Math 2.0

Mechanics of deformable solids Definition

Continuum damage mechanics

Builds on Constitutive equationElasticity tensorStrain tensorCauchy stress tensorClausius–Duhem inequality

Definition

Let a body be made of a linear elastic material with elasticity tensor CC in its intact state. An isotropic continuum damage model is a constitutive model with an internal variable D(x,t)∈[0,1]D(x, t) \in [0, 1], the damage, such that:

  1. the stress is σ=(1−D) C:ε\sigma = (1 - D) \, C : \varepsilon, where ε\varepsilon is the strain;
  2. the damage does not decrease: D˙≥0\dot D \ge 0;
  3. D˙\dot D is given by an evolution law in terms of the state variables, compatible with the Clausius–Duhem inequality.

The effective stress is σ~=σ/(1−D)\tilde\sigma = \sigma / (1 - D). Continuum damage mechanics is the theory of such models and of their anisotropic generalizations, in which DD is a tensor.

Remarks

Example: take the free energy ψ=12(1−D) ε:C:ε\psi = \frac{1}{2}(1 - D) \, \varepsilon : C : \varepsilon. The variable conjugate to DD is Y=−∂ψ/∂D=12ε:C:ε≥0Y = -\partial\psi/\partial D = \frac{1}{2} \varepsilon : C : \varepsilon \ge 0. The isothermal dissipation is YD˙Y \dot D, which is nonnegative whenever D˙≥0\dot D \ge 0, so clause 3 holds for any law with clause 2. A typical law lets DD grow when YY exceeds its largest past value and a threshold. Non-example: a model in which DD can decrease on unloading violates clause 2; it would describe healing, which is excluded for cracks and voids. The relation in clause 1 expresses the principle of strain equivalence: the damaged material under σ\sigma deforms like the intact material under the effective stress. In one dimension, measuring the reduced Young's modulus E~=(1−D)E\tilde E = (1 - D) E gives the damage as D=1−E~/ED = 1 - \tilde E / E.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Cauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptConstitutive equation, concept, no evidence levelConstitutive equationconceptElasticity tensor, concept, no evidence levelElasticity tensorconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptStrain tensor, concept, no evidence levelStrain tensorconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptDivergence, concept, no evidence levelDivergenceconceptClausius–Duhem inequality, statement, no evidence levelClausius–Duhem inequa…statementContinuum damage mechanics, definition, no evidence levelContinuum damage mech…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cauchy stress tensor concept
  2. Constitutive equation concept
  3. Elasticity tensor concept
  4. Euclidean space object
  5. Function concept
  6. Strain tensor concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Partial derivative concept
  10. Vector field concept
  11. Divergence concept
  12. Clausius–Duhem inequality statement
  13. Continuum damage mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Damage_mechanics
About this page
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Kinddefinition
WritingW0 stub
MSC74R05, 74A45, 74R20
DefinesContinuum damage mechanics
UsesConstitutive equation, Elasticity tensor, Strain tensor, Cauchy stress tensor, Clausius–Duhem inequality
Changed3 November 2026

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