- Email: [email protected]
Modeling memory effects in amorphous polymers
International Journal of Engineering Science 84 (2014) 95–112
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Modeling memory effects in amorphous polymers R. Sujithra, S.M. Srinivasan ⇑, A. Arockiarajan Department of Applied Mechanics, Indian Institute of Technology, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 5 January 2014 Accepted 20 June 2014
Keywords: Shape memory polymers Amorphous polymers Memory effects Multiple natural configurations ABAQUS VUMAT subroutine Glass transition Constitutive modeling
a b s t r a c t A model that can describe the shape memory behavior of amorphous polymers based on theory of multiple natural configurations is presented and implemented for use in applications. The thermodynamic framework presented considers evolution of multiple natural configurations and models the key features, especially, the temperature and deformation dependent change in stiffness and shape memorizing characteristics. A single parameter called the degree of glass transition is used to describe the memory locking and releasing characteristics. Using this framework and specialization to small deformations, the developed model is implemented in ABAQUS through VUMAT subroutine feature and correlated with existing epoxy shape memory experiments. Various thermomechanical conditions on structural components such as rods under tension and compression and beams are analyzed. Simulations results confirm that the model is capable of predicting the memory characteristics including the multi-shape memory locking and release mechanisms. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Shape memory polymers (SMPs) are classified as smart material due to the capability to change their shape temporarily from a permanent shape by the application of a thermo-mechanical stimulus that renders this material can be tuned to possess numerous temporary shapes. While maintaining the operating temperature, it remains in the temporary shape (under low mechanical loads); and by reheating the SMP, it reverts back to the permanent shape. Since it remembers one shape at a time, this process is referred as one way actuation process or one way shape memory effect (Lendlein & Kelch, 2002). The shape change phenomena over the transition temperature makes this material to act as a sensor and also as an actuator. Shape memory polymers are inexpensive, light weight, malleable, damage tolerant, bio-degradable, can be molded into a variety of complex shapes by conventional processing technique and also provide large recoverable strains with distributed actuation for complex shape changes (Monkman, 2000). These highlighted features make shape memory polymers a unique material with wide range of applications in various fields such as deployable components in aerospace structures, bio-medical, smart textiles, self-healing composite systems, optical reflectors, morphing skins and automobile actuators (Lendlein & Langer, 2002; Leng, Lan, Liu, & Du, 2011). Shape memory polymers are classified based on the cross-links and the transition temperatures (Lendlein & Kelch, 2002; Liu & Mather, 2007). Some of the commonly used shape memory polymers are polyurethanes, polyolefin, epoxy and polyether esters; for an overview refer (Lendlein & Kelch, 2002). Based on the application, SMPs can be tailor-made based on the transition temperature, varying the chemical composition and the preparation method. Material design at the molecular level plays a prominent role in tuning these polymers. Though, lot of experiments are being carried out and reported in ⇑ Corresponding author. E-mail address: [email protected] (S.M. Srinivasan). http://dx.doi.org/10.1016/j.ijengsci.2014.06.009 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.
96
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
the literature for the successful development of shape memory polymer-based devices, it is necessary to develop theoretical models to predict the shape memory behavior; to be specific in the memory-dependent characteristics (Leng et al., 2011; Xie, 2011). Several models have been proposed in literature (Zhang & Qing-Sheng, 2012) to describe the behavior of shape memory polymer. The constitutive modeling can be distinguished based on the assumptions of different approaches namely: rheological (Tobushi, Okumura, Hayashi, & Ito, 2001; Bhattacharyya & Tobushi, 2000; Li & Wei, 2011), micromechanical (Liu, Gall, Dunn, Greenberg, & Diani, 2006; Chen & Lagoudas, 2008), phenomenological (Qi, Nguyen, Castrao, Yakacki, & ShandaSa, 2008; Kim, Kang, & Yu, 2010), plasticity (Pritha & Srinivasa, 2011; Baghani, Naghdabadi, Arghavani, & Sohrabpour, 2012; Ghosh & Srinivasa, 2013) and rate dependent (incorporating structural relaxation) models (Nguyen, Qi, Castrao, & Long, 2008; Nguyen, 2013; Westbrook, Kao, Castro, Ding, & Qi, 2011). A thermo-mechanical constitutive model was developed by modifying a standard linear viscoelastic model wherein a slip element is introduced to take into account of internal friction (Tobushi, Hashimoto, Hayashi, & Yamada, 1997). A micro-mechanical model was proposed using the free energy function (ClausiusDuhem inequality) wherein the thermo-mechanical behavior of SMPs can be described using the concept of entropy and internal energy (without explicitly considering the molecular interactions) (Liu et al., 2006). A simple thermo-visco-elastic model (neglect the effects of heat conduction and of pressure on the structural relaxation and inelastic behavior) was presented by incorporating the nonlinear Adam–Gibbs model of structural relaxation into a continuum finitedeformation (Nguyen et al., 2008). Recently, a simple one-dimensional model was proposed in which the hysteretic behavior during heating and cooling phenomena was captured by introducing a yield-stress function (Pritha & Srinivasa, 2011). While designing SMP based devices, it is important to understand the response of SMP under complex force and different temperature protocols which will help to get an optimal device design. To be specific, it is important to have a minimalistic model with the flexibility of being able to add the additional features such as rate effects, large deformations, cyclic damage and ageing. An attempt has been made by Barot, Rao, and Rajagopal (2008) and Rajagopal and Srinivasa (2004) to model the memory effects for the crystallizable polymers using a thermodynamic framework based on the theory of multiple natural configurations. The evolution equation for the natural configuration during phase change is obtained by maximizing the rate of entropy production. This natural configuration approach (Rajagopal & Srinivasa, 2004) is capable of describing a variety of responses ranging from traditional plasticity to smart materials under a single framework. However, it would be appropriate to choose parameters in the model that directly link to the physical basis of the function characterizing the response. From the designers perspective, for complex geometries, it may be necessary to introduce the simplistic model within finite element framework. Thus, objective of this work is to model the fundamental behavior of shape memory polymers and conduct numerical test for the memory locking and releasing characteristics by using glass transition temperature as the primary locking mechanism. Multiple natural configuration theory is used as the basic framework for the proposed model and simulation. Two unique features of shape memory polymer, namely the change in stiffness and shape memory characteristics rendered by controlling temperature and deformation (stress) are characterized using a single parameter that is chosen directly link to evolution of multiple natural configurations and therefore, the memory response. The proposed model is limited to small deformations and is implemented using a material subroutine in an existing commercial finite element software (ABAQUS) to analyze device related structural elements. Simulated results based on the developed model is compared with available experimental observations. In addition to that, thermo-mechanical history based memory characteristics, thermal expansion characteristics and multiple shape memory effect are simulated to show the performance of the model. The paper is organized as follows: Section 2 briefly reviews the structure–property relationship for the shape memory effect exhibited by amorphous polymers in order to highlight the importance of introducing the glass transition degree parameter in this model. The proposed modeling framework is elaborated in Section 3. Section 4 focusing on explaining the framework is specialized to small deformations by assuming that the strains of order 10% or less with linear elastic response in the rubbery as well glassy phases. Simulation results at the material point level as well as for a simple prismatic bar and beam of the material are presented in Section 5. The capabilities of simulating various features of a typical SMP are shown also discussed. Finally, various numerical simulations are carried to show that the model is capable of capturing typical memory characteristics of the SMPs. 2. Structure property relationship in SMPs The shape memory effect observed in shape memory polymer is represented by the shape memory cycle as shown in Fig. 1 and the corresponding thermo-mechanical loading cycle is shown in Fig. 2, in which the stress (or strain) and the temperature are controlled at different time steps. Depending on temperature, amorphous polymers exist in two different states, the rubbery state and the glassy state. Here, T g act as a transition switch between these two states. There are three representative temperatures that is generally used to describe the shape memory behavior through glass transition. A high temperature, T high that sees beyond the completion of the effect of glass transition and similarly, a low temperature, T low that sees near completion of the freezing effect of glass transition. The shape memory polymer network consists of chain segments which are interconnected by net points. At temperature above T g , it exists in rubbery phase. This is characterized by the long chain molecules, oriented in a random fashion and molecules vibrating vigorously with high kinetic energy. The effect of interactions between molecules are generally negligible that it exhibits a state akin to an ideal gas. At T high (step-1), the
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
97
Fig. 1. Typical dual shape memory cycle.
Fig. 2. A sample thermomechanical loading for shape memory effect.
permanent shape (A) is easy to deform to any temporary shape (B), as the stiffness of the material is very low. During deformation, the molecular chains are oriented in the direction of the stretch, thereby reducing the entropy of the system. This phenomenon is called entropic elasticity. Here, net points prevent the chain from slipping and contribute to the elasticity of these polymers at T high . Keeping the deformation fixed (step-2), it is gradually cooled to store the desired shape (C). Here, the chain segments act as molecular switches and enable the fixation of temporary shape. As the temperature is decreased below T g , the vibration of the molecules gradually decreases due to the freezing of newly formed bonds between the molecules. The polymer moves towards glassy behavior, which makes it stiffer and the interaction between the molecules increase due to internal energy. The elasticity exhibited at these temperatures is termed as energy elasticity. Due to thermal contraction and gradual stiffening of the polymer as it move towards the glassy behavior, results in the increased stress level. Unloading at low temperature results in a small spring back in strain, due to the high stiffness in the glassy state (step-3).
