PSI - Issue 72
Alexander Kamenskikh et al. / Procedia Structural Integrity 72 (2025) 252–259
255
u
u x
u
0 0
0 0
0 0
3
1
2
x
x
1
1
1
u x
u
u x
0
0 0
0 0
0
3
1
2
x
2
2
2
u x
u x
u x
0 0
0 0
0 0
3
1
2
3
3
3
S
(7)
,
2 x x u u u
1 u u x x
2 u u x x
3
0
0
0
3
1
2
2
1
2
1
1
u
u x
1 u u x x
u
0
0
0
3
3
1
2
2
x x
3 x u u x x
3
1
3
1
3
1
1 u u x x
2 u u x x
0
0
0
3
1
2
3
2
3
2
3
2
ε is the vector containing components of the linear part of strains, S is the matrix of linear multipliers, s is the vector of strains components, 0 0 , s X X are the matrices, whose elements are found from the solution of the corresponding static problem using the following expressions:
1 2
T * 1 0 S с ε X s 0 1
T S сS s X s . 1 0 0 0 1 s
,
The numerical implementation of the above mathematical formulation is based on the finite element method. After applying the known procedures, instead of equation (5) we obtain: 2 0. M K K K u (8)
The typical mass matrix M and stiffness matrix K for each finite element are formed as follows:
T MNN KBcB K VXV T T d , V d , S d , V e
0
e
e
e
V
V
V
T K V X V B сWV VWсB VWсWV T T T T T d d d d , V V V V
e
0 s
0
0
0
0
e
e
e
e
V
V
V
V
where N is the shape functions for the displacement vector, B is the matrix linking the linear part of the strain vector * with nodal values of displacements, V is the matrix linking the strains vector s with the nodal values of displacements; W 0 is the matrix consisting of strains components similar to expression (7). The finite element formulation of the nonlinear static problem, from which the initial stresses and strains are found, is written as: 1 1 2 2 , K K K u f (9)
where the typical vector of the right-hand part of an individual finite element is computed as follows:
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