Multi-cycle deformation of silicone elastomer: observations and constitutive modeling with finite strains

Observations are reported on a medical grade of silicone elastomer in uniaxial tensile tests up to breakage of specimens, short-term relaxation tests, and cyclic tests with monotonically increasing maximum elongation ratios. Experimental data in cyclic tests demonstrate the fading memory phenomenon: stress–strain diagrams for two specimens with different deformation histories along the first n?1 cycles and coinciding loading programs for the other cycles become identical starting from the nth cycle. A constitutive model is developed in cyclic viscoplasticity of elastomers with finite strains, and its adjustable parameters are found by fitting the experimental data. Ability of the stress–strain relations to predict the mechanical response in cyclic tests with various deformation programs is confirmed by numerical simulation.     

Cyclic thermo-viscoplasticity of high density polyethylene

Observations are reported in uniaxial cyclic tensile tests (loading–unloading with various maximum strains) on high density polyethylene at temperatures ranging from room temperature up to 90 °C. It is demonstrated that the maximum stress per cycle and an apparent residual strain (measured at the instant when the tensile force vanishes under retraction) strongly decrease with temperature. The latter seems unexpected as the interval of temperatures covers the α-relaxation temperature, which is conventionally associated with activation of additional mechanisms for inelastic flow. A model is developed that captures the decrease in residual strain with temperature. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. The effects of temperature and maximum strain per cycle on residual strains are studied numerically.     

Thermo-viscoelastic and viscoplastic behavior of high-density polyethylene

Observations are reported on high-density polyethylene in uniaxial tensile tests with constant strain rates and relaxation tests at various temperatures ranging from 25 to 90 °C. A constitutive model is derived for the nonlinear viscoelastic and viscoplastic behavior of semi-crystalline polymers at three-dimensional deformations. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. It is demonstrated that (i) the model correctly approximates the observations and (ii) material parameters are independent of strain rate and change consistently with temperature.     

Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress

Uniaxial compression stress–strain tests were carried out on three commercial amorphous polymers: polycarbonate (PC), polymethylmethacrylate (PMMA), and polyamideimide (PAI). The experiments were conducted under a wide range of temperatures (−40 °C to 180 °C) and strain rates (0.0001 s⁻¹ up to 5000 s⁻¹). A modified split-Hopkinson pressure bar was used for high strain rate tests. Temperature and strain rate greatly influence the mechanical response of the three polymers. In particular, the yield stress is found to increase with decreasing temperature and with increasing strain rate. The experimental data for the compressive yield stress were modeled for a wide range of strain rates and temperatures according to a new formulation of the cooperative model based on a strain rate/temperature superposition principle. The modeling results of the cooperative model provide evidence on the secondary transition by linking the yield behavior to the energy associated to the β mechanical loss peak. The effect of hydrostatic pressure is also addressed from a modeling perspective.     

Integral-based constitutive equation for rubber at high strain rates

High-speed experiments were conducted to characterize the deformation and failure of Styrene Butadiene Rubber at impact rates. Dynamic tensile stress–strain curves of uniaxial strip specimens and force–extension curves of thin sheets were obtained from a Charpy tensile impact apparatus. Results from the uniaxial tension tests indicated that although the rubber became stiffer with increasing strain rates, the stress–strain curves remained virtually the same above 280 s⁻¹. Above this critical strain rate, strength, fracture strain and toughness decreased with increasing strain rates. When strain rates were below 180 s⁻¹, the initial modulus, tensile strength and breaking extension increased as the strain rate increased. Between strain rates of 180 and 280 s⁻¹, the initial modulus and tensile strength increased with increasing strain rates but the extension at break decreased with increasing strain rates. A hyper-viscoelastic constitutive relation of integral form was used to describe the rate-dependent material behavior of the rubber. Two characteristic relaxation times, 5 ms and 0.25 ms, were needed to fit the proposed constitutive equation to the data. The proposed constitutive equation was implemented in ABAQUS Explicit via a user-defined subroutine and used to predict the dynamic response of the rubber sheets in the experiments. Numerical predictions for the transient deformation and failure of the rubber sheet were within 10% of experimental results.     

Cyclic viscoelastoplasticity and low-cycle fatigue of polymer composites

Observations are reported on a polymer composite (polyamide-6 reinforced with short glass fibers) in tensile relaxation tests with various strains, tensile creep tests with various stresses, and cyclic tests with a stress-controlled program (ratcheting with a fixed maximum stress and various minimum stresses). Constitutive equations are developed in cyclic viscoelastoplasticity of polymer composites. Adjustable parameters in the stress–strain relations are found by fitting observations in relaxation tests and cyclic tests (16 cycles of loading–unloading). It is demonstrated that the model correctly predicts experimental data in creep tests and dependencies of maximum and minimum strains per cycle on number of cycles up to fatigue fracture of specimens. The influence of strain rate and minimum stress on number of cycles to failure is studied numerically.     

