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Geometric continuum mechanics

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Abstract

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.

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Notes

  1. In differential geometry these are respectively denoted by low and high asterisks ∗;∗ [51]. This standard notation leads however to consider too many similar stars in the geometric sky, i.e. push, pull, duality, Hodge operator.

  2. The time-function is assumed to be a projection, i.e. surjective with a surjective tangent map.

  3. An immersion is a injective map whose tangent map is injective too.

  4. The introduction of the geometric notions of connection and parallel transport can be avoided by restricting oneself to transport by translation in Euclid space.

  5. The proof is due to Fréchet, von Neumann, Jordan, see [59]. Validity of parallelogram identity is an assumption stronger than the one of validity of Pythagoras’ theorem.

  6. For bodies of maximal dimension, the VPP is a proved theorem [27].

  7. The Whitney product of tensor bundles with projection \(\boldsymbol{\pi}_{\mathbb{M},\mathbb{N}}\in \mathrm{C}^{1}(\mathbb{N};\mathbb{M}) \) and \(\boldsymbol{\pi}_{\mathbb{M},\mathbb{H}}\in \mathrm{C}^{1}(\mathbb{H};\mathbb{M}) \) over the same base manifold \(\mathbb{M}\), is the product bundle fulfilling the condition \(\mathbb{N}\times_{\mathbb{M}}\mathbb{H}:=\{(\mathbf{n},\mathbf{h})\in \mathbb{N}\times \mathbb{H}\mid \boldsymbol{\pi}_{\mathbb{M},\mathbb{N}}(\mathbf{n})= \boldsymbol{\pi}_{\mathbb{M},\mathbb{H}}(\mathbf{h})\}\) [27].

  8. CFI substitutes the notion of Material Frame Indifference stated in [54] by the equality in which the change of constitutive operator due to the change of observer is not taken into account [31].

  9. An hypo-elastic model was introduced by Truesdell in [52] with a different definition. The new formulation of rate elasticity was first contributed in [29].

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Acknowledgements

The geometric formulation of CM has been the subject of a general lecture delivered by the first author at the AIMETA XX Congress, in Bologna, Italy, on Sept. 13, 2011. The kind invitation of the president of AIMETA and of the president of the local organizing committee is gratefully acknowledged. Useful hints and precious comments by an anonymous reviewer are also gratefully acknowledged.

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  1. Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, 80125, Naples, Italy

    Giovanni Romano, Raffaele Barretta & Marina Diaco

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  1. Giovanni Romano
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Correspondence to Giovanni Romano.

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Romano, G., Barretta, R. & Diaco, M. Geometric continuum mechanics. Meccanica 49, 111–133 (2014). https://doi.org/10.1007/s11012-013-9777-9

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