Internal Variable Theory Research Articles

A micromechanics constitutive model of transformation plasticity with shear and dilatation effect

B ased on micromechanics, thermodynamics and microscale t → m transformation mechanism considerations a micromechanics constitutive model which takes into account both the dilatation and shear effects of the transformation is proposed to describe the plastic, pseudoelastic and shape memory behaviors of structural ceramics during transformation under different temperatures. In the derivation, a constitutive element (representative material sample) was used which contains many of the transformed m-ZrO 2 grains or precipitates as the second phase inclusions embedded in an elastic matrix. Under some basic assumptions, analytic expressions for the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka method which takes into account the interaction between the transformed inclusions. The derived free energy is a function of externally applied macroscopic stress (or strain), temperature, volume fraction of transformed phase and the averaged stressfree transformation strain (eigenstrain) of all the transformed inclusions in the constitutive element, the latter two quantities being considered to be the internal variables describing the micro-structural rearrangement in the constitutive element. In the framework of the Hill-Rice internal variable constitutive theory, the transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for proportional and non-proportional loading histories are derived, respectively. The theoretical predictions are compared with the available experimental data of Mg-PSZ and Ce-TZP polycrystalline toughening ceramics.

On the constitutive modeling of transient creep in polycrystalline ice

The theoretical generality of constitutive models for the transient creep of polycrystalline ice and the ability of such models to represent knowledge derived from experimentation in a physically consistent manner are examined in this paper. The focus is on physically based but phenomenological models that employ evolving internal state variables to characterize the deformation resistance offered by the material. These include the models of Le Gac and Duval (1980) and Shyam Sunder and Wu (1989a, b). The widely used model of Sinha (1978), though not based on the theory of internal state variables, is also discussed briefly. A set of nine criteria are identified that should be used in evaluating transient-creep models for polycrystalline ice. These include the well-known correspondence between the creep and constant strain-rate responses observed experimentally by Mellor and Cole (1982, 1983) and the ability to reduce creep and constant strain-rate responses into non-dimensional forms that are independent of the loading parameters, viz., temperature and stress or strain rate, as proposed by Ashby and Duval (1985). The kinematic consistency between the response variables during transient creep, i.e., strain, strain rate and time, also emerges as an important consideration in constitutive modeling and the subsequent use of such models in a finite element framework. To satisfy this requirement, the model must accurately predict the experimentally observed relationship between all three pairs of kinematic variables (strain versus time, strain rate versus time, and strain rate versus strain) keeping the material parameters fixed. This paper finds that the formulations of Le Gac and Duval (1980) and Sinha (1978) are unable to satisfy the requirement of kinematic consistency. On the basis of the nine evaluation criteria, the formulation proposed by Shyam Sunder and Wu (1989a) emerges as the more complete creep model for polycrystalline ice. In conclusion, the paper identifies difficult problems that will benefit from future research.

A micromechanics constitutive theory for forward transformation plasticity with shear and dilatation effect: I, nonproportional loading history

A micromechanics constitutive theory which takes into account both the dilatation and shear ef fects of the transformation is proposed to describe the macroscopic plastic behavior of structure ceramics during forward transformation under different temperatures. Under some basic assumptions, the analytic expressions of the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka's method which takes into account the interaction between the transformed inclusions. In the framework of Hill-Rice's internal variable constitutive theory, the forward transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for non-proportional loading histories are obtained.

Combined Plasticity and Damage Mechanics Model for Plain Concrete

A combined plasticity and damage mechanics model for concrete is developed within the general framework of the internal variable theory of thermodynamics. The necessity of using both plasticity and...

An internal variable finite-strain theory of plasticity within the framework of convex analysis

An internal variable constitutive theory for elastic-plastic materials undergoing finite strains is presented. The theory is based on a corresponding study in the context of small strains [6], and has the following features: first, with a view to embracing the classical notions of convex yield surfaces and the normality law, the evolution law is developed within the framework of nonsmooth convex analysis, which proves to be a powerful unifying tool; secondly, the special case of elastic materials is recovered from the theory in a natural manner. After presentation of the theory a concrete example is discussed in detail.

Stress analysis of blood vessels by a viscoplastic constitutive model based on the internal variable theory.

Stress analysis of blood vessels is performed by means of a transversely isotropic viscoplastic constitutive model. This model is based on the concept that the hysteresis loops of stress-strain relationships under cyclic loading conditions are caused by loading path dependence and loading rate dependence. Then the method of analysis is discussed in detail on the basis of finite deformation theory for the case of axisymmetric deformation of a long circular cylindrical tube under the condition of constant axial stretch and repeated pressurization. Simulation results show that the model can describe the experimental results by Takamizawa et al. with a relatively high accuracy. The stress histories at some points in the wall and the stress distribution across the wall are also discussed with a special emphasis on the difference between viscoplastic and elastic predictions. The results show that the inelastic properties of blood vessels reduce the stress concentration at the inner wall.

