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An alternative mathematical theory of elastoplastic behaviour

This paper presents a theory for the behaviour of isotropic-hardening/softening elastoplastic materials that do not have a preferred reference configuration. In spite of important differences, many ingredients of classical plasticity are present. Main features are: the elastic and plastic responses are given by solutions of hypoelastic differential equations, no decomposition of the deformation into elastic and plastic parts is done from the start, the hardening rule is an outcome, and the principle of material objectivity is obeyed. An important result is the existence of a limit surface that divides the stress space into regions of hardening and softening and is composed of equilibrium points of the differential equation of plastic response.


INTRODUCTION
The theory presented in this paper differs from the modern theories of finite plasticity that arose in the last forty years (see, e.g. [1]) but has a similar objective, namely, to arrive at a mathematically clear and physically sound description of elastoplastic behaviour and to enlarge the range of applicability of the classical theory to large deformation problems.
It is known that classical plasticity lacks precision and obedience to the principle of material objectivity. Besides, in most expositions of classical plasticity, there is no clear definition of which stress, deformation tensor and rateof-deformation tensor is being used; time derivatives are treated as increments, and hypotheses are sometimes tacitly assumed (cf. comments in [2] and [3]).
These defects have been a motivation to development of the proposed theory.
The constitutive equations presented in this paper describe the behaviour of isotropic-hardening/softening elastoplastic materials. The theory is organised in a relatively simple manner, with a small number of axioms and constitutive functions. In spite of important differences, many ingredients of classical plasticity are present.
In what follows we state the main features of the theory.
a) The principle of material objectivity is obeyed. This theory is, therefore, in principle appropriate for large deformation problems.
b) The constitutive equations involve only quantities defined in the current configuration: the Cauchy stress T and the spatial velocity gradient L. Both the elastic and plastic responses are given by solutions of hypoelastic differential equations.
c) The problem of determining the material response (the Cauchy stress) arising in a given motion is treated with full generality. The way classical and some finite plasticity theories are usually constructed, this problem is not treated as simply as in this theory. Traditionally, the plastic part of the deformation is unknown and is related (through the so-called flow rule) to the stress and to the stress rate, which are also unknown, when the motion is given.
d) The constitutive equations for elastic and plastic responses are hypoelastic differential equations and as such they must be integrated in intervals. The yield criterion is established in intervals, as it must be since the equations involve derivatives, in contrast to the usual approach to rate-type plasticity, in which the yield criterion and the equations refer to isolated instants. Traditionally, the constitutive equations are treated as if they involved increments of stress and strain rather than time derivatives, and this hides general properties of solutions. e) Here there is no axiomatic decomposition of the deformation into elastic and plastic parts. In rate-type theories of plasticity the decomposition of the time derivative of the infinitesimal strain tensor (in classical plasticity) or of the stretching tensor (in modern theories) is imposed at the outset, but is rather artificial because, although it is true that in a cycle of stress the body does not end up in the same original configuration due to irreversible strains, it is not natural that at each stage of the process the rate of deformation should be seen as the sum of a reversible and an irreversible part. Recall that the two parts of the stretching are not the symmetric part of the velocity gradient of any motion as its sum is. In this work no decomposition is necessary. However, to make the comparison with other rate-type theories, we will show how the stretching tensor can be considered as the sum of elastic and plastic parts.
f) The yield surface is defined in stress space and the yield criterion involves both the Cauchy stress and the stretching tensor. g) No hardening rule is set from the start. It is obtained as an outcome.
h) The theory incorporates hardening, softening and perfectly plastic behaviour.
i) An interesting result is that there exists a limit surface that divides the stress space into regions of hardening and softening. This surface is composed of equilibrium points of the differential equation of plastic response for certain motions.
Characteristic (b) is important if one wishes to model the behaviour of materials that have no preferred reference configurations, like soils. For this kind of material it seems to be more consistent to work with stress and rate-ofdeformation tensors defined in the current configuration; we wish to know how the stress will change, from the current state, if a certain motion is imposed. The motive is the same that led Truesdell [4] to propose hypoelasticity. Property (f) is also compatible with materials that have no preferred configurations. For those materials it would not make sense to define the yield surface in strain space. Property (i) deserves special attention (sections 3 and 4). It allows modelling materials whose failure surface is different in form from the yield surface. It is especially interesting to model critical states of soils.
Throughout the paper the standard notation of modern continuum mechanics is employed.

