1. Introduction
Understanding the thermochemical and thermomechanical coupling states in materials is very important and necessary from various points of view, especially for designing smart materials and estimating the durability of various industrial products. This research area investigates the interaction of mechanical and chemical forces in metals and polymers, as well as temperature dependent effects.
Among the various industrial materials, polymers are more sensitive to such mechanical and chemical forces. This is due to their chemical structure and also their morphology. Amorphous and semi-crystalline polymers can be easily filled with low molecular weight compounds, such as water and other chemical solvents, when they come into contact with them. This would lead to morphology changes and structural defects, as the polymer chains would then be pulled apart by small molecules diffused into them. As a result, their characteristic temperatures, such as the glass transition temperature () or crystallization temperature (), will change and their effective functional temperature range varies clearly.
One of the most sensitive materials to even small thermochemical and thermomechanical coupling states are Shape-Memory-Polymers (SMPs). Shape-Memory Effect (SME) is an ability of a SMP to be deformed and manipulated to a fixed temporary shape until an appropriate trigger is utilized for transformation of temporary shape to a memorized original shape [
1,
2,
3]. There are a variety of shape storage and triggering mechanisms for different polymer systems [
4,
5]. However, polymers with appropriate chemical structure and morphology should be programmed and processed prior to triggering with, e.g., extrusion, electro-spinning, or 3-D printing, in order to achieve the Shape-Memory (SM) capability [
6]. The SME appears normally due to heating and deforming of the system above its transformation temperature (
), which could be glass transition (
) or meting temperature (
) and subsequent cooling while the deformation is kept constant for solidification of chain segments and shape-fixation. Next, this deformed SMP can be exposed to a temperature above its
to transform its temporary shape to permanent shape. This cycle is called Shape-Memory Creation Cycle (SMC) and could be repeated several times [
7,
8]. It should be noted that any diffusion of low molecular weight compounds into the polymer matrix would upset the thermochemical and thermomechanical coupling states in the materials and lead to SMC failure.
Thermoplastic phase-segregated multi-block copolymers, like our investigated polyether urethane Estane (Lubrizol, Ovele Westerlo, Belgium), are very interesting materials because of their mechanical stability and the capability of showing SME. Moreover, their permanent shape can be easily achieved through common processing approaches. However, computational modeling studies and predictive models are needed to design and optimize their SM-properties prior to any technical application. In this content, constitutive relations between the field variables, e.g., stress (), strain or stretch ( or ), and temperature (), are of interest.
Elastic and hyperelastic behavior of SMPs can be studied by simple (static) tensile testing, whereby their viscoelastic properties should be studied, e.g., by relaxation- or creep experiments. Both of these properties can also be described by mathematical models. As an example, the hyperelastic behavior of polymeric samples can be explained by strain-invariant-based models, like the one of Mooney-Rivlin [
9] or Yeoh [
10], or by principle-stretched-based models, like that of Ogden [
11]. Marckmann et al. [
12] proposed a thorough comparison of twenty hyperelastic models for rubberlike materials and analyzed their abilities to reproduce different types of loading conditions.
As a matter of course, choosing a correct mathematical model and precise determination of material parameters is very crucial as it has a significant impact on the accuracy and reliability of the results. The usual way to find the parameters of a model is as follows: a series of proper and relevant experiments is performed and then by mathematical describing the physical behavior in the test, the results of the experiments are fitted to the mathematical model. Here, normally, simple methods, like stochastic techniques and evolution strategies, optimization procedures for inherent parameter identification based on the Nedler-Mead simplex algorithms, and other procedures, are employed [
13,
14,
15]. Although no gradient information is needed in all these methods their performance is poor. Therefore, sometimes multi-axial tests are performed employing complex sample geometries and resulting inhomogeneities, which also have their own disadvantages since the inverse calculations are computationally demanding [
16,
17,
18].
