Fractional derivative modelling of adhesive cure

Applied Mathematical Modelling 77 (2020) 1041–1053

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Fractional derivative modelling of adhesive cure Harry Esmonde a,∗, Sverre Holm b a b

Dublin City University, Dublin, Ireland University of Oslo, Oslo, Norway

a r t i c l e

i n f o

Article history: Received 28 February 2019 Revised 10 July 2019 Accepted 19 August 2019 Available online 24 August 2019 Keywords: Fractional derivative model Adhesive Cure Viscoelastic Oscillatory Squeeze film

a b s t r a c t This paper considers the modelling of curing adhesive properties using fractional derivatives. A systematic approach is adopted where results can be related to a physical interpretation of the system rather than relying on a purely data-driven approach. The method relies on selecting standard integer order models based on the pre-cure and post-cure behaviour, from which fractional order derivative models are derived. Results from dynamic mechanical testing of two chemistries, a cyanoacrylate adhesive and a methacrylate resin are used to identify the parameter values for their respective fractional models. These results are then used to interpret behaviour of the adhesives during cure such as the onset of solidification. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Adhesives are commonly used to form solid mechanical structural bonds between surfaces that may be uneven or at least have a surface texture. To maximise the bond strength the adhesive is often capable of flowing when the surfaces are joined to account for the surface irregularities and fill any voids. It is clear that a material phase change occurs as the adhesive cures or moves from liquid-like to solid-like behaviour. In fact the adhesive material will have viscoelastic behaviour throughout the curing process with the phase changing as the viscous (liquid) character diminishes and elastic (solid) behaviour becomes more apparent. The dynamic behaviour of the adhesive during cure can be assessed using a suitable form of dynamic mechanical analysis where the adhesive is placed between two substrates that are slightly perturbed relative to each other while recording the movement and the induced force. Converting the force and displacement to stress and strain allows calculation of the complex modulus. The real component of the modulus is a measure of elasticity while the imaginary component describes viscous behaviour. Another way of representing the modulus is in terms of magnitude and phase. A Newtonian liquid has a purely imaginary modulus and consequently 90° phase. A Hookean solid has a purely real modulus and hence 0° phase. Viscoelastic materials will therefore exhibit a phase characteristic somewhere between these values depending on the ratio of viscous to elastic response. Typically the phase of an adhesive, which will be viscoelastic in nature, will start from somewhat less than 90° before cure commences and then move towards 0° as it transitions from liquid to solid. Complex phase is therefore a useful parameter when assessing the degree of adhesive cure, a process that may be regarded as a type of polymerisation. In some processes where phase change of a time dependent material occurs, the complex phase exhibits relatively constant values as cure progresses over wide frequency ranges at values that are not integer multiples of π /2, a feature that ∗

Corresponding author. E-mail address: [email protected] (H. Esmonde).

https://doi.org/10.1016/j.apm.2019.08.021 0307-904X/© 2019 Elsevier Inc. All rights reserved.

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Fig. 1. Fractal Network and equivalent system.

cannot be described by simple configurations made up from elastic and viscous components but instead points towards fractional derivative behaviour. Fractional viscoelastic behaviour has received widespread attention primarily as a convenient method for fitting experimental data to models that require only a few parameters to explain quite complicated behaviour [1]. The object of this paper is to try to understand the curing process and the associated fractional derivative behaviour. Fractional behaviour in relation to mechanical properties of materials was unknowingly first noted by Nutting [2] in 1921 when studying the behaviour of cheese as it matures. It was observed that the changing response function G(t) could best be described using time to a negative fractional exponent α

G(t ) ∝ t −α

(1)

