Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 477, Issue 2245, Pages -Publisher
ROYAL SOC
DOI: 10.1098/rspa.2020.0462
Keywords
Riemannian geometry; anelasticity; intermediate configuration; multiplicative decomposition
Categories
Funding
- NSF [CMMI 1561578]
- ARO [W911NF-18-1-0003]
- Engineering and Physical Sciences Research Council [EP/R020205/1]
- EPSRC [EP/R020205/1] Funding Source: UKRI
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This study derives a sufficient condition for the existence of global intermediate configurations by starting from a multiplicative decomposition of the deformation gradient, showing that these global configurations are unique up to isometry. For radially symmetric deformations, isometric embeddings of intermediate configurations are constructed and residual stress fields are computed explicitly.
A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.
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