An internal variable theory of elastoplasticity based on the maximum plastic work inequality

The methods of convex analysis are used to explore in greater depth the nature of the evolution equation in internal variable formulations of elastoplasticity. The evolution equation is considered in a form in which the thermodynamic force belongs to a set defined by a multi-valued map G G . It is shown that the maximum plastic work inequality together with the assumption that G G is maximal responsive (a term defined in Sec. 4), is necessary and sufficient to give a theory equivalent to that proposed by Moreau. Further consequences are investigated or elucidated, including the relationship between the yield function and the dissipation function; these functions are polars of each other. Examples are given to illustrate the theory.

Wave propagation in periodically layered elastic media based on an internal variable model

A phenomenological model of wave propagation in a periodically layered elastic media, with linear elastic constituents, which is based on an internal variable theory, is presented; the main result is a system of coupled second- and fourth-order partial differential equations which describe the motion of each constituent and which, in turn, appear to predict the correct qualitative behavior for both the composite (or main) wave and the precursor wave in the laminate.

On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects

Novel energy-based coupled elastoplastic damage theories are presented in this paper. The proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials. At variance with Lemaitre's work on damage-elastoplasticity, the present formulation renders rational thermodynamic potential and damage energy release rate. In contrast to previous work by Simo and Ju (featuring an additive split of the stress tensor), current formulation assumes an additive split of the strain tensor. It is shown that the “strain split” damage-elastoplasticity formulation leads to more robust tangent moduli than the “stress split” formulation. The plastic flow rule and hardening law are characterized in terms of the effective quantities; viz. the effective stress space plasticity. This mechanism is both physically well-motivated and computationally efficient. Further, a fourth-order anisotropic damage mechanism is proposed for brittle materials. Rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain-rate dependency of microcrack growth.Efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the “operator splitting” methodology. In particular, new three-step operator split algorithms are presented Application is made to a class of inviseid and rate-dependent cap-damage models for concrete and mortar. Experimental validations are also given to illustrate the applicability of the proposed damage models.

An anisotropic damage model with dilatation for concrete

An anisotropic damage model for concrete is developed within the general framework of the internal variable theory of thermodynamics. The rate of change of the compliance tensor is described in terms of kinetic relations involving a damage parameter whose increment is governed by the consistency equation associated with a pressure-dependent damage surface in stress space. The use of the compliance tensor implies that damage is reflected through a fourth-order tensor. Dilatation is obtained as a consequence of damage, and permanent deformation due to damage is addressed via a simple evolution equation. The theory is capable of accommodating the anisotropy induced by microcracking and is very suitable for computer implementation.

Thermodynamical formulation for coupled electromechanical hysteresis effects—I. Basic equations

In this work we present a thermodynamical phenomenological formulation of a theory capable of displaying electromechanical hysteresis effects in continous media, which should apply to ferroelectric ceramics. This is built within the scheme of the thermodynamical theory of internal variables. This theory produces both plastic and electric-hysteresis effects in the form of “plasticity”, i.e. rate-independent, evolution equations for the plastic strain, the residual electric polarization and both mechanical and electric “hardenings”. This paper develops in detail the electric hardening effects through the residual polarization vector and an internal variable representing an electric polarization. An electric version of Drucker's inequality of plasticity has been derived. Several examples of loading functions are studied including the influence of piezoelectric and electrostrictive couplings. In particular, the purely electric loading function has been studied in greater detail and the first polarization curve either based on experimental data or theoretically displayed is discussed. In all the paper provides all ingredients for a future discussion of more specific cases.

Inelastic constitutive equations of polycrystalline metals subjected to finite deformation part I: Effect of grain rotation

In this article, a set of inelastic constitutive equations of polycrystalline metals is derived by combining a finite deformation kinematics of single crystal component, and a shear stress-shear strain relation of slip system based on a thermoactivated motion of dislocation. Interactions among grains are incorporated by “constant deformation gradient assumption.” The forms of these equations are rather simple internal variable theory types. By using these equations, some fundamental effects of grain rotations on inelastic behaviors of polycrystalline metals in a finite deformation range under complex loading and elevated temperature conditions are demonstrated. Some comments are given on a problem of plastic spin tensor.

An overview of the constitutive behavior of crystalline solids

An overview of the general principles involved in the mechanics of crystalline solids is presented. The emphasis is on the inelastic behavior, and particularly on microcracking, in brittle crystalline solids. A phenomenological theory of microcracking which is based upon application of the internal variable theory of continuum thermodynamics (the so-called continuous damage theory) is suggested to be inadequate for a clear description of the phenomena that occur on the microscale. Also, it is emphasized that the phenomenological theories of plasticity are inapplicable at the very early stages of microplasticity prior to microfracture, when the inelastic deformations are highly heterogeneous on the scale of a grain size, and at the very late stages of plasticity, when either shear band localization or necking occurs as a result of the non-normality of the constitutive structure. It is, therefore, suggested that there is need for development of micromechanical constitutive theories which will embody the nature and the extent of microstructural changes that occur during inelastic deformation.

