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The destabilization of the steady-state regime of two semi-infinite half-spaces of different elastic coefficients sliding upon each other has been theoretically investigated when a rate-and-state friction constitutive law controls the sliding. In the framework of linear and isotropic elastodynamics, the effect of the frictional constitutive law has been investigated onto the development of self-excited oscillations as well as the influence of the shear modulus difference between the two materials. The possibility of existence of a stick–slip regime and the conditions for the loss-of-contact are finally discussed.

With the recent development of micromechanics in micropolar solids, it is now possible to characterize the macroscopic mechanical behavior of cellular solids as a micropolar continuum. The aim of the present article is to apply these methods to determine the micropolar constitutive relation of various cellular solids. The main focus will be on the hexagonal packed circular honeycomb to demonstrate how its constitutive relationship is obtained. In addition, the same method will be applied to determine the material properties of a grid structure and a regular hexagon honeycomb. Since we model the cellular solid as an assembly of Euler–Bernoulli beams, we know that the macroscopic material properties will depend on the cell wall thickness, length, and Young's modulus. From this, and in conjunction with nondimensional analysis, we can provide a closed form solution, up to a multiplicative constant, without resorting to analyzing the governing equations. The multiplicative constant is evaluated through a single numerical simulation. The predicted values are then compared against assemblies with different local properties, using the numerical result as a benchmark since it takes into account higher order thickness effects. It is concluded that our closed form expressions vary from the numerical predictions only when the thickness of the beams increase, as expected since shear effects must be taken into account. However, for most engineering applications, these expressions are practical since our closed form solution with the Euler–Bernoulli assumption only produces about 10% error for most extreme cases. Our results are also verified by comparing them against those reported in the literature.

In this article, a coupled experimental and numerical method is utilized for characterizing the elastic–plastic constitutive properties of ductile materials. Three-dimensional digital image correlation (DIC) is used to measure the full field deformation on two mutually orthogonal surfaces of a uniaxial tensile test specimen. The material’s constitutive model, whose parameters are unknown a priori, is determined through an optimization process that compares these experimental measurements with finite element simulations in which the constitutive model is implemented. The optimization procedure utilizes the robust dataset of locally observed deformation measurements from DIC in addition to the standard measurements of boundary load and displacement data. When the difference between the experiment and simulations is reduced sufficiently, a set of parameters is found for the material model that is suitable to large strain levels. This method of material characterization is applied to a tensile specimen fabricated from a sheet of 15-5 PH stainless steel. This method proves to be a powerful tool for calibration of material models. The final parameters produce a simulation that tracks the local experimental displacement field to within a couple percent of error. Simultaneously, the percent error in the simulation for the load carried by the specimen throughout the test is less than 1%. Additionally, half of the parameters for Hill’s yield criterion, describing anisotropy of the normal stresses, are found from a single tensile test. This method will find even greater utility in calibrating more complex material models by greatly reducing the experimental effort required to identify the appropriate model parameters.

We propose a constitutive relation for finite deformations of nearly incompressible isotropic viscoelastic rubbery adhesives assuming that the Cauchy stress tensor can be written as the sum of elastic and viscoelastic parts. The former is derived from a stored energy function and the latter from a hereditary type integral. Using Ogden’s expression for the strain energy density and the Prony series for the viscoelastic shear modulus, values of material parameters are estimated by using experimental data for uniaxial tensile and compressive cyclic deformations at different constant engineering axial strain rates. It is found that values of material parameters using the loading part of the first cycle, the complete first cycle, and the complete two loading cycles are quite different. Furthermore, the constitutive relation with values of material parameters determined from the monotonic loading during the first cycle of deformations cannot well predict even deformations during the unloading portion of the first cycle. The developed constitutive relation is used to study low-velocity impact of polymethylmethacrylate (PMMA)/adhesive/polycarbonate (PC) laminate. The three sets of values of material parameters for the adhesive seem to have a negligible effect on the overall deformations of the laminate. It is attributed to the fact that peak strain rates in the severely deforming regions are large, and the corresponding stresses are essentially unaffected by the long time response of the adhesive.