98
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
This deformed shape (D) is stored as long as the temperature is maintained at T low . By heating up the material in the stressfree state (step-4), the bonds that were formed during the cooling are broken. The net points help in regaining its original shape and due to entropy elasticity, the material is forced back to its permanent shape (A). It can be observed that most of the strain is released around the glass transition temperature. 3. A framework for modeling SMPs The framework proposed here is based on the theory of multiple natural configurations. There are two distinct but related phenomena that occur during the shape memory process pivoted around the glass transition state. One is the change in the stiffness of the material and the other, the change in the natural configuration of the body leading to locking up of strains. While modeling the shape memory phenomenon, the glass transition acts as a memory switch. As the first step in developing this model, the response of the polymer is assumed to be rate-independent and thermo-elastic, neglecting the viscous effects. The viscous effects are significant only in the glass transition region and for all practical purposes, it is assumed that the material resides around the glass transition temperature over a small period time only and thus, negligible. Moreover, the structural relaxation may not affect memory characteristics, since such a relaxation is not pronounced at the temperatures of operation in relation to shape memory utilization of polymer. But these features can also be accommodated later into the model. Many solid bodies are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion (Rajagopal & Srinivasa, 2004). These stress free configurations are known as natural configurations. The evolution of the multiple natural configurations during a typical shape memory cycle is shown in Fig. 3. The permanent shape at high temperature occupies a stress-free configuration (A), since the body is in an unload state. When it is deformed at high temperature, its configuration changes (B). When the external load is removed, the material regains its original configuration (A). In this work, this original configuration is assumed to be the reference natural configuration. The change in natural configuration of the body depends on the class of processes that the body is allowed to undergo (Rajagopal & Srinivasa, 2004). During cooling and heating, the SMP undergoes changes in natural configurations that depend on the strain it has undergone from a reference configuration. In order to model the shape memory characteristics, a reversible transformation is assumed to take place between the reference configuration and a natural configuration that is dependent on temperature and the current strained state. The process of transformation to the current configuration from the natural configuration and vice versa due to stress alone is taken to be purely elastic. The current deformed configuration (C) is an elastic detour from the temporary stress free natural configuration (D) at the low temperature. During this process, body possess multiple natural configurations (W1 ; W2 ; W3 ; . . .) and it has an instantaneous elastic response from each of these natural configuration. At any instant, the natural configuration at a particular temperature associated with the current configuration represents the current structural shape. The shape locking that occurs during freezing, depends on the current deformed configuration. From the current configuration (C), it reaches the stress-free configuration (D) by a rapid unloading process. While unloading, the body will take its configuration depending on the unloaded temperature. During reheating, the
Fig. 3. Stressed, stress free and multiple natural configurations in SMP cycle.
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
99
lock-in strain stored during cooling is released. Here a critical assumption is made that the freezing and release processes are strictly reversible. In other words, the response of the material during reheating depends upon the unloaded temperature, that the body will retrace its path from one natural configuration to other. More details are discussed in Sujithra, Kumaran, Arockiarajan, and Srinivasan (submitted for publication). To keep track of the natural configuration at any instance, a suitable parameter is introduced based on the experimental data. The typical test that is conducted on polymers is the dynamic mechanical analyzer (DMA) test. Usually, three curves are present in this: curves related to storage modulus, loss modulus and tan d. A typical DMA test result for an amorphous shape memory polymer is shown in Fig. 4. The storage modulus is directly related to the elastic stiffness of the material and tan d to the dissipative (viscous) nature of the material. In an amorphous shape memory polymer, the elastic stiffness goes from a very insignificant stiffness in the rubbery state to a stiffness that is orders of magnitude higher at the glassy state passing through the glass transition temperature. Being a higher order transformation, such a transition occurs over a band of temperature near the T g , with the peak activity at the T g . It can be seen the elastic stiffness goes through a near sigmoidal variation in the T g zone where the glass transition activity is the maximum. It can also be observed that elastic stiffness has two asymptotes – one at the rubbery end and the other at the glassy end to indicate that a parameter that is purely dependent on temperature can be introduced that takes the shear modulus from the rubbery modulus to the glassy modulus as,
GðWÞ ¼ ð1 WÞGr þ WGg
ð1Þ
where Gr is the rubber shear modulus and Gg is the glassy shear modulus. Since a linear variation in W is assumed for the stiffness change over temperature, W assumes a functional form for a sigmoidal curve which can be determined from the stiffness curve available from the DMA data. Since this change is primarily due to the change in natural configuration, W is taken to be the parameter that is provides information on the degree of glass transition and thus, is named the degree of glass transition parameter (dogt parameter). W is a purely temperature dependent function. This sigmoidal function hinges around the temperature T g asymptotically reaching the values of 0 at T high and 1 at T low in tune with the observation on the storage modulus curve shown in Fig. 4. It is now assumed based on the framework described above, that all property changes are associated with the dogt parameter that represents the natural configuration change. For example, the thermal expansion coefficient can also be taken to be changing with the dogt parameter as,
a ¼ ð1 WÞar þ Wag
ð2Þ
4. Specialization and simplifications for simulation 4.1. Modeling for small deformations Using the framework described in Section 3, a small deformation model is implemented by assuming the strains in the order of 10% or less. Within this region of strains, the material can be assumed to be practically linear elastic for both rubbery
Fig. 4. A typical DMA test data – storage modulus, loss modulus and tan-d curves.
100
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
and glassy phases. Because of the smallness in deformations, the total strain can be decomposed additively into elastic strain, ee , thermal strain, eT and stored strain es , respectively. Thus,
e ¼ ee þ eT þ es
ð3Þ
When the SMP goes through the glass transition, the deformed configuration is progressively frozen to form the natural configuration. The natural configuration in this model is categorized using the stored strain es . The natural configuration is obtained by applying a strain es on the original reference configuration. Deforming the current natural configuration by ee , one obtains the current deformed configuration. In a strain controlled experiment, for any given temperature the stress can be calculated from,
r ¼ EðWÞ : ee ¼ E : ðe es eT Þ
ð4Þ
where E is the effective modulus of the polymer at the given state (provided by the dogt parameter). Initially, in a typical shape memory cycle, at the reference state, shape memory polymer exists in the rubbery state at T high and is deform to temporary shape (step-1) holding the temperature. Here, the thermal strain and stored strain are zero. Thus, the stress–strain relation at T high is given as
r ¼ Er : ee
ð5Þ
Then keeping the total strain fixed, the temperature is reduced from T high to T low (step-2). The variation of the thermal strain with change temperature is given by (assuming isotropy of the SMP),
_ e_ T ¼ aTI
ð6Þ
During the cooling phase of the shape memory cycle, the dogt parameter provides the potential for locking up of the current configuration. The locking up of the current deformed configuration into natural configuration is expressed in terms of stored strain. The current natural configuration is represented by es and current deformed configuration from the current natural configuration is given by the current elastic strain, ee . For a change in the dogt parameter, a part of the current deformed configuration is locked so that the current natural configuration changes to a new one that includes the locked amount of current deformed configuration. Thus, the evolution of the stored strain is given by,
e_ s ¼ gðWÞee W_ ; W_ < 0
ð7Þ
where gðWÞ quantifies the freezing or locking capability at the appropriate temperature. This parameter is related to the current shear modulus, G and the reference shear modulus Gr . The update of the natural configuration is affected by the stored strain, which is updated appropriate to the strain stored in memory. During heating or reheating, i.e. dT > 0, the history of memory is retraced and the appropriate stored strain release to realize the updated natural configuration. Once the strain at a particular W is released, that memory is erased in the stored energy function and the functional values are available only for values above that W. This is incorporated using a continuous Preisach’s model as in Sujithra et al. (submitted for publication). Thus,
es ¼ es ðWÞ
ð8Þ
In the above model described, it must be noted that the total strain and the temperature are independent (controllable) variables. 4.2. Material functions and parameters for simulation Simulations have been carried out for an amorphous epoxy resin for which experimental data is available in Liu et al. (2006). As has been mentioned in Section 3, typically, a DMA test is conducted to determine the variation of storage modulus with temperature taken from T high to T low through the glass transition temperature, T g . For the DMA data in Liu et al. (2006), the glass transition temperature (T g ) is found to be 343 K (tan d peak) and the temperature range is chosen from T low ¼ 273 K to T high ¼ 358 K, to simulate the memory cycle (see Fig. 4). Poisson’s ratio is assumed to be 0.4 for glassy state in the elastic region and 0.499 for the rubbery state. It is varied between these two through the dogt parameter. The bulk modulus for epoxy is taken to be temperature independent value of 1 GPa. Most DMAs use bending tests while some use extension/contraction for determining the variation with temperature. Though there are differences in the results, they are often well within margins good for use in modeling. Since the glass transition phenomenon is the one that takes the amorphous cross-linked polymer from a rubbery nature to a glassy nature, the variation of the properties primarily hinge around a region of temperature around the glass transition temperature and then flatten out beyond that region. Eq. (1) provides the variation of the shear modulus with a dogt parameter which can found out by fitting with the storage modulus curve in the DMA result. Since the rubbery stiffness is usually orders of magnitude smaller than the glassy stiffness, it be a good idea to use expression for stiffness that arises from entropic elasticity,
Gr ¼ NKT
ð9Þ
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
101
where, N is the cross-link density, K is the Boltzmann’s constant and T is the absolute temperature. Notice that there is a variation of the stiffness with temperature that is captured by the equation. However, the slope of this stiffness curve with temperature is very small. In Fig. 5, the variation of the storage modulus with temperature from the DMA test seems to suggest that the glassy stiffness does not tend to a constant value but has a near constant slope at T low with respect to temperature. Hence, the variation of the glassy shear modulus is modeled using a linearly varying function with respect to temperature. Thus, the glassy shear modulus is given by
Gg ¼ G0 bðT T low Þ
ð10Þ
where G0 is the shear modulus at T low and b is the slope of the modulus curve in the glassy region. G0 and b are obtained directly from the DMA data. Since the dogt parameter follows a sigmoidal form with respect to temperature, a simple form is used (Liu et al., 2006):
W¼1
1 1 þ aðT r TÞn
ð11Þ
where a; T r and n are constants obtained from the curve fit of the variation of the storage modulus curve obtained from the DMA data. The variation of the transition parameter thus obtained vs. temperature is shown in Fig. 6. As can be seen, the parameter abruptly varies in the small region around the glass transition temperature. From the experimental uniaxial stress–strain results, the rubbery (Gr ) and glassy modulus (Gg ) are found to be 2.93 MPa and 268 MPa respectively. The temperature dependent modulus curve is fitted by using these end values as shown in Fig. 7. By using the change in natural configuration parameter, the variation of modulus with temperature is simulated using the Eq. (1). It is observed from the thermal strain behavior that the heating and the cooling curves coincide. There are two zones, one below T g (ag ¼ 0:9E 04) and other above T g (ag ¼ 1:8E 04), where the coefficient of thermal expansion (CET) is considered to be almost a constant. Using the transition parameter, the thermal strain curve is simulated using the Eq. (2). The variation of the thermal strain as function of temperature is shown in Fig. 8. The various constants used in this example are listed in Table 1. 5. Finite element simulation, results and discussion The simulations carried out at the 1-D material level for different test conditions on a specific SMP are provided in Sujithra et al. (submitted for publication). It is important to test the model under conditions of external loading on 3-D device components under boundary conditions that cannot be idealized to perform at the material level. Moreover, it is generally necessary to carry out a finite element or a similar numerical simulation to obtain results for conditions other than specification of strain and temperature. For example, a simple uniaxial testing of an SMP specimen would require a force or displacement condition at the axial direction and stress-free conditions at the side boundaries of the specimen. Such conditions are generally not simple when it comes to simulations at the material level. Therefore, the material model has to be incorporated within a numerical simulation framework, typically a finite element software. Numerical simulations have been carried out in ABAQUS by incorporating the model developed in a VUMAT subroutine. Explicit dynamics is chosen, because there are no simultaneous equations to solve and each increment is inexpensive. First,
300 Glassy modulus Gg = G0 − β (T − Tlow)
250
G (MPa)
200
150 Rubbery modulus G = G(f )T
100
r
f
CL
50
Tg
Tlow 0
280
CL
− Crosslinker density
290
300
310 320 330 Temperature (K)
340
Fig. 5. Variation of shear modulus with temperature.