Inelastic constitutive equations of polycrystalline metals at finite deformation. (1st report. Effect of grain rotation).

In this paper, a set of inelastic constitutive equations of poly-crystalline metals is derived by combining finite kinematics of a single crystal component and the shear stress-shear strain relation of a slip system based on a thermo-activated motions of dislocations. The interactions among grains are incorporated by deformation gradient constant assumption. The equations are rather simple internal variable theory type. By using these equations, some fundamental effects of grain rotations on the inelastic behaviors of polycrystalline metals in a finite range are clarified under complex loading and elevated temperature conditions. Some comments are made on the problem of plastic spin tensor.

Constitutive Laws for Ceramics Exhibiting Stress-Induced Martensitic Transformation

ABSTRACTThe theory of internal variables is used in order to develop multiaxial constitutive laws for ceramics undergoing martensitic stress-assisted transformation, such as partially stabilized zirconia or A12O3-ZrO2. The internal variable is identified with the volume concentration of transformed particles, and we assume that transformation occurs so that the change in potential energy due to the transformation is maximized. When the rate of transformation depends on the applied stresses only through the corresponding change in potential energy, it is shown that the inelastic strain rates are along the normal of a stress function in stress space. The constitutive law depends on all three stress invariants. We further discuss specific stress environments such as crack tip fields, the special case of homogeneous transforming particle distribution, and conditions under which normality is not obeyed.

Visco‐damage tension model for rocks in ground‐shock applications

AbstractA simple rate‐dependent tension model, tailored primarily for rocks in ground‐shock calculations, has been constructed. It is formulated as an internal state variable theory with spherical void development taken to approximate the underlying dissipative micromechanism. This simplifying assumption of ductile, rather than brittle, damage growth has been made to enhance the practicality of the model for large‐scale problems. The model has been restated within the fromework of (strain‐softening) viscoplastcity to associate it with the desirable mathematical properties of the latter theory. The tension option, extended to include healing in the post‐damage compressive phase, has been implemented in a standard cap constitutive routine and is referred to as the visco‐damage tension model. Model parameters are selected for a shale‐like material. Results of uniaxial shock‐loading strain‐rate calculations exhibit desired characteristics in tension. Damage‐related hysteretic effects are illustrated through cyclic loading calculations. It is planned to develop more effecient computational procedures to facilitate adoption of the model into large‐scale ground shock calculations. Anisotropic extensions of the model will then be considered.

The constitutive relations of elastic-inelastic materials at small strains

A set of constitutive relations is presented to describe the elastic-inelastic material behaviour of austenitic steel in the range of small strains, and up to 600°C. This model permits a description of the hardening behaviour in the case of mechanical loading as well as hardening and softening in the case of thermal loading. The loading path may be either monotonic or cyclic. For this purpose, the well-known concepts of isotropic and kinematic hardening are connected within the framework of an internal variable theory. A number of suitably defined internal variables is introduced to describe the transition from isotropic to kinematic hardening behaviour of the material. The model allows for a rate-dependence of the material behaviour when exceeding the yield limit. Therefore the concept of “overstresses” is introduced, where the (real) stresses during inelastic deformations are rate-dependent multiples of the (static) plastic stresses, thus that both stresses coincide in the limit of vanishing velocities (static limit). For some simple examples this model is applied to numerical calculations.

Some remarks on the application of a two-surface model in plasticity

An internal variable theory of rate-independent plasticity is presented incorporating a combination of classical models of kinematic hardening as well as isotropic hardening. In addition to the yield surface a second “bounding surface” has been introduced to accommodate to problems with non-radial loading paths. The behaviour of this model unter uniaxial and complex loading has been tested and compared with experimental results and other theoretical predictions.

Changes of invariance in internal variable theories

We show how to construct internal energy functions which have arbitrarily specified symmetries on the two sides of a phase transition.

Deformation kinetics of ageing materials

A constitutive equation of time-dependent, chemically stable materials, which stems from the basic ideas of the irreversible thermodynamics of an internal variable and Eyring's absolute reaction rate theory, has been extended to chemically unstable materials. This formulation is quite general and, in principle, can be applied to many types of materials. In this paper, the ageing behaviour of time-dependent network polymers undergoing chain scission is considered. In the network scission process, we postulate that the energy barrier is affected by a changing of the chemical crosslink density. An explicit equation to account for the energy barrier change, which influences the relaxation process, is formulated. For the purpose of illustration, the effect of different chemical crosslink density, ν, on the relaxation rate has been considered, from which the following theoretical expression of relaxation modulus Δ E(t) is obtained: ΔE = ΔE[t exp (γv/kT)] It can be seen that a change in v leads to an effective change in the time scale, usually denoted by a x. Here the analytical expression a x = exp( γν kT ) correlates quite well with the experimental data.