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Research Papers

# Decomposition of Elastic Stiffness Degradation in Continuum Damage Mechanics

[+] Author and Article Information

Boyd Professor
Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70803

Peter I. Kattan

Petra Books,
P.O. Box 1392,
Amman 11118, Jordan
e-mails: pkattan@orange.jo; info@PetraBooks.com

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received May 29, 2016; final manuscript received September 10, 2016; published online February 1, 2017. Assoc. Editor: Taehyo Park.

J. Eng. Mater. Technol 139(2), 021005 (Feb 01, 2017) (15 pages) Paper No: MATS-16-1152; doi: 10.1115/1.4035292 History: Received May 29, 2016; Revised September 10, 2016

## Abstract

The degradation of elastic stiffness is investigated systematically within the framework of continuum damage mechanics. Consistent equations are obtained showing how the degradation of elastic stiffness can be decomposed into a part due to cracks and another part due to voids. For this purpose, the hypothesis of elastic energy equivalence of order n is utilized. In addition, it is shown that the hypothesis of elastic strain equivalence is obtained as a special case of the hypothesis of elastic energy equivalence of order n. In the first part of this work, the formulation is scalar and applies to the one-dimensional case. The tensorial formulation for the decomposition is also presented that is applicable to general states of deformation and damage. In this general case, one cannot obtain a single explicit tensorial decomposition equation for elastic stiffness degradation. Instead, one obtains an implicit system of three tensorial decomposition equations (called the tensorial decomposition system). Finally, solution of the tensorial decomposition system is illustrated in detail for the special case of plane stress.

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## Figures

Fig. 1

Damaged and fictitious undamaged configurations

Fig. 2

Stress–strain curves of damage and undamaged material

Fig. 3

Illustration of the fourth-rank damage effect tensor M

Fig. 4

Illustration of the fourth-rank damage tensors L(1) andL(2)

Fig. 5

Illustration of the fourth-rank damage effect tensors Mc and Mv

Fig. 6

Illustration of the fourth-rank damage effect tensors Lc(1), Lv(1), Lc(2), and Lv(2)

## Errata

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