T
high
350
360
102
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 6. Variation of the dogt parameter with temperature.
Fig. 7. Variation of shear stiffness with temperature.
Fig. 8. Variation of thermal strain with temperature.
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
103
Table 1 Model constants used for the sample material. Constants
Symbol
Values
Rubbery shear modulus Glassy shear modulus Bulk modulus Poisson’s ratio (glassy)
Gr Gg k
2.93 MPa 268 MPa 1000 MPa 0.4
Poisson’s ratio (rubbery) Thermal expansion coefficient
lg lr ar ag
0.499 1:8 1004 (1/K) 0.9 1004 (1/K)
Cross link density
N
9:86 104 ðmol=cm3 Þ
Boltzmann’s constant
K
1:38 1023 J/K
a
2:76 1005 (1/K4) 4 1300 ðkg=m3 Þ 0.15 (W/m K) 1250 (J/Kg K)
Coefficients for curve fit
n Density Thermal conductivity Specific heat
q j Cp
to validate the model with the experimental results, the shape memory cycle is simulated for tension (9.1%), compression (9.1%) and undeformed for an epoxy SMP for which experimental results are available in Liu et al. (2006). A tension test or a compression test on an SMP specimen, should be realistically modeled. Therefore, in this study, the first example chosen is a axial bar undergoing tension or compression mechanical loading in addition to thermal loading pertaining to a shape memory cycle. Another example in the form of a beam undergoing a shape memory cycle of thermomechanical loading is also simulated to show the capability of the model. 5.1. Simulation results of the axial bar example A simple T-shaped specimen (SMP) is modeled with a gripper (rigid body) for tension that will allow for pulling in the first stage of the shape memory cycle, unloading in the third stage and importantly, the fourth stage which is a stress free heating stage. Similarly, rectangular block with a rigid plate is used for simulating a compression test with all the four stages shape memory cycle possible. The simplest to model being just a block for simulating the undeformed block of SMP. The appropriate boundary conditions are shown in Fig. 9. Frictionless contact is defined between surface of the deformed body and the rigid body. The deformed body is discretized using C3D8T element in ABAQUS. Based on the thermo-mechanical loading cycle shown in Fig. 2, the displacement of the gripper and the temperature are controlled by the time steps to simulate a typical uniaxial test in tension or compression. Here, simulation results for a typical shape memory cycle for the above finite element model are first presented. Then, the analysis on the effect of different parameters on the response of the SMP model is discussed. 5.1.1. Isothermal stress–strain response As the first step in this shape memory cycle simulation, SMP block is heated to high temperature T high (358 K). Now, the block is stretched by the gripper for 9.1% tension (AB) and compressed by a plate for about 9.1% (A00 B00 ). This gives the linear stress–strain plot (step-1) at T high , as shown in Fig. 10. At high temperature, it exists in the rubbery state and during deformation, stresses generated in the block are generally very low. 5.1.2. Stress evolution during cooling and heating In step-2, by keeping the displacement of the gripper and the plate at the same position for tension and compression, the temperature reduces to T low (273 K). For undeformed block, both ends are fixed. The simulated stress response for various pre-strain levels, with similarly varying trend is shown in Fig. 11. It shows a flat region until T g , due to low stiffness of
Fig. 9. FEA models of axial bar example with different boundary conditions for (a) tension (b) undeformed and (c) compression thermo-mechanical testing accommodating all the four stages of a shape memory cycle.
104
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 10. Stress–strain at T high temperature.
Fig. 11. Stress evolution during cooling at different predefined strains.
the rubbery state. As the temperature decreases further, thermal contraction takes place and the thermal stress gradually rises as the material stiffens. For the pre-defined compression above T g , the stress gradually decreases and there is transition to a state of tension. For the undeformed, due to constraints, thermal stress develops. At the end of the step-2, all the fixed strain is stored in the material as stored strain. After cooling, it is heated again with predefined strain constraint. Because of the reversibility assumption, the variation during heating curve coincides (CB, C0 B0 , C00 B00 ) with cooling curve (BC, B0 C0 , B00 C00 ) as shown in Fig. 12. 5.1.3. Evolution of various strains during cooling The evolution of various strains during cooling is shown in Fig. 13 for tension. It is seen that the total strain is pre-defined strain and is almost constant due to constraint. After loading at T high , the total strain is the equal to the elastic strain and there is no stored strain and thermal strain. During cooling, the elastic strain is transformed to the stored strain and almost equal to the pre-defined strain. This reflects the memory effect. Here, for simplicity, the thermal strain is neglected. 5.1.4. Free strain recovery In step-3, after cooling, the contact between the plate and the block is deactivated and the plate moves back to the initial position. Unloading at T low produces a small spring back in strain due to glassy stiffness. This spring-back strain is very small when compared to pre-strain levels. In step-4, the block is again reheated at the stress-free condition. The stored strain is gradually released as shown in Fig. 14. In both tension (DA) and compression (D00 A00 ) cases, most of the strain gets released within a small band of temperature around T g . For undeformed block (D0 A0 ), it shows only thermal strain. Initially, during heating, thermal expansion takes place, and after T g , releasing of stored strain is dominant. In the proposed model, it remembers the history of strain storage and retraces its path. But, VUMAT stores only two- state architecture; the initial values are
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 12. Stress evolution at subsequent heating after cooling at different strain levels.
Fig. 13. Evolution of various strains during cooling at constraint deformation.
Fig. 14. Evolution of stored strain during reheating at different fixed strains.
105
106
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
in old arrays and current values are in new arrays. So, at the end of step-3, the final amount of stored strain is stored in old array. By using this old stored array, the evolution for the release of new stored strain is determined:
e_ s ¼ es W_ ; dW > 0:
ð12Þ
5.1.5. Constrained stress recovery While the above 4-step process is a typical shape memory cycle, there is also an alternative shape memory cycle. In this alternative cycle, while the first three steps are the same as what has been explained above, before the start of step-4, in its unloaded state at T low , the strain is now kept fixed when heated to T high . This would change the stress level in the specimen because of the constraint effect with the initial stress being zero. The evolution of stress during heating for various fixed strain is shown in Fig. 15. Since there will be constraint for the release of stored strain and thermal expansion, compressive stress develops. As the temperature increases above T g , the material becomes soft and the stress shoots up to pre-defined stress (DE, D0 E0 , D00 E00 ). 5.1.6. Two types of cycles A complete cycle consists of: deformation at high temperature (AB), cooling to T low by fixing the strain or at constant stress, releasing the stress and heating in stress-free state to release of stored strain. The complete simulated shape memory cycle, by either strain-controlled (ABCDA) or stress-controlled mode (abcda) is shown in Fig. 16. 5.1.7. Effect of thermal expansion The thermal expansion is the temperature dependent behavior. The effect of thermal expansion in the shape memory cycle is simulated by keeping constant glassy coefficient, mixture of rubbery and glassy coefficient and zero coefficient thermal expansion. The stress responses are similar for constant glassy coefficient (ABCg DA) and mixture of both coefficients (ABCDA) (see Fig. 17). For the zero coefficients (ABC0 DA), it shows that the thermal coefficient has an impact on the final stress during cooling and also reduction in spring back strain. For the constant glassy coefficient, there is slight shift in strain recovery in the vicinity at T g (Fig. 18). 5.1.8. Simulations for memory characteristics For the memory characteristics studies, the shape memory cycle is simulated for tension deformation using displacement control method. The amount of strain stored is dependent on the loading and cooling temperature. The evolution of the stored strain at various loading temperature is simulated by loading at other than T high and subsequently it is cooled, unloaded and reheated to release the stored strain. The evolution of stresses at different loading temperatures (358 K, 348 K, 338 K, 328 K, 318 K, and 308 K) is shown in Fig. 19. It shows that significant amount of stress is needed for deformation below T g , due to increase in stiffness of the material. And also the memory activation is due to entropy and as well as by internal energy. The evolution of the different strain fixities at these various loading temperature is shown in Fig. 20. This graph shows the thermomechanical history based memory behavior. It is noticed that strain fixity drops down due to loading below T g . Due to storage limitations in VUMAT, strain recovery is not shown. Instead, evolution of stored strain during cooling is shown in Fig. 20. Instead of loading other than T high , now cooling is stopped above T low . The evolution of stresses at various unloading temperatures (358 K, 348 K, 338 K, 328 K, 318 K, and 308 K) is shown in Fig. 21. Unloading near T g , shows that most of the strain
Fig. 15. Evolution of stress during heating at fixed strain.
