Section 2.4

2.4Damage

Materials with a distribution of microcracks can suffer damage when loaded: cracks coalesce, forming voids and larger cracks, reducing the stiffness of the material and eventually leading to failure.

2.4.1Thermomechanical Model

The reduction in stiffness can be modeled by using a free energy function of the form

(2.4.1)

Here, is the free energy of the un-damaged material, the effective strain energy, and is a dimensionless scalar internal variable quantifying the amount of damage, called the damage variable. The factor is called the reduction factor.

Now

(2.4.2)

From the Clausius-Planck inequality

(2.4.3)

With the stress independent of the strain-rate,

(2.4.4)

where is the corresponding stress in the un-damaged material, the effective stress, and is ascalar dissipative stress:

(2.4.5)

Then

(2.4.6)

which is an effective stress power.

2.4.2Damage Criteria

In an analogy to the yield function of plasticity theory, define now an isotropicdamage function according to

(2.4.7)

where is some new damage variable, a critical dissipative stress. When , there is no damage. Damage occurs when . This defines the surface in strain-space with outward normal (for fixed ), Fig. 2.4.1. The double contraction of the strain and normal is

(2.4.8)

which, when positive, implies that the strain is directed outward from the damage syrface, Fig. 2.4.1 Thus damage accumulates when .

The evolution of damage can thus be described by

(2.4.9)

Figure 2.4.1: Damage surface in Principle Strain Space

Damage Models

At any given time t, the damage variable is a function of the complete history of loading. A simple model would be one where is the maximum value of the dissipative stress (or equivalently, of the effective strain energy):

(2.4.10)

The dissipative stress here is equivalent to the effective strain energy, and the maximum will occur at maximum strain. Therefore, in a strain-controlled cyclic test for this particular model, damage will occur on the first cycle only, during the loading phase.

It remains to relate the kinematic damage variables to the force damage variable .

A commonly used model is the function

(2.4.11)

which is plotted in Fig. 2.4.2. The maximum possible value of is , which must be less than 1.

Figure 2.4.2: Constitutive relation for damage parameter

2.4.3Example

An algorithm for the evaluation of the stress response to a strain history using the above damage mode is then

  1. Evaluate
  2. Evaluate
  3. Evaluate
  4. Evaluate
  5. Evaluate

Consider a simple one-dimensional non-linear elastic material with strain energy function and stress

with , so that

Damage occurs according to the parameters and . The resulting stress-strain to a loading/unloading cycle is as shown in Fig. 2.4.3, together with the corresponding (un-damaged effective stress).

Figure 2.4.3: Stress-Strain results for a material with Damage under cyclic loading

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Solid Mechanics Part IV Kelly