%. The parameters a, b and l are selected to control the stress response of the model during the initial high-temperature stretching (process a–b). The stress vs. strain graph will show how the non-linearity of the response comes into play. For higher strains this can be seen in Section 7, Fig. 9 for 10 % strain. The yield-stress functions are chosen to control the stress-rise during cooling (process b–c) and the strain recovery during heating (process d–e) as discussed in Section 6.1. The parameter E2 is selected to control the strain response i.e. the shape fixity, during low temperature stress relaxation (process c–d). The yield-stress rate function chosen for this material response is dependent on strain levels, and hence with change in the maximum strain applied in the thermomechanical cycle, the yield-stress changes slightly to accommodate the changing responses in the strain levels as can be seen in Figs. 7, 8 and 9 for 4%, 2.4% and 10% strain respectively. The non-linear response becomes obvious at this higher strain level as can be seen in the stress vs. strain plot in Fig. 9. The parameters a and b were chosen to exhibit non-linearity in the model. The stress rise during cooling, for the chose yieldstress function, is not satisfactory and this consequently affects the strain recovery during heating.
7.2. For shape memory epoxy resin thermal ¼ ^f ð A thermal strain function hÞ is fitted from experimental data, and used in the model in the rate form directly as previously. below, rather than in terms of the conductivity a
r_ E2 _ þ E2 _p þ E2 _ thermal ¼ 0
ð30Þ
The validation results for shape memory epoxy resin undergoing tension, compression and no load strain conditions are provided in Figs. 10–12 respectively.
stress vs temp
strain vs temp strain
stress
4 3 2 1 0.8
0.85
0.9 0.95 temp
1
0.8 0.6 0.4 0.2 0.85
0.9 0.95 temp
1
Fig. 10. Comparison of the experimental data of epoxy by Liu et al. (2006) with the prediction of the model developed here under tension of 9.1%.
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stress vs temp 3 strain
stress
2 1 0 −1 0.8
0.85
0.9 0.95 temp
strain vs temp
0 −0.2 −0.4 −0.6 −0.8
1
0.8
0.85
0.9 0.95 temp
1
Fig. 11. Comparison of the experimental data of epoxy by Liu et al. (2006) with the prediction of the model developed here under compression of 9.1%.
stress vs temp
2
−0.5
1 0 0.8
strain vs temp
0 strain
stress
3
0.85
0.9 0.95 temp
1
−1 0.8
0.85
0.9 0.95 temp
1
Fig. 12. Comparison of the experimental data of epoxy by Liu et al. (2006) with the prediction of the model developed here in the undeformed case.
8. Conclusions The yield-stress of the friction dashpot plays the lead role in controlling the response of the model during heating and cooling. Different functions of the yield-stress evolution during heating and cooling processes give improved results in strain recovery. This suggests that there is a hysteresis in the yield stress of the material when it undergoes a heating–cooling cycle. The yield stress rate, when made to be a function of the strain of the material, affects the recovery behavior during heating, which helps simulate the significantly different strain recoveries at the different strain levels. This suggests that the yielding of the material is influenced by the extent to which it is loaded. The variation of the thermal strain of the material during the heating and cooling processes assists in the stress rise during cooling, but opposes the strain recovery during heating. It was concluded that low temperature thermal expansion coefficient has strong influence on the stress response during cooling and strain recovery during heating. However, the significance of the contribution of the thermal strains on the material response reduces with increasing strain levels of the experiments. The model exhibits non-linearity of the response at high strain levels for the polyurethane experiments, which is accounted by the non-linear response of the rubbery spring. However, even though the non-linear behavior of the material is evident at higher applied strain levels during loading, it does not affect the recovery of the material significantly. This may suggest that the recovery responses at high strain levels are governed mainly by the yielding of the material, and since the recovery patterns are not monotonic with increasing applied strain levels, as can be seen in the shape recovery factors R% in Table 5, there may be some other mode of yielding or failure that has to be taken into consideration. Acknowledgement The authors would like to acknowledge the support of the National Science Foundation CMMI Grant 1000790 in carrying out this research. References Abrahamson, E., Lake, M., Munshi, N., & Gall, K. (2003). Shape memory mechanics of an elastic memory composite resin. Journal of Intelligent Material Systems and Structures, 14(10), 623–632. Barot, G., & Rao, I. (2006). Constitutive modeling of the mechanics associated with crystallizable shape memory polymers. Zeitschrift f ur¨ Angewandte Mathematik und Physik (ZAMP), 57(4), 652–681. Barot, G., Rao, I., & Rajagopal, K. (2008). A thermodynamic framework for the modeling of crystallizable shape memory polymers. International Journal of Engineering Science, 46(4), 325–351. Beloshenko, V., Varyukhin, V., & Voznyak, Y. (2005). The shape memory effect in polymers. Russian Chemical Reviews, 74(3), 265–283. Bhattacharyya, A., & Tobushi, H. (2000). Analysis of the isothermal mechanical response of a shape memory polymer rheological model. Polymer Engineering and Science, 40(12), 2498–2510. Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics. New York: John Wiley & Sons.
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