Keywords: boundary element method, contact, creep, time-dependent material behaviour, quadratic elements.

The boundary element (BE) method has become an established numerical technique for linear and non-linear stress analysis problems. In elastic contact problems, the BE method, with its surface-only modelling capability, offers several advantages over the FE method in solving contact problems. A higher resolution of the BE surface stresses is obtained and the surface tractions are treated as independent solution variables which are calculated to the same degree of accuracy as the displacements. The contact relationships are directly incorporated into the BE system of equations to satisfy the equilibrium and compatibility relationships, without the need for gap or interface elements across the contact interface. Previous applications of the BE method have demonstrated its suitability for elastic contact problems, see for example [1,2].

The BE advantage of surface-only modelling is lost when modelling material non-linearity, such as plasticity and creep. This is due to the presence of an additional volume integral which is required to represent the non-linear volume effects. Therefore, it becomes necessary to model both the surface and the interior volume in non-linear BE formulations, making the BE meshes similar in appearance to FE meshes.

In creep problems, the BE formulation is usually based on an initial strain approach using two creep power laws, time hardening and strain hardening, based on the Prandtl-Reuss flow rule (see, for example, [3]). To construct the BE system of linear algebraic equations, the surface and volume integrations are performed numerically. The surface is discretised into 'boundary' elements, while the volume is discretised into 'domain' cells. The BE solution matrix, however, is only of size 2N x 2N for 2D problems or 3N x 3N for 3D problems, where N is the total number of boundary nodes. However, unlike the FE stiffness matrices, the BE solution matrices are fully populated.

Since in most practical contact problems, the contact area is unknown in advance, it is necessary to use iterative schemes and/or load incrementation to arrive at the final contact solution. In addition, a suitable automatic time incrementation scheme is required to model the creep behaviour. To incorporate the contact iterations within the time-dependent BE creep formulation, it is assumed that the contact and creep processes are separable. The contact algorithms can be incorporated within the time-stepping creep algorithms as follows:

(i)

At creep time = 0, the contact algorithms are used to arrive at the elastic contact solutions.

(ii)

The creep strain rates are calculated using time hardening or strain hardening assumptions.

(iii)

For a small time step, the BE equations for creep are solved and the stress and strain rates are obtained at all nodes.

(iv)

Convergence is checked, i.e. the change in strains or stresses being below a specified tolerance. If the solution is not converged, the current time step is reduced by a suitable factor and the analysis repeated until convergence is achieved.

(v)

The variables are updated using a creep time-marching scheme, such as the Euler method.

(vi)

The next time step is selected by multiplying the previous time step by a constant factor based on the strain and stress change in that time step.

(vii)

After a few time steps, the contact algorithms are used to update the contact status.

(viii)

Steps (ii)-(vii) are repeated until the final creep time is reached.

This paper presents an assessment of the application of the BE formulation to two-dimensional contact problems under creep conditions. The boundary of the domain is discretised using 3-node quadratic surface elements, while the interior domain is discretised using 8-node quadratic quadrilateral cells. Two contact examples under creep conditions are presented to demonstrate the validity and accuracy of the BE solutions; a flat punch and a cylinder on deformable foundations. The BE solutions are compared to the corresponding FE solutions.

1

Andersson, T. and Allan-Persson, B.G., "The boundary element method applied to two-dimensional contact problems", Progress in Boundary Element Methods - Vol. 2, Brebbia, C.A. (Editor), Pentech Press, London, 136-157, 1983.

2

Olukoko, O.A., Becker, A.A. and Fenner, R.T., "A review of three alternative approaches to modelling frictional contact problems using the boundary element method", Proc. Royal Society of London, Series A, 444, 37-51, 1994. doi:10.1098/rspa.1994.0003

3

Chandenduang, C. and Becker, A.A., "Boundary element formulation for two-dimensional creep problems using isoparametric quadratic elements", Computers & Structures, 81, 1611-1619, 2003. doi:10.1016/S0045-7949(03)00182-2

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