Shakedown limit theorems for frictional contact on a linear elastic body

• Cyclic loading of an elastic body in contact with a Coulomb frictional surface is considered.
• Extensions of classical shakedown theorems are derived for the non-associated flow conditions of frictional slip.
• Kinematic and static bounds are derived that separate conditions of cyclic slip and shakedown.
• A programming method is described based on the Linear Matching Method for evaluating the kinematic bound.

The paper considers the problem of a body composed of a linear elastic material in contact with a planar rigid surface with an inter-surface coefficient of Coulomb friction μ. The body is subjected to a cyclic history of loading, λPi(xi,t) where λ denotes a scalar multiplier. The objective is to assess the conditions when movement occurs between the elastic body and the surface. The problem has a close analogy with classical plasticity, where shakedown and limit load bounds exist. However, existing plasticity theory is not generally applicable to frictional slip as it obeys a non-associated flow rule. In this paper upper and lower bound shakedown theorems are derived in terms of the Coulomb coefficient of friction μ. It is shown that optimal kinematic and static bounds do not coincide. This implies that for a prescribed λPi(xi,t) there are ranges of μ for which shakedown definitely occurs and for which shakedown definitely does not occur, independent of the state of slip at the beginning and end of the cycle. However there exists an intermediate range of μ for which it is not possible to say whether shakedown or ratchetting occurs without detailed knowledge of the slip displacements at the beginning and end of the cycle of loading. This observation accords with simulations reported by Flecek R.C., Hills D.A., Barber J.B. and Dini D., (2015).A programming method for the shakedown limit is developed, based on the Linear Matching Method. The method is illustrated by a simple example. The theory derived in this paper paves the way for a new theory of limit and shakedown analysis for structures and materials with a non-associated flow rule.