Multi-cycle deformation of semicrystalline polymers: Observations and constitutive modeling

Experimental data are reported on isotactic polypropylene in multi-cycle uniaxial tensile tests where a specimen is stretched up to some maximum strain and retracted down to the zero minimum stress, while maximum strains increase with number of cycles. Fading memory of deformation history is observed: when two samples are subjected to loading programs that differ along the first n − 1 cycles only, their stress–strain diagrams coincide starting from the nth cycle. Constitutive equations are developed in cyclic viscoelasticity and viscoplasticity of semicrystalline polymers, and adjustable parameters in the stress–strain relations are found by fitting the experimental data. Results of numerical simulation demonstrate that the model predicts the fading memory effect quantitatively. To confirm that the observed phenomenon is typical of semicrystalline polymers, experimental data are presented in tensile cyclic tests with large strains on low density polyethylene and compressive cyclic tests on poly(oxymethylene).     

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an elevated temperature and subsequent cooling to a lower temperature. When reheated, the original shape can be recovered. Relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial compression. After initial pre-deformation at a high temperature, the strain is held constant for shape storage while the stress evolution is monitored. Three cases of heated recovery are selected: unconstrained free strain recovery, stress recovery under full constraint at the pre-deformation strain level (no low temperature unloading), and stress recovery under full constraint at a strain level fixed at a low temperature (low temperature unloading). The free strain recovery results indicate that the polymer can fully recover the original shape when reheated above its glass transition temperature (T_g). Due to the high stiffness in the glassy state (T < T_g), the evolution of the stress under strain constraint is strongly influenced by thermal expansion of the polymer. The relationship between the final recoverable stress and strain is governed by the stress–strain response of the polymer above T_g. Based on the experimental results and the molecular mechanism of shape memory, a three-dimensional small-strain internal state variable constitutive model is developed. The model quantifies the storage and release of the entropic deformation during thermomechanical processes. The fraction of the material freezing a temporary entropy state is a function of temperature, which can be determined by fitting the free strain recovery response. A free energy function for the model is formulated and thermodynamic consistency is ensured. The model can predict the stress evolution of the uniaxial experimental results. The model captures differences in the tensile and compressive recovery responses caused by thermal expansion. The model is used to explore strain and stress recovery responses under various flexible external constraints that would be encountered in applications of SMPs.     

Evaluation of rheological constitutive equations for branched polymers in step shear strain flows

The pom-pom rheological constitutive equation for branched polymers proposed by McLeish and Larson is evaluated in step shear strain flows. Semianalytic expressions for the shear-stress relaxation modulus are derived for both the integral and approximate differential versions of the pom-pom model. Predictions from the thermodynamically motivated differential pompon model of ?ttinger are also examined. Single-mode integral and differential pom-pom models are found to give qualitatively different predictions, the former displays time–strain factorability after the backbone stretch is relaxed, while the latter does not. We also find that the differential pompon model gives quantitatively similar predictions to the integral pom-pom model in step strain flows. Predictions from multimode integral and differential pom-pom models are compared with experimental data on a widely characterized, low-density polyethylene known as 1810H. The experiments strongly support time–strain factorability, while the multimode pom-pom model predictions show deviations from this behavior over the entire range of time that is experimentally accessible.     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations

The so-called viscoplastic consistency model, proposed by Wang, Sluys and de Borst, is extended here to the integration of a thermoviscoplastic constitutive equation for J₂ plasticity and adiabatic conditions. The consistency condition in this case includes not only strain rate but also the effect of temperature on the yield function. Using the backward Euler integration scheme to integrate the constitutive equations, an implicit algorithm is proposed, leading to generalized expressions of the classical return mapping algorithm for J₂ plasticity, both for the iterative calculation of the plastic multiplier increment and for the consistent tangent operator when strain rate and temperature are considered also as state variables of the hardening equation. The model was implemented in a commercial finite element code and its performance is demonstrated with the numerical simulation of four Taylor impact tests.     

