Application Of Amendable T.L Method To The FEA Of Rubber Parts

In recent years we have seen a tremendous growth in both theory and application of finite element method. The gradual improvement of the computing abilities makes many problems of structures solvable, which are unable to be analyzed in the past. This paper with the beginning of studying the mechanical nature of rubber, comprehensively and concretely describes geometrical nonlinear and dual nonlinear combining physical nonlinear with geometrical nonlinear. With respect to geometrical nonlinear, this paper present a different method from traditional T.L and U.L, which seems like a shortcut between linear elastic little deformation finite element method and nonlinear finite element method. Meanwhile, this paper also makes a general discussion on the expansion of rubber nonlinear constitutive law, and this part is seldom seen in other papers by possible reason of some abstract conception involved in it such as tensor, tensor derivative of tensor etc.. We think it is undoubted that our work can provide lots of valuable references to the programmers of rubber design and computing, meanwhile, also can give some enlightenment to those beginners of nonlinear continuum mechanical and finite element method. In nonlinear continuum mechanical theories, in order to describe the large reformation of continuum, it generally must introduce two coordinates and two configurations. The two coordinates refers separately to Lagrange Coordinate and Euler Coordinate; while the two configurations are separately Practical Configuration and Reference Configuration. Lagrange Coordinate, also called intrinsic coordinate, is a kind of coordinate that coordinate value is unchanged while basic vector is variant, so the coordinate value of material point is constant in Lagrange Coordinate, whose use is to make label for each material point, which looks like that person's name or ID card is unchanged wherever. On the contrary, Euler coordinate, also called spatial coordinate which, is a sort of coordinate that basic vector is unchanged while coordinate value is variant, so the coordinate value is changed in Euler coordinate, whose use is to depict the spatial point location of each material point at different time. Practical Configuration is an unknown status of deformation at t time, while Reference Configuration, compared with practical configuration, is any known configuration before t time, and the one at t=0 is called Initial Configuration. This paper assumes that Initial Configuration is Reference Configuration, and Euler Coordinate is Cartesian Coordinate, under this case, Lagrange coordinate overlaps with Euler spatial coordinate at initial status, that means the Lagrange coordinate value of material point is the same as Euler coordinate one of that at t=0,therefore, the displacement of material point at t time is just the difference of the two coordinate values, which we call Global Displacement, in chapter two, we discuss the method of solving the global displacement of discrete continuum unit nodes. The basic idea of traditional both T.L method and U.L method is that the process of continuum deformation is taken as a continuously variant process with time, and under the same assumption that the deformation at t is known and the solved object is displacement of the unit node from t to t+Δt .The difference of them is that T.L takes initial configuration as reference configuration and U.L makes the t configuration as reference configuration. Like T.L, the Global Displacement method we provided also takes process of continuum deformation as a continuously variant process with time, and makes the t configuration as reference configuration, however unlike traditional methods, it directly takes the unit node replacement at t as the solved object, which is the very important difference from other known methods, it is just because the difference that we deduce completely different unit rigidity matrix from the two traditional methods, and our rigidity matrix deduced by Global Displacement method is a function of replacement of unit node at t, and it is also a symmetrized matrix. Because rigidity matrix deduced by T.L and U.L methods is a function of replacement of unit node from t to t+Δt, it is an asymmetrized matrix. You can see more description about this in chapter two and six in detail. There is an outstanding strength about the new method that it makes easier to understand pure geometrical nonlinear finite element method, and more smooth transition from little deformation to large deformation. After obtaining global replacement, we immediately get Green strain tensor of any material point in continuum at t, then get the second class Piola-Kirchhoff stress according to the relationship of the second class Piola-Kirchhoff stress assumed in geometrical nonlinear and Green strain tensor, at last get the Cauchy stress tensor, that means the stress tensor of each material point at t, according to the relationship of the second class Piola-Kirchhoff stress and Cauchy stress tensor. All of above is about geometrical nonlinear. However, undoubtedly geometrical nonlinear is not enough to describe the constitute law of material, for instance, the...