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Journal of Elasticity

Erschienen in:

10.01.2017

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

verfasst von: Ashkan Golgoon, Arash Yavari

Erschienen in: Journal of Elasticity | Ausgabe 1/2017

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Abstract

In this paper we analyze the stress field of a solid torus made of an incompressible isotropic solid with a toroidal inclusion that is concentric with the solid torus and has a uniform distribution of pure dilatational finite eigenstrains. We use a perturbation analysis and calculate the residual stresses to the first order in the thinness ratio (the ratio of the radius of the generating circle and the overall radius of the solid torus). In particular, we show that the stress field inside the inclusion is not uniform. This is in contrast with the corresponding results for infinitely-long and finite circular cylindrical bars and spherical balls with cylindrical and spherical inclusions, respectively. We also show that for a solid torus of any size made of an incompressible linear elastic solid with an inclusion with uniform (infinitesimal) pure dilatational eigenstrains the stress inside the inclusion is not uniform.

Vorheriger Artikel Waves in Nonlocal Elastic Solid with Voids

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Similar constructions have been discussed in [25, 34, 36, 42‐44, 46, 48] to address problems in growth mechanics, thermoelasticity, and the nonlinear mechanics of distributed defects.

We use Mathematica [41] for the symbolic computations.

Note that \(\sigma^{r\phi }_{ ( 0 ) }=0\) and \(\hat{\sigma }^{\theta \theta }=\hat{\sigma }^{\theta \theta }_{ ( 0 ) }+ \hat{\sigma }^{\theta \theta }_{ ( 1 ) }+\mathcal{O}( \varepsilon^{2})\).

For the sake of simplicity of calculations, here we do not consider the dependence of \(W\) on \(X\), which would be needed in the case of an inhomogeneity. Instead, we model inhomogeneities by assuming different energy functions in different regions of the body.

Here, using (3.32), we find the zero-order pressure field in the inclusion and the matrix separately. Alternatively, the discontinuous eigenstrain distribution (3.34) may be treated as a step function defined in the entire region, and the pressure field is found from (3.32). In both cases, continuity of the traction vector on the inclusion-matrix interface is needed to find the unknown constants.

Solutions with a similar form were discussed in [15, 17].

Note that one can easily verify that (3.55) and (3.56) are indeed the homogeneous solutions of (3.45b) for \(R\in [ R_{i},R_{o} ] \), and therefore, using the power series method is justified.

The dilogarithm function is defined as: \(\text{Li}_{2}(z)=\sum\limits_{n=1}^{\infty }\frac{z ^{n}}{n^{2}}=-\int_{0}^{z}\ln (1-\zeta )\frac{d\zeta }{\zeta }\) for \(|z|<1\).

Note that \(|u(R)|<1\) for \(R\in [R_{i},R_{o}]\), and therefore, \(\text{Li}_{2}(\pm u(R))\) is well-defined.

Note that the system of equations for the unknown constants is nonlinear in \(k\) and linear with respect to the other constants.

Note that positive and negative eccentricity values correspond to the inclusion moving to the left and right relative to the matrix, respectively.

This immediately follows from the symmetry of the problem for radially-symmetric eigenstrain distributions.

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Titel: On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion
verfasst von: Ashkan Golgoon
Arash Yavari
Publikationsdatum: 10.01.2017
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 1/2017
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-016-9620-3

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