Keywords: singular velocity field, upper bound method, limit load, highly undermatched welded joints.

This present paper is concerned with an approach to determine a semi-analytical limit load for a class of highly undermatched welded joints. The definition for highly undermatched welded joints is that the weld is much softer than the base material. Therefore, plastic deformation is confined within the weld whereas the base material is elastic. A consequence of this feature of flow pattern is that the yield stress of the base material is not involved in the formulation of the boundary value problem. Since elastic properties have no effect on the limit load, the base material can be regarded as rigid. The approach proposed is based on two principles. First, it is known that the real velocity field is singular in the vicinity of maximum shear stress surfaces [1]. In particular, the equivalent strain rate involved in the formulation of the upper bound theorem of plasticity follows an inverse square root rule near such surfaces and, therefore, approaches infinity. In the case of highly undermatched welded joint surfaces the maximum shear stress coincides with the bi-material interface. It is advantageous to account for the singular behaviour of the real velocity fields in kinematically admissible velocity fields. Second, the thickness of the weld is usually much smaller than other geometric dimensions of specimens. It is therefore natural to assume a linear through-thickness distribution on the velocity component normal to the weld.

Using the aforementioned assumptions concerning the velocity field the expression for the upper bound limit load for tensile panels is derived. Assuming one of the simplest functions accounting for the real singular velocity field in the vicinity of the bi-material interface the upper bound limit load is determined. This solution is compared with an accurate numerical solution found with the use of the slip-line technique. It is shown that that the difference between these two solutions is negligible. It is also shown how to extend the approach proposed in this paper to a wider class of welded joints for which no slip-line solutions are available.

1

S. Alexandrov, O. Richmond, "Singular plastic flow fields near surfaces of maximum friction stress", International Journal of Non-Linear Mechanics, 36, 1-11, 2001. doi:10.1016/S0020-7462(99)00075-X

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