Thermodynamical deduction of constitutive equations of a plastic material with combined work-hardening

Summary The constitutive equations of a plastic material are deduced by the Clausius-Duhem inequality and assumed yield restrictions. A scalar and a tensor internal state variable are introduced and we have naturally two sets of the constitutive equations, that is, those in the elastic state and those in the yield state. The latter includes the isotropic and the kinematic work-hardening case. As a simple example an isotropic linear plastic material is discussed, which follows the generalized Hooke's law with thermal expansion in the elastic state, and the generalized Huber-von Mises yield condition in the yield state.

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Thermodynamische Herleitung der Materialgleichungen eines plastischen Materials mit kombinierter Verfestigung

Zusammenfassung Die Materialgleichungen eines plastischen Werkstoffes werden aus der Clausius-Duhemschen Ungleichung und angenommenen Fließbeschränkungen hergeleitet. Eine skalare und eine tensorielle innere Zustandsvariable werden eingeführt, und wir haben natürlich zwei Mengen von Materialgleichungen, eine im elastischen Zustand und eine im Fließzustand. Letztere enthält den isotropen Fall und den Fall der kinematischen Verfestigung. Als ein einfaches Beispiel wird ein isotropes, linear plastisches Material diskutiert. Das Material gehorcht dem allgemeinen Hookeschen Gesetz mit Wärmedehnung im elastischen Zustand und dem allgemeinen Huber-von Misesschen Gesetz im Fließzustand.

     

Finite plastic constitutive laws for finite deformations

Summary The paper investigates special forms of the constitutive equations of the flow theory and extends the concept of proportional loading to finite deformations for the interpretation of the finite laws of the deformation theory of plasticity as the integral of the differential laws of the flow theory.     

Rate-type constitutive equations and elastic-plastic materials

This paper is concerned with a class of rate-type constitutive equations, which characterize the behavior of elastic-plastic solids under large deformation. The theory is developed without à priori introduction of the concept of plastic strain and involves only the stress and the total strain as basic ingredients, apart from the thermodynamic variables. The relationship between the results obtained here and those of a more familiar form of a theory of elastic-plastic solids is indicated.     

Theoretical and experimental constitutive equations of superplastic behaviour: discussion

Two basic models of superplastic deformation are considered. The stress strain-rate curves obtained by combining these two models for a material of different grain sizes at various temperatures are shown to be very different. This is due to the introduction of physically ill-defined parameters in the theoretical relations. The experimental results obtained from mechanical tests do not help in clarifying the situation since the microstructure is continually changing during such experiments. Consequently, the validity of the deformation mechanisms in dealing with models of constant grain-size cannot be demonstrated by fitting theoretical curves to the experimental results of mechanical tests.     

On the derivation of the constitutive equations of ideal plasticity

Summary TheLèvy-Mises equations for an ideal plastic material are derived from a general implicit relation between the stress and rate of deformation tensors. This derivation provides an alternative to the usuala priori postulates of a yield criterion and flow rule.

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Zur Herleitung der Materialgleichungen ideal plastischer Stoffe

Zusammenfassung DieLèvy-Mises-Gleichungen für ein ideal plastisches Material wurden aus einer allgemeinen impliziten Beziehung zwischen dem Spannungstensor und dem Tensor der Deformations-geschwindigkeiten hergeleitet. Diese Herleitung bietet eine Alternative zu den üblichena priori Postulaten einer Fließbedingung und eines Fließgesetzes.

     

Analysis of creep curve by constitutive equations

Abstract

Creep curves of 0·5Cr–0·5Mo–0·25V steel have been analysed based on a simple power function of time. The long term creep properties, namely, the creep rate and the rupture lifetimes at low stress levels can be very well predicted by this simple power function. The Monkman–Grant relationship can also be obtained based on the predicted values.

MST/1411     

Symbolic derivation of potential based constitutive equations

Structural alloys used in high temperature applications exhibit complex thermomechanical behavior that is inherently time dependent and hereditary, as the current behavior depends not only on current conditions but on the thermomechanical history. Derivation of mathematical expressions (constitutive equations) which describe this high temperature material behavior can be quite time consuming, involved, and error-prone, thus intelligent application of symbolic systems to facilitate this tedious processes can be of significant benefit. Here a computerized package, running under MACSYMA, capable of efficiently deriving potential based constitutive models, in analytical form (involving tensors, partial differentiation, invariants, and the like) is presented. Special purpose utility algorithms are designed and implemented to perform partial differentiation (chain rule), tensor manipulation, case distinction and simplification. Four constitutive theories reported in the literature are utilized to verify implementation accuracy. It is expected that this symbolic package can and will provide a significant incentive to the development of new constitutive theories.     

Symmetry restrictions on the constitutive equations for magnetic materials

A group-theoretical procedure is given for the decomposition of magnetic material tensors with intrinsic symmetry into basic symmetry types. This decomposition is used in finding the total number of independent components of magnetic material tensors in linear constitutive equations, and it simplies the determination of polynomial integrity bases in non-linear constitutive equations. Next, a method is devised to obtain the independent components of magnetic material tensors explicitly. Using the isomorphism between a classical point group and the corresponding magnetic point groups together with the property of basic quantities, determination of polynomial integrity bases is made immediate.     