Scott Blair et al. [3] realised in 1947 that this in fact implies a fractional derivative since in the case of the Riemann– Liouville fractional derivative, the derivative is described as

dα f 1 d = dt α (1 − α ) dt



t 0

f (τ ) dτ t ( − τ )α

(2)

where 0 < α < 1. In the Laplace domain a derivative in the time domain is equivalent to a multiplication by the Laplace operator s. A fractional derivative of order α is therefore equivalent to a multiplication by sα and the phase associated with this term is α × π2 . This phase is independent of frequency and is constant at a value that is not an integer multiple of π2 . Fractional behaviour can be established by observation with the fraction α fitted to experimental data, a practice that is widely used when modelling fractional systems [4,5]. While convenient for modelling purposes this does not provide much insight into the process of maturation of curing materials other than the obvious fact that solidification is occurring. The first instance where the fractional derivative power α was related to physical properties of the viscoelastic system is where authors have shown that there is a link between fractal networks and fractional derivative behaviour [6,7] In Fig. 1 an infinite network consisting of Hookean elastic elements with elastic constant E and viscous dashpots with viscosity η is equivalent to a fractional system with complex modulus X(s). The analysis to derive the equivalent system is performed as follows [7]. The infinite network on the left of Fig. 1 can be condensed to the finite systems shown on the right of the figure. The recurrent pattern can be represented as a parallel network of a spring in series with an element X and a damper in series with an element X. This in turn must be equivalent to the overall complex modulus of the system G(s).

G (s ) = X =

1 1 E

+

G(s ) = (E ηs )0.5

1 X

+

1 1 ηs

+

1 X

(3)

(4)

Of note is the fact that the Laplace operator s is raised to the power 0.5 showing that a differential order of 0.5 describes the modulus of the system [8]. The hierarchal models typically result in fractional power of 0.5. This value has been shown by Bagley and Torvik [9] to be close to experimental values of polymer solutions and was related back to molecular theory developed by Rouse [10]. The fractional nature of viscoelastic systems can be related to a distribution of first order systems with different time constants [11]. Depending on the distribution of values it is possible to adjust the fractional power to a value other than 0.5. An example of this is the more sophisticated network developed by Schiessel and Blumen [6] shown in Fig. 2 consisting of a combination of springs E1 …En and dampers η1 …ηn .

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Fig. 2. Fractal network with varying stiffness and damping.

Fig. 3. Maxwell system with variable viscosity.

With the appropriate choice of variation in the stiffness and damping coefficients the authors arrived at a modulus described by Eq. (5).

E0

G (s ) = 1+

E0 η0 s

 E0

(5)

− EE1

η0 s + 1

0

E

In this case the power of the Laplace operator depends on a stiffness ratio E1 . If one is to employ this as a model for a 0 curing adhesive one can consider the stiffness ratio to change from 0 at the start of cure to unity at full cure. Thus at the start, t = 0, E1 = 0 and the system behaves as a Maxwell viscoelastic fluid

G (s ) =

1 1 E0

(6)

+ η10 s

At full cure one could assume the adhesive becomes stiffer in nature with E1 = E0 resulting in a complex modulus that describes a standard linear solid (SLS) otherwise known as the Zener model [1].

G ( s ) = E0

E 0 + η0 s 2 E 0 + η0 s

(7)

Between these stages the non-integer value for E1 /E0 would give rise to a fractional derivative characteristic for the adhesive. It is interesting to note that if all the springs have the same elasticity E, and all the dashpots have the same viscous coefficient η, then using the same approach to model the fractal system as used in [7] one obtains

G (s ) =

   ηs −1 ± 1 + 4E/ηs 2

(8)

When E/ηs  1 and taking the positive root

G (s ) ≈



E ηs

(9)

which is the same as the fractal network in Fig. 1 despite the difference between the topologies. Another approach that results in fractional derivative behaviour is to include a time varying viscosity when modelling material properties. Pandey and Holm [12] have considered a Maxwell system with a constant stiffness E0 and where the viscosity is described by a constant η0 and a linear time dependent component θ t as shown in Fig. 3.

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35

2

20 mins

|G| [Pa]

25 20 15

sim

10

Exp

Phase [rads]

30

20 mins

2 sim 1

exp ex

1

5 0

0

0

20

40 60 80 Frequency [rad/s]

100

120

0

20

40 60 80 Frequency [rad/s]

100

120

Fig. 4. Magnitude and phase plot of curing adhesive after 20 min.

