Computational Mechanics

, Volume 62, Issue 4, pp 783–801

# Numerical analysis of a main crack interactions with micro-defects/inhomogeneities using two-scale generalized/extended finite element method

Original Paper

## Abstract

Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner–Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.

## Keywords

Generalized/extended FEM Fracture mechanics Two-scale analysis Micro-defects Stress intensity factor

## Nomenclature

$$\varvec{{\bar{t}}}$$

External traction vector

$$\varvec{{\bar{u}}}$$

Prescribed displacement

$$\varvec{{\hat{n}}}$$

Unit outward normal

$$\varvec{b}$$

Body force vector

$$\varvec{m}_j, n_j$$

Outward unit normal vectors in contour integral

$$\varvec{u}$$

Displacement field vector

$$\varvec{v}$$

Test function

$$\delta$$

Kronecker delta

$$\epsilon _b, \epsilon _s$$

Bending and shear strains

$$\epsilon _{ij}$$

Strain tensor

$$\Gamma$$

Inner J-integral boundary

$$\kappa$$

Material constant

$$\mu$$

Shear modulus

$$\mu _d$$

Inhomogeneities shear modulus

$$\mu _m$$

Main problem shear modulus

$$\nu _d$$

Inhomogeneities Poisson’s ratio

$$\Omega$$

Problem domain

$$\omega _j$$

Cloud of node j

$$\partial \Omega$$

Problem boundary

$$\partial \Omega _c$$

Crack surface

$$\partial \Omega _t$$

Surface traction boundary

$$\partial \Omega _u$$

Displacement boundary

$$\phi _i$$

Angle of micro-defect i and main crack-tip connection line with respect to the x-axis

$$\psi$$

Section rotation of the plate middle plane

$$\sigma$$

Cauchy stress tensor

$$\sigma _{ij}$$

Stress tensor

$$\theta _i$$

Micro-defect i direction angle

$$\varepsilon$$

Linear strain tensor

$$a_{mc}$$

Main crack length

$$C_+, C_-$$

Upper and lower crack surfaces

$$C_0$$

Outer J-integral boundary

$$d_i$$

Distance of the micro-defect i to the main crack-tip

detJ

Determinant of the Jacobian

$$h_{elem}$$

Square root of the crack-tip element area

I

Interaction integral

J

J-integral

$$l_{md}$$

Characteristic length of the micro-defect

M

Bending moment

$$N_{gp}$$

Number of Gauss points

Q

q

Weighting function

$$q_j$$

A set of linearly independent functions defined at each nodal cloud

$$r_m$$

Interaction integral scalar multiplier

W

Strain energy density

w

Transverse displacement of the plate

$$w_{gp}$$

Weight of each Gauss point

$$\varvec{b}_{ji}$$

Nodal parameters associated with G/XFEM

$$\varvec{D}$$

Hook’s tensor

$${\mathbb {R}}^{2}$$

Bi-dimensional domain

$${\mathcal {N}}_j$$

FE Shape function

$$\nu$$

Poisson ratio

$$\Omega _{G}$$

Global domain

$$\Omega _{L}$$

Local domain

$$\phi _{ji}$$

G/XFEM shape function

$$\theta$$

Angle between x coordinate and direction of the crack front

$${\tilde{\varvec{u}}}(\varvec{x})$$

G/XFEM displacement approximation

E

Young’s modulus

$$K_I, K_{II}, K_{III}$$

Mode-I, II and III stress intensity factors

$$L_{ji}$$

Local approximation function

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