By G. Fowles, G. Cassiday

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**Example text**

Example of a thermo-elastic problem in plane strain state tions may take if non-linear eﬀects appear. The practical relevance of hyperelasticity is due to the fact that it allows for a good description of large deformations of rubber-like materials. As an example, Fig. 15 shows the corresponding deformation of a hexahedral rubber block under compression. Fig. 15. Deformation of a rubber block under compression A hyperelastic material is characterized by the fact that the stresses can be expressed as derivatives of a strain energy density function W with respect to the components Fij = ∂xi /∂aj of the deformation gradient tensor F: Tij = Tij (F) = ∂W (F) .

Tb ✻ ✻ ✻ ✒ ✻ ✒ tb ✒ ✻ ✒ ✒ x2 ✻ x1 ✒✲x 3 ✠ f ✠ tb ✠ ❄ ✠ ❄ ❄ ❄tb Fig. 12. Disk in plane strain state The plane strain state is characterized by ε13 = ε23 = ε33 = 0 . The normal stress in x3 -direction T33 does not necessarily vanish in this case. The essential diﬀerence from a disk in plane stress state is the diﬀerent strainstress relation. 28) must be used. 49) results. The deformation of a thin plate, which is subjected to a vertical load (see Fig. 13), can under certain conditions also be formulated as a twodimensional problem.

As can be seen from the above equations, in the case of hyperelasticity one is faced with a rather complex non-linear system of partial diﬀerential equations together with usually also non-linear boundary conditions. 5 Fluid Mechanical Problems The general task in ﬂuid mechanical problems is to characterize the ﬂow behavior of liquids or gases, possibly with additional consideration of heat and species transport processes. For the description of ﬂuid ﬂows usually the Eulerian formulation is employed, because one is usually interested in the properties of the ﬂow at certain locations in the ﬂow domain.