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 16. Shape memory cycles for tension.
Fig. 17. Stress response due to effect of thermal expansion coefficient.
Fig. 18. Evolution of strain for various thermal expansion coefficients.
107
108
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 19. Evolution of stresses at different loading temperatures.
Fig. 20. Evolution of stored strain after cooling at various pre-straining temperatures.
Fig. 21. Stress response where cooling stops at above T low .
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
109
Fig. 22. Evolution of strain during cooling at various pre-straining temperatures.
spring backs due to entropy. The evolution of strain during the SMP cycle for various unloading temperature is shown in Fig. 22. While reheating, it follows the same path from its unloading temperature. 5.1.9. Multi shape memory effects Recent studies show that Nafion has multiple shape memory effect in a single shape memory cycle and possesses broad glass transition (Xie, 2010). But, amorphous polymer has narrow transition region. In order to show effect of multiple natural configurations in our proposed framework, multiple shape memory effect is simulated using the same epoxy material parameters. The thermomechanical cycle for fixing multiple shapes is simulated as shown in Fig. 23. For multiple shapes, the sample is deformed to various strain levels (e1 ; e2 ; e3 , and e4 ) at subsequent temperatures (T 1 ; T 2 ; T 3 and T 4 ). For clarity and to show the capability, after each loading and subsequent cooling, specimen is unloaded at each programming step to show changes in the temperature based stored strain. The simulation is done as follows. In the first step, at high temperature (T 1 ) the permanent shape is subjected to e1 (3%). This strain is kept fixed during cooling till temperature T 2 ¼ 348 K reached and then unloaded. Unloading above T g involves nearly complete spring back. In the second step, the specimen is deformed to e2 (7%) at T 2 and cooled till temperature T 3 ¼ 338 K is reached. This cycle repeated subsequently for the strain e3 (2.84%) and e4 (2.1%) at T 3 ¼ 338 K and at T 4 ¼ 328 K respectively. And finally, it is cooled till T 5 ¼ 318 K and unloaded. The stored strain is the locking strain at the each end of cooling process is represented as e1s ; e2s ; e3s and e4s . From the stored strain curve, it indicates that using this framework multiple shape memory effect can also be simulated. During recovery, the SMP is heated under a stress-free condition to release the thermomechanical history based stored strain. Here the shape recovery is not simulated, due to the limitation of storage values in VUMAT.
Fig. 23. Simulated multiple shape fixing process and evolution of stored strain at subsequent pre-strain temperatures.
110
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 24. FE model of the beam example.
5.2. Simulation results of the beam example The shape memory behavior for bending deformation is simulated using a cantilever beam (100 10 10 mm). The finite element model used for this is shown in Fig. 24. The simulation of a typical shape memory cycle is carried out on this beam model. At the T high temperature, the beam is horizontal in configuration. This is the permanent shape of the SMP beam and also taken as the reference configuration. In the first step of the cycle, at T high the beam is deflected by applying a uniform pressure at the top surface of the beam. Then in the second step of the cycle, keeping the load constant, the temperature is lowered from T high to T low passing through the
Fig. 25. Stressed, stress-free and multiple natural configurations simulated when the beam undergoes an SMP cycle (configurations shown are directly taken from simulation results).
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
111
Fig. 26. Simulated shape memory cycle in terms of tip-deflection, load and temperature axes.
glass transition temperature T g . Then in the next step, the beam is unloaded keeping the temperature at T low . The beam is then reheated at the unloaded state to T high where the original configuration is expected to be regained given that the beam is a shape memory beam. The results of the above simulation are shown in Fig. 25 in the form of deflected shapes (and the corresponding natural configurations) as in Fig. 3. In the figure, (a) is the reference configuration at T high ; (b) refers to the deflected shape at T high ; (c) corresponds to stressed configuration at T low and (d) is the unloaded configuration at the low temperature stage. Also, the tip deflection-load-temperature diagram obtained from the simulation is shown in Fig. 26. As can be seen, the model is capable of simulating shape memory cycle under bending deformations also. 6. Conclusions A general framework for modeling shape memory behavior in amorphous polymer is presented. This framework is built such the additional features can be incorporated into model with least effort. The small deformation model under that framework is implemented in ABAQUS through user material subroutines (VUMAT) which allows for modeling complex geometries and for complex loading conditions. The results of the model simulation on an amorphous shape memory polymer are verified with existing experimental results in literature. Further, various thermomechanical loading conditions are applied to show the capability of the model to simulate the behavior with reasonably well. The memory characteristics studied show the potential of this framework to use for modeling the multi-shape memory effect also. The encouraging results from the simulation on a beam for a shape memory cycle provides confidence in the use of this model for the analysis of SMP devices designed for use in various practical applications. Acknowledgments The authors wish to thank Prof. Abhijit Deshpande, Prof. Arun Srinivasa, Dr. V. Buravalla and Dr. P.D. Mangalgiri for many helpful technical inputs. References Baghani, M., Naghdabadi, R., Arghavani, J., & Sohrabpour, S. (2012). A thermodynamically-consistent 3D constitutive model for shape memory polymers. International Journal of Plasticity, 35, 13–30. Barot, G., Rao, I. J., & Rajagopal, K. R. (2008). A thermodynamic framework for the modelling of crystallisable shape memory polymers. International Journal of Engineering Science, 46(4), 325–351. 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. Chen, Y. C., & Lagoudas, D. C. (2008). A constitutive theory for shape memory polymers. Part I: Large deformations. Journal of the Mechanics and Physics of Solids, 56(5), 1752–1765. Ghosh, P., & Srinivasa, A. R. (2013). A two-network thermomechanical model and parametric study of the response of shape memory polymers. Mechanics of Materials, 60, 1–17. Kim, H. J., Kang, T. J., & Yu, W. R. (2010). Thermo-mechanical constitutive modelling of shape memory polyurethanes using a phenomenological approach. International Journal of Plasticity, 26(2), 204–218. Lendlein, A., & Kelch, S. (2002). Shape memory polymers. Angewandte Chemie International Edition in English, 41, 2034–2057. Lendlein, A., & Langer, R. (2002). Biodegradable, elastic shape-memory polymers for potential biomedical applications. Science, 296(5573), 1673–1676. Leng, J., Lan, X., Liu, Y., & Du, S. (2011). Shape-memory polymers and their composites: Stimulus methods and applications. Progress in Materials Science, 56(7), 1077–1135.
