Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates

A robust physically consistent three-dimensional constitutive model is developed to describe the finite mechanical response of amorphous polymers over a wide range of temperatures and strain rates, including the rubbery region and for impact loading rates. This thermomechanical model is based on an elastic–viscoplastic rheological approach, wherein the effects of temperature, strain rate, and hydrostatic pressure are accounted for. Intramolecular, as well as intermolecular, interactions under large elastic–inelastic behavior are considered for the mechanisms of deformation and hardening. For a wide range of temperatures and strain rates, our simulated results for poly(methyl methacrylate) (PMMA) and polycarbonate (PC) are in good agreement with experimental observations.     

Thermomechanical formulation of ductile damage coupled to nonlinear isotropic hardening and multiplicative viscoplasticity

In this paper, we present a thermomechanical framework which makes use of the internal variable theory of thermodynamics for damage-coupled finite viscoplasticity with nonlinear isotropic hardening. Damage evolution, being an irreversible process, generates heat. In addition to its direct effect on material's strength and stiffness, it causes deterioration of the heat conduction. The formulation, following the footsteps of Simó and Miehe (1992), introduces inelastic entropy as an additional state variable. Given a temperature dependent damage dissipation potential, we show that the evolution of inelastic entropy assumes a split form relating to plastic and damage parts, respectively. The solution of the thermomechanical problem is based on the so-called isothermal split. This allows the use of the model in 2D and 3D example problems involving geometrical imperfection triggered necking in an axisymmetric bar and thermally triggered necking of a 3D rectangular bar.     

A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers

The stress response of amorphous polymers exhibits tremendous change during the glass transition region, from soft viscoelastic response to stiff viscoplastic response. In order to describe the temperature-dependent and rate-dependent stress response of amorphous polymers, we extend the one-dimensional small strain fractional Zener model to the three-dimensional finite deformation model. The Eyring model is adopted to represent the stress-activated viscous flow. A phenomenological evolution equation of yield strength is used to describe the strain softening behaviors. We demonstrate that the stress response predicted by the three-dimensional model is consistent with that of one-dimensional model under uniaxial deformation, which confirms the validity of the extension. The model is then applied to describe the stress response of an amorphous thermoset at various temperatures and strain rates, which shows good agreement between experiments and simulation. We further perform a parameter study to investigate the influence of the model parameters on the stress response. The results show that a smaller fractional order results in a larger yield strain while has little effect on the yield stress when the temperature is below the glass transition temperature. For the stress relaxation tests, a smaller fractional order leads to a slower relaxation rate.     

Thermomechanical cyclic behavior modeling of Cu-Al-Be SMA materials and structures

This paper concerns the behavior of Cu-Al-Be polycrystalline shape memory alloys under cyclic thermomechanical loadings. Sometimes, as shown by many experimental observations, a permanent inelastic strain occurs and increases with the number of cycles. A series of cyclic thermomechanical tests has been carried out and the origin of the residual strain has been identified as residual martensite. These observations have been used to develop a 3D macroscopic model for the superelasticity and stress assisted memory effect of SMAs able to describe the evolution of permanent inelastic strain during cycles. The model has been implemented in a finite elements code and used to simulate the behavior of antagonistic actuators based on SMA springs under cyclic thermomechanical loading with a residual displacement appearance.     

Predicting thermal shape memory of crosslinked polymer networks from linear viscoelasticity

The viscoelastic behavior of an amorphous shape memory polymer network and its dependence on time and temperature were measured by dynamic mechanical analysis. The resulting thermo-mechanical behavior was modeled and implemented in a commercial finite element code. The ability of the resulting thermomechanical model to simulate and, eventually, predict the shape storage and shape recovery of the material was evaluated against experimental shape memory thermomechanical torsion data in a large deformation regimen. The simulations showed excellent agreement with experimental shape memory thermomechanical cycle data. This demonstrates the dependence of the shape recovery on time and temperature. The results suggest that accurate predictions of the shape recovery of any amorphous polymer networks under any thermomechanical conditions combination solely depends on considering the material viscoelasticity and its time–temperature dependence.     

Development and implementation of a beam theory model for shape memory polymers

In this paper we focus on the development of a beam theory for a small strain continuum model of thermoviscoelastic shape memory polymers (SMP). Rather than a history integral model that is common for viscoelastic materials, a thermodynamically based state evolution model developed by Ghosh and Srinivasa (2011a) is used as the basis for the beam model based on the Euler–Bernoulli beam theory. An example of a three-point bend test is simulated using the beam theory model. The numerical solution is implemented by using an operator split technique that utilizes an elastic predictor and dissipative corrector. The key idea is that the elastic predictor is based on the solution to a beam theory boundary value problem while the dissipative corrector is entirely local (and hence can be parallelized) and is applied by considering the beam as a two or three dimensional body. This enables a very rapid solution of the problem yet maintaining fidelity of the distribution of inelastic strains across the cross-section. A displacement based convergence criterion is used in each time step. This algorithm is validated by using a three-point bending experiment for three different material cases: elastic, plastic and thermoplastic response. Time step convergence and mesh density convergence studies are carried out for the thermoviscoelastic FEM model. Finally, we implement and study this model for a SMP beam undergoing three-point bending strain recovery and stress recovery thermomechanical loading.     

