PSI - Issue 14

Manish Kumar et al. / Procedia Structural Integrity 14 (2019) 839–848

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Manish Kumar et. al/ Structural Integrity Procedia 00 (2018) 000–000

where, s  is the symmetric gradient operator. A set of discrete equations is obtained using strain displacement relation (  ε u ), constitutive relation (  σ Dε ) and displacement approximation ( u = Nv and δ δ  u N v ) as, (6) where, D is the constitutive matrix, v is the nodal displacement, N is the vector of shape functions and B is the matrix having derivatives of shape functions. By solving the discrete equations (Eq. (6)) simultaneously, the displacement field is obtained. The stress and strain fields are evaluated from the displacement field using constitutive and strain displacement relations. After the elasto-plastic analysis, converged continuum state is used to initiate the creep analysis. The combinations of different functions of stress state, strain state or temperature are used to describe the creep deformations via creep laws like Power law and Theta projection model, etc. Incremental effective creep strain for a particular time step is calculated from creep law using converged continuum state of last time step. A pseudo body force (creep force, c F ) vector based on the creep strain field is assembled to account the effect of the creep as, 0 T B DBu T N b T N t  e T e V V dV dV d         During creep analysis, the set of discrete equations i.e. Eq. (6) is modified as, 0 T T T T c e e T e V V V dV dV d dV           B DBu N b N t B Dε  (8) The incremental displacement and strain fields are obtained similarly as in elasto-plastic analysis, but for trial stress field some modifications are made to capture the effect of relaxation and redistribution of stress field (Hsu, 1986) as, (9) For obtained trial stress field, the yielding criterion is checked if a particular integration point yields then plastic treatment is done to calculate stress field and plastic strain otherwise, the trial stress field will act as stress field for that particular integration point. If the solution is converged then proceed for next time step, otherwise the newly obtained continuum state is used to estimate the creep strain using creep law. This process is repeated until convergence is achieved. If the solution starts diverging, then the time step is reduced and the entire process of creep strain and stress evaluation is repeated with the reduced time step. Terminate the simulation when the time step reaches to a very small value. This usually occurs in the tertiary stage. 2.2. XFEM implementation To capture the effect of discontinuity two type of enrichment functions are added to the standard finite element approximation in XFEM (Kumar et al. , 2018b). An XFEM displacement approximation for two-dimensional problem (Moës et al ., 1999) is written as, ℎ ( ) = ∑ � ( ) � + � ��� ∑ � [ ( ) − ( � )] � ������������� Shiftedjump enrichment � � ��� + ∑ � ∑ � � ( ) − � ( � )� � � � ����� ����� Shifted tip enrichment � ��� � � ��� ���������������������������������������� XFEMapproximation (10) where, n is the set of all nodes in the mesh, i u is a nodal displacement vector of the standard finite element, p n is the set of nodes associated with those elements which are completely cut by the crack, q n is the set of nodes associated with those elements which are partially cut by the crack. (x) Η is Heaviside function, (x) j  are the crack tip enrichment function, i α is the enriched nodal degree of freedom associated with Heaviside function (x) Η , j i β ( trial c   σ D ε ε ) T c   F B Dε c e V dV (7)