Continuum Damage Mechanics (CDM)

Category: Structural Analysis | Integrated 2026-04-06
CAE visualization for damage continuum theory - technical simulation diagram
Continuum Damage Mechanics (CDM)

Continuum Damage Mechanics (CDM): Theoretical Foundations

What is CDM?

🧑‍🎓

Professor, what is Continuum Damage Mechanics (CDM)?


🎓

CDM (Continuum Damage Mechanics) is a theory that describes material degradation (damage) using a continuous variable $D$. Proposed by Kachanov (1958) and Rabotnov (1968).


$$ \tilde{\sigma} = \frac{\sigma}{1-D} $$

$D = 0$ for intact material, $D = 1$ for complete failure. $\tilde{\sigma}$ is the effective stress.


CDM Framework

🎓

Stress-strain relation:

$$ \sigma = (1-D) C \varepsilon $$

Damage evolution law (example: creep damage):

$$ \dot{D} = A \left(\frac{\sigma}{1-D}\right)^n $$

🧑‍🎓

As damage progresses, the effective stress increases, leading to further acceleration of damage—a positive feedback loop.


🎓

This "chain reaction" leads to final failure ($D \to 1$). CDM provides a unified framework for creep fracture, fatigue, and ductile fracture.


Summary

🎓
  • Damage variable $D$ (0~1) describes material degradation — within a continuum framework
  • Effective stress $\tilde{\sigma} = \sigma/(1-D)$ — effective stress increases with damage
  • Creep damage, fatigue damage, ductile damage — applications of CDM
  • Hashin damage (composites) is also a type of CDM

    • Coffee Break Yomoyama Talk

    Kachanov's 1958 Paper

    The foundation of Continuum Damage Mechanics (CDM) lies in a mere 8-page Russian paper "On the Creep Fracture Time" published by L.M. Kachanov in 1958. He introduced the concept of "effective stress" and represented the accumulation of microcracks with a single scalar damage variable ω. Although Kachanov was marginalized in the Soviet Union at the time, after being introduced to the West in the 1980s, his work rapidly gained attention and led to the systematization of CDM by Lemaitre and Chaboche.

    Computational Methods for Continuum Damage Mechanics (CDM)

    CDM in FEM

    🎓

    Abaqus's composite Hashin damage and Ductile Damage are CDM-based. Custom CDM models can be implemented via user subroutines (UMAT/VUMAT).


    Summary

    🎓
    • Abaqus's *DAMAGE family is CDM-based — Hashin, Ductile, Shear, etc.
    • Custom CDM via UMAT/VUMAT — arbitrary damage evolution laws
    • Regularization via fracture energy — eliminates mesh dependency

      • Coffee Break Yomoyama Talk

      Lemaitre's Damage-Plasticity Coupling Model

      Jean Lemaitre, born in Chambon, France, extended Kachanov's uniaxial theory to three-dimensional elastoplastic damage mechanics in his 1984 paper "How to Use Damage Mechanics". Damage is defined as an isotropic scalar D, and by replacing stress with effective stress σ̃ = σ/(1−D), it can be easily coupled with existing plasticity models. The Lemaitre model is now included in the standard material libraries of France's Code_Aster and SYSTUS (by ESI).

      Continuum Damage Mechanics (CDM) in Practice

      CDM in Practice

      🎓

      Used for metal ductile fracture (damage index), progressive damage in composites, and damage plasticity in concrete.


      Practical Checklist

      🎓
      • [ ] Is the damage initiation criterion based on material tests?
      • [ ] Is damage evolution regularized by fracture energy?
      • [ ] Is element deletion set for elements where damage variable $D$ reaches 1.0?
      • [ ] Has mesh convergence been verified? (Damage tends to localize)

        • Coffee Break Yomoyama Talk

        Application to Automotive Crash Analysis

        CDM also plays an important role in automotive crash safety (crash) analysis. Since the early 2000s, Volkswagen has adopted a combination of LS-DYNA's Gurson-Tvergaard-Needleman (GTN) model and CDM for sheet metal fracture prediction, utilizing it in occupant protection performance evaluation for frontal crash tests (Euro NCAP). It is reported that introducing CDM enabled predicting perforation fracture of high-strength steel (780 MPa grade) with 30% higher accuracy compared to the conventional maximum strain criterion.

        Continuum Damage Mechanics (CDM): Software & Solver Comparison

        CDM Tools

        🎓
        • Abaqus — *DAMAGE family (Hashin, Ductile, Shear, FLD). Most extensive.
        • LS-DYNA — *MAT_ADD_EROSION. Johnson-Cook fracture, etc.
        • UMAT/VUMAT — Custom CDM

          • Coffee Break Yomoyama Talk

          Implementation History of SIMULIA Damage Mechanics

          CDM was first standardly implemented in Abaqus in version 6.2 (2001), with a simplified version of the Lemaitre model implemented as "Ductile Damage". Subsequently, a Gurson-type void model was added in 6.14 (2014), and rebranding progressed as Abaqus 2019 from 2019, also adding a sheet metal fracture model coupled with FLD (Forming Limit Diagram). In the current Abaqus 2024, five types of ductile damage criteria and three types of shear damage criteria are available for selection.

          Advanced Technologies

          Advanced CDM

          🎓
          • Phase-Field Method — Tracks damage as a continuous field. A natural extension of CDM.
          • Non-local CDM — Fundamental solution to mesh dependency.
          • Multi-scale CDM — Micro-scale damage (voids, cracks) → Macro-scale damage variable.

            • Coffee Break Yomoyama Talk

            Non-localization Theory and Eliminating Mesh Dependency

            Since CDM involves strain localization, standard local theory shows a pathological dependency where fracture energy approaches zero as mesh size becomes finer. In 1987, Bažant and Pijaudier-Cabot proposed "non-local damage mechanics", evaluating the damage variable as a weighted average of surrounding elements (influence radius l ≈ 3× maximum aggregate size) to eliminate this dependency. This method has been implemented as an extension in diaFEA and the latest Abaqus 2024.

            Continuum Damage Mechanics (CDM): Common Issues & Debugging

            CDM Troubles

            🎓
            • Mesh dependency → Regularize with fracture energy. Consider non-local CDM.

            • Related Topics

              StructuralCreep bucklingGlossaryDamage Mechanics — CAE GlossaryError Solution DBMaterial Damage Exceeds Limit — Element DeletionStructuralCreep-fatigue interactionStructuralGurson model (ductile fracture)StructuralCreep analysis using Norton's law
              Related Simulators

              Experience the theory firsthand with the interactive simulator for this field

              All Simulators
              Rate this article
              Thank you for your feedback!
              Helpful
              More details
              Report error
              Helpful
              0
              More details
              0
              Report error
              0
              Written by NovaSolver Contributors
              Anonymous Engineers & AI — Sitemap