A constitutive equation for nonlinear stress–strain curves of crystalline polymers

We present a new method for the analysis of stress–strain curves for crystalline polymers such as polypropylene and polyethylene. A nonlinear constitutive equation that includes terms that cover the plastic deformation and anharmonicity of the spring is developed. In order to quantitatively characterize the nonlinear viscoelasticity using this equation, data on the transient moduli during elongation at a constant rate of strain are required. Hence, the simultaneous measurements of linear oscillatory viscoelastic moduli during a constant rate of elongation were investigated. It was found that the present method makes possible the evaluation of the plastic deformation fraction and the Gruneisen constant for crystalline polymers. © 1998 Chapman & Hall     

A constitutive model for the viscoplastic behavior of rubbery polymers at finite strains

Summary. A constitutive model is derived for the viscoplastic behavior of rubbery polymers at finite strains. A polymer is treated as an equivalent network of chains bridged by permanent junctions. The elastic response of the network is attributed to the elongation of strands, whereas its plastic behavior is associated with the sliding of nodes with respect to their initial positions. Unlike conventional stress–strain relations in finite viscoplasticity, the rate-of-strain tensor for the sliding of junctions is expressed in terms of the rate-of-strain tensor for macro-deformation. Constitutive equations are developed by using the laws of thermodynamics. These relations are simplified for simple shear of an incompressible medium with finite strains. The governing equations are determined by 3 material constants. To verify the model, a series of shear tests is performed on polycarbonate melts reinforced with short glass fibers. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is shown that the material constants change with the filler content in a physically plausible way.Department of Chemical Engineering, Kuwait University, Safat 13060, Kuwait     

A constitutive equation for concrete

A more rational approach to strength criterion development for concrete is proposed to cover the composite nature and complex failure mechanism of concrete materials. The use of scalar valued function theory as applied to concrete failure prediction is demonstrated. The results are applicable to general brittle materials     

A constitutive relation for the deformation of snow

In this paper a constitutive equation which describes the uniaxial deformation of snow is developed. The basic assumption underlying this work is that the stress-strain response can be derived by considering the structure of the material. The equation which describes the plastic portion of the deformation is developed by considering the relationship between three fundamental variables: the mean spacing between ice grains, the relative velocity between grains, and the fraction of the total number of grains which participate in the deformation process.The mean distance between ice grains is determined by a stereological investigation of the snow structure, and the velocity component is found by empirically characterizing the relaxation of the snow. To determine the mobility of the ice grains acoustic emission data are used. An equation describing the pattern of acoustic emissions for constant rates of deformation is derived and applied to a number of tests. Combining the above variables produces a compressive and tensile constitutive equation which reflects the behavior of the snow under both uniaxial deformations.     

Mg-9Al-3Si-0.375Sr-0.78Y合金的热变形行为及本构模型

利用Gleeble-3500热模拟试验机对Mg-9Al-3Si-0.375Sr-0.78Y合金试样进行等温恒应变速率压缩实验,研究其在温度250~400℃、应变速率0.001~10s~(-1)条件下的热变形行为。结果表明:在热变形过程中,峰值应力随着应变速率的降低和温度的升高而减小,且峰值应力对应变速率的敏感性随着变形温度的下降而增强。建立了考虑应变的热变形Arrhenius本构模型,模型精度良好,在300,350℃及0.001~10s~(-1)范围内,模型的平均绝对误差分别为1.57%和1.76%;合金的平均变形激活能为183.58k J/mol,平均应变速率敏感指数为0.1616。热变形过程中,α-Mg相呈现明显的动态再结晶特征,β-Mg₁₇Al₁₂相尺寸减小且分布均匀,初生Mg_2Si相较小。在低温(250~300℃)变形时,动态再结晶仅发生在晶界处。在高温(350~400℃)变形时,初生α-Mg晶粒发生了明显的动态再结晶。随着温度的增加和应变速率的降低,再结晶程度提高,再结晶晶粒逐渐长大。     

A constitutive equation for non-linear electro-active solids

Summary Electro-active solids are solids that are either infused with electrorheological fluids or embedded with electrically conducting particles, the body as a whole however conducting negligible current. In this paper, we provide a mathematical framework, within the context of continuum mechanics, for the study of electro-active solids. The theory assumes that the body can be considered as a continuum, in the sense of homogenization, which is isotropic, incompressible, elastic and is capable of responding to an electric field. Appealing to standard techniques in continuum mechanics, we obtain a constitutive relation for the stresses in terms of the deformation and electric field. This is used in a study of triaxial extension, simple shear and anisotropy induced by the electric field.     

A simple novel constitutive equation for concrete

A simple novel constitutive equation, three parameter strength criterion for concrete is proposed to represent the composite nature and complex failure mechanism of material of concrete. In this paper, the study is to demonstrate the use of scalar valued function, invariant theory, as applied to concrete failure prediction. Without the loss of accuracy of prediction, a three parameter strength criterion is developed.     