Convected derivatives for differential constitutive equations

To date, no differential constitutive equation has been proposed that agrees with each of four important experimental observations in relaxation after step shear strains: that the stress is often factorable into time and strain-dependent functions, that the strain-dependent function is shear thinning, that the ratio of first normal stress difference to shear stress equals the shear strain—that is, the Lodge-Meissner relationship holds, and that there is a negative second normal stress difference. The Johnson-Segalman model satisfies three of these, but fails to satisfy the Lodge-Meissner relationship, because in step strains the principal stress and strain axes do not rotate together. Using a mathematical technique for forcing co-rotation of stress and strain axes in an arbitrary deformation, we here present an explicit differential constitutive equation that satisfies all four of the above experimental observations.     

On constitutive equations for electrorheological materials

Constitutive equations for electrorheological (ER) fluids have been based on experimental results for steady shearing flows and constant electric fields. The fluids have been modeled as being rigid until a yield stress is reached. Additional stress is then proportional to the shear rate. Recent experimental results indicate that ER materials have a regime of solid-like response when deformed from a rest state. They behave in a viscoelastic-like manner under sinusoidal shearing and exhibit time-dependent response under sudden changes in shear rate or electric field. In this work, a constitutive theory for ER materials is presented which accounts for these recent experimental observations. The stress is given by a functional of the deformation gradient history and the electric field vector. Using the methods of continuum mechanics, a general three-dimensional constitutive equation is obtained. A sample constitutive equation is introduced which is then used to determine the response of an ER material for different shear histories. The calculated shear response is shown to be qualitatively similar to that observed experimentally.     

Inelastic constitutive equations for complex flows

A new class of inelastic constitutive equations is presented and discussed. In addition to the rate-of-strain tensor, the stress is assumed to depend also on the relative-rate-of-rotation tensor, a frame-indifferent quantity that brings information about the nature of the flow. The material functions predicted by these constitutive equations are given for simple shear and uniaxial extension. A special case of these equations takes the Newtonian form, except that the viscosity is a function of the invariants of both kinematic tensors on which the stress depends. This simple constitutive equation has potential applications in liquid flow process simulations, since it combines simplicity with the capability of responding independently to shear and extension, as real liquids seem to do. Finally, possible forms for the new viscosity function are discussed.     

A constitutive equation of elastic-visco-plasticity for polymers

The rheological behavior and a constitutive relationship of elastic-visco-plasticity for polymers are investigated in this paper. Several sets of experiments have been carried out to determine the material constants and to test the validity of constitutive formulation. It is shown that the theoretical profiles are in good agreement with experimental results. The rheological characteristic and the strain-rate effect of model are analyzed by computer simulation.The project is supported by the National Natural Science Foundation of China     

Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers

Under certain conditions, such as sufficiently low temperatures, high loading rates and/or highly triaxial stress states, glassy polymers display an unfavorable characteristic—brittleness. A technique used for reducing the brittleness (increasing the fracture toughness) of these materials is rubber toughening. While there is significant qualitative understanding of the mechanical behavior of rubber-toughened polymers, quantitative modeling tools for the large-strain deformation of rubber-toughened glassy polymers are largely lacking.In this paper, we develop a suite of numerical tools to investigate the mechanical behavior of rubber-toughened glassy polymers, with emphasis on rubber-toughened polycarbonate. The rubber particles are modeled as voids in view of their deformation-induced cavitation early during deformation. A three-dimensional micromechanical model of the heterogeneous microstructure is developed to study the effects of initial rubber particle (void) volume fraction on the underlying elasto-viscoplastic deformation mechanisms in the material, and how these mechanisms influence the macroscopic response of the material. A continuum-level constitutive model is developed for the large-strain elasto-viscoplastic deformation of porous glassy polymers, and it is calibrated against micromechanical modeling results for porous polycarbonate. The constitutive model can be used to study various boundary value problems involving rubber-toughened (porous) glassy polymers. As an example, the case of an axisymmetric notched bar is simulated for the case of polycarbonate with varying levels of initial porosity. The quality of the constitutive model calibration is assessed using a multi-scale modeling approach.     