The stress in the damper is equal to the stress in the spring

(η0 + θ t )˙ d = E0 s

(10)

For a step strain input and t > 0

˙ d = −˙ s

(11)

where ˙ d is the strain rate in the damper or viscous element, ˙ s is the strain rate in the spring or elastic element. Using this substitution, solving for the elastic strain in response to a step input and converting to the Laplace domain, the relaxation modulus G(s) is determined as

G ( s ) = E0

 η  Eθ0 0

θ

e

η0 θ s

 E  E 0  1− 0 sθ θ

(12)

where  () represents the gamma function. Here we see that the fractional behaviour depends on the ratio of the stiffness E divided by the growth rate of the viscosity, θ0 and thus related to a physical interpretation of system behaviour. This technique can be extended to include varying stiffness E0 (1 − γ t) (see appendix) resulting in the following expression for the relaxation modulus



G ( s ) = kE0 1 −

γ s

E0 γ

e η0 θ s

+ θs

E0

s η0 θ

(13)

where k, γ , θ are constants. This model was fitted to experimental data for a methacrylate adhesive as shown in Fig. 4. The magnitude and phase of the complex modulus of curing adhesive after 20 min shows that while it is possible to get a good fit for the magnitude, the exponent s/θ in the model means that the phase of the modulus will increase with frequency, a feature that was not present in the experimental data. The objective of the current paper is to model the curing process of adhesives using fractional derivatives and therefore a new modelling technique is introduced next that will be compared to experimental results. The method is a combination of structural modelling and temporal variation and falls somewhere between the techniques mentioned previously. 2. Theory To begin the analysis an assumption is made on the structure of the material at the start and end of the phase transition. In Fig. 5a material is assumed to have the properties of a Newtonian fluid initially with viscosity η0 and after transition to have the properties of an elastic solid with stiffness E1 . The complex modulus G(s) for the system in transition varies as G(s): s η0 → E1 . For most material phase changes the transition takes a finite amount of time and therefore there will be interstitial stages during the process. One way of determining the state of the material during transition is to formulate a transition function HT (s) and then multiply the initial modulus by fractional powers of this function. The transition function is obtained by formulating the ratio of the final state to the initial state. In the case of the system in Fig. 5 this is

HT (s ) =

E1 s

(14)

η0

Thus if the transition has progressed by a fraction β (0 ≤ β ≤ 1) then the state of the material is given by the complex modulus Gβ (s)

Gβ (s ) = [HT (s )]β sη0 =

 E β 1 s η0 = E 1 β ( s η0 ) 1 − β s η0

(15)

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Fig. 5. Material transition model.

Fig. 6. Cyanoacrylate adhesive curing model.

The value of β will vary as time progresses starting at 0 and eventually reaching 1 when full transition has occurred. Between these values the fractional value will give rise to fractional derivative behaviour since the Laplace operator is raised to a fractional power. If one considers the mid point of the transition when β = 0.5 Eq. (15) is identical to Eq. (4) which was derived using the infinite fractal network. This makes sense since there would be an equal contribution from elastic and viscous elements that would be the case for both the transition model and the infinite network. If one can say that there is a combination of first order systems in the model in Fig. 2, then, if one looks at the reduced model in that paper [6], their Eq. (38), it is the same as Eq. (15) here (albeit their 1 − γ is equivalent to β here). As such the transition from a purely viscous to a purely elastic state can be considered to incorporate a multiplicity of first order systems that give rise to the fractional nature of the physical phenomenon that is occurring in solidification. This multiplicative operation in the Laplace domain is consistent with a gradual change in state where the index β is a function of time. In the time domain this corresponds to a scenario where the modulus changes by convolving the time domain version of the transition function to the power β with the initial modulus. As such, unlike the method used to derive Eq. (13) where only the elasticity and viscosity parameters change with time, the system changes by a complicated process so that the impulse response function, the memory of the material, evolves based on a weighted sum of its values where the weighting is controlled by the filter values of the time domain version of the transition function. This dependence on current and previous states is an appropriate modelling technique that will account for the variable level of cure in different parts of the adhesive network. For this technique to be relevant there has to be a phase change in the system, in other words a change in the ratio of real to imaginary components in the complex modulus. This will obviously be the case for a curing adhesive but the technique could be applied to any system described by a changing complex parameter. To use the procedure outlined above, an appropriate choice of start and finish structure is required. This is done by examining the dynamic modulus of the material at the start of transition and at the end. Two adhesive chemistries were investigated in the study. The first chemistry investigated was that of a cyanoacrylate adhesive, a fast curing material that was substantially cured within two minutes as tested for this study. The second was a methacrylate anaerobic retaining adhesive a material that undergoes the bulk of its transition from liquid-like to solid-like behaviour over the course of one hour under the conditions tested in this work. For the cyanoacrylate adhesive a Kelvin Voigt model is selected for the start condition while a purely elastic material is selected for the end condition, see Fig. 6. The complex modulus for the system in transition varies as G(s): E0 + s η0 → E1 . The transition function is then

