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Fracture Mechanics: an Introduction, Second Edition

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Fracture Mechanics: an Introduction, Second Edition Chapter 6 J-Integral and Crack Opening Displacement Fracture Criteria 6.1. Introduction A number of investigators have proposed the mathematical formulation of elastostatic conservation laws as path independent integrals of some functionals of the elastic field over the bounding surface of a closed region. For notch problems, Rice [6.1] introduced the two-dimensional version of the conservation law, a path independent line integral, known as the J-integral. The present chapter is devoted to the theoretical foundation of the path independent J-integral and its use as a fracture criterion. The critical value of the opening of the crack faces near the crack tip is also introduced as a fracture criterion. First we present the definition of the J-integral in two-dimensional crack problems, and its physical interpretation in terms of the rate of change of potential energy with respect to an incremental extension of the crack. The path independent nature of the integral allows the integration path to be taken close or far away from the crack tip. Although the J-integral is based on purely elastic (linear or nonlinear) analysis, its use for plasticity-type materials has been supported by experimentation or numerical analysis. We present experimental methods for the evaluation of the integral. We introduce a failure criterion based on the J-integral and describe the standard ASTM method for the experimental determination of the critical value 4, of the J-integral. The chapter concludes with a brief presentation of the crack opening displacement fracture criterion. 6.2. Path-independent integrals First we introduce some path-independent surface integrals in three-dimensional space. We assume that the solid body is linearly or nonlinearly elastic, homogeneous, ranisotropicand in a state of static equilibrium under the action of a system of traction Tk. Denote by C the bounding surface of the region R occupied by the body and 154 Chapter 6 refer all quantities to a fixed Cartesian coordinate system 0x1~2~3.TO simplify the analysis we assume that the deformation is small. The stress tensor uij is obtained from the elastic strain energy w by where ~ijdenotes the strain tensor. The strain energy density w is considered to be continuously differentiable with respect to strain. For elastic behavior Equation (6.1) where the integral is path independent in the strain space. In the absence of body forces the equations of equilibrium take the form and the traction vector Tjon surface S is given by where ni denotes the normal vector. For small deformation the strain tensor is derived from the displacement field by 1 E.. - &.. - - v - ,z - (."j + Uj,i). Consider the integrals where C is a closed surface bounding a region R which is assumed to be free of singularities. Equation (6.6) can be put in the form and Gauss's divergence theorem gives The integrand takes the form J-IMegral and Crack Opening Displacement Fracture Criteria = -uklTkl,j = 0. In these derivations we used Equations (6.1) to (6.2). T~Ldenotes the nonsymmet- rical rotation tensor for small deformation given by Thus, we have We can prove in a similar manner that Equation (6.10) holds when the elastic strain energy density w in Equation (6.6) is replaced by the complementary elastic strain energy density R given by The integral in Equation (6.1 1) is path independent in the stress space. The strain is obtained from R by in the same way that the stress is obtained from R by Equation (6.1). (a)Dejinition For the particular case of the two-dimensional plane elastic problem, consider the integral Chapter 6 Fig. 6.1. A closed contour r and paths rl and r2between two points 0, and O2 in a continuum. where r is a closed contour bounding a region R (Figure 6.1). With Equation (6.14) becomes and defines the J-integral along a closed contour in the two-dimensional space. From Equation (6.10) it follows that J = 0. Since J is zero for any closed paths, the J- integrals along any two paths I'l, F2 connecting any two points 01 and 02, are equal (Figure 6.1). We have (b)Application to notches and cracks Consider a notch or a crack, with flat surfaces parallel to the x-axis, which may have an arbitrary root radius (Figure 6.2). The J-integral defined from Equation (6.16) is calculated along a path I' in a counterclockwise sense starting from an arbitrary point on the flat part of the lower notch surface and ending at an arbitrary point on the flat part of the upper surface of the notch. The region R bounded by the closed J-Integral and