Intrinsic incremental evolution of hypoelastic discrete mechanical systems

This paper provides an explicit geometric and coordinate-free formulation of incremental discrete mechanics in the framework of potentially non integrable hypoelasticity. First, the general framework is developed in order to tackle hypoelasticity as an Ehresmann connection on the cotangent bundle T*M. Two types of incremental evolutions may be distinguished, the weak or integrable incremental evolutions and the strong or non integrable incremental evolutions, according to the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle TT*M is fully used in order to get the geometric counterpart kappa of the so-called tangent stiffness matrix. Subject to specific conditions in TT*M, the incremental evolution is then a well-founded question. An hypoelastic four-grains granular system illustrates in detail these general results.

Automated summary

This paper presents an explicit and coordinate-free formulation of incremental discrete mechanics in potentially non-integrable hypoelasticity. It develops a general framework that treats hypoelasticity as an Ehresmann connection on the cotangent bundle T*M, distinguishing between weak or integrable incremental evolutions and strong or non-integrable incremental evolutions based on the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle TT*M is utilized to obtain the geometric counterpart kappa of the tangent stiffness matrix. The validity of the incremental evolution is established under specific conditions in TT*M, and a four-grains hypoelastic granular system is used to illustrate the general results in detail.