J.S. Bergström

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.

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.

MECHANICAL BEHAVIOR OF PARTICLE FILLED ELASTOMERS

The strong influence of relatively small amounts of filler particles, such as carbon black, on the mechanical properties of elastomers has been well known for decades and has significantly contributed to increased use of elastomeric materials in many commercial applications. Even though the use of fillers is ubiquitous, satisfactory understanding and modeling of the micromechanisms by which fillers alter the mechanical behavior of elastomers has still not been realized. In this work the influence of filler particles on the equilibrium stress-strain response has been investigated. First, an experimental investigation probing the behavior of a Chloroprene rubber with varied filler content has been performed. The experimental data allowed for a direct evaluation of both a newly developed constitutive model based on the amplification of the first strain invariant, and a number of other models proposed in the literature. A direct comparison with experimental data suggest that the new model generates superior predictions, particularly for large strain deformations. Then, in an eort to examine some of the assumptions that are common in the constitutive modeling and also to try to determine which of the input parameters are most important, a detailed series of micromechanical models were constructed using two- and three-dimensional finite element simulations. The results indicate that the eect of filler particles on the equilibrium behavior of elastomers can be accurately predicted using stochastic three-dimensional simulations suggesting that successful modeling mainly requires a rigorous treatment of the composite nature of the microstructure and not molecular level concepts such as alteration of mobility or eective crosslinking density in the elastomeric phase of the material.

Thermomechanical behavior of virgin and highly crosslinked ultra-high molecular weight polyethylene used in total joint replacements

Three series of uniaxial tension and compression tests were conducted on two conventional and two highly crosslinked ultra-high molecular weight polyethylenes (UHMWPEs) all prepared from the same lot of medical grade GUR 1050. The conventional materials were unirradiated (control) and gamma irradiated in nitrogen with a dose of 30 kGy. The highly crosslinked UHMWPEs were gamma irradiated at room temperature with 100 kGy and then thermally processed by either annealing below the melt transition at 100 degrees C or by remelting above the melt transition at 150 degrees C. The true stress-strain behavior of the four UHMWPE materials was characterized as a function of strain rate (between 0.02 and 0.10 s(-1)) and test temperature (20-60 degrees C). Although annealing and remelting of UHMWPE are primarily considered as methods of improving oxidation resistance, thermal processing was found to significantly impact the crystallinity, and hence the mechanical behavior, of the highly crosslinked UHMWPE. The crystallinity and radiation dose were key predictors of the uniaxial yielding, plastic flow, and failure properties of conventional and highly crosslinked UHMWPEs. The thermomechanical behavior of UHMWPE was accurately predicted using an Arrhenius model, and the associated activation energies for thermal softening were related to the crystallinity of the polymers. The conventional and highly crosslinked UHMWPEs exhibited low strain rate dependence in power law relationships, comparable to metals. In light of the unifying trends observed in the true stress-strain curves of the four materials investigated in this study, both crosslinking (governed by the gamma radiation dose) and crystallinity (governed by the thermal processing) were found to be useful predictors of the mechanical behavior of UHMWPE for a wide range of test temperatures and rates. The data collected in this study will be used to develop constitutive models based on the physics of polymer systems for predicting the thermomechanical behavior of conventional and crosslinked UHMWPE used in total joint replacements.

Constitutive modeling of the time-dependent and cyclic loading of elastomers and application to soft biological tissues

Abstract The stress–strain behavior of both elastomeric materials and soft biological tissues exhibits time-dependence and hysteresis when subjected to cyclic loading. This paper discusses a new constitutive model capable of capturing the experimentally observed behavior under different general multiaxial loading conditions. The proposed model is a modification of the Bergstrom–Boyce model [J. Mech. Phys. Solids 46 (1998) 931] in which predictions of cyclic loading states have been improved by augmenting a reptation-based scaling law to include one additional material parameter that limits the maximum flow rate at any given deformation state. By direct comparison with experimental data on two rubber compounds and two soft biological tissues, the new material model is shown to capture the rate-dependence and the cyclic loading (ranging from positive to negative axial and shear loadings) in both elastomers and soft biological tissues.