Crack Opening Displacement Fracture Criteria Fig. 6.2. (a) Path r starting from the lower and ending up to the upper face of anotch in a two-dimensional body. The flat notch surfaces are parallel to the x-axis. (b) Paths r1and r2around a notch tip. contour ABrlCDr2Ais free of singularities and the J-integral calculated along ABrl CDr2A is zero. We have When parts AB and CD of the notch surfaces that are parallel to the x-axis are traction free, dy = 0 and Tk= 0, and Equation (6.16) implies that Equation (6.18) takes the form 158 Chapter 6 when the contour Ar2D is described in a counterclockwise sense. Equation (6.19) establishes the path independence of the J-integral defined by Equation (6.16) for notch problems. Note that path independence of the J-integral defined by Equation (6.16) for an arbitrary path I? (Figure 6.2a) is based on the assumption that the notch surfaces are traction free and parallel to the x-axis. The integration path may be taken close or far away from the crack tip, and can be selected to make the calculation of the J-integral easy. As an example problem, let us study the case of a crack in a mixed-mode stress field governed by the values of the three stress intensity factors KI,Kn, I(m. For a circular path of radius r encompassing the crack tip, J from Equation (6.16) becomes J = r [w(r,B) cos B - Tx(r,B) / dx -7r Letting r -+ 0 we see from Equation (6.20) that only the singular terms of the stress field around the crack tip contribute to J. Using the equations of the singular stresses and displacement for the three modes of deformation given in Section 2.4, we obtain the following equation for the J-integral after lengthy algebra. where 77 = 1 for plane stress and 77 = 1 - u2 for plane strain. 6.4. Relationship between the J-integral and potential energy We look for a physical interpretation of the J-integral in terms of the rate of change of potential energy with respect to incremental change of crack size. The derivation concerns a linear or nonlinear elastic plane body, with a crack of length a subjected to prescribed tractions and displacements along parts of its boundary. Tractions and displacements are assumed to be independent of crack length. The body is referred to a fixed system of Cartesian coordinates XI xz with the XI -axis parallel to the crack faces (Figure 6.3). It is also assumed that the crack extends in a self-similar manner. The potential energy n(a)of the body is given by where A is the area of the body and r its boundary. Under the previous assumptions differentiation of Equation (6.22) with respect to crack length a yields J-Integral and Crack Opening Displacement Fracture Criteria Fig. 6.3. A two-dimensional cracked body. A new coordinate system XlX2attached to the crack tip is introduced: XI = XI - a ; X2 = 22 (6.24a) 'and Thus, Equation (6.23) takes the form We have and by applying the principle of virtual work we obtain Furthermore, the divergence theorem yields Chapter 6 Introducing Equations (6.26) and (6.27) into Equation (6.25) we get (6.28) (6.29) for any path of integration surrounding the crack tip. Equation (6.29) expresses the J-integral as the rate of decrease of potential energy with respect to the crack length, and holds only for self-similar crack growth. From Equations (6.29) and (4.10) we obtain for "fixed-grips" or "dead-load loading during crack growth. For a crack in a mixed-mode stress field, the value of the J-integral given by Equation (6.30) represents the elastic strain energy release rate only for self-similar crack growth. However, a crack under mixed-mode loading does not extend along its own plane and therefore the value of energy releaserate G given from Equation (6.30) is physically unrealistic. 6.5. J-integral fracture criterion The J-integral can be viewed as a parameter which characterizes the state of affairs in the region around the crack tip. This argument is supported by the following fundamental properties of J: (i) J is path independent for linear or nonlinear elastic material response; (ii) J is equal to -dH/da for linear or nonlinear elastic material response; (iii) J is equal to G; (iv) J can easily be determined experimentally; (v) J can be related to the crack-tip opening displacement 6 by a simple relation of the form J = MuyS (for the Dugdale model M = 1 (Example 6.2)). Because of these characteristic properties Jhas been proposed [6.2] as an attractive candidate for a fracture criterion. Under opening-mode loading, the criterion for crack initiation takes the form J-Integral and Crack Opening Displacement Fracture Criteria 161 where Jcis a material property for a given thickness under specified environmental conditions.
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