HT (s ) =

E1 E0 + s

η0

(16)

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Fig. 7. Methacrylate adhesive curing model.

Fig. 8. Schematic of squeeze film geometry (not to scale).

and the complex modulus during transition is then

G β ( s ) = [E 1 ]β [E 0 + s η 0 ]1 − β

(17)

For the methacrylate adhesive a Maxwell system is chosen for the initial configuration while a SLS model is selected for the end state. Thus the adhesive curing process can be regarded as a transition from a Maxwell to SLS system, see Fig. 7. Using these start and end points corresponds with the fractal network approach described by Eq. (5) when it is reduced to Eqs. (6) and (7) with the appropriate values of elasticity and viscosity. The complex modulus of the liquid GL (s) at the start is

GL ( s ) =

E 0 η0 s E 0 + η0 s

(18)

The complex modulus of the solid Gs (s) at full cure is

Gs ( s ) =

E 2 ( E 1 + η1 s ) E 2 + E 1 + η1 s

(19)

The transition from liquid to solid can then be described using a transition function as

HT (s ) =

Gs ( s ) E 2 ( E 1 + η1 s ) ( E 0 + η0 s ) = GL ( s ) (E2 + E1 + η1 s)E0 η0 s

(20)

The complex modulus of the intermediate state Gβ (s) as the transition progresses can be described by



Gβ ( s ) =

Gs ( s ) GL ( s )

β

GL (s ) = [Gs (s )]β [GL (s )]1−β

(21)

3. Testing The dynamic mechanical analysis of adhesive is carried out using the Micro Fourier Rheometer (GBC Scientific). It is an oscillatory squeeze film rheometer that uses an axisymmetric geometry where flat circular plates of radius R move relative to one another along their axes (z axis) with amplitudes up to 20 μm. A schematic of the test geometry is shown in Fig. 8. The top plate moves with displacement zp , squeezing the adhesive between it and the fixed lower plate. The displacement in the upper plate is measured to determine the instantaneous height h as well as the force induced in the lower plate from which the complex modulus of the material under test, in this case the adhesive, is determined. One of the issues when testing adhesives is that they will eventually bond the rheometer together rendering it unusable. To preserve the integrity of the rheometer a removable system using neodymium magnets is used to hold the top and

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Fig. 9. Design of removable upper platen. Table 1 Parameter values for curing plots in Fig. 9. Cure time

7s

34 s

61 s

88 s

115 s

E0 [Pa]

0.36 0.1 0.49 0.25

1.09 0.15 0.54 0.31

1.86 0.43 0.77 0.46

2.03 7.38 2.73 0.77

2.1 10.02 5.30 0.81

η0 [Pas] E1 [Pa]

