Geometry of finite deformations and time-incremental analysis

• Solid mechanics as a simple Lagrangian system.
• Space of deformation tensors Sym+ as a configuration manifold.
• Riemannian geometry of Sym+ and time-linearization of deformation processes.
• Evolution equation of Lie type for finite deformations.
• Riemannian geometry of Sym+ and time-integration of deformation processes.

In connection with the origin of computational mechanics and consequent progress of incremental methods, a fundamental problem came up even in solid mechanics – namely how to correctly time-linearize and time-integrate deformation processes within finite deformations. Contrary to small deformations (actually infinitesimal), which represent a correction of an initial configuration in terms of tensor fields and so a description by means of a linear vector space of all symmetric matrices sym(3,R)sym(3,R) is well-fitting, a situation with finite deformations is rather more complicated. In fact, while the position and shape of a deformed body take place in the usual three-dimensional Euclidean space R3R3, a corresponding progress of deformation tensor makes up a trajectory in Sym+(3,R)Sym+(3,R) – a negatively curved Riemannian symmetric manifold. Since this space is not a linear vector space, we cannot simply employ tools from the theory of small deformations, but in order to analyze deformation processes correctly, we have to resort to the corresponding tools from the differential geometry and Lie group theory which are capable of handling the very geometric nature of this space. The paper first briefly recalls a common approach to solid mechanics and then its formulation as a simple Lagrangian system with configuration space Sym+(3,R)Sym+(3,R). After a detailed exposition of the geometry of the configuration space, we finally sum up its consequences for the time-incremental analysis, resulting in clear and unambiguous conclusions.