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Non-metric Connection and Metric Anomalies in Materially Uniform Elastic Solids

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Abstract

Metric anomalies arising from a distribution of point defects (intrinsic interstitials, vacancies, point stacking faults), thermal deformation, biological growth, etc. are well known sources of material inhomogeneity and internal stress. By emphasizing the geometric nature of such anomalies we seek their representations for materially uniform crystalline elastic solids. In particular, we introduce a quasi-plastic deformation framework where the multiplicative decomposition of the total deformation gradient into an elastic and a plastic deformation is established such that the plastic deformation is further decomposed multiplicatively in terms of a deformation due to dislocations and another due to metric anomalies. We discuss our work in the context of quasi-plastic strain formulation and Weyl geometry. We also derive a general form of metric anomalies which yield a zero stress field in the absence of other inhomogeneities and any external sources of stress.

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Notes

  1. The domain of the partial function \(\hat{W}(\cdot,\boldsymbol{X})\), for \(\boldsymbol{X}\in\mathcal{B}\), is customarily assumed to be \(\mathit{InvLin}^{+}\), the space where the deformation gradients reside [27, 36], which, under the Principle of Material Frame Indifference, gets restricted to its subset \(\mathit{Sym}^{+}\). In presence of certain material inhomogeneities (e.g., disclinations), a well-defined element in \(\mathit{InvLin}^{+}\) may not exist to appear in the constitutive function. Our treatment bypasses this limitation, as it is always guaranteed that a well-defined element in \(\mathit{Sym}^{+}\) exists to appear in \(\hat{W}(\cdot,\boldsymbol{X})\) as an argument. This well-defined element could be, in our context, any of the standard measures of strain.

  2. Apart from the point symmetry group \(\mathcal{G}\), which essentially describes rotational symmetries, the material structure presently under consideration possesses, due to its expanse in the Euclidean 3-space, spatial translational symmetries.

  3. A \(\mathit{Unim}\)-principal bundle is a principal fibre bundle whose structure group is the group \(\mathit{Unim}\).

  4. Within the realm of the classical solutions of the PDEs that we are considering here, the existence and uniqueness theorems of Cauchy-Kowalevski and Holmgren do no extend to the class of smooth functions which are not analytic; in this context, we would like to refer to the well-known Lewy’s example that demonstrates a linear PDE with smooth coefficients which has no solution [23].

  5. This is with reference to Weingarten’s classical theorem [37] in linear elasticity and the subsequent construction of elementary dislocations and disclinations by Volterra [35] as the fundamental line singularities in a linear elastic solid. The same construction also holds in non-linear elasticity, cf. [44, Chap. 1] and [40].

  6. Recall, from our discussion of quasi-plastic deformation in the last section, the tensor \(\boldsymbol{H}\) which is now identified with \(\operatorname{Grad} \boldsymbol{\chi}\) in the absence of material inhomogeneities.

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  1. Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, 208016, India

    Ayan Roychowdhury & Anurag Gupta

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Roychowdhury, A., Gupta, A. Non-metric Connection and Metric Anomalies in Materially Uniform Elastic Solids. J Elast 126, 1–26 (2017). https://doi.org/10.1007/s10659-016-9578-1

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