Uniaxial Softening Models

After explaining the basic concepts in crack band models, we can now discuss in detail the uniaxial version of various simple constitutive models for strain softening. Depending on the behavior at unloading, one may distinguish three basic types of models :

1. Continuum damage mechanics, in which strain-softening is due solely to degradation of elastic moduli and no other inelastic behavior takes place. The basic characteristic of such theory is that the material unloads along a straight line pointed toward the origin (Fig. 8.4.1a).

2. Plasticity with yield limit degradation, in which the constitutive relation is the same as in plasticity except that the yield limit is decreasing, rather than increasing. The elastic moduli remain constant (Fig. 8.4.1b). The basic characteristic is that the unloading slope is constant, equal to the elastic modulus E.

3. A mixed theory, in which both the elastic moduli and the yield limit suffer degradation. This behavior, for which the unloading slope is intermediate (as shown Fig. 8.4.1c), is normally a better description of experimental reality.

The foregoing classification neglects the fact that the unloading-reloading response is actually nonlinear, as discussed in Section 8.3.3 and depicted in Fig. 8.4. Id. Models including such behavior can be generated, but they are considerably more complex, and are a subject for specialized studies that will not be treated here. As an exception, the microplane model, which implements this kind of behavior naturally, will be discussed at length in Chapter 14.

8.4.1 Elastic-Softening Model with Stiffness Degradation

As a simple continuum model of a material fracturing in tension, we can adopt the elastic-softening model whose behavior for monotonic stretching was described in Fig. 8.2.2. To give a physical content to the model, we can assume that this behavior corresponds to an elastic matrix with an array of densely distributed cracks normal to the load direction (Fig. 8.4.2a). Thus, we assume that the total strain is the sunt of the elastic strain of the elastic matrix, єе1, and the strain contributed hy the crack opening, :

є = єа + є! = ~ + є! (8.4.1)

Jb

where is is the elastic modulus of the matrix (i. e., of the virgin material between the cracks). For monotonic straining the assumed behavior is that shown in Fig. 8.2.2. Consider now the unloading behavior after the specimen has been loaded until a certain maximum inelastic strain є? (Figs. 8.4.2b – d). Let us further assume that unloading is straight to the origin. This means that during unloading, the cracks close so that they are completely closed at zero stress. As depicted in (Fig. 8.4.2b), єf represents the maximum cracking strain reached before unloading, and£^ the actual cracking strain. Obviously, for such unloading

Uniaxial Softening Models

Uniaxial Softening Models
Uniaxial Softening Models

(8.4.2)

Подпись: Figure 8.4.2 (a) Densely cracked elastic material, (b) Model with cracks completely closed after full unloading, (c) Model with totally prevented crack closure, (d) Mixed type model with partial crack closure upon unloading.

But this equation also holds if the loading is monotonic (i. e., if fracture is taking place) because then єf —■ if and a — ф(є?). However, we must impose the condition that the line that corresponds to monotonic loading can never be crossed. This can be expressed in various ways, but two are most useful: one in terms of a and if, and the other in terms of є? and еЛ

a – ф{є?) <0 or ef – < 0 (8.4.3)

Note that while e. f can decrease, if is a nondecreasing variable.

Uniaxial Softening Models Подпись: Eif Eif + ф{є{) Подпись: (8.4.4)

Now, the foregoing results can be reformulated so as to look as a genuine continuum damage model. To this end, one may define a derived variable u, the damage, as

Then we just insert this definition into (8.4.2) and the result into (8.4.1), which yields the classical form of continuum damage mechanics for an clastic matrix:

Подпись: (8.4.5)a

We will introduce this expression in a more standard way after we analyze the strength degradation model and the mixed model for this elastic softening model.