112
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Liu, Y. P., Gall, K., Dunn, M. L., Greenberg, A. R., & Diani, J. (2006). Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modelling. International Journal of Plasticity, 22(2), 279–313. Liu, H. Q., & Mather, P. T. (2007). Review of progress in shape-memory polymers. Journal of Materials Chemistry, 17(16), 1543–1558. Li, G., & Wei, X. (2011). Thermomechanical behavior of thermoset shape memory polymer programmed by cold-compression: Testing and constitutive modeling. Journal of the Mechanics and Physics of Solids, 59(6), 1231–1250. Monkman, G. J. (2000). Advances in shape memory actuation. Mechatronics, 10(4–5), 489–498. Nguyen, T. D. (2013). Modeling shape-memory behavior of polymers. Polymer Reviews, 53(1), 130–152. Nguyen, T. D., Qi, H. J., Castrao, F., & Long, K. N. (2008). A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation. Journal of the Mechanics and Physics of Solids, 56(9), 2792–2814. Pritha, G., & Srinivasa, A. R. (2011). A two network thermomechanical model of a shape memory polymer. International Journal of Engineering Science, 49(9), 823–838. Qi, H. J., Nguyen, T. D., Castrao, F., Yakacki, C. M., & ShandaSa, R. (2008). Finite deformation thermo-mechanical behaviour of thermally induced shape memory polymers. Journal of Mechanics and Physics of Solids, 56(5), 1730–1751. Rajagopal, K. R., & Srinivasa, A. R. (2004). On the thermomechanics of materials that have multiple natural configurations. Part 1: Viscoelasticity and classical plasticity. Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 55(5), 861–893. Sujithra, R., Kumaran, S. H. S., Arockiarajan, A., & Srinivasan, S. M. (submitted for publication). A framework for modeling memory characteristics in SMPs, Acta Mechanica. 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, 9, 711–718. Tobushi, H., Okumura, K., Hayashi, S., & Ito, N. (2001). Thermomechanical constitutive model of shape memory polymer. Mechanics of Materials, 33(10), 545–554. Westbrook, K. K., Kao, P. H., Castro, F., Ding, Y., & Qi, J. H. (2011). A 3D finite deformation constitutive model for amorphous shape memory polymers: a multi-branch modeling approach for nonequilibrium relaxation processes. Mechanics of Materials, 43(12), 853–869. Xie, T. (2010). Tunable polymer multi-shape memory effect. Nature, 464, 267–270. Xie, T. (2011). Recent advances in polymer shape memory. Polymer, 52(22), 4985–5000. Zhang, Q., & Qing-Sheng, Y. (2012). Recent advance on constitutive models of thermal-sensitive shape memory polymers. Journal of Applied Polymer Science, 123(3), 1502–1508.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Modeling memory effects in amorphous polymers R. Sujithra, S.M. Srinivasan ⇑, A. Arockiarajan Department of Applied Mechanics, Indian Institute of Technology, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 5 January 2014 Accepted 20 June 2014
Keywords: Shape memory polymers Amorphous polymers Memory effects Multiple natural configurations ABAQUS VUMAT subroutine Glass transition Constitutive modeling
a b s t r a c t A model that can describe the shape memory behavior of amorphous polymers based on theory of multiple natural configurations is presented and implemented for use in applications. The thermodynamic framework presented considers evolution of multiple natural configurations and models the key features, especially, the temperature and deformation dependent change in stiffness and shape memorizing characteristics. A single parameter called the degree of glass transition is used to describe the memory locking and releasing characteristics. Using this framework and specialization to small deformations, the developed model is implemented in ABAQUS through VUMAT subroutine feature and correlated with existing epoxy shape memory experiments. Various thermomechanical conditions on structural components such as rods under tension and compression and beams are analyzed. Simulations results confirm that the model is capable of predicting the memory characteristics including the multi-shape memory locking and release mechanisms. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Shape memory polymers (SMPs) are classified as smart material due to the capability to change their shape temporarily from a permanent shape by the application of a thermo-mechanical stimulus that renders this material can be tuned to possess numerous temporary shapes. While maintaining the operating temperature, it remains in the temporary shape (under low mechanical loads); and by reheating the SMP, it reverts back to the permanent shape. Since it remembers one shape at a time, this process is referred as one way actuation process or one way shape memory effect (Lendlein & Kelch, 2002). The shape change phenomena over the transition temperature makes this material to act as a sensor and also as an actuator. Shape memory polymers are inexpensive, light weight, malleable, damage tolerant, bio-degradable, can be molded into a variety of complex shapes by conventional processing technique and also provide large recoverable strains with distributed actuation for complex shape changes (Monkman, 2000). These highlighted features make shape memory polymers a unique material with wide range of applications in various fields such as deployable components in aerospace structures, bio-medical, smart textiles, self-healing composite systems, optical reflectors, morphing skins and automobile actuators (Lendlein & Langer, 2002; Leng, Lan, Liu, & Du, 2011). Shape memory polymers are classified based on the cross-links and the transition temperatures (Lendlein & Kelch, 2002; Liu & Mather, 2007). Some of the commonly used shape memory polymers are polyurethanes, polyolefin, epoxy and polyether esters; for an overview refer (Lendlein & Kelch, 2002). Based on the application, SMPs can be tailor-made based on the transition temperature, varying the chemical composition and the preparation method. Material design at the molecular level plays a prominent role in tuning these polymers. Though, lot of experiments are being carried out and reported in ⇑ Corresponding author. E-mail address: [email protected] (S.M. Srinivasan). http://dx.doi.org/10.1016/j.ijengsci.2014.06.009 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.
96
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
the literature for the successful development of shape memory polymer-based devices, it is necessary to develop theoretical models to predict the shape memory behavior; to be specific in the memory-dependent characteristics (Leng et al., 2011; Xie, 2011). Several models have been proposed in literature (Zhang & Qing-Sheng, 2012) to describe the behavior of shape memory polymer. The constitutive modeling can be distinguished based on the assumptions of different approaches namely: rheological (Tobushi, Okumura, Hayashi, & Ito, 2001; Bhattacharyya & Tobushi, 2000; Li & Wei, 2011), micromechanical (Liu, Gall, Dunn, Greenberg, & Diani, 2006; Chen & Lagoudas, 2008), phenomenological (Qi, Nguyen, Castrao, Yakacki, & ShandaSa, 2008; Kim, Kang, & Yu, 2010), plasticity (Pritha & Srinivasa, 2011; Baghani, Naghdabadi, Arghavani, & Sohrabpour, 2012; Ghosh & Srinivasa, 2013) and rate dependent (incorporating structural relaxation) models (Nguyen, Qi, Castrao, & Long, 2008; Nguyen, 2013; Westbrook, Kao, Castro, Ding, & Qi, 2011). A thermo-mechanical constitutive model was developed by modifying a standard linear viscoelastic model wherein a slip element is introduced to take into account of internal friction (Tobushi, Hashimoto, Hayashi, & Yamada, 1997). A micro-mechanical model was proposed using the free energy function (ClausiusDuhem inequality) wherein the thermo-mechanical behavior of SMPs can be described using the concept of entropy and internal energy (without explicitly considering the molecular interactions) (Liu et al., 2006). A simple thermo-visco-elastic model (neglect the effects of heat conduction and of pressure on the structural relaxation and inelastic behavior) was presented by incorporating the nonlinear Adam–Gibbs model of structural relaxation into a continuum finitedeformation (Nguyen et al., 2008). Recently, a simple one-dimensional model was proposed in which the hysteretic behavior during heating and cooling phenomena was captured by introducing a yield-stress function (Pritha & Srinivasa, 2011). While designing SMP based devices, it is important to understand the response of SMP under complex force and different temperature protocols which will help to get an optimal device design. To be specific, it is important to have a minimalistic model with the flexibility of being able to add the additional features such as rate effects, large deformations, cyclic damage and ageing. An attempt has been made by Barot, Rao, and Rajagopal (2008) and Rajagopal and Srinivasa (2004) to model the memory effects for the crystallizable polymers using a thermodynamic framework based on the theory of multiple natural configurations. The evolution equation for the natural configuration during phase change is obtained by maximizing the rate of entropy production. This natural configuration approach (Rajagopal & Srinivasa, 2004) is capable of describing a variety of responses ranging from traditional plasticity to smart materials under a single framework. However, it would be appropriate to choose parameters in the model that directly link to the physical basis of the function characterizing the response. From the designers perspective, for complex geometries, it may be necessary to introduce the simplistic model within finite element framework. Thus, objective of this work is to model the fundamental behavior of shape memory polymers and conduct numerical test for the memory locking and releasing characteristics by using glass transition temperature as the primary locking mechanism. Multiple natural configuration theory is used as the basic framework for the proposed model and simulation. Two unique features of shape memory polymer, namely the change in stiffness and shape memory characteristics rendered by controlling temperature and deformation (stress) are characterized using a single parameter that is chosen directly link to evolution of multiple natural configurations and therefore, the memory response. The proposed model is limited to small deformations and is implemented using a material subroutine in an existing commercial finite element software (ABAQUS) to analyze device related structural elements. Simulated results based on the developed model is compared with available experimental observations. In addition to that, thermo-mechanical history based memory characteristics, thermal expansion characteristics and multiple shape memory effect are simulated to show the performance of the model. The paper is organized as follows: Section 2 briefly reviews the structure–property relationship for the shape memory effect exhibited by amorphous polymers in order to highlight the importance of introducing the glass transition degree parameter in this model. The proposed modeling framework is elaborated in Section 3. Section 4 focusing on explaining the framework is specialized to small deformations by assuming that the strains of order 10% or less with linear elastic response in the rubbery as well glassy phases. Simulation results at the material point level as well as for a simple prismatic bar and beam of the material are presented in Section 5. The capabilities of simulating various features of a typical SMP are shown also discussed. Finally, various numerical simulations are carried to show that the model is capable of capturing typical memory characteristics of the SMPs. 2. Structure property relationship in SMPs The shape memory effect observed in shape memory polymer is represented by the shape memory cycle as shown in Fig. 1 and the corresponding thermo-mechanical loading cycle is shown in Fig. 2, in which the stress (or strain) and the temperature are controlled at different time steps. Depending on temperature, amorphous polymers exist in two different states, the rubbery state and the glassy state. Here, T g act as a transition switch between these two states. There are three representative temperatures that is generally used to describe the shape memory behavior through glass transition. A high temperature, T high that sees beyond the completion of the effect of glass transition and similarly, a low temperature, T low that sees near completion of the freezing effect of glass transition. The shape memory polymer network consists of chain segments which are interconnected by net points. At temperature above T g , it exists in rubbery phase. This is characterized by the long chain molecules, oriented in a random fashion and molecules vibrating vigorously with high kinetic energy. The effect of interactions between molecules are generally negligible that it exhibits a state akin to an ideal gas. At T high (step-1), the
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
97
Fig. 1. Typical dual shape memory cycle.
Fig. 2. A sample thermomechanical loading for shape memory effect.
permanent shape (A) is easy to deform to any temporary shape (B), as the stiffness of the material is very low. During deformation, the molecular chains are oriented in the direction of the stretch, thereby reducing the entropy of the system. This phenomenon is called entropic elasticity. Here, net points prevent the chain from slipping and contribute to the elasticity of these polymers at T high . Keeping the deformation fixed (step-2), it is gradually cooled to store the desired shape (C). Here, the chain segments act as molecular switches and enable the fixation of temporary shape. As the temperature is decreased below T g , the vibration of the molecules gradually decreases due to the freezing of newly formed bonds between the molecules. The polymer moves towards glassy behavior, which makes it stiffer and the interaction between the molecules increase due to internal energy. The elasticity exhibited at these temperatures is termed as energy elasticity. Due to thermal contraction and gradual stiffening of the polymer as it move towards the glassy behavior, results in the increased stress level. Unloading at low temperature results in a small spring back in strain, due to the high stiffness in the glassy state (step-3).
