Shakedown of unidirectional composites

The shakedown problem for a composite lamina made of an elastic-plastic matrix and elastic cylindrical fibers is studied. The plastic deformation modes of the lamina are reviewed, and it is concluded that significant shakedown effects can be caused only by the I₁ = 1/2(T₁₁ + T₂₂) and I₂ = T₃₃ components of the remotely applied stress field which are symmetric about the axis x₃ of the fiber; T₁₁ and T₂₂ are the normal composite stresses in the transverse plane. It is shown that the I₁I₂ stress system is needed also to represent thermal loads caused by a uniform change of temperature in the composite.Two methods for evaluation of shakedown limits in the I₁I₂-plane are described. First, the classical approach involving the determination of parametric families of self-stress fields and the solution of mathematical programming problems is used. Results are presented for selected B-Al, Be-Al, B-Ti and B-Mg composites.In the second method, the shakedown problem is related to the recently developed kinematic hardening rules for fibrous composites. It is shown that the composite will shake down for any loading program within a prescribed domain in the I₁I₂-plane, providing that the domain can be contained within a translated initial yield surface. This approach leads to a closed-form evaluation of shakedown limits for any arbitrary combination of mechanical and thermal variable cyclic loads in fibrous composites with temperaturedependent matrix yield strengths.The relationship between shakedown and fatigue in metal matrix composites is discussed.