A thermo-viscoelastic constitutive model for compressible amorphous polymers

We propose a compressible thermo-viscoelastic constitutive model at finite deformation for amorphous polymers. Firstly, the compressible hyperelastic deformation energy is extended to include thermal effect. Secondly, in order to describe the viscous property of the materials, internal variables are introduced into the above thermo-hyperelastic constitutive relation. As the Maxwell model is equivalent to the transient network model under certain condition, the rationality of introducing the internal variables for a general deformation mode is proved. The model only contains a few material parameters, just necessary for a complicated thermal-mechanical coupling system. Besides, this model provides an explanation for the deformation energy function from the microscopic mechanism. Based on this model, the influences of the loading rate, compressibility and the thermal expansion coefficient on the coupled thermo-mechanical behavior of polymers are discussed.     

A 3D finite deformation constitutive model for amorphous shape memory polymers: A multi-branch modeling approach for nonequilibrium relaxation processes

Shape memory polymers (SMPs) are materials that can recover a large pre-deformed shape in response to environmental stimuli. For a thermally activated amorphous SMP, the pre-deformation and recovery of the shape require the SMP to traverse its glass transition temperature (T_g) to complete the shape memory (SM) cycle. As a result, the recovery behavior of SMPs shows strong dependency on both the pre-deforming temperature and recovery temperature. Generally, to capture the multitude of relaxation processes, multi-branch models (similar to the 1D generalized viscoelastic model or Prony series) are used to model the time-dependent behaviors of polymers. This approach often requires an arbitrary (usually numerous) number of branches to capture the material behavior, which results in a substantial number of material parameters. In this paper, a multi-branch model is developed to capture the SM effect by considering the complex thermomechanical properties of amorphous SMPs as the temperature crosses T_g. The model utilizes two sets of nonequilibrium branches for fundamentally different modes of relaxation: the glassy mode and Rouse modes. This leads to a significant reduction in the number of material parameters. Model simulation comparisons with a range of thermomechanical experiments conducted on a tert-butyl acrylate-based SMP show very good agreement. The model is further utilized to explore the intrinsic recovery behavior of an SMP and the size effects on the free recovery characteristics of a magneto-sensitive SMP composite.     

A constitutive theory for the inelastic behavior of concrete

A general theory for the inelasticity of concrete is proposed, the main constituents being a new, rate independent model of distributed damage for mortar and the application of mixture theories to account for the composite nature of concrete. The proposed theory of damage is capable of accommodating fully anisotropic elastic degradation, both in tension and in compression, in a manner which is ideally suited for computation. Mixture theories, on the other hand, are found to provide a simple yet effective tool for characterizing the values of the phase stresses that act on mortar and aggregate and which drive damage and plastic flow. This uneven distribution of stresses between mortar and aggregate is seen to lie at the foundation of effects such as the characteristic splitting failure modes in uniaxial compression and the unloading hysteretic loops that arise during cyclic loading. Further to furnishing useful insights into the physical mechanisms underlying the inelastic behavior of concrete, the proposed model provides a simple means of quantifying such behavior in a way which can be readily implemented in any standard finite element code. Possible generalizations of the theory are suggested. In particular, it is noted how rate and rheological effects can be incorporated into the proposed framework by extending it into the viscoplastic range and through the use of Eyring's theory of thermal activation.     

A non linear integral constitutive equation for viscoelastic fluids

An intégral constitutive equation is written using a particular reference frame, built with unit vectors along the principal axes of the rate-of-deformation tensor, and using the associated intrinsic rate-of-rotation. This equation is easier to handle in calculations than corotational or codeformational models. The material functions for a rheological model including the first six terms of the constitutive equation have been studied in steady and unsteady shear flows, as well as in elongational flows. Material functions are readily written from six memory functions and no inconsistency comes out.     

A constitutive model for the deformation induced anisotropic plastic flow of metals

A mathematical framework is established for the equations governing inelastic deformation under multi-dimensional stress states and for the associated evolution equations of the internal state variables. The formulation is based on a generalization of the Prandtl-Reuss flow law. In the evolution equations for the inelastic state variables that control plastic flow, it is assumed that part of the rate of change is isotropic and the remaining part varies according to the sign and orientation of the current rate of deformation vector. This leads to a minimum of twelve components of the internal state tensor which represents resistance to inelastic deformation. In this manner, both initial and load history induced plastic anisotropy can be modeled. A specific set of equations for anisotropic plastic flow is developed consistent with the inelastic state variables.     