Prediction of the softening and damage effects with permanent set in fibrous biological materials

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical response of soft biological tissues. Inelastic phenomena which occur in mechanical testing of biological tissues are very likely to be associated with alterations in the internal structure of these materials.In this study, a novel structural constitutive model is formulated to describe the inelastic effects in soft biological tissues such as Mullins type behavior, damage and permanent set as a result of residual strains after unloading. Anisotropic softening is considered by evolution of internal variables governing the anisotropic properties of the material. We consider two weight factors w_i (softening) and s_k (discontinuous damage) as internal variables characterizing the structural state of the material. Numerical simulations of several soft tissues are used to demonstrate the performance of the model in reproducing the inelastic behavior of soft biological tissues.     

Planar Flexural Vibrations and Dissipative Heating of Laminated Rectangular Plates

The coupled thermomechanical dynamic behavior of a heterogeneous solid is investigated by an example of laminated inelastic rectangular plates subjected to forced vibration and dissipative heating. The problem is solved in two formulations: complete and approximate. The complete formulation uses generalized flow theory to describe the inelastic properties of the material. The approximate formulation employs a model based on the idea of complex moduli. Special distributions of the field variables and amplitude–frequency responses of the rectangular plate are analyzed. The effect of the loading level on the mode shape and stress distribution is studied as well. A good agreement between the results obtained in the complete and approximate formulations is pointed out.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

Long time response of aging glassy polymers

Aging amorphous polymeric materials undergo free volume relaxation, which causes slowing down of the relaxation dynamics as a function of time. The resulting time dependency poses difficulties in predicting their long time physical behavior. In this work, we apply an effective time domain approach to the experimental data on aging amorphous polymers and demonstrate that it enables prediction of long time behavior over the extraordinary timescales. We demonstrate that, unlike the conventional methods, the proposed effective time domain approach can account for physical aging that occurs over the duration of the experiments. Furthermore, this procedure successfully describes time–temperature and time–stress superpositions. It can also allow incorporation of varying dependences of relaxation time on aging time as well as complicated but known deformation history in the same experiments. This work strongly suggests that the effective time domain approach can act as an important tool to analyze the long time physical behavior of aging amorphous polymeric materials.     

Formulation of a damage internal state variable model for amorphous glassy polymers

The following article proposes a damage model that is implemented into a glassy, amorphous thermoplastic thermomechanical inelastic internal state variable framework. Internal state variable evolution equations are defined through thermodynamics, kinematics, and kinetics for isotropic damage arising from two different inclusion types: pores and particles. The damage arising from the particles and crazing is accounted for by three processes of damage: nucleation, growth, and coalescence. Nucleation is defined as the number density of voids/crazes with an associated internal state variable rate equation and is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing as an internal state variable rate equation that is a function of stress state, rate sensitivity, and strain rate. The coalescence internal state variable rate equation is an interactive term between voids and crazes and is a function of the nearest neighbor distance of voids/crazes and size of voids/crazes, temperature, and strain rate. The damage arising from the pre-existing voids employs the Cocks–Ashby void growth rule. The total damage progression is a summation of the damage volume fraction arising from particles and pores and subsequent crazing. The modeling results compare well to experimental findings garnered from the literature. Finally, this formulation can be readily implemented into a finite element analysis.     

Rate (time)-dependent deformation behavior: an overview of some properties of metals and solid polymers

The inelastic deformation behaviors of metals and polymers are discussed with the aim of finding a common base that would simplify academic and engineering analyses. Only monotonic loading conditions at room temperature are considered. For loading at different rates, nonlinear relations between loading rate and stress level, creep stress level and creep strain, and relaxation rate and stress were common to both type of materials. There are, of course, significant differences in elastic properties, strength levels and the strains involved. Special properties such as relaxation behaviors and creep anomalies can be qualitatively and quantitatively reproduced by the state variable model VBO (viscoplasticity theory based on overstress). Since experimental investigations typically concentrate on one particular aspect of inelastic deformation behavior such as creep or strain-rate dependence, it is often difficult to gather a comprehensive data set for a given material. In spite of this, considerable similitude in the deformation behavior of metals and polymers in various test conditions has nevertheless been established.     