β

bottom plates in place. A representation of the top plate is shown in Fig. 9 and a similar set up is used for the bottom plate. The forces incurred during testing are well below the magnetic holding force on the removable plate but once testing is finished it is then possible to slide the bonded removable plates out from the rheometer and separate them. 4. Experimental and simulated results The adhesives cure in the absence of air and were tested between mild steel surfaces. The adhesives were placed between 25 mm diameter plates on the rheometer and the initial gap was set to 100 μm. The top plate was excited at amplitudes below 1 μm and the force induced was maintained at less than 1 N throughout the test period. A random displacement signal was used with frequency content between 0.5 Hz and 20 Hz. Data was sampled at regular intervals during cure. The data was not ensemble averaged since the system was changing over time. The simulated results were obtained based on Eq. (17) for the cyanoacrylate adhesive and Eq. (21) for the methacrylate adhesive after determining the parameter values for the models using the generalised reduced gradient solver in Excel. The comparison between experimental and simulated complex modulus for the cyanoacrylate adhesive is presented in Fig. 10 and that for the methacrylate adhesive is presented in Fig. 11. 5. Discussion For the cyanoacrylate adhesive at seven seconds (Fig. 9, Table 1) the magnitude of the modulus at zero radians per second is not zero, which was the reason a Kelvin Voigt (KV) model was selected as the start condition rather than a Newtonian fluid for this adhesive. Also the phase is seen to increase with frequency, which is consistent with a KV model. The data for the cyanoacrylate adhesive is relatively noisy by comparison with that for the methacrylate adhesive (Fig. 11). During testing, 400 points are acquired at 40 Hz so that 10 s elapses during the sampling process. The cure speed for the cyanoacrylate is of the order of tens of seconds and therefore there is appreciable change in material behaviour during sampling leading to noisy results. There is an increase in the parameter values E0 , η0 , E1 and β used to model the cyanoacrylate adhesive as time and the cure progress as shown in Table 1. If the curing process were linear, one would expect the same values for all parameters across the columns except for β , which would increase with time. The fact that this is not the case would suggest that the curing process is better described by an affine function where the start and end conditions have different parameter settings. Other than E0 , the values show a substantial change after 61 s that would indicate a major change in material behaviour has occurred due to solidification. In particular the value of η0 has increased by an order of magnitude that can be explained

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Fig. 10. Magnitude and phase plots of cyanoacrylate adhesive as cure progresses.

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Fig. 11. Magnitude and phase plots of methacrylate adhesive as cure progresses.

by substantial crosslinking leaving little space between polymer chains and therefore causing much higher viscous shear stresses between the chains. The constant phase typically associated with fractional systems becomes most apparent at 88 and 115 s when the force induced on the bottom plate of the viscometer leads the induced displacement of the top plate by roughly 0.4 and 0.3 radians, respectively.

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H. Esmonde and S. Holm / Applied Mathematical Modelling 77 (2020) 1041–1053 Table 2 Parameter values for curing plots in Fig. 11. Cure time

0 min

20 min

40 min

60 min

E0 [Pa]

72.4 0.098

70.0 0.229

0.777 48.0 0.682

2.14 33.3 0.581

38.1 0.384 4.16 2.19 9.79 0.673

42.6 0.001 4.03 9.79 15.9 0.912

η0 [Pas] E1 [Pa]

η1 [Pas] E2 [Pa]

β

For the methacrylate adhesive (Fig. 11, Table 2), the behaviour at 0 min is that of a Maxwell system with zero magnitude at 0 rad/s and a phase that decreases linearly from π /2 as frequency increases. The data is relatively noise free since the curing process is of the order of minutes rather than seconds and is therefore not affected by the 10 s sampling time to the same extent as the cyanoacrylate adhesive. From Table 2, E1 = 0 at 0 and 20 min indicating that the curing process at these times is best modelled as a transition from one Maxwell system to another. The general characteristic at these times is one of a Maxwell system with high stiffness and low viscosity moving towards a system with lower stiffness and higher viscosity. This can be understood by considering the polymerisation process where polymer chains lengthen rather than crosslink, lowering the stiffness while at the same time reducing the free space in the matrix thereby increasing the shear stress and the effective viscosity. Again the changing values of E0 , η0 , E1 , η1 and E2 suggest an affine transformation rather than a simple linear transformation best describes the transition from liquid-like to solid-like behaviour. At 40 and 60 min E1 = 0 and the cure is modelled by a transition from a Maxwell to a SLS system and the magnitude plots show non-zero values at 0 rad/s indicating that solid-like behaviour is present. At 60 min the value for β is greater than 0.9 indicating that the solidification has occurred to a large extent. Depending on the type of adhesive and the curing conditions β may not necessarily attain a value of unity. Similar to the cyanoacrylate adhesive, it is at this later stage of cure that the methacrylate adhesive exhibits a flat phase readily associated with fractional derivative behaviour. By and large there is a good match between experimental (black trace) and simulated values (circles) for both adhesive types except at close to 0 rad/s where the experimental data cannot be relied upon because the piezoelectric force transducer has a lower limit of about 3 rad/s.