Finite torsional deformations of an incompressible viscoelastic circular cylinder are studied with its material modeled by two constitutive relations. One of these is a linear relation between the determinate part of the second Piola-Kirchhoff stress tensor and the time history of the Green-St. Venant strain tensor, and the other a linear relation between the deviatoric Cauchy stress tensor and the left Cauchy-Green tensor, its inverse, and the time history of the relative Green-St. Venant strain tensor. It is shown that the response predicted by the latter constitutive relation is in better agreement with the test data, and this constitutive relation is used to compute energy dissipated during torsional oscillations of the cylinder. [S0021-8936(00)00502-X]

An integrable Eulerian rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational rate of the Kirchhoff stress with material spin to the same rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integrated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann rate of stress is employed.

A constitutive model for finite elastoplastic deformations is presented. This model incorporates two novel features: first, a strain-hardening law that is applicable to complex loading paths and histories; and second, an objective stress-rate measure that is based on the spin of an orthogonal triad of material unit vectors which instantaneously coincides with the principal directions of the stress tensor. Problems of shear superposed on triaxial tension, cyclic shear deformation, and biaxial nonproportional loading are studied. It is shown that realistic predictions for the aforementioned problems are obtained by using the proposed constitutive model.

Void formation in materials under isotropic tension is considered within the theory of finite elasticity. This phenomenon is described as the bifurcation of the solution containing a spherical cavity from the state of homogeneous deformation of a solid hyperelastic sphere. The equation giving the bifurcation point is derived, and the critical stretch and stress are numerically calculated for a special class of compressible nonlinear elastic materials for which the strain energy function was proposed by Hill. The effects of constitutive parameters on the post-bifurcation behavior as well as on the critical stretch and stress are discussed.

The Perzyna viscoplastic constitutive theory, which contains a scalar variable for description of material damage, is used to study material behavior at high strain rates. The damage parameter for materials which undergo ductile fracture by nucleation, growth, and coalescence of microvoids, is taken to be the void volume fraction. The linear hardening law in both the constitutive equation and the derivation of the void growth rate equation has been replaced by a nonlinear hardening law that allows for the saturation of the hardening with increase of strain. The modified constitutive equations are then specialized to uniaxial deformation with multiaxial stress, which is typical of that occurring in flyer plate impact experiments. Calculations are performed showing the rate dependence of the material response and the effects of the growth of the void volume (damage). The change in the predicted response due to the modification of the hardening law is illustrated. Ductile spall fracture is modeled by considering the response to a simulated compressive-tensile wave using a critical value of the void volume as the local criteria for fracture.

Constitutive equations of elastoplastic materials with anisotropic hardening and elastic-plastic transition are presented by introducing three similar surfaces, i.e., a loading surface on which a current stress exists, a subyield surface limiting a size of the loading surface and a distinct-yield surface representing a fully plastic state. The assumption of similarity of these surfaces leads the derived equations to remarkably simple forms. Also a more general rule of the kinematic hardening for the distinct-yield surface is incorporated into the constitutive equations. While they seem to be applicable to various materials, special constitutive equations of metals, for example, are derived from them and are compared with experimental data on a cyclic uniaxial loading of aluminum. A close correlation between theory and experiment is observed in this comparison.

A rate-independent constitutive theory for finite inelastic deformation is formulated in terms of the symmetric Piola-Kirchhoff stress, the Lagrangian strain, and a kinematic tensor which describes inelastic or microstructural effects. Assumptions of (a) continuity in the transition from loading to neutral loading, (b) consistency, and (c) nonnegative work in closed cycles of deformation, lead to simplification of the theory. The response is described by two scalar functions — a stress potential and a loading function. The theory can describe isotropic or anisotropic response, and allows for hardening, softening, or ideal behavior. It may also be appropriate to describe the response of porous materials, such as metals, rocks and ceramics, and also the evolution of damage.