Study of the Resonant Vibrations and Dissipative Heating of Thin-Walled Elements Made of a Physically Nonlinear Material

An approximate formulation is given to a dynamic coupled thermomechanical problem for physically nonlinear inelastic thin-walled structural elements within the framework of a geometrically linear theory and the Kirchhoff–Love hypotheses. A simplified model is used to describe the vibrations and dissipative heating of inhomogeneous physically nonlinear bodies under harmonic loading. Nonstationary vibroheating problem is solved. The dissipative function obtained from the solution for steady-state vibrations is used to simulate internal heat sources. For the partial case of forced vibrations of a beam, the amplitude–frequency characteristics of the field quantities are studied within a wide frequency range. The temperature characteristics for the first and second resonance modes are compared.     

Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers

A self-consistent model for semi-crystalline polymers is proposed to study their constitutive behavior, texture and morphology evolution during large plastic deformation. The material is considered as an aggregate of composite inclusions, each representing a stack of crystalline lamellae with their adjacent amorphous layers. The deformation within the inclusions is volume-averaged over the phases. The interlamellar shear is modeled as an additional slip system with a slip direction depending on the inclusion's stress. Hardening of the amorphous phase due to molecular orientation and, eventually, coarse slip, is introduced via Arruda-Boyce hardening law for the corresponding plastic resistance. The morphology evolution is accounted for through the change of shape of the inclusions under the applied deformation gradient. The overall behavior is obtained via a viscoplastic tangent self-consistent scheme. The model is applied to high density polyethylene (HDPE). The stress-strain response, texture and morphology changes are simulated under different modes of straining and compared to experimental data as well as to the predictions of other models.     

Photomechanics of light-activated polymers

Light-activated polymers are an exciting class of modern materials that respond mechanically when irradiated by light at particular wavelengths. While details of the mechanisms that connect the optical excitation to mechanical behavior are complex and differ from material to material, there is sufficient commonality among them to permit the development of a generalized modeling framework to describe the photomechanics. The features shared by light-activated polymers involve light interacting with the material, which triggers photochemical reactions that alter the structure of the crosslinked polymer network. Many such structural alterations result in an evolution of the polymer network, and subsequent macroscopic deformation. When this process is appropriately executed it can enable a photomechanical shape-memory effect. In this paper, we develop a three-dimensional finite-deformation modeling framework to describe the photomechanical response of light-activated polymer systems. This framework integrates four coupled phenomena that contribute to macroscopic photomechanical behavior: photophysics, photochemistry, chemomechanical coupling, and mechanical deformation. The chemomechanical coupling consists of chemically induced structural alterations of the crosslinked network that result in subsequent deformation. We describe this behavior through a decomposition of the crosslinked network into two components consisting of an original network and a photochemically altered network; both evolve during photomechanical deformation. The modeling framework presented in this paper is sufficiently general that it is applicable to light-activated polymer systems that operate with various mechanisms in each of the four areas. Using this modeling approach, we develop constitutive models for two recently developed light-activated polymer systems [Lendlein, A., Hongyan, J., Junger, O., Langer, R., 2005. Light-induced shape-memory polymers. Nature 434 (7035) 879; Scott, T.F., Schneider, A.D., Cook, W.D., Bowman, C.N., 2005. Photoinduced plasticity in crosslinked polymers. Science 308 (5728) 1615]. For the material developed by Scott and his co-workers we validate our model by measuring and numerically simulating photo-induced stress relaxation and bending deformation and obtain good agreement between measurements and predictions. Finally, we use the model to study the effects of photomechanical parameters (applied strain magnitude, irradiation time and intensity, and photoabsorber concentration) and the behavior of the network evolution rule on the material response.     

Thermomechanical response of metallic sandwich tubes with prismatic cores considering active cooling

In this paper, the transient temperature distribution and the thermomechanical response of sandwich tubes with prismatic cores are analyzed considering active cooling. The effective thermal conductivities of prismatic cores with active cooling are derived. By using the effective thermal conductivities, the transient temperature fields of the tubes are predicted and are found to be very close to the results of finite element simulations, which confirms the correctness of the effective thermal conductivity. Based on the high-order sandwich shell theory, the thermal structural responses of sandwich structures are studied and compared with the results of finite element simulations. The reduction of the thermal structural responses as a result of active cooling is studied to demonstrate the advantages of prismatic cellular materials. The design of replacing the solid metal with cellular materials which has the capability of active cooling can reduce the temperature and the thermal structural response of the structure.     

Analysis of deformation process characteristics in amorphous-crystalline polymers

In the framework of a two-component medium, the phenomenological approach is used to develop a system of constitutive equations describing the thermomechanical behavior of amorphous-crystalline polymers. This model is designed to describe the stress-strain states in the temperature range comprising the intervals of phase and relaxation transitions. The results of numerical experiments demonstrating the possibility of describing the characteristic properties of deformation processes typical of amorphous-crystalline polymers are given.