# Coupled Theory of Damage Mechanics and Finite Strain Elasto-Plasticity

This pioneering work by Professor Voyiadjis uses damage mechanics in metals and metal matrix composites to address anisotropic degradation of the material behavior. An anisotropic damage model is proposed for fibrous metal matrix composites with a ductile matrix. The model incorporates damage mechanics with micromechanical behavior. An anisotropic damage tensor is used to describe both the overall degradation of the material as well as that of its constituents. The proposed micromechanical damage composite model used is such that separate local constitutive damage relations are used for each of the matrix and the fiber. This is coupled with the interfacial damage between the matrix and the fiber exclusively. The damage relations are linked to the overall response through a certain homogenization procedure. Two local damage tensors are used to for the damage in the ductile matrix and the brittle fibers. An additional tensor is incorporated in the overall formulation that represents interfacial damage between the matrix and the fiber. An overall damage tensor, M, is introduced that accounts for all these separate local damage tensors. This formulation allows the damage model to directly use the elastoplastic stiffness tensor obtained for the undamaged effect configuration. An explicit expression is obtained for the elastoplastic stiffness tensor for the damaged composite material. Numerical solutions are obtained for different types of laminate layups compared with experimental results. Finally, the coupled theory of damage with inelastic behavior is presented for both room and elevated temperatures. This is accomplished for both rate dependent and rate independent plasticity and damage.

The contributions in this work include:

- The formulation is incorporates both isotropic and kinematic hardening,
- the von Mises yield criterion is modified to include the effects of damage through the use of the hypothesis of elastic energy equivalence,
- a modified elasto-plastic stiffness tensor that includes the effects of damage is derived within the framework of the proposed model.
- the problem of crack initiation is addressed for a thin elasto-plastic plate with a center crack that is subjected to inplane tension,
- an explicit matrix representation is derived for the damage effect tensor for a general state of deformation and damage,
- a linear transformation is shown to exist between the effective deviatoric Cauchy stress tensor and the total Cauchy stress tensor,
- an effective elasto-plastic stiffness tensor is derived that includes the effects of damage,
- the proposed model is applied to void growth through the use of Gurson’s yield function. It is also shown how a modified Gurson function can be related to the proposed model. Some interesting results are obtained in this case.
- numerical solutions are obtained using the proposed theory for two types of laminate layups (0/90)s and (+- 45)s each consisting of four plies. These are compared with experimental results where good correlation is obtained between the experimental and numerical results.
- anisotropic low and high cyclic damage with anisotropic plasticity is also addressed,
- the generalized cells model is applied to damage models as an alternate approach to the homogenization procedures that use the averaging scheme.

Some results of Voyiadjis’s work appeared in the following books/book chapters:

Books:

**Voyiadjis, G. Z.**, and Kattan, P., *Advances in Damage Mechanics: Metals and Metal Matrix Composites With an Introduction to Fabric Tensors (2nd edition)*, 742 p., Elsevier, Oxford, ISBN: 0-08-044688-4, 2006.

**Voyiadjis, G. Z.**, and Song, C. R*., The Coupled Theory of Mixtures in Geomechanics with Applications*, 438 p., Springer-Verlag GmbH & Co.KG, Heidelberg, ISBN: 3-540-25130-8, 2006.

**Voyiadjis, G. Z.**, and Kattan, P., *Damage Mechanics*, 257 p., CRC Press – A Taylor & Francis Company, Florida, USA, ISBN: 082472756, 2005.

**Voyiadjis, G. Z.**, and Kattan, P., *Mechanics of Composite Materials with MATLAB*, 336 p., with CD-ROM, Springer-Verlag GmbH & Co.KG, Heidelberg, ISBN: 3540243534, 2005.

Kattan, P., and **Voyiadjis, G. Z.**, *Damage Mechanics with Finite Elements: Practical Applications with Computer Tools*, 114p., with CD-ROM, Springer-Verlag GmbH & Co.KG, ISBN: 3-540-42279-X, 2001.

**Voyiadjis, G. Z.**, and Kattan, P., *Advances in Damage Mechanics: Metals and Metal Matrix Composites*, 542 p., Elsevier, Oxford, ISBN 0-08-043601-3, 1999.

Book Chapters in Handbooks:

**Voyiadjis, G. Z.**, “Continuum Damage Mechanics,” Handbook of Materials Modeling, Vol. 1 Fundamental Models and Methods, Chapter 3: Mesoscale/Macroscale Computational Methods, Section 3.8, Editor Sidney Yip, Springer, The Netherlands, ISBN 1-4020-3287-0, 2005, pp. 1183-1192.

**Voyiadjis, G. Z.**, “Model of Inelastic Behavior Coupled to Damage,” Handbook of Materials Behavior Models, Chapter 9, Section 9.4, Editor J. Lemaitre, Academic Press, New York, 2001, pp. 814-820.

CDs for Computational Codes Associated with this work:

**Voyiadjis, G. Z.**, and Kattan, P. I, *Supplementary CDROM of Mechanics of Composite Materials with MATLAB Book*, Springer-Verlag GmbH & Co.KG, Heidelberg, ISBN: 3540243534, June 2005.

Kattan, P. I., and **Voyiadjis, G. Z.**, *Supplementary CDROM of Damage Mechanics with Finite Elements Book*, Springer-Verlag GmbH & Co.KG, ISBN: 3-540-42279-X, 2001.

Numerous parts of the above codes have been implemented in commercial codes and Federal Laboratories affiliated codes.

Professor Voyiadjis was invited to edit/co-edit the following special issues of journals:

**Voyiadjis, G. Z.**, and Antonio Rinaldi, Co-Editors of the special issue “In Honor of Professor Dusan Krajcinovic,” in the International Journal of Plasticity, Vol. 23, No. 10 and 11, 2007, pp. 1826-1937.

**Voyiadjis, G. Z.**, Pijaudier-Cabot, G., and Haj-Ali, R., Co-Editors of the special issue “Multi-Scale Modeling of Damage, and Material Characterization with Microstructure” in the Journal of Engineering Mechanics, ASCE, Vol. 127, No. 7, 2001, pp. 635-746.

**Voyiadjis, G. Z.**, and Ju, J. W., Co-Editors of the special issue “Advances in Computational Methods for Damage Mechanics,” in the Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 183, No. 3-4, 2000, pp. 157-362.