CONSTITUTIVE EQUATIONS AND MATERIAL RESPONSE
Let the deformation gradient at a point of a body, at each instant t of an interval I, be given by a piecewise continuously differentiable function 1 F:I⊂R→Lin + .
The response of the elastoplastic material subject to the deformation gradient function F is represented by T:I→Sym, which associates to each instant t the Cauchy stress T(t). The functions F and T are linked by hypoelastic differential constitutive equations, as we shall see. In the description of the material response there will appear the function k:I→R, which associates to each instant the hardening parameter k(t).
The spatial velocity gradient will enter into the constitutive equations through its symmetric part, the stretching tensor D, and its skew-symmetric part, the spin tensor W. The corotational or Jaumann rate of T, defined , will also be used.
The definition of the material response will refer to closed sub-intervals of I on which the restriction of F is continuously differentiable. In this respect, it is important to state clearly a convention regarding derivatives at the extremes of a closed interval. Suppose that F fails to be continuously differentiable at a and b but is continuously differentiable on [a,b]. In studying the material response in we mean the right-hand derivative of F at a and the left-hand derivative of F at b, respectively. The same convention applies to T and other functions calculated from F.
The elastoplastic material is characterised by three constitutive functions: The symbol L(Sym) stands for the set of all linear operators of Sym (a fourthorder tensor space).
The functions A, B and f will take part in the differential constitutive equations and have such smoothness properties so as to ensure existence and uniqueness of solution of initial value problems. The principle of material objectivity places restrictions upon these functions, as we shall see.
At each t∈I we define the elastic domain as the set and its boundary, is the yield surface.
As an axiom, we require that

Material Response
Let the restriction of F to [a,b]⊂I be continuously differentiable. Suppose that at t 0 ∈[a,b) the stress is T 0 , the hardening parameter is k 0 , and that they are such as to satisfy the axiom established in (1). We will now specify the material To characterise the material response consider four possible initial situations.
Case IV: f(T 0 )=k 0 and ψ(T 0 , D 0 )>0. a) In cases I, II and III, the response is given by T satisfying In such intervals the response is said to be elastic and the hardening parameter is constant: In cases III and IV, the response is given by T satisfying T(t 0 )=T 0 and in any interval [t 0 ,τ], such that ψ(T(t),D(t))>0, ∀t∈(t 0 ,τ). In these intervals the response is said to be plastic and the hardening parameter varies according to The function B appearing in (3) is such that B(T)≠0, for all T.
Proposition 1 below shows that this definition of material response is complete and definite in the sense that all possible situations are taken into account and that the response cannot be simultaneously elastic and plastic. In short, after t 0 either one type of response or the other will follow. This is a minimum requirement that a reasonable theory should satisfy.
Proposition 1: In cases I and II intervals as specified in (a) exist. In case III there exist either intervals as specified in (a) or intervals as specified in (b). In case IV intervals as specified in (b) exist.
Proof: That the specified intervals exist in cases I and IV is clear from the continuity of the functions involved.
In cases II and III, intervals as specified in (a) exist if and only if foT is a non-increasing function in a right-neighbourhood of t 0 . This certainly happens in case II, but it may not happen in case III, as is shown next.
Let us compute the derivative of foT in a right-neighbourhood of t 0 , when the response is elastic (2): Thus, in case II, foT decreases in a right-neighbourhood of t 0 , since ψ(T 0 , D 0 )<0.
In case III, as ψ(T 0 , D 0 )=0, nothing can be said about the increase of foT.
According to the definition of material response, case III can be the beginning of elastic or plastic response. It will now be shown that case III gives rise either to elastic or to plastic response.
If, considering the solution of (2) in (a), does not increase (remains 0 or becomes negative) in a right-neighbourhood of t 0 , then the response is elastic in a right-neighbourhood of t 0 , for foT will not increase as well.
If, considering the solution of (3) (becomes positive) in a right-neighbourhood of t 0 , then the response is plastic in that neighbourhood.
near t 0 will now be studied by means of its derivative. Let us introduce P(T)=A(T) T [∇f(T)], so that we can write ψ(T,D)=D:P(T), and compute the following derivative: in which dP indicates the Fréchet-derivative of P. As both equations (2) and (3) give, in this case, the same value for & T ( ) t 0 , then, by the above equation, Suppose now that t 0 =a and that F is not continuosly differentiable at a.
Then, any of the four cases may happen (at t 0 =a, D and, hence, ψ(T,D) have different limits from the left and from the right of t 0 ).

Restrictions imposed by the principle of material objectivity upon the constitutive functions
The fact that a certain T belongs or not the elastic domain should not depend on the observer. So, we impose that f(QTQ T )=f(T), for all T∈Sym and for all Q∈Orth. Hence, the yield function must be isotropic. It can be shown that the gradient of the yield function, ∇f, is isotropic too: ∇f(QTQ T )= Q∇f(T)Q T .