6. Conclusion The complex modulus of curing adhesive can be described using a fractional derivative model. The principal novelty in this paper lies in a systematic approach that allows one to develop fractional derivative models from the basis of wellunderstood standard integer order models and in turn allows for a physical understanding of the curing process. By comparison with a data-driven fractional model, the number of parameters in the fractional model derived with the current method will tend to be larger, however with this technique it is possible to relate results to a physical interpretation of the system dynamics. From model values it is possible to infer when a transition from liquid-like to solid-like behaviour has occurred. In the tests here this occurred after 61 s for the rapid curing cyanoacrylate and after 20 min for the slower methacrylate resin. The fractional power will depend on how the system is restructuring and the stage of the restructuring and is therefore dependent on topology and time. The method used to establish the fractional model is capable of accurately predicting both the amplitude and phase behaviour of phase transition and should be an appropriate method for modelling change of complex modulus in any process that does not exhibit instantaneous phase change. The methodology consists of three steps as follows: 1. Choose complex models for the start and end points of the transition that are based on standard models of integer powers in s. 2. Formulate the transition function based on these boundary models. 3. Calculate the interstitial states of the complex modulus by multiplying the initial state by a fractional power of the transition function.

Acknowledgments This paper is based upon work from COST Action Fractional (CA15225), supported by COST (European Cooperation in Science and Technology).

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Appendices A1. Maxwell model of curing adhesive with time varying viscosity and stiffness Using the same spirit as the model developed in [12] a Maxwell system shown in Fig. A1 is used here to model a curing adhesive in the early stage of cure. In this case both the elasticity E(t) and damping η(t) are time varying To model the variation of stiffness and damping a relatively simple formulation will be used. The elasticity will be modelled on the premise that bonds are initially formed between molecules and as the cure continues the molecular chains lengthen. If one likens this to a spring where springs are connected in series, then the overall stiffness of the polymer chain decreases with time. The elasticity E(t) can therefore be described as

E (t ) = E0 (1 − γ t )

(A1.1)

where E0 is the initial elasticity and γ is a positive constant. A simple approach is also adopted when describing how the variable damper behaves. As polymerization occurs the volume of liquid material in the matrix decreases or put another way, the volume fraction ϕ of solid material increases. This gives rise to an increase in the apparent viscosity η of the liquid/solid material and will be modelled as

η = η0 ( 1 + φ )

(A1.2)

where η0 is the viscosity of the liquid phase. This is a similar form to the model devised by Einstein [13] in his work on the viscosity of a suspension of spheres in a liquid. If one assumes that ϕ = θ t describes a time changing solid volume fraction, where θ is a constant then

η (t ) = η0 (1 + θ t )

(A1.3)

More sophisticated models of apparent viscosity have been developed depending on the volume fraction [14] and one might argue that for both the elastic and viscous models chosen here the power of t in Equations A1 and A3 is still arbitrary. While this is the case the experimental work described in the Sections 3 and 4 will rely on data that are sampled over short time periods at instances during the curing process and therefore using a linear evolution for the increase in chain length and volume fraction is a reasonable approach. Stress in an elastic (spring) element can be determined using the Boltzmann superposition principal



σs (t ) = s (0 )E (t ) +

t

0

˙ s (τ )E (t − τ )dτ

(A1.4)

This can also be described by the convolution between the stiffness and the rate of strain.