Twizell et al. [
19] used the optimization algorithm method of Levenberg-Marquardt and determined the material constants of the Ogden model. Saleeb et al. [
20] introduced an issue of developing effective and robust schemes to implement a class of the Ogden type hyperelastic constitutive models, for large strain analysis of rubber-like materials.
Constitutive relations for viscoelastic materials can be obtained from elastic and viscous elements. In order to elucidate the viscoelastic and viscoplastic responses of polymers, two hypotheses have been used: (I) the split of the free energy of the viscoelastic solid into an equilibrium and non-equilibrium part and (II) the multiplicative decomposition of the deformation gradient into an elastic and viscous part [
21]. In this framework, Huber et al. [
22], proposed a rheological three-parameter model to describe the mechanical behavior of materials in a limited range of small deformations and extended it to large strains.
Park and Schapery [
23] used the Prony series to describe the relaxation and creep behavior of a viscoelastic material. Diebels et al. [
24] identified the elastic and viscoelastic material parameters from constitutive equations by means of a Tikhonov regularization and inspired an extra penalty term from the stress-strain relationships to expect better results. Haupt et al. [
25] used a relatively simple identification method based on the concept of fractional calculus and obtain the model-inherent material parameters. Yoshida et al. [
26] suggested a constitutive model comprised of two elastoplastic and hyperelastic parts. The elastoplastic part includes a strain-dependent isotropic hardening law and the hyperelastic part incorporates the damage model. Amin et al. [
27] introduced a hyperviscoelastic model to explain the mechanical behavior of rubbers. Their model consists of a nonlinear viscous coefficient to represent the rate dependent behavior and is validated through relevant tests for compression and shear regimes.
The second method for determination of material constants involves the use of Finite Element Methods (FEM). Some researchers simulated the experimental investigations with Finite Element Analysis and so identified the material parameters. This method has the advantage that (I) the complexitiy of heterogeneous problems are more in consideration than in homogeneous analytic investigations, and (II) the obtained parameters are often more accurate. As an example, Petera et al. used Finite Element Analysis for improvement of viscometry results obtained by a cone-plate rheometer [
28]. Ghoreishy [
29] has also benefited from Finite Element Method and determined the parameters of the Prony series in a hyperviscoelastic material model.
Aside from the above mentioned methods, Huang et al. [
30] used nanoindentation tests to measure the complex moduli of linearly viscoelastic materials through an indentation process with a spherical indenter. Beake [
31] has also utilized this novel testing technique to investigate the creep behavior of thin semi-crystalline and amorphous polymers. Experimental data were adapted to a logarithmic equation relating the fractional increase in penetration depth during creep and predicted the extension and creep ratio for different maximum loads. The influence of temperature on viscoelastic behavior of SMPs is especially important. A polymer system is said to be thermo-rheological simple if all relaxation times are affected by temperature in the same way. Thus, by application of the Time-Temperature Superposition Principle (TTSP) and the Williams-Landel-Ferry (WLF) equations, it is possible to emerge mastercurves using a reduced time variables or shift factors to obtain a broader time (frequency) domain for the data of the system [
32]. In previous contributions, we have performed frequency sweep tests under torsion and determined storage and loss shear moduli mastercurves and computed the Prony constants for the tested material as solutions of a minimization problem for Tikhonov functionals [
33,
34]. Pacheco et al. [
35] proposed a methodology for characterization of material parameters of thermo-rheologically and piezorheologically simple systems and determined the Prony series based on a mixed optimization technique of Genetic Algorithms and Nonlinear Programming.
Nevertheless, based on our comprehensive experimental investigations [
34], it is now clear that considering Estane as a thermo-rheological simple material is a bad assumption. We have shown that this presupposition leads to inadequate results. Therefore, a new approach should be established to represent the temperature dependence of the viscoelastic properties [
36]. In the following contribution, we show that based on the presumption of thermo-rheological complexity and finding the right material parameters through uni-axial relaxation tests for finite strains, the functional properties can be very well simulated. Here, the experimentally observed effects are exhibited by a finite viscoelastic and incompressible material model and enhanced by new approach of temperature-dependency. The material parameters are strategically identified by Levenberg-Marquardt algorithm, and results are validated through stress relaxation experiments under torsion.
