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Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohesive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form.

The material behaviour at the tip of the Mode I crack in a homogenous body is in general very complex and difficult to describe by continuum mechanical models. The crack tip region where the material undergoes degradation or damage is known as the process region. Refer

In a bimaterial comprising elasticity identical but plasticity and strength mismatched constituents (like steels), the Mode I crack near the interface has the characteristics similar to the one in homogeneous parent body as long as the cohesive zone is in the parent body alone. The effect of approaching interface body of different strength is not felt by the crack in such a stage because of similar elastic properties across the interface. But as the crack grows and reaches nearer to the interface, the increasing magnitude of crack tip stress field causes the cohesive zone to develop in the interface body. Consequently, the part of cohesive zone in the interface body is subjected to cohesive stress different from that acting over its portion in the parent body that triggers the effect of strength mismatch across the interface over the crack tip. The effect continues with increasing intensity as the cohesive zone spreads deeper into the interface body with crack growth and reaches the maximum when the crack tip touches the interface body with the cohesive zone fully in the interface body.

Cases of thin and thick welds between the steels are examined. Thin weld, obtained by non-fusion, solid state like friction welding between dissimilar steels leads to a single thin interface whereas a thick weld by fusion bonding from electron or laser beam welding results in two interfaces, one between the parent body and the weld and the other between the weld and the interface body. The parent body, the weld and the interface body have similar elastic properties but variable strengths of comparable magnitudes.

Solution for load line opening of the crack is obtained by modeling its cohesive zone. Complex potentials are used

for this purpose in linear elastic regime.

Half load line crack opening, v, in the cohesive zone of size, r, in parent body A in stage I,

where, , is the modulus of elasticity of parent body, A. On integrating Equation (1), the expression for v, as the function of distance x from the crack tip in the cohesive zone is obtained as

Stage I is valid till the cohesive zone lies in the parent body i.e. the distance of the crack tip from the interface, a, fulfills the condition,.

Refer

Using similar principles as in thin weld, the expressions for assume the following forms in Stages I, II and III in the case of thick weld:

All the integrals, especially non-singular ones, in Equation (1) to Equation (6) need to be solved for obtaining a usable form of.

Equations in important stages of crack growth towards the interface in bodies with thin and thick welds are solved as follows.

In Stage II,

where

and

Integral I_{1 }is easily solved as

Integrals I_{2} and I_{3} are singular in nature and are evaluated as follows:

Use of Cauchy’s principal value theorem [

.

On using the following substitutions

in integral I_{2}, one obtains

On evaluating the inner integral, one obtains

Since for +ve, term causes singularity in the solution and is not considered further. Therefore

which can also be written as

On defining, one obtains Therefore

where constant. Solution of the integral is obtained as

Similarly, other term in integral I_{2} is obtained as

Integral I_{2} is finally written as

On re-substituting and, one obtains

Applying the limits of integration results in the following

Similarly, Integral I_{3 }is written as

and is evaluated in similar manner as integral I_{2}.

On applying the limits of integration, one obtains

Substitution of integrals I_{1},_{ }I_{2} and I_{3} in Equation (7) on using, b = a + l, results in closed form expression for load line displacement, v(x), in the cohesive zone spread across the interface of bodies A and B as

Solution in Stage II is obtained upon replacing by and by in Equation (8).

In Stage III,

where and are similar as in Equation (7).

is written as

or

or

is written as

or

or

Substitution of integrals I_{1} to I_{4} in Equation (9) results in closed form expression for displacement, v(x), in the cohesive zone spread across parent body, A, weld, W and interface body, B, as

Solution of integrals is validated by once again reverting to Stages I and II of crack near thin weld. Refer Equation

(2). In Stage I, solution of integral

is.

Lower limit is disregarded since value of at upper limit acts over the specific location in the cohesive zone. Singular integral

is evaluated as

which simplifies to

Crack tip opening displacement (CTOD), at x = 0i.e., is obtained as. On applying Dugdale’s cohesive zone criterion,

, reduces to

which is the well known solution for half crack tip opening displacement in homogenous body A in linear elastic regime.

In Stage II, J integral at crack tip, , expressed as

, is equal to whereas J integral at interface, , is equal to. Since

is given by, the following equation is obtained upon using conservation of energy release rate,

Equation (8) leads to the following expressions

and

Stress intensity parameter applied over the crack with its cohesive zone split across the interface in Stage II [

With Equation (12) and Equation (13), R.H.S. of Equation (11) results in the expression

which equals i.e. L.H.S upon using Equation (14) that validates the solutions of singular integrals.

Singular integrals in mathematical model of load line opening of Mode I crack near the interface of elasticity identical but strength and plasticity mismatched materials in linear elastic regime are solved. The solution is well validated.

Support received from the School of Mechanical Building Sciences, VIT University, Vellore, India during the course of this work is gratefully acknowledged.