The Approximation Of Some Variables And The Comparison Of Elasto-plastic Large Deformation Definitions In Continuum Mechanics

The physical quantity in continuum mechanics is usually expressed as a tensor function. So the research of the tensor functions and their derivatives is a very important issue in continuum mechanics and computational mechanics. The principal axis representations and abstract representations are two kinds of representations of tensor functions. An important character of the abstract representations is coordinate-free, which makes the derivations clear and formulations concise. Therefore, it has been the interest of many scientists working in the fields of theroretical and applied mechanics. But the representation of many tensor functions and their derivatives can't be applied in engineering directly, so the approximate expressions for tensor functions and their derivatives which can be applied in engineering calculations get more and more attentions.With the development of industrial technology, more precise estimates of mechanical responses of materials are required and more effective and realistic constitutive relations are needed. Then, many different elasto-plastic large deformation constitutive definitions appeared. Facing so many elasto-plastic large deformation definitions, the differences of different definitions are widely concerned.The approximate expressions of tensor functions and their derivatives and the difference of three elasto-plastic large deformation definitions are studied in this paper. The main work and achievements are as follows1) The isotropic expression of square root tensors, logarithmic strain tensors and exponential tensor (which are three commonly met tensors in continuum mechanics) are got by Taylor series expanding, the reminders of expansions are analysed and the expanding points with the minimum errors are got.2) Using the conclusions drawn above, the approximate expressions of right stretch tensor U, rotation tensor R, Hencky logarithmic strain tensor H and exponential tensor are deduced. These approximate expressions not only have simple representations and high precision, but also calculate much faster than exact expressions.3) The approximate expressions of derivatives of right stretch tensor U, Hencky logarithmic strain tensor H and exponential tensor on right Cauchy-Green strain tensor are deduced. These approximate expressions also have simple representations, high precisions and faster calculating rates. And do not need to consider the eigenvalues of independent variable tensors weather equal or not.4) Comparing the Simo-Ortiz definition, Moran-Ortiz-Shih definition and the large deformation definition generalized from small elasto-plastic deformation, the quantitative relationships of the three definitions are given by applying tensor function to a simple shear deformation.