2. Materials and Methods
In the following essay, a polyether-based thermoplastic polyurethane, commercially available under the name of Estane (Lubrizol, Ovele Westerlo, Belgium), is used without further purification. Estane is a block-copolymer, synthesized from Methylendiphenylisocyanate (MDI) and 1,4- Butanediol with a polyether. The chemical structure of each reacting components is depicted in
Figure 1. For Estane, a number average molecular weight of about 132 kg mol
was reported using gel permeation chromatography (GPC) [
8].
For sample preparation, Estane granules are processed using an injection molding machine (Arburg Allrounder 270M 500–210, Lossburg, Germany) with an injection temperature of about 204 C and the outer temperature of the injection barrel of about 30 C. Furthermore, the samples are molded with an injection rate of 26 mm, an injection pressure of 60 MPa and a holding pressure of 55 MPa for 15 s. After processing, the plates were kept in a vacuum desiccator to keep them dry. At the end, prior to quasi-static experiments and Dynamic Mechanical Thermal Analysis (DMTA), the samples are punched either to rectangular samples or dumbbell-shaped specimens, using a manual knuckle joint press.
2.1. Dynamic Mechanical Thermal Analysis
Using rectangular samples with dimensions of W × H × L: 2 mm × 10 mm × 50 mm (
Figure 2), DMTA experiments were accomplished in torsion mode with a torque-controlled rheometer and integrated Peltier-based temperature chamber (Anton Paar Physica MCR 702 Twin Drive plus CTD 180, Graz, Austria). A small uni-axial tensile force of around 0.5 N is applied to maintain the specimen under net tension. Thereupon, with a constant heating rate of 0.25
C min
, temperature sweep tests with prescribed amplitude (0.01%) and different constant frequencies ranging from 0.5 to 16 Hz were performed. Here, the temperatures in the range of −20 to 120
C could be adjusted and kept constant with a precision of
°C. Such experiments provide important insights into the effective viscoelastic properties of the investigated material. For frequency sweep tests, samples have been tensioned as before under isothermal conditions in a frequency range of [0.1–100] Hz.
Figure 2 demonstrates schematically the investigated rectangular sample under torsional load as performed here. In an attempt of adequately characterize the temperature and the time dependency of the material response, transient stress relaxation experiments were performed. To conduct such quasi-static experiments, the dumbbell-shaped specimens of the type DIN EN ISO 527-2 and dimensions of W × H × L: 2 × 4 × 75 mm
(25 mm parallel sample length) were pneumatically fixed along their stretching axis on the rig in displacement-driven control mode.
To calibrate the displacement of the applied testing device with the local strains of the samples at ambient conditions (out of the environmental chamber), an own custom-made optical strain measurement system has been used. This leads to a one-to-one calibration of local strains and global displacements of the test rig. To do so, we performed our experiments on vertically positioned screw-driven electromechanical test frame Schenk-Trebel RM 50 operated in displacement-driven control mode. Digital control of the test rig and data acquisition was performed using the acquisition system DOLI EDC580 (DOLI Elektronik GmbH, Münsingen, Germany) which is interfaced to the host software DOLI “Test & Motion” (version 3.0) using a proprietary networking protocol. In particular, a properly calibrated S-Type load cell was used during this work, namely, a large strain-gouge-type load cell with a maximum application range of up to 500 N. The samples are fixed along their longitudinal axis, where the bottom end is fixed stationary and the upper end is moved during deformation. The motion of the upper end will be referred to as the machine displacement
U (mm). The load cell was attached to the upper clamp. Pneumatic grips with a grip pressure of 5 bar are used to clamp the samples. The grips allow for a force-controlled clamping. The grip faces are serrated and incorporated fences such that the samples can be positioned at the same location and relative motion between sample and grip face during deformation would be minimized. By using this custom-made real-time optical extensometer, the principal stretches and the corresponding principal strains in longitudinal and transversal directions are quantified. This optical strain measurement is a tool chain of image processing programs. Explicitly, a Charge-Coupled Device (CCD)-type industrial camera with resolution of 1024 × 768 pixel is used to acquire gray-scale raster-based images of the polymer sample during experiments. Mounted on a stepper-motor-driven mechanical stage, the camera is moved with half of the velocity and in the same direction of the machine traverse such that the relative motions between the center of the sample and the camera are minimized. Prior to that, a circle-shaped mark with a radius of r = 1.5 mm was painted on the center of the sample using laser-cut stencils and proper printing marker. The printing color is chosen such that the contrast between sample color and marking paint is as high as possible after gray-scale conversion. In this particular case, black color was used on our transparent Estane. As the sample is stretched, an initially circle-shaped mark will undergo a continuous affine transformation in accordance to the motion of the samples’ geometry, such that, for any spatial configuration, the set of points on the boundary of the mark, i.e., pixel data points, will satisfy the canonical ellipse equation
with (
) denoting the center of the ellipse.