98
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
This deformed shape (D) is stored as long as the temperature is maintained at T low . By heating up the material in the stressfree state (step-4), the bonds that were formed during the cooling are broken. The net points help in regaining its original shape and due to entropy elasticity, the material is forced back to its permanent shape (A). It can be observed that most of the strain is released around the glass transition temperature. 3. A framework for modeling SMPs The framework proposed here is based on the theory of multiple natural configurations. There are two distinct but related phenomena that occur during the shape memory process pivoted around the glass transition state. One is the change in the stiffness of the material and the other, the change in the natural configuration of the body leading to locking up of strains. While modeling the shape memory phenomenon, the glass transition acts as a memory switch. As the first step in developing this model, the response of the polymer is assumed to be rate-independent and thermo-elastic, neglecting the viscous effects. The viscous effects are significant only in the glass transition region and for all practical purposes, it is assumed that the material resides around the glass transition temperature over a small period time only and thus, negligible. Moreover, the structural relaxation may not affect memory characteristics, since such a relaxation is not pronounced at the temperatures of operation in relation to shape memory utilization of polymer. But these features can also be accommodated later into the model. Many solid bodies are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion (Rajagopal & Srinivasa, 2004). These stress free configurations are known as natural configurations. The evolution of the multiple natural configurations during a typical shape memory cycle is shown in Fig. 3. The permanent shape at high temperature occupies a stress-free configuration (A), since the body is in an unload state. When it is deformed at high temperature, its configuration changes (B). When the external load is removed, the material regains its original configuration (A). In this work, this original configuration is assumed to be the reference natural configuration. The change in natural configuration of the body depends on the class of processes that the body is allowed to undergo (Rajagopal & Srinivasa, 2004). During cooling and heating, the SMP undergoes changes in natural configurations that depend on the strain it has undergone from a reference configuration. In order to model the shape memory characteristics, a reversible transformation is assumed to take place between the reference configuration and a natural configuration that is dependent on temperature and the current strained state. The process of transformation to the current configuration from the natural configuration and vice versa due to stress alone is taken to be purely elastic. The current deformed configuration (C) is an elastic detour from the temporary stress free natural configuration (D) at the low temperature. During this process, body possess multiple natural configurations (W1 ; W2 ; W3 ; . . .) and it has an instantaneous elastic response from each of these natural configuration. At any instant, the natural configuration at a particular temperature associated with the current configuration represents the current structural shape. The shape locking that occurs during freezing, depends on the current deformed configuration. From the current configuration (C), it reaches the stress-free configuration (D) by a rapid unloading process. While unloading, the body will take its configuration depending on the unloaded temperature. During reheating, the
Fig. 3. Stressed, stress free and multiple natural configurations in SMP cycle.
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
99
lock-in strain stored during cooling is released. Here a critical assumption is made that the freezing and release processes are strictly reversible. In other words, the response of the material during reheating depends upon the unloaded temperature, that the body will retrace its path from one natural configuration to other. More details are discussed in Sujithra, Kumaran, Arockiarajan, and Srinivasan (submitted for publication). To keep track of the natural configuration at any instance, a suitable parameter is introduced based on the experimental data. The typical test that is conducted on polymers is the dynamic mechanical analyzer (DMA) test. Usually, three curves are present in this: curves related to storage modulus, loss modulus and tan d. A typical DMA test result for an amorphous shape memory polymer is shown in Fig. 4. The storage modulus is directly related to the elastic stiffness of the material and tan d to the dissipative (viscous) nature of the material. In an amorphous shape memory polymer, the elastic stiffness goes from a very insignificant stiffness in the rubbery state to a stiffness that is orders of magnitude higher at the glassy state passing through the glass transition temperature. Being a higher order transformation, such a transition occurs over a band of temperature near the T g , with the peak activity at the T g . It can be seen the elastic stiffness goes through a near sigmoidal variation in the T g zone where the glass transition activity is the maximum. It can also be observed that elastic stiffness has two asymptotes – one at the rubbery end and the other at the glassy end to indicate that a parameter that is purely dependent on temperature can be introduced that takes the shear modulus from the rubbery modulus to the glassy modulus as,
GðWÞ ¼ ð1 WÞGr þ WGg
ð1Þ
where Gr is the rubber shear modulus and Gg is the glassy shear modulus. Since a linear variation in W is assumed for the stiffness change over temperature, W assumes a functional form for a sigmoidal curve which can be determined from the stiffness curve available from the DMA data. Since this change is primarily due to the change in natural configuration, W is taken to be the parameter that is provides information on the degree of glass transition and thus, is named the degree of glass transition parameter (dogt parameter). W is a purely temperature dependent function. This sigmoidal function hinges around the temperature T g asymptotically reaching the values of 0 at T high and 1 at T low in tune with the observation on the storage modulus curve shown in Fig. 4. It is now assumed based on the framework described above, that all property changes are associated with the dogt parameter that represents the natural configuration change. For example, the thermal expansion coefficient can also be taken to be changing with the dogt parameter as,
a ¼ ð1 WÞar þ Wag
ð2Þ
4. Specialization and simplifications for simulation 4.1. Modeling for small deformations Using the framework described in Section 3, a small deformation model is implemented by assuming the strains in the order of 10% or less. Within this region of strains, the material can be assumed to be practically linear elastic for both rubbery
Fig. 4. A typical DMA test data – storage modulus, loss modulus and tan-d curves.
100
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
and glassy phases. Because of the smallness in deformations, the total strain can be decomposed additively into elastic strain, ee , thermal strain, eT and stored strain es , respectively. Thus,
e ¼ ee þ eT þ es
ð3Þ
When the SMP goes through the glass transition, the deformed configuration is progressively frozen to form the natural configuration. The natural configuration in this model is categorized using the stored strain es . The natural configuration is obtained by applying a strain es on the original reference configuration. Deforming the current natural configuration by ee , one obtains the current deformed configuration. In a strain controlled experiment, for any given temperature the stress can be calculated from,
r ¼ EðWÞ : ee ¼ E : ðe es eT Þ
ð4Þ
where E is the effective modulus of the polymer at the given state (provided by the dogt parameter). Initially, in a typical shape memory cycle, at the reference state, shape memory polymer exists in the rubbery state at T high and is deform to temporary shape (step-1) holding the temperature. Here, the thermal strain and stored strain are zero. Thus, the stress–strain relation at T high is given as
r ¼ Er : ee
ð5Þ
Then keeping the total strain fixed, the temperature is reduced from T high to T low (step-2). The variation of the thermal strain with change temperature is given by (assuming isotropy of the SMP),
_ e_ T ¼ aTI
ð6Þ
During the cooling phase of the shape memory cycle, the dogt parameter provides the potential for locking up of the current configuration. The locking up of the current deformed configuration into natural configuration is expressed in terms of stored strain. The current natural configuration is represented by es and current deformed configuration from the current natural configuration is given by the current elastic strain, ee . For a change in the dogt parameter, a part of the current deformed configuration is locked so that the current natural configuration changes to a new one that includes the locked amount of current deformed configuration. Thus, the evolution of the stored strain is given by,
e_ s ¼ gðWÞee W_ ; W_ < 0
ð7Þ
where gðWÞ quantifies the freezing or locking capability at the appropriate temperature. This parameter is related to the current shear modulus, G and the reference shear modulus Gr . The update of the natural configuration is affected by the stored strain, which is updated appropriate to the strain stored in memory. During heating or reheating, i.e. dT > 0, the history of memory is retraced and the appropriate stored strain release to realize the updated natural configuration. Once the strain at a particular W is released, that memory is erased in the stored energy function and the functional values are available only for values above that W. This is incorporated using a continuous Preisach’s model as in Sujithra et al. (submitted for publication). Thus,
es ¼ es ðWÞ
ð8Þ
In the above model described, it must be noted that the total strain and the temperature are independent (controllable) variables. 4.2. Material functions and parameters for simulation Simulations have been carried out for an amorphous epoxy resin for which experimental data is available in Liu et al. (2006). As has been mentioned in Section 3, typically, a DMA test is conducted to determine the variation of storage modulus with temperature taken from T high to T low through the glass transition temperature, T g . For the DMA data in Liu et al. (2006), the glass transition temperature (T g ) is found to be 343 K (tan d peak) and the temperature range is chosen from T low ¼ 273 K to T high ¼ 358 K, to simulate the memory cycle (see Fig. 4). Poisson’s ratio is assumed to be 0.4 for glassy state in the elastic region and 0.499 for the rubbery state. It is varied between these two through the dogt parameter. The bulk modulus for epoxy is taken to be temperature independent value of 1 GPa. Most DMAs use bending tests while some use extension/contraction for determining the variation with temperature. Though there are differences in the results, they are often well within margins good for use in modeling. Since the glass transition phenomenon is the one that takes the amorphous cross-linked polymer from a rubbery nature to a glassy nature, the variation of the properties primarily hinge around a region of temperature around the glass transition temperature and then flatten out beyond that region. Eq. (1) provides the variation of the shear modulus with a dogt parameter which can found out by fitting with the storage modulus curve in the DMA result. Since the rubbery stiffness is usually orders of magnitude smaller than the glassy stiffness, it be a good idea to use expression for stiffness that arises from entropic elasticity,
Gr ¼ NKT
ð9Þ
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
101
where, N is the cross-link density, K is the Boltzmann’s constant and T is the absolute temperature. Notice that there is a variation of the stiffness with temperature that is captured by the equation. However, the slope of this stiffness curve with temperature is very small. In Fig. 5, the variation of the storage modulus with temperature from the DMA test seems to suggest that the glassy stiffness does not tend to a constant value but has a near constant slope at T low with respect to temperature. Hence, the variation of the glassy shear modulus is modeled using a linearly varying function with respect to temperature. Thus, the glassy shear modulus is given by
Gg ¼ G0 bðT T low Þ
ð10Þ
where G0 is the shear modulus at T low and b is the slope of the modulus curve in the glassy region. G0 and b are obtained directly from the DMA data. Since the dogt parameter follows a sigmoidal form with respect to temperature, a simple form is used (Liu et al., 2006):
W¼1
1 1 þ aðT r TÞn
ð11Þ
where a; T r and n are constants obtained from the curve fit of the variation of the storage modulus curve obtained from the DMA data. The variation of the transition parameter thus obtained vs. temperature is shown in Fig. 6. As can be seen, the parameter abruptly varies in the small region around the glass transition temperature. From the experimental uniaxial stress–strain results, the rubbery (Gr ) and glassy modulus (Gg ) are found to be 2.93 MPa and 268 MPa respectively. The temperature dependent modulus curve is fitted by using these end values as shown in Fig. 7. By using the change in natural configuration parameter, the variation of modulus with temperature is simulated using the Eq. (1). It is observed from the thermal strain behavior that the heating and the cooling curves coincide. There are two zones, one below T g (ag ¼ 0:9E 04) and other above T g (ag ¼ 1:8E 04), where the coefficient of thermal expansion (CET) is considered to be almost a constant. Using the transition parameter, the thermal strain curve is simulated using the Eq. (2). The variation of the thermal strain as function of temperature is shown in Fig. 8. The various constants used in this example are listed in Table 1. 5. Finite element simulation, results and discussion The simulations carried out at the 1-D material level for different test conditions on a specific SMP are provided in Sujithra et al. (submitted for publication). It is important to test the model under conditions of external loading on 3-D device components under boundary conditions that cannot be idealized to perform at the material level. Moreover, it is generally necessary to carry out a finite element or a similar numerical simulation to obtain results for conditions other than specification of strain and temperature. For example, a simple uniaxial testing of an SMP specimen would require a force or displacement condition at the axial direction and stress-free conditions at the side boundaries of the specimen. Such conditions are generally not simple when it comes to simulations at the material level. Therefore, the material model has to be incorporated within a numerical simulation framework, typically a finite element software. Numerical simulations have been carried out in ABAQUS by incorporating the model developed in a VUMAT subroutine. Explicit dynamics is chosen, because there are no simultaneous equations to solve and each increment is inexpensive. First,
300 Glassy modulus Gg = G0 − β (T − Tlow)
250
G (MPa)
200
150 Rubbery modulus G = G(f )T
100
r
f
CL
50
Tg
Tlow 0
280
CL
− Crosslinker density
290
300
310 320 330 Temperature (K)
340
Fig. 5. Variation of shear modulus with temperature.