σs (t ) = ˙ s (t )  E (t )

(A1.5)

where  represents a convolution. Using a similar approach to Pandey [9], the overall modulus for the time varying system can be determined as



G ( s ) = kE0 1 −

γ s

E0 γ 1 s s+θ

e η0 θ

E0

s η0 θ

(A1.6)

where k is a constant. A2. Time domain descriptions of models derived from complex moduli Three complex moduli derived in Section 2 and are represented here in the their time domain form. The first is that for the transition from a Newtonian liquid to a Hookean elastic model represented by Eq. (15) as

Gβ (s ) = E1 β (sη0 )1−β

(A2.1)

Fig. A1. Maxwell system with variable elasticity and variable damping.

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The stress (σ (t)) strain (ε (t)) equation in the time domain is then

σ (t ) = E1 β η0 1−β

∂β  (t ) ∂tβ

(A2.2)

The second model is that used to describe the transition from a Kelvin Voigt to a Hookean solid represented by Eq. (17)

G β ( s ) = [E 1 ]β [E 0 + s η 0 ]1 − β

(A2.3)

This must be transformed to the time domain by taking the inverse Laplace transform L−1 {}

Gβ (t ) = L

−1



E1 β η0 1−β L−1

β

[E 1 ] [E 0 + s η 0 ]

 E

0

η0

+s

1 −β

1−β 

=e

= E1 E t

− η0 0

β

η0

1 −β

L

−1



E0

η0

E1 β η0 1−β L−1 s1−β

+ s

1−β 

(A2.4)

(A2.5)

The stress strain equation in the time domain is then E0 t

σ (t ) = e− η0 E1 β η0 1−β

d 1 −β  (t ) d t 1 −β

(A2.6)

The third model is that used to describe the transition from a Maxwell to a Standard Linear Solid represented by Eq. (21)

Gβ ( s ) =

 E E + η s β  E η s 1 −β 2( 1 1 ) 0 0 E 2 + E 1 + η1 s E 0 + η0 s

(A2.7)

Representing this as

Gβ (s ) = A(s ).B(s ) so that

(A2.8)

A (s ) =

 E E + η s β 2( 1 1 ) E 2 + E 1 + η1 s

B (s ) =

 E η s 1 −β 0 0 E 0 + η0 s

and

(A2.9)

(A2.10)

This must be transformed to the time domain by taking the inverse Laplace transform

A(t ) = L

−1



E 0 η0 s E 0 + η0 s

1 −β 



= η0

1 −β

L

−1

s 1 + ηE00 s

1 − β 

(A2.11)

It is known that the complex modulus for the Davidson Cole model, GDC (s) is given by Garrappa et al. [15]

GDC (s ) =

1

( 1 + sτ ) γ

0≤γ ≤1

(A2.12)

and that the response function GDC (t), that is the inverse Laplace transform is given by

GDC (t ) =

1 (t/τ )γ −1 (−t/τ ) e

(A2.13)

τ (γ )

therefore

⎡ 

− β

−

E0 t

E0 t e η0 d1−β E0 η0 A(t ) = η0 1−β 1−β ⎣ (1 − β ) dt

⎤ ⎦

(A2.14)

Now considering the inverse Laplace transform of B(s)

B(t ) = L−1

B(t ) = η1



βe

⎧

⎨  −E1 t E 2 ( E 1 + η1 s ) β = e η1 L−1 E 2 + E 1 + η1 s ⎩

−E1 t

η1



L

−1

s 1 + ηE21 s

E2 s E

2 η1 + s

β ⎫ ⎬ ⎭

(A2.15)

β  (A2.16)

H. Esmonde and S. Holm / Applied Mathematical Modelling 77 (2020) 1041–1053

B(t ) = η1 β e

−E1 t

η1

⎡   ⎤ β −1 − Eη2 t E2 Eη21t e 1 ⎣ ⎦ η1 (β ) dt β dβ

1053

(A2.17)

The stress strain equation in the time domain is then

σ (t ) = A(t )  B(t )   (t )

(A2.18)

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