Multiscale constitutive modeling for plastic deformation of nanocrystalline materials

Macroscopic and microscopic constitutive modeling that can display large plastic deformation with shear band were presented for nanocrystalline materials subjected to uniaxial load over a wide strain rate range. The macroscopic model implemented with parameters microscopic meaning was established based on the theory of plastic dissipation energy. The microscopic model based on deformation mechanisms was composed of two parts: hardening and softening stages. In the hardening stage, the phase mixture model was used and a shear band deformation mechanism was proposed in the softening stage. Numerical simulations shown that the predications were in good agreement with experimental data. Finally, a parameter of normalized softening rate was proposed and its characteristics were evaluated quantitatively. It can be concluded that the failure strain could be prolonged when the normalized softening rate decrease through changing the softening path.     

Non-local constitutive equations for functionally graded materials

In this paper, composites with a graded distribution of heterogeneities are considered. The heterogeneities vary in statistically non-uniform fashion since in a finite layer (or region) properties such as local volume fraction vary gradually. In order to study this class of composites, a procedure of analysis which leads to the effective constitutive non-local operator of the medium is proposed. For two-phase composites, an approximation of Hashin–Shtrikman type for this operator has been obtained in real space and this has been developed explicitly in the case of laminates.     

On approximate constitutive equations in nonlinear viscoelasticity

Summary In this paper a simplified constitutive equation for nonlinear slightly viscoelastic materials is derived by a method similar to that ofColeman andNoll. This equation is also valid for short times, slow motions, or small deformations superposed on a large steady deformation for all viscoelastic materials. An experimental method for determining the material parameters contained in this constitutive equation for dynamic conditions is also presented.

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Über genäherte Materialgleichungen in der nichtlinearen Viskoelastizität

Zusammenfassung Mittels einer Methode ähnlich der vonColeman undNoll wird eine vereinfachte Materialgleichung für nichtlineare, schwach viskoelastische Stoffe hergeleitet. Diese Gleichung gilt auch für das Kurzzeitverhalten, für langsame Bewegungen, oder für die Überlagerung kleiner Deformationen auf große stationäre Verformungen bei allen viskoelastischen Materialien. Eine experimentelle Methode zur Bestimmung der in der Materialgleichung auftretenden Stoffkonstanten für dynamische Bedingungen wird gleichfalls angegeben.

     

Numerical testing of the stability criterion for hypoplastic constitutive equations

An outline of so-called hypoplastic constitutive equations is presented. The stability criterion proposed by Hill is applied to a hypoplastic constitutive equation for granular materials. The response envelope as proposed by Gudehus is used to check the stability criterion numerically.     

Computational plasticity of mixed hardening pressure-dependency constitutive equations

In the present study, two semi-implicit schemes, based on the exponential maps method, are derived for integrating the pressure-sensitive constitutive equations. In spite of the fact that the consistent tangent operator is necessary to preserve the quadratic rate for the asymptotic convergence of the Newton-Raphson solution in the finite element analyses, there exists no derivation of this operator for the exponential-based integrations of the pressure-sensitive plasticity in the literature. To fulfill this need, the algorithmic tangent operators are extracted for the new semi-implicit as well as the former exponential-based integrations. Moreover, for the accurate integration presented by Rezaiee-Pajand et al. (Eur J Mech A Solids 30:345–361, 2011), the consistent tangent operator is obtained. Eventually, all the investigations are assessed by a broad range of numerical tests.     

A note on the Stouffer-Bodner constitutive model for anisotropic plastic flow

Stouffer and Bodner present constitutive equations governing the multi-dimensional behavior of Inelastic materials. The equations feature an Internal state variable, defined by its evolution equation, related to work hardening. Three different classifications of mechanical properties are defined in terms of restrictions upon this Internal variable. The authors' assumptions, however, render these three classes mutually Indistinguishable. A modification of the generalization to three dimensions and a redefinition of the material classes are proposed in this note.     

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 general antisymmetry property of the constitutive equations of non-equilibrium thermodynamics

The constitutive equations of non-equilibrium thermodynamics which describe the exchange of heat, work and mass between a discontinuous thermodynamic system Σ and its environment Σ¹, show a general antisymmetry property. It is due to the fact that if the combined system Σ + Σ¹ forms an isolated thermodynamic system, the constitutive equations should be invariant against the substitution Σ Σ¹. This requirement is called the principle of environs invariance, some consequences of which are discussed especially in view of engineering applications.     

Development of constitutive equations for modelling of hot rolling

Abstract

Constitutive equations which describe the flow stress behaviour of materials during hot deformation are used to model forming processes. Since the flow stress depends on both temperature and strain rate, the Zener–Hollomon parameter which combines these factors, is frequently used to describe the shape of such curves. For materials which dynamically recover, (only) the flow stress reaches a steady state value at high strains and methodologies which enable such behaviour to be modelled have previously been presented. Beyond the onset of dynamic recrystallisation, the flow stress of materials, such as steels, reach a peak in flow stress before gradual softening. The relative position of the peak in the flow stress shifts as a function of the Zener–Hollomon parameter further complicating such analyses. The present paper describes the development and application of a methodology for modelling the flow stress of microalloyed steels: materials which exhibit dynamic recrystallisation behaviour.