T
high
350
360
102
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 6. Variation of the dogt parameter with temperature.
Fig. 7. Variation of shear stiffness with temperature.
Fig. 8. Variation of thermal strain with temperature.
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
103
Table 1 Model constants used for the sample material. Constants
Symbol
Values
Rubbery shear modulus Glassy shear modulus Bulk modulus Poisson’s ratio (glassy)
Gr Gg k
2.93 MPa 268 MPa 1000 MPa 0.4
Poisson’s ratio (rubbery) Thermal expansion coefficient
lg lr ar ag
0.499 1:8 1004 (1/K) 0.9 1004 (1/K)
Cross link density
N
9:86 104 ðmol=cm3 Þ
Boltzmann’s constant
K
1:38 1023 J/K
a
2:76 1005 (1/K4) 4 1300 ðkg=m3 Þ 0.15 (W/m K) 1250 (J/Kg K)
Coefficients for curve fit
n Density Thermal conductivity Specific heat
q j Cp
to validate the model with the experimental results, the shape memory cycle is simulated for tension (9.1%), compression (9.1%) and undeformed for an epoxy SMP for which experimental results are available in Liu et al. (2006). A tension test or a compression test on an SMP specimen, should be realistically modeled. Therefore, in this study, the first example chosen is a axial bar undergoing tension or compression mechanical loading in addition to thermal loading pertaining to a shape memory cycle. Another example in the form of a beam undergoing a shape memory cycle of thermomechanical loading is also simulated to show the capability of the model. 5.1. Simulation results of the axial bar example A simple T-shaped specimen (SMP) is modeled with a gripper (rigid body) for tension that will allow for pulling in the first stage of the shape memory cycle, unloading in the third stage and importantly, the fourth stage which is a stress free heating stage. Similarly, rectangular block with a rigid plate is used for simulating a compression test with all the four stages shape memory cycle possible. The simplest to model being just a block for simulating the undeformed block of SMP. The appropriate boundary conditions are shown in Fig. 9. Frictionless contact is defined between surface of the deformed body and the rigid body. The deformed body is discretized using C3D8T element in ABAQUS. Based on the thermo-mechanical loading cycle shown in Fig. 2, the displacement of the gripper and the temperature are controlled by the time steps to simulate a typical uniaxial test in tension or compression. Here, simulation results for a typical shape memory cycle for the above finite element model are first presented. Then, the analysis on the effect of different parameters on the response of the SMP model is discussed. 5.1.1. Isothermal stress–strain response As the first step in this shape memory cycle simulation, SMP block is heated to high temperature T high (358 K). Now, the block is stretched by the gripper for 9.1% tension (AB) and compressed by a plate for about 9.1% (A00 B00 ). This gives the linear stress–strain plot (step-1) at T high , as shown in Fig. 10. At high temperature, it exists in the rubbery state and during deformation, stresses generated in the block are generally very low. 5.1.2. Stress evolution during cooling and heating In step-2, by keeping the displacement of the gripper and the plate at the same position for tension and compression, the temperature reduces to T low (273 K). For undeformed block, both ends are fixed. The simulated stress response for various pre-strain levels, with similarly varying trend is shown in Fig. 11. It shows a flat region until T g , due to low stiffness of
Fig. 9. FEA models of axial bar example with different boundary conditions for (a) tension (b) undeformed and (c) compression thermo-mechanical testing accommodating all the four stages of a shape memory cycle.
104
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 10. Stress–strain at T high temperature.
Fig. 11. Stress evolution during cooling at different predefined strains.
the rubbery state. As the temperature decreases further, thermal contraction takes place and the thermal stress gradually rises as the material stiffens. For the pre-defined compression above T g , the stress gradually decreases and there is transition to a state of tension. For the undeformed, due to constraints, thermal stress develops. At the end of the step-2, all the fixed strain is stored in the material as stored strain. After cooling, it is heated again with predefined strain constraint. Because of the reversibility assumption, the variation during heating curve coincides (CB, C0 B0 , C00 B00 ) with cooling curve (BC, B0 C0 , B00 C00 ) as shown in Fig. 12. 5.1.3. Evolution of various strains during cooling The evolution of various strains during cooling is shown in Fig. 13 for tension. It is seen that the total strain is pre-defined strain and is almost constant due to constraint. After loading at T high , the total strain is the equal to the elastic strain and there is no stored strain and thermal strain. During cooling, the elastic strain is transformed to the stored strain and almost equal to the pre-defined strain. This reflects the memory effect. Here, for simplicity, the thermal strain is neglected. 5.1.4. Free strain recovery In step-3, after cooling, the contact between the plate and the block is deactivated and the plate moves back to the initial position. Unloading at T low produces a small spring back in strain due to glassy stiffness. This spring-back strain is very small when compared to pre-strain levels. In step-4, the block is again reheated at the stress-free condition. The stored strain is gradually released as shown in Fig. 14. In both tension (DA) and compression (D00 A00 ) cases, most of the strain gets released within a small band of temperature around T g . For undeformed block (D0 A0 ), it shows only thermal strain. Initially, during heating, thermal expansion takes place, and after T g , releasing of stored strain is dominant. In the proposed model, it remembers the history of strain storage and retraces its path. But, VUMAT stores only two- state architecture; the initial values are
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 12. Stress evolution at subsequent heating after cooling at different strain levels.
Fig. 13. Evolution of various strains during cooling at constraint deformation.
Fig. 14. Evolution of stored strain during reheating at different fixed strains.
105
106
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
in old arrays and current values are in new arrays. So, at the end of step-3, the final amount of stored strain is stored in old array. By using this old stored array, the evolution for the release of new stored strain is determined:
e_ s ¼ es W_ ; dW > 0:
ð12Þ
5.1.5. Constrained stress recovery While the above 4-step process is a typical shape memory cycle, there is also an alternative shape memory cycle. In this alternative cycle, while the first three steps are the same as what has been explained above, before the start of step-4, in its unloaded state at T low , the strain is now kept fixed when heated to T high . This would change the stress level in the specimen because of the constraint effect with the initial stress being zero. The evolution of stress during heating for various fixed strain is shown in Fig. 15. Since there will be constraint for the release of stored strain and thermal expansion, compressive stress develops. As the temperature increases above T g , the material becomes soft and the stress shoots up to pre-defined stress (DE, D0 E0 , D00 E00 ). 5.1.6. Two types of cycles A complete cycle consists of: deformation at high temperature (AB), cooling to T low by fixing the strain or at constant stress, releasing the stress and heating in stress-free state to release of stored strain. The complete simulated shape memory cycle, by either strain-controlled (ABCDA) or stress-controlled mode (abcda) is shown in Fig. 16. 5.1.7. Effect of thermal expansion The thermal expansion is the temperature dependent behavior. The effect of thermal expansion in the shape memory cycle is simulated by keeping constant glassy coefficient, mixture of rubbery and glassy coefficient and zero coefficient thermal expansion. The stress responses are similar for constant glassy coefficient (ABCg DA) and mixture of both coefficients (ABCDA) (see Fig. 17). For the zero coefficients (ABC0 DA), it shows that the thermal coefficient has an impact on the final stress during cooling and also reduction in spring back strain. For the constant glassy coefficient, there is slight shift in strain recovery in the vicinity at T g (Fig. 18). 5.1.8. Simulations for memory characteristics For the memory characteristics studies, the shape memory cycle is simulated for tension deformation using displacement control method. The amount of strain stored is dependent on the loading and cooling temperature. The evolution of the stored strain at various loading temperature is simulated by loading at other than T high and subsequently it is cooled, unloaded and reheated to release the stored strain. The evolution of stresses at different loading temperatures (358 K, 348 K, 338 K, 328 K, 318 K, and 308 K) is shown in Fig. 19. It shows that significant amount of stress is needed for deformation below T g , due to increase in stiffness of the material. And also the memory activation is due to entropy and as well as by internal energy. The evolution of the different strain fixities at these various loading temperature is shown in Fig. 20. This graph shows the thermomechanical history based memory behavior. It is noticed that strain fixity drops down due to loading below T g . Due to storage limitations in VUMAT, strain recovery is not shown. Instead, evolution of stored strain during cooling is shown in Fig. 20. Instead of loading other than T high , now cooling is stopped above T low . The evolution of stresses at various unloading temperatures (358 K, 348 K, 338 K, 328 K, 318 K, and 308 K) is shown in Fig. 21. Unloading near T g , shows that most of the strain
Fig. 15. Evolution of stress during heating at fixed strain.
