Dynamic Behavior of Concrete: Constitutive Models - Dynamic Behavior of Concrete and Seismic Engineering - page 74

2.5.4. Damage

The subject of damage mechanics seeks to represent the irreversible micro
cracking and stiffness-degradation processes which occur in some materials. The
threshold above which this phenomenon appears can be represented by one criterion,
as in the case of plasticity. The criterion can be expressed in terms of strains instead
of stresses, which is not the same in dynamics in the presence of viscosity. It can
also be asymmetric, i.e. a function of extensions and not compressions. For example,
Mazars' criterion [MAZ 84], which has been used for concrete, is a function of the
principal strains if they are positive. The most widely used model is the scalar
isotropic damage model [LEM 85]. The behavior of damaged real material is
expressed by the law used for the undamaged material, except the usual stress is
replaced by the effective stress.

V

H [2.28]

ED (1 )

There are also damage models that depend on time.

H

"

[2.29]

DD ,,

This type of formulation derives its justification from the fact that the creation
and propagation of cracks are phenomena that cannot be instantaneous [SUA 84].

2.5.5. Notion of a state law

A common hypothesis on which to build up a material behavior model involves
splitting the stress tensor and the strain tensor into their spherical and deviatoric
parts.

c

V p s I D and H T I e [2.30]

Then we can define separately the relationship between the spherical parts
(pressure and volume variation) called isotropic behavior or state law by abusing the
language, and the relationship between the deviatoric parts (which represents the
deviatoric behavior). This approach has proved quite interesting in the case of
metals, as both relationships are then independent, and two independent experiments