Computational Fracture Mechanics Anderson’s book, third ed. , chap 12

Elements of Theory Energy domain integral method: - Formulated by Shih et al. (1986): CF Shih, B. Moran and T. Nakamura, “Energy release rate along a three-dimensional crack front in a thermally stressed body”, International Journal of Fracture 30 (1986), pp. 79-102 - Generalized definition of the J- integral (nonlinear materials, thermal strain, dynamic effects). - Relatively simple to implement numerically, very efficient. Finite element (FE) code ABAQUS

Energy Domain Integral : In 2D, under quasistatic conditions, J may be expressed by The contour G surrounds the crack tip. The limit indicates that G shrinks onto the crack tip. n : unit outward normal to G. x1 , x2 Cartesian system q : unit vector in the virtual crack extension direction. w : strain energy density and, s : Cauchy stress tensor displacement gradient tensor H : Eshelby’s elastic energy-momentum tensor (for a non-linear elastic solid) For details see Abaqus Theory Manual, section 2.16.1

With q along x1 and the field quantities expressed in Cartesian components, i.e. Thus, In indexed form, we obtain The expression of J (see eq. 6.45) is recovered with The previous equation is not suitable for a numerical analysis of J. Transformation into a domain integral

Following Shih et al. (1986), (*) m : outward normal on the closed contour : the surface traction on the crack faces. A is a sufficiently smooth weighting function in the domain A. m = -n on G with Note that, A includes the crack-tip region as

(*) Derivation of the integral expression = = q → Line integral along the closed contour enclosing the region A. Noting that, since since

Using the divergence theorem, the contour integral is converted into the domain integral Under certain circumstances, H is divergence free, i.e. indicates the path independence of the J-integral. In the general case of thermo-mechanical loading and with body forces and crack face tractions: the J-integral is only defined by the limiting contour Introducing then the vector, in A or Using next the relationship, Contributions due to crack face tractions.

- Different contours are created: In Abaqus: - This integral is evaluated using ring elements surrounding the crack tip. - Different contours are created: First contour (1) = elements directly connected to crack-tip nodes. The second contour (2) are elements sharing nodes with the first, … etc Refined mesh Contour (i) nodes outside nodes inside Crack 2 1 8-node quadratic plane strain element (CPE8) Exception: on midside nodes (if they exist) in the outer ring of elements

J-integral in three dimensions Local orthogonal Cartesian coordinates at the point s on the crack front: J defined in the x1- x2 plane crack front at s L Point-wise value For a virtual crack advance l(s) in the plane of a 3D crack, T L : length of the crack front under consideration. : surface element on a vanishingly small tubular surface enclosing the crack front along the length L.

Numerical application (bi-material interface): SEN specimen geometry (see Appendix III.1): s a = 40 mm b = 100 mm h = 100 mm a/b = 0.4 Material 1 and h/b = 1 s = 1 MPa. Remote loading: 2h a b Materials properties (Young’s modulus, Poisson’s ratio): Material 1: E1 = 3 GPa n1 = 0.35 Material 2 x y Material 2: E2 = 70 GPa n2 = 0.2 s Plane strain conditions.

Typical mesh: Material 1= Material 2 Material 1 Material 2 Refined mesh around the crack tip Number of elements used: 1376 Type: CPE8 (plane strain)

Results: Remarks: Material 1 Material 2 Bi-material J (N/mm) Abaqus 0.1641 0.0077 0.0837 (*) (*) same values on the contours 2-8 Isotropic Bi-material KI KII Annex III 0.746 0. / Abaqus 0.748 0.752 0.072 SIF given in Remarks: for the isotropic case (i =1,2). One checks that:

Relationship between J and the SIF’s for the bi-material configuration: - Extracted from Abaqus Theory Manual, section 2.16.2. - For an interfacial crack between two dissimilar isotropic materials (plane strain), where and plane strain, i = 1,2 KI and KII are defined here from a complex intensity factor, such that with