Application of Dual Kriging to the Construction of a General Phenomenological Material Law for Shape Memory Alloys

A large number of applications could benefit from the remarkable properties of shape memory alloys; but up to now, a relatively limited number have been brought to the market. This can be attributed, in part, to the lack of numerical tools dedicated to the computer aided design of shape memory devices. The development of a general material law is the first important step before reliable design calculations can be carried out. This paper presents a new phenomenological constitutive law based on dual kriging, which is a powerful mathematical tool used here as an interpolation method. The model was initially developed for a particular and limited purpose, namely, to simulate the macroscopic behavior of simple shape memory devices. From a few isothermal experimental forcedeflection curves at different temperatures, two surfaces are constructed which describe the loading and unloading behavior of the device.The response of the material subjected to complex thermomechanical loadings is calculated by successive interpolations on these surfaces via dual kriging. For hysteretic subcycles, the response is calculated through the volume delimited by the two surfaces in such a way that any recursive thermomechanical subcycles can be simulated. This methodology yields a uniaxial material law for shape memory alloys that includes in a single formulation superelasticity, pseudo-plasticity and shape memory effect. Preliminary validations on a set of simple examples show the potential of this approach.