PSI - Issue 70

Varsha S et al. / Procedia Structural Integrity 70 (2025) 51–58

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where J is the Jacobean function

( , ) = ( , ) = 0∑ ∑ ≤ 2=0 ≤ 2 =0 .( − ) + cos ( )sin ( )

(22)

1 ( ) = √2 2 √ ∫ ∫ 2 ( ) = √2 2 √ ∫ ∫

√ ( − ) ( − ) 2 + 2 −2 ( − ) cos( − ) = 0 = 0 √ ( − ) ( − ) 2 + 2 −2 ( − ) cos( − ) ( ) = 0 =0

(23)

(24)

3 ( ) = √2 2 √ + ∫ ∫ √ ( − ) + +1 ( − ) 2 + 2 −2 ( − ) cos( − ) = 0 cos ( )sin ( ) = 0 4 ( ) = √2 2 √ + ∫ ∫ √ ( − ) + +1 ( − ) 2 + 2 −2 ( − ) cos( − ) ( )cos ( )sin ( ) = 0 = 0

(25)

(26)

The variation of I 1 , I 2 , I 3 , and I 4 in terms of θ is shown in a novel function as: ( )= + sin( ) cos ( ) 1+ sin( )+ sin 2 ( )

(27)

For i =1,2,3 and 4 .The presented function contains four coefficients which are identified as Bi, Ci, Di, and Ei respectively. When both m and n equal zero so that the functions of I11 and I2 become equivalent to those of I3 and I4 then the related coefficients remain equivalent. Table 1 contains the complete list of coefficients from I3 and I4 operational functions. ( ) = ∑ ∑ ≤ 2=0 ≤ 2 =0 . [ 3 + 3 sin( ) cos ( ) 1+ 3 sin( )+ 3 sin 2 ( ) + 4 + 4 sin( ) cos ( ) 1+ 4 sin( )+ 4 sin 2 ( ) ( − 21++ 2 1s s i 1 i n n + ( ( ) 1 ) c + s o i s n 1 ( s ( i ) n )2( ) 1+ 2 sin( )+ 2 sin2( ) )] (28) Substituting Eqs. (12), (23), and (28) into Eq. (15) provides an explicit formulation for calculating SIF at various points along the semi-circular crack front under a given two-dimensional Mode I stress distribution.

Fig. 1. Boundary Conditions and Loading Setup for the FEM Model

Fig.1 shows the finite element model configuration implemented in ANSYS to compute stress intensity factors