Prediction of multiaxial mechanical behavior for conventional and highly crosslinked UHMWPE using a hybrid constitutive model

The development of theoretical failure, fatigue, and wear models for ultra-high molecular weight polyethylene (UHMWPE) used in joint replacements has been hindered by the lack of a validated constitutive model that can accurately predict large deformation mechanical behavior under clinically relevant, multiaxial loading conditions. Recently, a new Hybrid constitutive model for unirradiated UHMWPE was developed Bergström et al., (Biomaterials 23 (2002) 2329) based on a physics-motivated framework which incorporates the governing micro-mechanisms of polymers into an effective and accurate continuum representation. The goal of the present study was to compare the predictive capability of the new Hybrid model with the J(2)-plasticity model for four conventional and highly crosslinked UHMWPE materials during multiaxial loading. After calibration under uniaxial loading, the predictive capabilities of the J(2)-plasticity and Hybrid model were tested by comparing the load-displacement curves from experimental multiaxial (small punch) tests with simulated load-displacement curves calculated using a finite element model of the experimental apparatus. The quality of the model predictions was quantified using the coefficient of determination (r(2)). The results of the study demonstrate that the Hybrid model outperforms the J(2)-plasticity model both for combined uniaxial tension and compression predictions and for simulating multiaxial large deformation mechanical behavior produced by the small punch test. The results further suggest that the parameters of the HM may be generalizable for a wide range of conventional, highly crosslinked, and thermally treated UHMWPE materials, based on the characterization of four material properties related to the elastic modulus, yield stress, rate of strain hardening, and locking stretch of the polymer chains. Most importantly, from a practical perspective, these four key material properties for the Hybrid constitutive model can be measured by relatively simple uniaxial tension or compression tests.

Miniature specimen shear punch test for UHMWPE used in total joint replacements.

Despite the critical role that shear is hypothesized to play in the damage modes that limit the performance of total hip and knee replacements, the shear behavior of ultra-high molecular weight polyethylene (UHMWPE) remains poorly understood, especially after oxidative degradation or radiation crosslinking. In the present study, we developed the miniature specimen (0.5 mm thickness x 6.4mm diameter) shear punch test to evaluate the shear behavior of UHMWPE used in total joint replacement components. We investigated the shear punch behavior of virgin and crosslinked stock materials, as well as of UHMWPE from tibial implants that were gamma-irradiated in air and shelf aged for up to 8.5 years. Finite element analysis, scanning electron microscopy, and interrupted testing were conducted to aid in the interpretation of the shear punch load-displacement curves. The shear punch load-displacement curves exhibited similar distinctive features. Following toe-in, the load-displacement curves were typically bilinear, and characterized by an initial stiffness, a transition load, a hardening stiffness, and a peak load. The finite element analysis established that the initial stiffness was proportional to the elastic modulus of the UHMWPE, and the transition load of the bilinear curve reflected the development of a plastically deforming zone traversing through the thickness of the sample. Based on our observations, we propose two interpretations of the peak load during the shear punch test: one theory is based on the initiation of crystalline plasticity, the other based on the transition from shear to tension during the tests. Due to the miniature specimen size, the shear punch test offers several potential advantages over bulk test methods, including the capability to directly measure shear behavior, and quite possibly infer ultimate uniaxial behavior as well, from shelf aged and retrieved UHMWPE components. Thus, the shear punch test represents an effective and complementary new tool in the armamentarium of miniature specimen mechanical testing methods for UHMWPE used in total joint replacement components.