Then, in the absence of rotation, by determining the ellipse’s major and minor axes (
and
, respectively), which now coincide with the Cartesian axes, the principal stretches and strains could be determined. Here, an optimization of least-square type using Equation (
1) and the pixel data points of the boundary of ellipse is performed to compute the parameters of interests.
Samples have been heated to the experimental temperature with a heating rate of 3 C min. Once the temperature has been achieved, the specimens were equilibrated for 20 min and then deformed with a speed of 25 mm min to a local maximum stretch of . This maximum stretch was kept constant for a holding time (relaxation time) of = 90 min. The decay of stress over time was then investigated at different temperatures ranging from 10 to 70 C. Assuming a linear viscoelastic behavior and as long as no viscous flow is presented, the stress will ultimately decrease to an equilibrium stress S, following multiple superimposed relaxation processes. To determine material parameters for further simulations, these experiments have been used explicitly.
2.2. Torsional Relaxation Experiments
There is no single experimental procedure which would permit to follow the viscoelastic behavior of a polymer over the whole range of time and temperature. In the present work, validation of identified material parameters and numerics considering theoretical background from continuum mechanics is performed by stress relaxation experiments under torsion.
To do so, rectangular samples with dimensions W × H × L: 2 mm × 4 mm × 50 mm were bounded to the upper and lower clamps of the rheometer, respectively, and equilibrated for 20 min at different experimental temperatures: and 105 C. Once the experimental temperature was reached, the samples were twisted up to a twist angle of with the twist rate of °/s. The torsional relaxation was then performed by fixing the twist angle at isothermal conditions up to 2 h.
2.3. Thermal Expansion Coefficient Experiments
Complementary to DMTA and quasi-static experiments in uni-axial or torsional modes, the thermal behavior of Estane and in particular its Coefficient of Thermal Expansions (CTE) have been studied by thermal mechanical Analysis (TMA). The CTE was determined by Thermal Mechanical Analyzer (TMA) Metler-Toledo TMA/SDTA 841 (Wien, Austria). A 2-mm thick sample was placed between two thin quartz disks and positioned on the TMA holder. Afterwards, a ball-point probe of thickness of 3 mm was positioned on the top of the sample to ensure a uniform distribution of the exerted force over the entire probe surface. Then, a very small net force of about 0.02 N was implemented to maintain a good contact between the sample and probe surface without deforming it. Finally, the probe was three times heated from 5 to 140 C and cooled down subsequently, whereby the third run was used for data validation. It should be noted that the both first runs were performed to eliminate any relaxation effects or eigenstresses of the specimen.
