# Strain Intensity Factor and Interaction of Parallel Rigid Line Inclusion in Elastic Matrix Using FEA

• Prataprao Patil
• M. Ramji
Conference paper

## Abstract

When a rigid line inclusion embedded in an elastic matrix is subjected to an external load, stress singularity is generated at the tips. The magnitude of this singularity can be quantified in terms of a strain intensity factor rather than a stress intensity factor. The principal reason is that the strain intensity factor is independent of the material properties of the matrix. The strain intensity factor can be analytically calculated for the case of single inclusion. However, for two parallel rigid line inclusions, the strain intensity factor analytical expression is not readily available. Numerical calculation of the strain intensity factor for two parallel rigid line inclusions embedded in an infinite elastic matrix and the effect of distance between rigid line inclusions on the strain intensity factor is the objective of this paper. To this end, first the stress and displacement fields near the inclusions are calculated using the finite element method (FEM). Then, we use a numerical method based on the reciprocal theorem to calculate the strain intensity factor. It is found that the strain intensity factor is equal to that of the single rigid line inclusion case when the distance between the two parallel rigid inclusions is more than twice their length.

## Keywords

Strain intensity factor Rigid line inclusion FEM Singularity

## Nomenclature

E

Young’s modulus of the matrix material

2l

Length of rigid line inclusion

2w

Width of the matrix containing rigid line inclusions

2h

Height of the matrix containing rigid line inclusions

nj

Unit vector normal to contour

xi

Coordinates in the ith direction, i = 1, 2

r, $$\theta$$

Variables defining cylindrical coordinate reference system at an inclusion tip

$$\lambda$$

Order of singularity

$$\nu$$

Poisson’s ratio of the matrix material

σij,ui

Components of the stress tensor and displacement vector

$$\sigma_{ij}^{*} ,u_{i}^{*}$$

Components of the auxiliary stress tensor and displacement vector

$$L_{\text{cr}}$$

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