Modelization of extensional experiments under constant force using memory-integral constitutive equations and stream-tube analysis

In this paper, we propose a numerical simulation of axisymmetric extensional experiments on a viscoelastic polydimethylsiloxane (PDMS) material, using a falling-weight extensional rheometer. The polymer behaviour is represented by a K-BKZ memory-integral constitutive equation, involving a damping function of the Wagner type. Under the assumption of a homogeneous flow zone in the sample, a numerical model is set up, using the stream-tube method and approximating functions. The governing equations of the problem, associated to a limited number of unknowns, are solved by means of the Levenberg-Marquardt optimization algorithm. The numerical results are found to be consistent with the experimental data and reveal the importance of the non-homogeneous flow zone, in relation to the estimation of the extensional strain rate. The calculations involve the sensitivity of the model on the fluid parameters and those concerning the size of the initial column of fluid. The limited computing (CPU) time of the code is also to be underlined.     

Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements

The theory of Doi and Edwards for entangled polymers has been recently modified for the case of fast flows to account for convective contributions to molecular dynamics. The flow-induced relative motion between neighboring chains removes constraints and speeds up relaxation. Convective constraint release (CCR) may thus explain why the shear stress is seen to approach a plateau at high shear rates instead of decreasing as predicted by the basic theory. In slow flows, as well as in step strain, another discrepancy between theory and observations can be found in the normal stress ratio in shear Ψ=−N₂/N₁. The theoretical value for Ψ at low deformations is 1/7 whereas measured values for well-entangled systems are systematically larger. We have recently considered the possibility that this discrepancy arises because force balance requirements at the entanglement nodes are ignored in the classical theory. Accordingly, we have proposed a change in the orientational tensor Q. Here, we sum up on these recent findings by proposing single-relaxation-time constitutive equations of the integral or rate type incorporating those concepts in a simple way. Such equations should be suitable for numerical simulation of complex flows. Received: 1 January 2000 Accepted: 8 August 2000     

Some tests of constitutive equations for entangled polymers based on planar-extension flow histories in a periodically constricted channel

Stress and velocity were determined locally by birefringence measurements and laser Doppler velocimetry for a mildly entangled polystyrene solution flowing at steady state in a rectangular channel with sinusoidally varying wall spacing. Having measured both the velocity and stress fields, we were able to test constitutive equations locally, i.e., without solving the equations of motion for the entire flow. Four were examined for the periodic planar extensions on the channel centerplane: the Newtonian model, the Lodge network model, the Doi-Edwards tube model, and the Wagner-Schaeffer modification of Doi-Edwards. High enough Weissenberg and Deborah numbers were reached to produce sizable departures from the Newtonian predictions. The Doi-Edwards model underpredicted the stress, as did Wagner-Schaeffer, although to a lesser extent. Predictions of the Lodge model were best of all, a surprising result in view of its inadequacy for simple shear deformations. The predictions of the Lodge model, without parameter adjustment, agreed remarkably well with the planar extension data over the accessible range for our apparatus: Deborah numbers up to 2.0, extensional Weissenberg numbers up to 6.5, and a maximum extension ratio of about 2.3.     

Restrictions on nonlinear constitutive equations for elastic rods

Constitutive equations for the resultant forces and moments applied to a rod-like body necessarily couple the influences of the rod geometry and the constitutive nature of the three-dimensional material from which the rod was constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the influence of the rod geometry on the constitutive response of the rod is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic rods which ensure that exact solutions of the rod equations are consistent with exact nonlinear solutions of the three-dimensional equations for all homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of rods. Also, an example of a straight beam clamped at one end and subjected to a shear force at the other end is used to examine the validity of the proposed value for the transverse shear deformation coefficient.     

On constitutive equations for fiber-reinforced nonlinearly viscoelastic solids

In this paper, we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation. These invariants are analyzed, and we obtain restrictions such as positivity of some of them.