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 16. Shape memory cycles for tension.
Fig. 17. Stress response due to effect of thermal expansion coefficient.
Fig. 18. Evolution of strain for various thermal expansion coefficients.
107
108
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 19. Evolution of stresses at different loading temperatures.
Fig. 20. Evolution of stored strain after cooling at various pre-straining temperatures.
Fig. 21. Stress response where cooling stops at above T low .
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
109
Fig. 22. Evolution of strain during cooling at various pre-straining temperatures.
spring backs due to entropy. The evolution of strain during the SMP cycle for various unloading temperature is shown in Fig. 22. While reheating, it follows the same path from its unloading temperature. 5.1.9. Multi shape memory effects Recent studies show that Nafion has multiple shape memory effect in a single shape memory cycle and possesses broad glass transition (Xie, 2010). But, amorphous polymer has narrow transition region. In order to show effect of multiple natural configurations in our proposed framework, multiple shape memory effect is simulated using the same epoxy material parameters. The thermomechanical cycle for fixing multiple shapes is simulated as shown in Fig. 23. For multiple shapes, the sample is deformed to various strain levels (e1 ; e2 ; e3 , and e4 ) at subsequent temperatures (T 1 ; T 2 ; T 3 and T 4 ). For clarity and to show the capability, after each loading and subsequent cooling, specimen is unloaded at each programming step to show changes in the temperature based stored strain. The simulation is done as follows. In the first step, at high temperature (T 1 ) the permanent shape is subjected to e1 (3%). This strain is kept fixed during cooling till temperature T 2 ¼ 348 K reached and then unloaded. Unloading above T g involves nearly complete spring back. In the second step, the specimen is deformed to e2 (7%) at T 2 and cooled till temperature T 3 ¼ 338 K is reached. This cycle repeated subsequently for the strain e3 (2.84%) and e4 (2.1%) at T 3 ¼ 338 K and at T 4 ¼ 328 K respectively. And finally, it is cooled till T 5 ¼ 318 K and unloaded. The stored strain is the locking strain at the each end of cooling process is represented as e1s ; e2s ; e3s and e4s . From the stored strain curve, it indicates that using this framework multiple shape memory effect can also be simulated. During recovery, the SMP is heated under a stress-free condition to release the thermomechanical history based stored strain. Here the shape recovery is not simulated, due to the limitation of storage values in VUMAT.
Fig. 23. Simulated multiple shape fixing process and evolution of stored strain at subsequent pre-strain temperatures.
110
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Fig. 24. FE model of the beam example.
5.2. Simulation results of the beam example The shape memory behavior for bending deformation is simulated using a cantilever beam (100 10 10 mm). The finite element model used for this is shown in Fig. 24. The simulation of a typical shape memory cycle is carried out on this beam model. At the T high temperature, the beam is horizontal in configuration. This is the permanent shape of the SMP beam and also taken as the reference configuration. In the first step of the cycle, at T high the beam is deflected by applying a uniform pressure at the top surface of the beam. Then in the second step of the cycle, keeping the load constant, the temperature is lowered from T high to T low passing through the
Fig. 25. Stressed, stress-free and multiple natural configurations simulated when the beam undergoes an SMP cycle (configurations shown are directly taken from simulation results).
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
111
Fig. 26. Simulated shape memory cycle in terms of tip-deflection, load and temperature axes.
glass transition temperature T g . Then in the next step, the beam is unloaded keeping the temperature at T low . The beam is then reheated at the unloaded state to T high where the original configuration is expected to be regained given that the beam is a shape memory beam. The results of the above simulation are shown in Fig. 25 in the form of deflected shapes (and the corresponding natural configurations) as in Fig. 3. In the figure, (a) is the reference configuration at T high ; (b) refers to the deflected shape at T high ; (c) corresponds to stressed configuration at T low and (d) is the unloaded configuration at the low temperature stage. Also, the tip deflection-load-temperature diagram obtained from the simulation is shown in Fig. 26. As can be seen, the model is capable of simulating shape memory cycle under bending deformations also. 6. Conclusions A general framework for modeling shape memory behavior in amorphous polymer is presented. This framework is built such the additional features can be incorporated into model with least effort. The small deformation model under that framework is implemented in ABAQUS through user material subroutines (VUMAT) which allows for modeling complex geometries and for complex loading conditions. The results of the model simulation on an amorphous shape memory polymer are verified with existing experimental results in literature. Further, various thermomechanical loading conditions are applied to show the capability of the model to simulate the behavior with reasonably well. The memory characteristics studied show the potential of this framework to use for modeling the multi-shape memory effect also. The encouraging results from the simulation on a beam for a shape memory cycle provides confidence in the use of this model for the analysis of SMP devices designed for use in various practical applications. Acknowledgments The authors wish to thank Prof. Abhijit Deshpande, Prof. Arun Srinivasa, Dr. V. Buravalla and Dr. P.D. Mangalgiri for many helpful technical inputs. References Baghani, M., Naghdabadi, R., Arghavani, J., & Sohrabpour, S. (2012). A thermodynamically-consistent 3D constitutive model for shape memory polymers. International Journal of Plasticity, 35, 13–30. Barot, G., Rao, I. J., & Rajagopal, K. R. (2008). A thermodynamic framework for the modelling of crystallisable shape memory polymers. International Journal of Engineering Science, 46(4), 325–351. 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. Chen, Y. C., & Lagoudas, D. C. (2008). A constitutive theory for shape memory polymers. Part I: Large deformations. Journal of the Mechanics and Physics of Solids, 56(5), 1752–1765. Ghosh, P., & Srinivasa, A. R. (2013). A two-network thermomechanical model and parametric study of the response of shape memory polymers. Mechanics of Materials, 60, 1–17. Kim, H. J., Kang, T. J., & Yu, W. R. (2010). Thermo-mechanical constitutive modelling of shape memory polyurethanes using a phenomenological approach. International Journal of Plasticity, 26(2), 204–218. Lendlein, A., & Kelch, S. (2002). Shape memory polymers. Angewandte Chemie International Edition in English, 41, 2034–2057. Lendlein, A., & Langer, R. (2002). Biodegradable, elastic shape-memory polymers for potential biomedical applications. Science, 296(5573), 1673–1676. Leng, J., Lan, X., Liu, Y., & Du, S. (2011). Shape-memory polymers and their composites: Stimulus methods and applications. Progress in Materials Science, 56(7), 1077–1135.
112
R. Sujithra et al. / International Journal of Engineering Science 84 (2014) 95–112
Liu, Y. P., Gall, K., Dunn, M. L., Greenberg, A. R., & Diani, J. (2006). Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modelling. International Journal of Plasticity, 22(2), 279–313. Liu, H. Q., & Mather, P. T. (2007). Review of progress in shape-memory polymers. Journal of Materials Chemistry, 17(16), 1543–1558. Li, G., & Wei, X. (2011). Thermomechanical behavior of thermoset shape memory polymer programmed by cold-compression: Testing and constitutive modeling. Journal of the Mechanics and Physics of Solids, 59(6), 1231–1250. Monkman, G. J. (2000). Advances in shape memory actuation. Mechatronics, 10(4–5), 489–498. Nguyen, T. D. (2013). Modeling shape-memory behavior of polymers. Polymer Reviews, 53(1), 130–152. Nguyen, T. D., Qi, H. J., Castrao, F., & Long, K. N. (2008). A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation. Journal of the Mechanics and Physics of Solids, 56(9), 2792–2814. Pritha, G., & Srinivasa, A. R. (2011). A two network thermomechanical model of a shape memory polymer. International Journal of Engineering Science, 49(9), 823–838. Qi, H. J., Nguyen, T. D., Castrao, F., Yakacki, C. M., & ShandaSa, R. (2008). Finite deformation thermo-mechanical behaviour of thermally induced shape memory polymers. Journal of Mechanics and Physics of Solids, 56(5), 1730–1751. Rajagopal, K. R., & Srinivasa, A. R. (2004). On the thermomechanics of materials that have multiple natural configurations. Part 1: Viscoelasticity and classical plasticity. Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 55(5), 861–893. Sujithra, R., Kumaran, S. H. S., Arockiarajan, A., & Srinivasan, S. M. (submitted for publication). A framework for modeling memory characteristics in SMPs, Acta Mechanica. 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, 9, 711–718. Tobushi, H., Okumura, K., Hayashi, S., & Ito, N. (2001). Thermomechanical constitutive model of shape memory polymer. Mechanics of Materials, 33(10), 545–554. Westbrook, K. K., Kao, P. H., Castro, F., Ding, Y., & Qi, J. H. (2011). A 3D finite deformation constitutive model for amorphous shape memory polymers: a multi-branch modeling approach for nonequilibrium relaxation processes. Mechanics of Materials, 43(12), 853–869. Xie, T. (2010). Tunable polymer multi-shape memory effect. Nature, 464, 267–270. Xie, T. (2011). Recent advances in polymer shape memory. Polymer, 52(22), 4985–5000. Zhang, Q., & Qing-Sheng, Y. (2012). Recent advance on constitutive models of thermal-sensitive shape memory polymers. Journal of Applied Polymer Science, 123(3), 1502–1508.