A set of constitutive equations has been formulated to represent elastic-viscoplastic strain hardening material behavior for large deformations and arbitrary loading histories. An essential feature of the formulation is that the total deformation rate is considered to be separable into elastic and inelastic components which are functions of state variables at all stages of loading and unloading. The theory, therefore, is independent of a yield criterion or loading and unloading conditions. The deformation rate components are determinable from the current state which permits an incremental formulation of problems. Strain hardening is considered in the equations by introducing plastic work as the representative state variable. The problem of uniaxial straining has been examined for a number of histories that included straining at various rates, rapid changes of strain rate, unloading and reloading, and stress relaxation. The calculations were based on material constants chosen to represent commercially pure titanium. The results are in good agreement with corresponding experiments on titanium specimens. (Author)

Thermoelastic constitutive equations are derived for a material undergoing solidification or hardening as the result of a chemical reaction. The derivation is based upon a two component model whose composition is determined by the degree of hardening, and makes use of strain-energy considerations. Constitutive equations take the form of stress rate-strain rate relations, in which the coefficients are time-dependent functions of the composition. Specific results are developed for the case of a material of constant bulk modulus which undergoes a transition from an initial liquidlike state into an isotropic elastic solid. Potential applications are discussed.

Constitutive equations of elastoplastic materials with an elastic-plastic transition observed in the loading state after a first yield are presented by introducing a new parameter denoting the ratio of the size of a loading surface in the transitional state to that of a yield surface in the classical idealization which ignores the transitional state. These equations involve a reasonably simplified rule for the kinematic hardening. They would describe reasonably not only the hardening behavior but especially the softening behavior which requires our careful consideration about the elastic-plastic transition. From these equations, moreover, we derive plastic constitutive equations specifically of metals and granular media which exhibit very different plastic behaviors. Besides, brief discussions are provided concerning the existing constitutive equations describing the elastic-plastic transition.

Along the two contours of a doubly connected region, mixed boundary values of slipless nature are prescribed. The problem is formulated on a complex z-plane in terms of a Hilbert functional equation involving a holomorphic function φ(z). Based on plane theory of elasticity, the solution is found in the form of a double series. An effective method of successive approximation is shown for the computation of the coefficients of the series. Numerical examples of the field of stress and deformation are illustrated with diagrams. By superposition, the solution presented can be used to solve another class of mixed problems in elasticity involving frictionless indentation.

This paper is concerned with a special class of response functions for some of the constitutive equations in the nonlinear isothermal theory of elastic-plastic materials. Detailed attention is given to the development of special forms for the free energy and the stress response, motivated mainly by the mechanical behavior of ductile metals in the plastic range and in the presence of finite strains. After obtaining a properly invariant representation for the free energy response (and hence also for the stress) as a function of certain (easily interpretable) measures of deformation, the results are specialized to isotropic materials and are expressed in terms of the invariants of kinematic measures. Some special cases are elaborated upon and, by way of illustration, the influence of plastic deformation on the material properties of the stress response in a simple tension test is discussed.

Derivation of the constitutive equations of elastic-plastic and elastic-viscoplastic solids at finite deformations is discussed. The deformation is uncoupled by using the Lee-Freund three-configuration deformation model. By assuming elastic properties to be independent of plastic deformation, the elastic and plastic (or viscoplastic) constitutive equations are essentially uncoupled. The normality condition of the plastic strain-rate vector to the yield surface in stress space is obtained by incorporating the concept of internal variables in the energy equation.

Approximate constitutive equations for nonlinear viscoelastic incompressible materials under small finite deformation and for short time ranges are derived. The error bound of such a constitutive equation is investigated. Problems of nonlinear stress relaxation and creep are analyzed based on the proposed equation. Also, the problem of a pressurized viscoelastic hollow cylinder bonded to an elastic casing is solved. Numerical solutions, evaluated by assuming particular forms of kernel functions in the constitutive equation, are obtained by means of an inverse interpolation technique and the effects of nonlinearity of material properties are discussed. An experimental procedure is also proposed for measuring kernel functions from uniaxial tension tests for real materials. (Author)

The influence of related constitutive equations upon the stress-redistribution times for a simple two-bar structure is such that the time-hardening model is unlikely to provide reliable estimates of the stress-redistribution time for structures composed of a wider class of materials.