2.4. Finite Strain Maxwell-Zener Model
First, let us consider a specific version of the finite strain Maxwell-Zener model, which is an essential part of the constitutive model used in this study. Although a great amount of different formulations has been proposed for the Maxwell-Zener model, the preference is given to the multiplicative approach, due to numerous advantages (cf. Reference [
37]). The multiplicative Maxwell model is covered as a special case by the viscoplasticity model presented by Simo and Miehe [
38]. Interestingly, the Simo and Miehe model can be obtained in a number of different ways, using constitutive assumptions, which may seem unrelated [
39,
40,
41,
42]. In the current study, we employ the standard Lagrangian version of the Simo and Miehe model. We follow the presentation given by Lion [
43]. For simplicity, thermostatic conditions are assumed here, where the temperature is assumed given. A thermodynamically consistent generalization to a fully coupled thermo-mechanical state can be carried out [
44]. Let
be the deformation gradient mapping a line element
of the reference configuration to the line element
of the current configuration. We start with the multiplicative split of the deformation gradient
into an elastic part
and the inelastic part
In the context of large strain viscoelasticity, this split is known as the Sidoroff decomposition [
45]. Next, we consider the right Cauchy-Green tensor
and its inelastic counterpart
, both operating on the reference configuration:
The Helmholtz free energy per unit mass is assumed to be of neo-Hookean type [
46]
where
stands for the shear modulus,
is the mass density in the reference configuration,
is the trace operator, and
denotes the unimodular part of a tensor. According to the Coleman-Noll procedure, the 2nd Piola-Kirchhoff stress tensor
is computed through
The evolution equation for
takes the form
where
stands for viscosity, and
is the inherent relaxation time. The initial conditions at time instance
are specified as
The model Equations (
5) and (
6) are objective, thermodynamically consistent, and w-invariant (For a general definition of the w-invariance the reader is referred to Reference [
47]). The model exhibits a fading memory behavior. More precisely, the exact solution is exponentially stable with respect to small perturbations of the initial data [
48]. The exact solution exhibits the following important geometrical property
where the manifold
constitutes the set of symmetric unimodular tensors
In the following modifications, the relaxation time may depend on the temperature:
. It follows from (
8) that
remains positive definite if
.
2.5. Generalized Viscoelasticity
Now, we proceed to a model of generalized viscoelasticity, known as generalized Maxwell model (also known as Wiechert or Maxwell-Zener model) [
49]. The corresponding rheological interpretation contains a spring element (Hookean body) and
n Maxwell bodies; see
Figure 3b. All the Maxwell bodies are connected in parallel. The Hookean body is used to represent the equilibrium stresses and the Maxwell bodies are introduced to capture viscous effects. The total free energy is given by a sum of isotropic functions (cf. Reference [
50])
Here,
is the equilibrated part of the free energy stored in the spring element and
is the energy of the
mth Maxwell body, which corresponds to the non-equilibrated part. The equilibrium spring is modeled by the neo-Hookean ansatz reinforced by a volumetric contribution (cf. Reference [
51])
Here,
represents the shear modulus of the material in the equilibrium state and
k stands for the bulk modulus. Each of the Maxwell bodies is modeled by Equations (
5) and (
6) presented in the previous subsection. Thus, the Helmholtz free energy for the
Maxwell body is given by a potential of the neo-Hookean type
Here,
is the shear modulus of the
element and
is the corresponding inelastic tensor of right Cauchy-Green type. The evolution of each of these variables is governed by equations of type (
6), which takes the form
where
is a material parameter related to the relaxation time, and
is the initial value.
Equation (
9) implies that the overall second Piola-Kirchhoff stress
is given by the sum
The generalized viscoelasticity model inherits some properties of the Simo and Miehe formulation of the Maxwell model, discussed in the previous subsection. In particular, it is objective and thermodynamically consistent; it also exhibits fading memory. The exact solution of the evolution equations exhibits the geometric properties of type (
8):
for all
. In general, the relaxation times
depend on temperature
. To explain this dependency, one should assume the system as either thermo-rheological simple or complex. As explained before, in the case of thermo-rheologically simple materials, the Time-Temperature Superposition Principle (TTSP) can be used (see
Section 3.2):
where
are the relaxation times at the reference temperature
.
