Generalized Constitutive Equations with Softening

It is possible to put the constitutive equations with softening into a very general framework that embraces most known models. The most general thermomechanical approach is outside the scope of this book, and the reader may refer to the book by Lemaitre and Chaboche (1985).

A general constitutive equation may be based on three fundamental concepts:

1. A set of independent internal variables, pk, which together with the infinitesimal strain tensor є

(or the stress tensor a) are assumed to characterize uniquely the instantaneous state of the body at a given point. The internal variables may represent a physical magnitude or be abstract in nature. They can be related to kinematic events or to structural features. For example, the vector 0 in the smeared crack models is intended to represent the internal kinematics of cracks and n, the crack orientation, is a structural internal variable. It must be noted that, when a set of internal variables is chosen, any other set, uniquely related to the first, is strictly equivalent and, consequently, can be used instead of the first. This makes the physical interpretation of a given set of internal variables somewhat ambiguous. *

2. A system of equations relating the stress to the strain and to the internal variables:

e = E(cr, pfc) (8.5.37)

where E( ) is a symmetric tensor-valued function. In modern thermodynamic formulations E(-) is derived from a free energy function, which is a scalar function to be specified instead of E(-). Usually, E(-) is assumed to be linear in the infinitesimal strain tensor, that is,

є – C{jpk)cr + є?(рк) ‘ (8.5.38)

where C(pk) is the secant fourth-order compliance tensor, depending only on the internal variables; єр(рк) is the irrecoverable or plastic strain tensor, which again depends only on the internal variables. When ep(pk) 0 and C(pk) ts Cel = constant, the elastic behavior is obtained. When £p(pk) varies, and C(pk) = Cel = constant, a model displaying strength degradation is obtained. When єр(рк) = 0 and C{pk) is variable, one obtains a model displaying stiffness degradation, which always unloads to the origin (a = 0 for є = 0, and vice versa). When both C{pk) and єр(рі.) are variable, a general damage model with mixed behavior is obtained.

3. A set of flow rules, which specify the way in which the internal variables increase during loading. This is a delicate yet essential point, since assigning different flow rules to models having the same set of internal variables and the same structure for the stress-strain relation will lead to different behaviors. Moreover, the flow rules must be consistent with the irreversibility condition posed by the second law of thermodynamics. A detailed discussion of this important aspect is outside the scope of this book, so only general aspects will be mentioned. (By analogy with plasticity, the term “flow rule” is used even though “cracking rule” would be more logical in models in which cracking dominates.)

The flow rules may be specified at many different levels of generality. One relatively simple way is to use one or more loading functions obtained by direct generalization of the theory of classical plasticity. For the simplest case of a single yield surface, a loading function F(pk, cr, p) is specified, in which p is one further internal variable that governs hardening and softening (it can be singled out from the beginning). The loading function defines the region in which the behavior is elastic (i. e., in which dp = 0 and dp к — 0 for any k) which can be written as

F(pk, a,p) < 0 (8.5.39)

The associated flow rules are:

Pk = Hk(pk, o-,p)(i with ft > 0 (8.5.40)

where/(is the hardening-softening variable which takes the place of the plastic mult iplier and a, p)

are the hardening-softening functions. The flow rule for p itself is deduced from the consistency condition requiring that F{pk, cr, p) remain equal to 0 if p > 0.

Although this is a rather general formulation, it is not the only one possible. Formulations of the entlochronic type and multi-yield surface type are also possible.

In the literature, one can find many models meant to describe more or less general softening behaviors, from the simplest, with a single scalar internal variable (plus the hardening-softening variable), to very sophisticated models in which the internal variables are tensors of second or fourth order (even eighth-order tensors have been proposed as internal variables). For example, a very interesting model was proposed by Ortiz (1985) in which the full fourth-order compliance tensor is one internal variable (equivalent to a set of 21 internal variables, the independent components of the compliance tensor), and the second-order tensor of plastic strain a further internal variable (equivalent to a set of six scalar variables). However, this model is too complex to be described here in detail. Only two very simple models will be briefly discussed: Mazars’ scalar (isotropic) damage model and Rankine’s associated plastic model with strain softening.