Keywords: creep, BEM, singular elements, cracks, metals, viscoplasticity.

In the search for an accurate, yet generalized, computational method for evaluating singular crack tip stress and strain fields, the singular element approach in conjunction with boundary element method (BEM) has been properly used in various fracture mechanics applications. Several researchers have contributed to this field: Blandford et al [1] was the first who introduced the traction singular quarter- point boundary element approach in combination with a multi-domain formulation to the solution of both symmetrical and non-symmetrical crack problems. Thereafter, this approach has been extensively used in the application of the boundary element method to two- and three- dimensional crack problems. An extension of the quarter-point element technique was used by Hantschel et al [2] who made an attempt to model crack tip fields arising in two-dimensional elastoplastic cracked panels by introducing some special singular boundary elements which took into account the HRR singularity field (Hutchinson [3]; Rice and Rosengren [4] near the crack tip.

In the present paper, a creep strain-traction singular element (CR-STSE) is implemented in the direct boundary element formulation to evaluate the time- dependent inelastic stress and strain singularity field distribution involved in creeping cracked two-dimensional plates. This CR-STSE is produced by using the technique presented in Maiti [5] to simulate power type singularities around crack tips arising in various fracture problems of linear elasticity.

Then, after applying a boundary nodal point collocation procedure to the discretized versions of the resulted boundary integral equations an algebraic system of equations is produced. However, the vector of creep strain rates is known at any time through the constitutive equations and the stress rates. Half of the total number of components in the algebraic system of equations are prescribed through the boundary condition while the other half are unknowns.

Thus, by taking into account that the only existed strains at time step are elastic, the thermal and initial stresses and displacements can be obtained from the solution of the corresponding elastic problem. By the use of resulted integral equations, the displacement and stress rates can be obtained at time step while the rates of change of the nonelastic strains can be computed from constitutive equations. Thus, the initial rates of all the relevant variables are now known and their values at a new time can be obtained by integrating forward in time. The rates are then obtained at time and so on, and finally the time histories of all the variables can be computed. The energy rate contour integral is then evaluated numerically as a function of time through the computed values of stress, strain and displacement rates from the above mention algorithm and the use of integral equation for different paths around the crack tip. Each path is decomposed into sufficient number of straight segments and the integral over each segment is obtained by Gaussian quadrature (ten Gauss points).

Numerical examples are presented for cracked plates and the results obtained by the present BEM methodology are further compared with available finite element solutions. The creep constitutive model used in the numerical calculations is the Norton power law creep model but any other creep constitutive model having similar mathematical structure can be easily implemented in the proposed algorithm

1

Blandford, G.E., Ingraffea, A.R., Ligget J.A., "Two dimensional stress intensity factor computations using the boundary element method", Int. J. Num. Meth Engng, 17, pp. 387-404, 1981. doi:10.1002/nme.1620170308

2

Hantschel, T., Busch, M., Kuna M., Maschke, H.G., "Solution of elastic- plastic crack problems by an advanced boundary element method". in "Numerical Methods in Fracture Mechanics", A.R. Luxmoore and D.R.J. Owen (Eds.), Pineridge Press, Swansea, pp. 29-40, 1990.

3

Hutchinson, J.W., "Singular behavior at the end of a tensile crack in a hardening material", J. Mech Phys Solids, 16, pp. 13-31, 1968. doi:10.1016/0022-5096(68)90014-8

4

Rice, J.R., Rosengren, G.F., "Plane strain deformation near a crack tip in a power-law hardening material", J. Mech Phys Solids, 16, pp. 1-12, 1968. doi:10.1016/0022-5096(68)90013-6

5

Maiti, S.K., "Finite element for variable order singularities based on the displacement formulation", Int. J. Num Meth Engng, 33, pp. 1955-1974, 1992. doi:10.1002/nme.1620330912

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