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5.6.3 Damage Criterion
At each material point in the continuum, damage occurs when a scalar damage (or failure) measure achieves a critical value over the loading history, The scalar damage measure must be invariant to orthogonal transformations that preserve material symmetry, or else the damage formulation would not be observer-independent. For example, for isotropic materials, must be an isotropic function of the deformation, in which case it should be expressed as or , where is the right stretch tensor in the polar decomposition of the deformation gradient, and is the Green-Lagrange strain tensor. It follows that For anisotropic materials where is the unit normal to a symmetry plane and , the damage measure must satisfy for transformations that satisfy (or ). We may replace with in the above expression.
We assume that the amount of damage (the fraction of broken bonds) is given by the function of state where . As shown in , the Clausius-Duhem inequality imposes the constraint that must be a monotonically increasing function of its argument. Therefore, we may understand to represent a cumulative density function (CDF), whose derivative is a probability distribution function (PDF) that describes the probability of bonds breaking at the specific threshold .