Mazars Damage

Description

The MazarsDamage model is an implementation of the isotropic damage model for concrete described in Mazars (1984) and Pijaudier-Cabot and Mazars (2001). It is based on the theory of elasticity coupled with the isotropic damage. It describes the elastic stiffness degradation and the softening behavior of concrete under both uniaxial tension and uniaxial compression using an isotropic scalar damage variable . The damage depends only on the effective stresses applied to an undamaged area.

The evolution of the damage is determined by an equivalent strain that quantifies the amount of the local extension state during the mechanical loading. It is calculated from the positive eigenvalues of the strain tensor as follows: where are the components of principal strain and is the Macualay bracket, defined as

The damage loading function is: where is the parameter defining the threshold of damage growth. Initially, is equal to , where is the tensile strength and is the initial Young's modulus of the material. During subsequent steps, it takes on the maximum value of the equivalent strain during the entire load history:

To account for the differing behavior of the concrete in tension and in compression, the scalar damage is expressed as a combination of two damage modes: where and are dimensionless coefficients and represent the contribution of each loading mode such that and under uniaxial tension, and and under uniaxial compression. These coefficients are defined as functions of the principal components of the strain tensors and that result from the tensile and compressive components of the stress tensor, respectively, defined as: and are then computed as: The exponent in these equations is a parameter that can be tuned to fit experimental data for shear response, and is recommended initially to be set to 1. The current implementation of the model has this parameter hard-coded to 1, and it cannot be set by the user.

The scalar damage variables and for tension and compression, respectively, are computed as: where , , , and are material parameters that control the shape of the nonlinear response and are determined using compression and tension tests.

Implementation and Usage

Damage models such as this do not directly compute the stress, but are used in conjunction with a stress calculator to modify the stress computation that takes other phenomena into account as well. The damage models derive from a common base class that defines functionality that a damage model must supply. The ComputeDamageStress model is a stress calculating model that computes the behavior of materials that only exhibit damaged isotropic elastic behavior. The example below shows how the damage and stress models are used in conjunction with each other.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [damage]
    type = MazarsDamage<<<{"description": "Mazars scalar damage model", "href": "MazarsDamage.html"}>>>
    tensile_strength<<<{"description": "Tensile stress threshold for damage initiation"}>>> = 1e6
    a_t<<<{"description": "A_t parameter that controls the shape of the response in tension"}>>> = 0.87
    a_c<<<{"description": "A_c parameter that controls the shape of the response in compression"}>>> = 0.65
    b_t<<<{"description": "B_t parameter that controls the shape of the response in tension"}>>> = 20000
    b_c<<<{"description": "B_c parameter that controls the shape of the response in compression"}>>> = 2150
  []
[]

MazarsDamage is run in conjunction with a stress calculator that supports the use of a damage model, (ComputeDamageStress in this case), as shown:

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [stress]
    type = ComputeDamageStress<<<{"description": "Compute stress for damaged elastic materials in conjunction with a damage model.", "href": "ComputeDamageStress.html"}>>>
    damage_model<<<{"description": "Name of the damage model"}>>> = damage
  []
[]

Input Parameters

References