The function ( , ) ( )[ ] T D T D a A must be isotropic in order that (2) be objective: A(QTQ T )[QTQ T ]=QA(T)Q T , for all T, D∈Sym and for all Q∈Orth.
Note that ) , In order for (3) to be objective, the function B must also be isotropic:

LIMIT SURFACE
We can obtain more information on how k changes in plastic response differentiating both members of (4) with respect to time and using (3): Introducing µ(T)=1+∇f(T):B(T), we get a more compact equation: We say that hardening, softening or perfect plasticity takes place, in an interval [t 0 , t 1 ] in which the response is plastic, according to whether & k ) is positive, negative or null in (t 0 ,t 1 ), which is determined by µ(T), since ψ(T(t):D(t))>0, ∀t∈(t 0 ,t 1 ). Thus, the surface S defined by µ(T)=0 divides the stress space into regions where either hardening or softening may happen. S will be called the limit surface.
An important conclusion is that, in plastic response, a stress path such that at all of its points cannot cross the limit surface. In section 4 we shall investigate what happens as stress paths of this kind approach S.
It is interesting to see the relation between cases II, III and IV (in the definition of material response) and ) ( : In case II: 0 ) ( : , and elastic response follows.
In case III: 0 ) ( : , and one or the other response takes place depending on whether )) ( ), increases or not in a rightneighbourhood of t 0 .
In case IV: is negative, zero or positive depending on µ(T) and plastic response follows.
Note that the definition of material response could not be established in , since, if it is negative and f(T 0 )=k 0 , then the response may be elastic or plastic with softening in a right-neighbourhood of t 0 . However, if it is positive and f(T 0 )=k 0 , the response is certainly plastic with hardening.
is associated to neutral loading. This terminology is not appropriate since plastic response may follow.
In view of what has been presented so far we can condense equations (2) and (3) in just one equation, if µ(T)≠0: Recall that, in elastic response, 0 = k & and, in plastic response, (5) holds.

FURTHER PROPERTIES OF THE LIMIT SURFACE
As explained in section 2, given the deformation gradient at each instant, the corresponding Cauchy stress, in sub-intervals, is the solution of a differential equation: where

h(T,D)=A(T)[D],
if the response is elastic, and

h(T,D)=C(T)[D].
We pass now to the investigation of the equilibrium points of the autonomous differential equation To the stress path defined by the restriction of this curve to [0,1) there corresponds a plastic response with D given by (7), whose norm tends to infinity as t tends to 1. Moreover holds, or with softening, if (b) holds. In either case, as t tends to 1, the norm of D in (7) tends to infinity, since µ(T 1 )=0 and 0 (1) : An easy computation, using (7), gives surface S c , an important concept in soil mechanics (see, e.g., [5]), because it is possible to keep the stress constant on S c while the material suffers an isochoric plastic deformation (trD=0). In this case, the limit (8), which represents the limit of the ratio between the distortion rate and the volumetric strain rate, becomes infinity, in agreement with what is observed in experiments with soils.

NORMALITY CONDITION
The purpose of this section is to show that some elements of classical plasticity can be obtained in the present theory.
If A(T) is invertible, then, in elastic response, from (2), if the response is plastic, and D p =0, if the response is elastic.
Next step is to establish a relation between the evolution of k and the plastic stretching: the so-called hardening rule. From (5) and (9) we get Traditionally, the hardening rule is a primitive element of the theory whereas here it was derived. The above expression shows us which form the hardening rule must have as well as its relation with other functions of the theory.
Note that & k is given by an isotropic function of T and D p that is positively homogeneous of degree one in D p .
We will now see how the normality condition, an important ingredient of classical plasticity, can be introduced in this theory. It suffices to choose B given

by B(T)=β(T)A(T)[∇f(T)]
, with β:Sym→R, isotropic, β(T)<0, for all T. Thus, from (9): In this case it is said that the normality condition holds: at any instant of a plastic

FINAL REMARKS
It was shown that it is possible to construct a relatively simple theory of plasticity with a small number of elements (three constitutive functions) that describes the material response to any imposed motion, obeys the principle of material objectivity, represents hardening and softening, and yet is compatible with many characteristics of the classical theory. The hardening rule, for instance, was obtained as an outcome.
The definition of the material response deserved special attention. As the constitutive equations are differential equations, the material response must be specified in intervals, rather than at isolated instants.
A particularly interesting result is the existence of a limit surface, which divides the stress space into regions of hardening and softening. Interpreted in the light of the theory of ordinary differential equations, the limit surface is composed of equilibrium points of the equation for plastic response for certain values of the stretching tensor. This means that a state of perfect plasticity occurs on the limit surface for certain motions.
Finally, it should be noted that also elastic-perfectly plastic behaviour can be modelled within this framework. It suffices to impose that the initial yield surface and the limit surface S coincide, which can be attained by an appropriate choice of the functions f and B. It follows that the yield surface is fixed and that, in plastic response, the stress remains on S.