On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

Golgoon, Ashkan; Yavari, Arash

doi:10.1007/s10659-016-9620-3

Cited by 18 publications

(5 citation statements)

References 43 publications

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“…Similar constructions using nontrivial material manifolds with the explicit dependence of the material metric on the type of anelasticity were discussed in[Yavari, 2010, Ozakin and Yavari, 2010, Yavari and Goriely, 2013, Sadik and Yavari, 2015, Golgoon et al, 2016, Golgoon and Yavari, 2017.5 All the symbolic computations in this paper were performed usingMathematica [Wolfram Research, 2016]. 6 Note that N = e −ω R (R) E R is the unit vector defining the material preferred direction, where E R = ∂ ∂R is a radial basis vector for T X B such that E R , E R G = G RR .…”

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confidence: 84%

Nonlinear Elastic Inclusions in Anisotropic Solids

Golgoon

Yavari

2017

J Elast

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In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.

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confidence: 84%

Nonlinear Elastic Inclusions in Anisotropic Solids

Golgoon

Yavari

2017

J Elast

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“…Sass et al [9] first studied the elastic fields of a cuboidal inclusion, for which they offered a Fourier series solution; this problem was later reworked by Faivre [10], who achieved the first explicit solution to the elastic fields of a parallelepipedal inclusion in isotropic elasticity. Further geometries have been studied both in the context of homogeneous eigenstrains and isotropic elasticity and for anisotropic materials [11], with different elastic moduli [12], including solutions for the fields of square plate inclusions [13], cuboids [14], cylinders [15], rod-like inclusions [16], concave inclusions [17], toroids [18], generalized formulations for arbitrary closed geometries [19] and polyhedra [6,20].…”

Section: Introductionmentioning

confidence: 99%

The multipolar elastic fields of ellipsoidal and polytopal plastic inclusions

Gurrutxaga-Lerma

2023

Proc. R. Soc. A.

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The article offers a general formulation for the multipolar field expansion of an inclusion. The multipolar moments of arbitrary order for the ellipsoidal inclusion are derived, showing known features of their elastic fields in both the far and near field. Using integer point enumeration theory, the same formulation is extended to all polytopal inclusions, which serve to model faceted inclusions found in, e.g. igneous and metamorphic rocks, particle reinforced composite materials, or as intermetallic precipitates in alloys. A general algorithm for the calculation of the multipolar moments of polytopes of any number of vertices is derived. A number of examples for tetrahedral, cuboidal and dodecahedral inclusions are given. It is shown that the parity of the axial moment function of a polytopal inclusion mirrors the symmetry of the inclusion. This represents the main properties of their elastic fields, including the long- and near-field decay. If the inclusion is symmetric with respect to a given axes, the fields will decay with 1 / r 2 in the far field and 1 / r 4 in the near field. If symmetry along that direction is lost, as is expected in most real inclusions, the decay rate progresses with 1 / r 2 but the near field with 1 / r 3 .

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“…In the case of some special constitutive equations one can mention [6, 22-24, 35, 36]. There are several more recent works that use geometric techniques [13,14,16,43,[45][46][47].…”

Section: Introductionmentioning

confidence: 99%

Universality in Anisotropic Linear Anelasticity

Yavari

Goriely

2022

J Elast

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In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify the universal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of second-order linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear second-order PDEs with certain arbitrary forcing terms that depend on the symmetry class.

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On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

Cited by 18 publications

References 43 publications

Nonlinear Elastic Inclusions in Anisotropic Solids

Nonlinear Elastic Inclusions in Anisotropic Solids

The multipolar elastic fields of ellipsoidal and polytopal plastic inclusions

Universality in Anisotropic Linear Anelasticity

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