A phenomenological constitutive model for the nonlinear viscoelastic responses of biodegradable polymers

We formulate a constitutive framework for biodegradable polymers that accounts for nonlinear viscous behavior under regimes with large deformation. The generalized Maxwell model is used to represent the degraded viscoelastic response of a polymer. The large-deformation, time-dependent behavior of viscoelastic solids is described using an Ogden-type hyperviscoelastic model. A deformation-induced degradation mechanism is assumed in which a scalar field depicts the local state of the degradation, which is responsible for the changes in the material’s properties. The degradation process introduces another timescale (the intrinsic material clock) and an entropy production mechanism. Examples of the degradation of a polymer under various loading conditions, including creep, relaxation and cyclic loading, are presented. Results from parametric studies to determine the effects of various parameters on the process of degradation are reported. Finally, degradation of an annular cylinder subjected to pressure is also presented to mimic the effects of viscoelastic arterial walls (the outer cylinder) on the degradation response of a biodegradable stent (the inner cylinder). A general contact analysis is performed. As the stiffness of the biodegradable stent decreases, stress reduction in the stented viscoelastic arterial wall is observed. The integration of the proposed constitutive model with finite element software could help a designer to predict the time-dependent response of a biodegradable stent exhibiting finite deformation and under complex mechanical loading conditions.     

A large deformation anisotropic thermoviscoplastic constitutive model

Summary An anisotropic large deformation thermoviscoplastic constitutive model has been formulated in which flow is regarded as a dissipative process characterized by a driving energy, a threshold energy, and a retardation time. Thermodynamic state functions such as internal energy are derived explicitly as functions of the state variables, namely temperature, elastic strain and flow strain. The model developed here is relevant to applications such as temperature rise and stress relaxation in metals under shock loading.     

Instrumented anvil-on-rod impact experiments for validating constitutive strength model for simulating transient dynamic deformation response of metals

Instrumented anvil-on-rod impact experiments were performed to access the applicability of this approach for validating a constitutive strength model for dynamic, transient-state deformation and elastic–plastic wave interactions in vanadium, 21-6-9 stainless steel, titanium, and Ti–6Al–4V. In addition to soft-catching the impacted rod-shaped samples, their transient deformation states were captured by high-speed imaging, and velocity interferometry was used to record the sample back (free) surface velocity and monitor elastic–plastic wave interactions. Simulations utilizing AUTODYN-2D hydrocode with Steinberg–Guinan constitutive equation were used to generate simulated free surface velocity traces and final/transient deformation profiles for comparisons with experiments. The simulations were observed to under-predict the radial strain for bcc vanadium and fcc steel, but over-predict the radial strain for hcp titanium and Ti–6Al–4V. The correlations illustrate the applicability of the instrumented anvil-on-rod impact test as a method for providing robust model validation based on the entire deformation event, and not just the final deformed state.     

A constitutive model for finite deformation response of layered polyurethane-montmorillonite nanocomposites

A constitutive model is developed to predict the finite deformation response of multilayered polyurethane (PU)-montmorillonite (MTM) nanocomposites. In PU-MTM nanocomposites, the PU matrix in the vicinity of the MTM nanoparticles is modified leading to an interphase region, and its effect on the finite deformation response of these nanocomposites is largely neglected in many existing models. In this work, the entire spatial volume is considered to be occupied by multi-layers of bulk PU and effective particles which consist of MTM nanoparticles and the modified PU interphase region. A Langevin chain based eight chain model is used to capture the large stretch hyperelastic behavior of bulk PU. The effective particle component of the model consists of a linear elastic spring to capture the initial elastic response, a non-linear viscoplastic dash-pot for the strain-rate dependent yield strength of nanocomposites, and a non-linear spring element in parallel to the dash-pot for the strain-hardening response. The model adopts the concept of amplified strain of the confined PU chains to accommodate the applied strain owing to the limited strain in the MTM nanoparticles. The constitutive model predicts all the major features of the stress-strain constitutive response of a family of PU-MTM nanocomposites including the initial linear elastic response, yield strength and post yield strain hardening for all volume fractions of MTM nanoparticles, thus confirming the efficacy of the proposed constitutive model.     

A fully coupled model for diffusion-induced deformation in polymers

Polymers that mechanically respond to the presence of a diffusing fluid/solvent have found various applications in drug delivery, tissue scaffolding, sensors and actuators. These applications involve understanding of both the diffusion process and the evolution of the deformation of the polymers during the diffusion process. For example, in a polymeric actuator one might be interested in the extent of deformation one can achieve given a solvent environment and the time in which it can be achieved. There are two key aspects in modeling such behavior. First, the displacement gradients involved are usually large, especially in problems such as “self-assembly.” Second, since the diffusion occurs in a deforming polymeric medium, an appropriate diffusion model that includes the effect of the deformed state of the body as well as the interaction between the polymeric medium and the diffusing fluid has to be considered. In effect, this results in the diffusion and equilibrium equation being fully coupled and nonlinear. In this work, we model diffusion-induced deformation in an elastic material including large deformations based on thermodynamics framework. For the chemical potential, we use the Flory–Huggins potential adapted to include the effect of stress in the polymers. Using the model, we simulate folding and bending of a rectangular polymeric strip by simultaneous solution of the diffusion equation as well as the equilibrium equation using the finite element method. Parametric studies are also conducted in order to examine the effect of material parameters on the diffusion and deformation behaviors. Finally, using the coupled diffusion–deformation model we simulate deformations of composite domains comprising of polymeric constituents with different diffusion–deformation behaviors in order to achieve various interesting “self-assembly” shapes.