Geometric continuum mechanics

Geometric Continuum Mechanics (GCM) is a new formulation of Continuum Mechanics (CM) based on the requirement of Geometric Naturality (GN). According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory. The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm (GP) stating that push-pull transformations are the natural comparison tools for material fields. This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport. The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues. As a consequence, the notion of Material Frame Indifference (MFI) is corrected to the new Constitutive Frame Invariance (CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains. Geometrically consistent notions of Rate Elasticity (RE) and Rate Elasto-Visco-Plasticity (REVP) are formulated and consistent relevant computational methods are designed.