Mullins’ effect in semicrystalline polymers

Observations are reported on isotactic polypropylene in uniaxial cyclic tensile tests at room temperature. A model is derived for the viscoplastic response of semicrystalline polymers at three-dimensional deformations with small strains. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. It is shown that polypropylene reveals some characteristic features of the Mullins effect that can be quantitatively predicted by the constitutive equations.     

Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test

This paper derives a new three-dimensional (3-D) analytical solution for the indirect tensile tests standardized by ISRM (International Society for Rock Mechanics) for testing rocks, and by ASTM (American Society for Testing and Materials) for testing concretes. The present solution for solid circular cylinders of finite length can be considered as a 3-D counterpart of the classical two dimensional (2-D) solutions by Hertz in 1883 and by Hondros in 1959. The contacts between the two steel diametral loading platens and the curved surfaces of a cylindrical specimen of length H and diameter D are modeled as circular-to-circular Hertz contact and straight-to-circular Hertz contact for ISRM and ASTM standards respectively. The equilibrium equations of the linear elastic circular cylinder of finite length are first uncoupled by using displacement functions, which are then expressed in infinite series of some combinations of Bessel functions, hyperbolic functions, and trigonometric functions. The applied tractions are expanded in Fourier–Bessel series and boundary conditions are used to yield a system of simultaneous equations. For typical rock cylinders of 54 mm diameter subjected to ISRM indirect tensile tests, the contact width is in the order of 2 mm (or a contact angle of 4°) whereas for typical asphalt cylinders of 101.6 mm diameter subjected to ASTM indirect tensile tests the contact width is about 10 mm (or a contact angle of 12°). For such contact conditions, 50 terms in both Fourier and Fourier–Bessel series expansions are found sufficient in yielding converged solutions. The maximum hoop stress is always observed within the central portion on a circular section close to the flat end surfaces. The difference in the maximum hoop stress between the 2-D Hondros solution and the present 3-D solution increases with the aspect ratio H/D as well as Poisson’s ratio ν. When contact friction is neglected, the effect of loading platen stiffness on tensile stress in cylinders is found negligible. For the aspect ratio of H/D = 0.5 recommended by ISRM and ASTM, the error in tensile strength may be up to 15% for both typical rocks and asphalts, whereas for longer cylinders with H/D up to 2 the error ranges from 15% for highly compressible materials, and to 60% for nearly incompressible materials. The difference in compressive radial stress between the 2-D Hertz solution or 2-D Hondros solution and the present 3-D solution also increases with Poisson’s ratio and aspect ratio H/D. In summary, the 2-D solution, in general, underestimates the maximum tensile stress and cannot predict the location of the maximum hoop stress which typically locates close to the end surfaces of the cylinder.     

Simple constitutive modelling of nonlinear viscoelasticity under general extension

We study the behaviour of a single integral constitutive equation, capable of providing analytic expressions for the viscoelastic stress in extensional flows of a variety of deformation histories and geometries, ranging from uniaxial to equibiaxial. It is based on the use of a stress damping function, with a power-law dependence on the elongation, λ: h(λ) = 1/λⁿ. The parameter n (0 ≤ n ≤ 2) signifies the nonlinear viscoelastic character of the material and, therefore, is an inverse measure of network connectivity strength of the underlying microstructure. This renders the constitutive approach applicable to incompressible polymers of a variable degree of branching, strain hardening and stress thinning behavior. Methods of connecting n with the macromolecular architecture and the alignment strength of the flow are also explored.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

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.     

Modeling and measuring visco-elastic properties: From collagen molecules to collagen fibrils

Collagen is the main structural protein in vertebrate biology, determining the mechanical behavior of connective tissues such as tendon, bone and skin. Although extensive efforts in the study of the origin of collagen exceptional mechanical properties, a deep knowledge of the relationship between molecular structure and mechanical properties remains elusive, hindered by the complex hierarchical structure of collagen-based tissues. Understanding the viscoelastic behavior of collagenous tissues requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar and molecular level. Hence, we investigate the viscoelastic properties at the molecular level by using an atomistic modeling approach, performing in silico creep tests of a collagen-like peptide. The results are compared with creep and relaxation tests at the level of isolated collagen fibrils performed previously using a micro-electro-mechanical systems platform. Individual collagen molecules present a non-linear viscoelastic behavior, with a Young's modulus increasing from 6 to 16 GPa (for strains up to 20%), a viscosity of 3.84±0.38 Pa s, and a relaxation time in the range of 0.24–0.64 ns. At the fibrils level, stress–strain–time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell–Weichert model. The fibrils showed an elastic modulus of 123±46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell–Weichert model with a fast time response of 7±2 s and a slow time response of 102±5 s.     