As will be shown in the following, modeling of Estane as a thermo-rheological complex material leads to much better results. The specific dependency of relaxation times upon temperatures will be discussed in
Section 3.2.1.
2.6. Numerical Implementation
For simplicity, we consider here only the processes with a prescribed temperature. By
, denote the corresponding relaxation times. Let us consider a typical time step
,
. Assume that the right Cauchy-Green tensor at
equals
. Within the context of a displacement-based finite element method,
is known at each point of the Gauss integration. Let the previous values of the inelastic Cauchy-Green tensor for different Maxwell branches be equal to
. Our goal is to integrate the evolution Equation (11) within the time step. Since, in many applications, the time step size
may be very large compared to some of the relaxation times
, an implicit time stepping is needed. For the Simo-Miehe version of the Maxwell model with the energy storage of neo-Hookean type, an explicit update formula was presented in Reference [
52]. In our notation, it reads
Although this formula corresponds to an implicit discretization,
it is iteration free. It can be used for large time steps, if needed. In particular,
as
. A non-iterational generalization of this formula to cover the energy storage of the Mooney-Rivlin type is presented in Reference [
53].
The corresponding numerical scheme is first order accurate; it exactly preserves the geometric property
even for very large time steps. The exact preservation of the incompressibility, in turn, suppresses the accumulation of the numerical error [
48]. Since
is known, the evolution equations for the different Maxwell branches are integrated separately. After the current values
have been found, the overall stress is computed according to (
12)–(
14). The presented algorithm is implemented into the commercial finite element code MSC.MARC, employing the user subroutine HYPELA2. Obviously, any alternative forms of the equilibrium elastic potential
can be implemented without essential changes of the presented procedure.
4. Discussion and Conclusions
Apart from the accuracy and flexibility, the geometrically nonlinear approach of Simo and Miehe is highly practical. Owing to the explicit update Equation (
16), the time stepping is robust and efficient. This computational efficiency becomes important when dealing with large-scale FEM simulations: Depending on the number of finite elements, stress integration points per each element, time steps, and the average number of iterations, the number of calls of the material subroutine ranges up to
. If the model contains 20 Maxwell branches, this means
calls of the Maxwell material’s subroutine. Another advantage of the implemented non-iterational time-stepping method is the exact preservation of the incompressibility condition. For such algorithms, the accumulation of the numerical error is suppressed even when working with large time steps and strain increments [
48].
From the theoretical standpoint, the model is thermodynamically consistent, objective and free from spurious shear oscillations. The stress response exhibits a pure split into volumetric and deviatoric parts [
37], which is useful when modeling incompressbile or nearly incompressible materials. Interestingly, both thermo-rheological simple and complex approaches yield similar numerical schemes. The main difference lies in calibration and validation of the modeling assumptions, as well as their applicability domains.
A relatively large number of rheological Maxwell units is implemented to represent the relaxation behavior over a broad range of temperatures and strain rates. To solve this problem, fractional time rates are usually implemented [
2], at a cost of more complex numerical methods. However, a simpler way to reduce the number of Maxwell branches is to incorporate stress-dependent viscosities [
1].
The main conclusion of this study is that in a certain range of temperature, strain, and strain rate Estane behaves as a thermo-rheological simple material, which can be described by a generalized finite strain Maxwell-Zener model.
When modeling Estane as a thermo-rheological complex medium, a pragmatic approach is advocated, assuming temperature-dependent shear moduli; see Equation (
22). This temperature-dependence can occasionaly violate the 2nd law of thermodyamics. Therefore, the model should be restricted to a specific range of strains and temperatures. The development of an accurate material model which a priori satisfies the 2nd law of thermodynamics still remains an open problem. A promising line of research is to incorporate a chemical potential
into the additive decomposition of the free energy (
9). Thus, the evolution of the stiffness (Equation (
22)) can be brought into compliance with the energy balance.