Mary C. Boyce

A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials

Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.

Large inelastic deformation of glassy polymers. part I: rate dependent constitutive model

Abstract Glassy polymers constitute a large class of engineering solids. In order to successfully analyze the warm (near the glass transition temperature) mechanical processes by which many glassy polymeric products are manufactured, as well as to ascertain the response of the resulting part to service life loading conditions, a constitutive law that properly accounts for the large, inelastic deformation behavior of these materials is required. Such behavior is known to exhibit strain rate, temperature, and pressure dependent yield, as well as true strain softening and hardening after yield. This paper develops a three-dimensional constitutive model based on the macromolecular structure of these materials and the micromechanism of plastic flow which encompasses these above dependencies. The experiments necessary to determine the material properties used in the model are also identified. The model predictions for the true stress-strain behavior of PMMA are then compared with experimental data reported in the literature.

Constitutive Models of Rubber Elasticity: A Review

Abstract A review of constitutive models for the finite deformation response of rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible rubber elasticity are discussed and compared to experimental data. A hybrid of the Flory—Erman model for low stretch deformation and the Arruda—Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloars classical data over the entire stretch range for all deformation states. The modeling of compressibility is also addressed.

Constitutive modeling of the large strain time-dependent behavior of elastomers

Abstract The mechanical behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic loading. Although these features of the rubbery constitutive response are well-recognized and important to its function, few models attempt to quantify these aspects of response perhaps due to the complex nature of the behavior and its apparent inconsistency with regard to current reasonably successful models of rubber elasticity. In this paper a detailed experimental investigation probing the material response of carbon black filled Chloroprene rubber subjected to different time-dependent strain histories is presented. Some of the key observations from the experiments are: (1) both filled and unfilled elastomers show significant amounts of hysteresis during cyclic loading; (2) the amount of carbon black particles does not strongly influence the normalized amount of hysteresis; (3) both filled and unfilled elastomers are strain-rate dependent and the rate dependence is higher during the uploading than during the unloading; (4) at fixed strain, the stress is observed to approach the same equilibrium level with relaxation time whether loading or unloading. Based on the experimental data a new constitutive model has been developed. The foundation of the model is that the mechanical behavior can be decomposed into two parts: an equilibrium network corresponding to the state that is approached in long time stress relaxation tests; and a second network capturing the non-linear rate-dependent deviation from the equilibrium state. The time-dependence of the second network is further assumed to be governed by the reptational motion of molecules having the ability to significantly change conformation and thereby relaxing the overall stress state. By comparing the predictions from the proposed three-dimensional constitutive model with experimental data for uniaxial compression and plane strain compression we conclude that the constitutive model predicts rate-dependence and relaxation behavior well.

A general anisotropic yield criterion using bounds and a transformation weighting tensor

Abstract A gspeneral Expression for the yield surface of polycrystalline materials is developed. The proposed yield surface can describe both isotropic and anisotropic materials. The isotropic surface can be reduced to either the Tresca or von Mises surface if appropriate, or can be used to capture the yield behavior of materials (e.g. aluminum) which do not fall into either category. Anisotropy can be described by introducing a set of irreducible tensorial state variables. The introduced linear transformation is capable of describing different anisotropic states, including the most general anisotropy (triclinic) as opposed to existing criteria which describe only orthotropic materials. Also, it can successfully describe the variation of the plastic strain ratio (R-ratio), where polycrystalline plasticity models usually fail. A method for obtaining the material constants using only uniaxial test data is described and utilized for the special case of orthotropic anisotropy. Finally, the use of tensorial state variables together with the introduced mathematical formulation make the proposed yield function a very convenient tool for numerical implementation in finite element analysis.

Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers

Abstract The effects of strain rate and temperature on the inelastic response of a glassy polymer have been studied. Deformation tests in uniaxial compression to strains of −1.0 were conducted on polymethylmethacrylate (PMMA) over a range of temperatures at a strain rate of −0.001/s providing nearly isothermal test conditions and thus documenting the temperature dependence of yield, strain softening, and strain hardening. The specimen surface temperatures were monitored using an infrared detector. Room temperature environment tests were then conducted over a range in strain rates and revealed a significant temperature rise at the strain rates of −0.01/s and −0.1/s. The increase in temperature has a dramatic effect on the stress-strain behavior producing a thermal softening of the material. The moderate rate tests thus underline the importance of understanding the effects of thermo-mechanical coupling during polymer deformations as occurs during impact loading condiyions and deformation processing. The experimental results have been simulated using a fully three-dimensional constitutive model of the large strain inelastic response of glassy polymers in conjunction with a thermo-mechanically coupled finite element analysis. The strain rate and temperature dependence of initial yield is included in the material model as well as temperature dependence of evolving anisotropy and its associated strain hardening. The material model considers that part of the work of inelastic deformation responsible for strain hardening to be stored as an internal back stress and therefore is not dissipative. the remaining dissipative plastic work acts as a heat source in the test simulations where conduction between the specimen and steel platens is modelled as well as convection with the surroundings. Excellent agreement between simulation and experiment is found where the stress-strain curves and temperature-strain curves are well predicted over a range in strain rate and temperature.

Evolution of plastic anisotropy in amorphous polymers during finite straining

Abstract The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce , and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran , and Boyce [in press].

Bioinspired Structural Materials

Materials scientists are seeking to create synthetic materials based on the mechanical design principles found in biological materials such as seashell nacre.

Large strain time-dependent behavior of filled elastomers

Abstract The stress–strain behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic loading. Although these features of the rubbery constitutive response are well-recognized and important to its function, few models attempt to quantify these aspects of response. Experiments have acted to isolate the time-dependent and long term equilibrium components of the stress–strain behavior (Bergstrom, J.S., Boyce, M.C., 1998. J. Mech. Phys. Solids 46, 931–954). These data formed the foundation of a constitutive model for the time-dependent, hysteretic stress–strain behavior of elastomers where the behavior is decomposed into an equilibrium molecular network acting in parallel with a rate-dependent network (cf. loc. cit.). In this paper, the Bergstrom and Boyce constitutive model is extended to specifically account for the effect of filler particles such as carbon black on the time-dependent, hysteretic stress–strain behavior. The influence of filler particles is found to be well-modeled by amplification of scalar equivalent values of the stretch and the shear stress thus providing effective measures of matrix stretch and matrix shear stress. The amplification factor is dependent on the volume fraction and distribution of filler particles; three-dimensional stochastic micromechanical models are presented and verify the proposed amplification of stretch and stress. A direct comparison between the new model and experimental data for two series of filled elastomers (a chloroprene rubber series and a natural rubber series) indicates that the new model framework successfully captures the observed behavior. The success of the model implies that the effects of filler particles on the equilibrium, rate and hysteresis behavior of elastomers mainly requires a treatment of the composite nature of the microstructure and not micro-level concepts such as alteration of mobility or effective crosslinking density of the elastomeric phase of the material.

Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition

Abstract A constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) (PET) at temperatures above the glass transition temperature is presented. In this temperature regime, the behavior of PET is strongly dependent on strain rate and temperature; PET also experiences strain-induced crystallization at these temperatures. The constitutive model accounts for the rate and temperature dependence of the stress–strain behavior by modeling the competition between molecular orientation processes and molecular relaxation processes. The model is fully three-dimensional and is shown to be in good agreement with experimental data over a wide range in strain rates and temperatures as well as under both uniaxial compression and plane strain compression loading conditions.