Research Papers

Experimental Evaluation of the M-Integral in an Elastic-Plastic Material Containing Multiple Defects

[+] Author and Article Information
Q. Li

e-mail: qunli@mail.xjtu.edu.cn

Y. H. Chen

State Key Laboratory for Mechanical Structural
Strength and Vibration
School of Aerospace
Xi'an Jiaotong University
710049, P.R. China

1Corresponding author.

Manuscript received November 22, 2011; final manuscript received June 16, 2012; accepted manuscript posted July 6, 2012; published online November 19, 2012. Assoc. Editor: John Lambros.

J. Appl. Mech 80(1), 011021 (Nov 19, 2012) (8 pages) Paper No: JAM-11-1445; doi: 10.1115/1.4007083 History: Received November 22, 2011; Revised June 16, 2012; Accepted July 06, 2012

An experimental technique for evaluation of the M-integral in an elastic-plastic material containing multiple defects is proposed by using digital image correlation (DIC). This technique makes direct use of the definition of M by experimentally evaluating the integrand of M at various points along a square contour and determining the integral by numerical integration. The nonlinear Ramberg–Osgood model is used to capture the elastic-plastic behavior such as the elastic-plastic stress and the total strain energy density in terms of the measured displacements by DIC used in an ARAMIS 4M instrument. Compared with the previous experimental method proposed by King and Herrmann (King and Herrmann, 1981, “Nondestructive Evaluation of the J and M Integrals,” ASME J. Appl. Mech., 48, pp. 83–87), the present technique could be suitable to measure the M-integral for the various complicated damages, specimen geometries, loading conditions, and material behaviors. The path-independence or path-dependence of the M-integral is investigated under small-scale and large-scale yielding conditions, respectively. It is found that the values of M are path independent when the contours entirely enclose the nonlinear plastic region near the multiple defects. In contrast, the path-dependence is concluded for an elastic-plastic solid under large-scale yielding condition when the contours have to pass through the plastic zone. This interesting path-dependence of the M-integral is consistent with numerical prediction via the finite element method and theoretical analysis developed in this paper.

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Grahic Jump Location
Fig. 3

The measured values of the displacements (a) ux; (b) uy

Grahic Jump Location
Fig. 5

Strain-stress data from the uniaxial tensile test and the fitting data for aluminum LY12. Calculation technique of the total strain energy is also shown including the linear part and the nonlinear one by numerical integration of the strain-stress data.

Grahic Jump Location
Fig. 4

The evaluated values of the strains (a) ɛxx; (b) ɛyy; the evaluated displacement gradient (c) ∂ux/∂y; (d) ∂uy/∂x

Grahic Jump Location
Fig. 2

Schematic configuration of the ARAMIS 4M used in the experiment

Grahic Jump Location
Fig. 1

(a) Experimental specimen made of aluminum LY12 containing 60 voids; (b) powder sprays on the surface of specimen to obtain the good high contrast stochastic pattern; (c) experiment arrangement of specimens, MTS machine, and ARAMIS

Grahic Jump Location
Fig. 6

The evaluated stresses by Ramberg–Osgood theory (a) σxx; (b) σyy

Grahic Jump Location
Fig. 7

The evaluated strain energy density by numerical integration of stress-strain curve

Grahic Jump Location
Fig. 8

Schematic of square integral contours for calculating the M-integral

Grahic Jump Location
Fig. 9

Evaluated values of the M-integral versus the tensile loadings for 10 different square contours

Grahic Jump Location
Fig. 10

Finite element mesh of the experimental specimen for calculating the M-integral in an elastic-plastic plate: (a) full mesh; (b) fine mesh near the defect zone

Grahic Jump Location
Fig. 11

Comparisons of the M-integral between experimental evaluations and FEM calculations under three different loads



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