The characteristics of plastic flow and a physically-based model for 3003 Al–Mn alloy upon a wide range of strain rates and temperatures

The uniaxial compressive responses of 3003 Al–Mn alloy upon strain rates ranging from 0.001/s to about 10⁴/s with initial temperatures from 77 K to 800 K were investigated. Instron servohydraulic testing machine and enhanced split Hopkinson bar facilities have been employed in such uniaxial compressive loading tests. The maximum true strain up to 80% has been achieved. The following observations have been obtained from the experimental results: 1) 3003 Al–Mn alloy presents remarkable ductility and plasticity at low temperatures and high strain rates; 2) its plastic flow stress strongly depends on the applied temperatures and strain rates; 3) the temperature history during deformation strongly affects the microstructure evolution within the material. Finally, paralleled with the systematic experimental investigations, a physically-based model was developed based on the mechanism of dislocation kinetics. The model predictions are compared with the experimental results, and a good agreement has been observed.     

Influence of temperature on a carbon–fibre epoxy composite subjected to static and fatigue loading under mode-I delamination

In this paper, interlaminar crack initiation and propagation under mode-I with static and fatigue loading of a composite material are experimentally assessed for different test temperatures. The material under study is made of a 3501-6 epoxy matrix reinforced with AS4 unidirectional carbon fibres, with a symmetric laminate configuration [0°]_16/S. In the experimental programme, DCB specimens were tested under static and fatigue loading. Based on the results obtained from static tests, fatigue tests were programmed to analyse the mode-I fatigue behaviour, so the necessary number of cycles was calculated for initiation and propagation of the crack at the different temperatures. G–N curves were determined under fatigue loading, N being the number of cycles at which delamination begins for a given energy release rate. G_ICmax–a, a–N and da/dN–a curves were also determined for different G_cr rates (90%, 85%, 75%, etc.) and different test temperatures: 90 °C, 50 °C, 20 °C, 0 °C, ?30 °C and ?60 °C.     

Extrudate swell of linear and branched polyethylenes: ALE simulations and comparison with experiments

Extrudate swell is a common phenomenon observed in the polymer extrusion industry. Accurate prediction of the dimensions of an extrudate is important for appropriate design of dies for profile extrusion applications. Prediction of extrudate swell has been challenging due to (i) difficulties associated with accurate representation of the constitutive behavior of polymer melts, and (ii) difficulties associated with the simulation of free surfaces, which requires special techniques in the traditionally used Eulerian framework. In a previous work we had argued that an Arbitrary Lagrangian Eulerian (ALE) based finite element formulation may have advantages in simulating free surface deformations such as in extrudate swell. In the present work we reinforce this argument by comparing our ALE simulations with experimental data on the extrudate swell of commercial grades of linear polyethylene (LLDPE) and branched polyethylene (LDPE). Rheological behavior of the polymers was characterized in shear and uniaxial extensional deformations, and the data was modeled using either the Phan–Thien Tanner (PTT) model or the eXtended Pom–Pom (XPP) model. Additionally, flow birefringence and pressure drop measurements were done using a 10:1 contraction–expansion (CE) slit geometry in a MultiPass Rheometer. Simulated pressure drop and contours of the principal stress difference were compared with experimental data and were found to match well. This provided an independent test for the accuracy of the ALE code and the constitutive equations for simulating a processing-like flow. The polymers were extruded from long (L/D = 30) and short (L/D = 10) capillaries dies at 190 °C. ALE simulations were performed for the same extrusion conditions and the simulated extrudate swell showed good agreement with the experimental data.     

Elastic–plastic behavior of steel sheets under in-plane cyclic tension–compression at large strain

Elastic–plastic behavior of two types of steel sheets for press-forming (an aluminum-killed mild steel and a dual-phase high strength steel of 590 MPa ultimate tensile strength) under in-plane cyclic tension–compression at large strain (up to 25% strain for mild steel and 13% for high strength steel) have been investigated. From the experiments, it was found that the cyclic hardening is strongly influenced by cyclic strain range and mean strain. Transient softening and workhardening stagnation due to the Bauschinger effect, as well as the decrease in Young's moduli with increasing prestrain, were also observed during stress reversals. Some important points in constitutive modeling for such large-strain cyclic elasto-plasticity are discussed by comparing the stress–strain responses calculated by typical constitutive models of mixed isotropic–kinematic hardening with the corresponding experimental observations.     

Mullins�� effect in semicrystalline polymers: experiments and?modeling

Experimental data are reported on isotactic polypropylene in uniaxial cyclic tensile tests with various maximum strains at room temperature. It is demonstrated that polypropylene reveals all characteristic features (hysteresis of energy, damage accumulation, and strain-hardening) of the Mullins effect. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at three-dimensional deformations with small strains. Adjustable parameters in the stress?Cstrain relations are found by fitting the observations. Numerical simulation shows that the model adequately predicts the viscoplastic response of polypropylene in uniaxial and biaxial cyclic tests.