Compressive failure of rocks by shear faulting

A model for the compressive failure of rocks via the process of shear faulting is presented. The model addresses the progressive growth of damage that leads to the formation of a critical fault nucleus which grows unstably in its own plane by fracturing the grain boundaries in an increasingly rapid succession. The model uses a two-parameter Weibull-type shear strength distribution for defining the nucleation of initial damage, followed by the use of stress enhancement factors for addressing the increased probability of failure in the vicinity of already cracked grain boundaries. These factors essentially involve surface averaging of enhanced stresses in the neighboring grains with the appropriate strength distribution as the weighting function. As the stress is further increased, similar correlated fracturing events get preferentially aligned to the crack cluster, resulting in en echelon of cracks. This crack cluster is modeled as an elliptical inhomogeneity within which the cracks interact and lead to material pulverization, the effect of which, mechanistically, is to lower the shear modulus compared with the uncracked material on the outside. The shear stress concentration resulting from this moduli mismatch is calculated and used to compute the stress enhancement factors for defining the nucleation of additional cracking events near the crack cluster. Eventually, the size of the crack cluster becomes sufficiently large that it carries a stress concentration high enough to fracture all grain boundary elements in from of it in an increasingly rapid succession. The stress associated with this event is taken as the failure stress. Since the model allows other cracking events to occur within the material volume in accordance with the assumed strength distribution function, formation of other competing but subcritical shear faults naturally occurs. Besides the faulting stress for a prescribed confinement, the model is able to predict the angle of the shear fault fairly well. The model is applied to a variety of rock types, including granite, eclogite, gabbro, aplite, rock salt, sandstone, dunite, limestone, and marble, and the predicted failure envelopes compare rather well with the failure data available in the literature.

A progressive damage model for failure by shear faulting in polycrystalline ice under biaxial compression

Abstract A model for the compressive failure of polycrystalline ice via the process of shear faulting is presented. The model addresses the progressive growth of damage that leads to the formation of a critical fault nucleus, which grows unstably in its own plane by fracturing the grain boundaries in an increasingly rapid succession. Because the intrinsic grain boundary strength in ice polycrystals is fairly uniform, the variation in the boundary strength was simply due to their random orientation with respect to the loading axes. The latter was captured via an equivalent two parameter Weibull-type shear strength distribution for defining the nucleation of initial damage, followed by the use of stress enhancement factors for addressing the increased probability of failure in the vicinity of already cracked grain boundaries. These factors essentially involve surface averaging of enhanced stresses in the neighboring grains with the appropriate strength distribution as the weighting function. As the stress is further increased, similar correlated fracturing events gets preferentially aligned to the crack cluster, resulting in en echelon of cracks. This crack cluster is modeled as an elliptical inhomogeneity within which the cracks interact and lead to material pulverization, the effect of which, mechanistically, is to lower the shear modulus compared with the uncracked material on the outside. The shear stress concentration resulting from this moduli mismatch is calculated and used to compute the stress enhancement factors for defining the nucleation of additional cracking events near the crack cluster. Eventually, the size of the crack cluster becomes sufficiently large such that it carries a stress concentration high enough to fracture all grain boundary elements in front of it in an increasingly rapid succession. The stress associated with this event is taken as the failure stress. Since the model allows other cracking events to occur within the material volume in accordance with the assumed strength distribution function, formation of other competing but subcritical shear faults naturally occurs. Besides the faulting stress for a prescribed confinement, the model is able to predict the angle of the shear fault fairly well. The model is used to predict failure stress and understand observations of fault development in laboratory-grown freshwater columnar ice loaded under across-column biaxial compression.

Compressive failure of rocks

Abstract A model for the compressive failure of rocks via the process of shear faulting is presented. The model addresses the progressive growth of damage that leads to the formation of a critical fault nucleus, which grows unstably in its own plane by fracturing the grain boundaries in an increasingly rapid succession. The model uses a two parameter Weibull-type shear strength distribution for the defining nucleation of initial damage, followed by the use of stress enhancement factors for addressing the increased probability of failure in the vicinity of already cracked grain boundaries. As the stress is further increased, similar correlated fracturing events gets preferentially aligned to the crack cluster, resulting in an echelon of cracks. This crack cluster is modeled as an elliptical inhomogeneity with a lower shear modulus compared with the uncracked material on the outside. The shear stress concentration resulting from this moduli mismatch was calculated and used to compute the stress enhancement factors for defining the nucleation of additional cracking events near the crack cluster. Eventually, the size of the crack cluster becomes sufficiently large such that it carries a stress concentration high enough to fracture all grain boundary elements in front of it in an increasingly rapid succession. The stress associated with this event is taken as the failure stress. The model is applied to a variety of rocks, including granite, eclogite, gabbro, aplite, rocksalt, sandstone, dunite, limestone, and marble, and the results compare rather well with the failure data from the literature.