On the Nonlinear Eshelby Inclusion Problem and its Isomorphic Growth Limit

Abstract

In the late 1950's, Eshelby's linear solutions for the deformation field inside an ellipsoidal inclusion and, subsequently, the infinite matrix in which it is embedded were published. The solutions' ability to capture the behavior of an orthotropically symmetric shaped inclusion made it invaluable in efforts to understand the behavior of defects within, and the micromechanics of, metals and other stiff materials throughout the rest of the 20th century. Over half a century later, we wish to understand the analogous effects of microstructure on the behavior of soft materials; both organic and synthetic; but in order to do so, we must venture beyond the linear limit, far into the nonlinear regime. However, no solutions to these analogous problems currently exist for non-spherical inclusions. In this work, we present an accurate semi-inverse solution for the elastic field in an isotropically growing spheroidal inclusion embedded in an infinite matrix, both made of the same incompressible neo-Hookean material. We also investigate the behavior of such an inclusion as it grows infinitely large, demonstrating the existence of a non-spherical asymptotic shape and an associated asymptotic pressure. We call this the isomorphic limit, and the associated pressure the isomorphic pressure.

Figures (15)

• Figure 1: The infinitely large body $\mathcal{B}$ in the reference (bottom left), transformed (top center), and deformed (bottom right) configurations. In each configuration the region occupied by the inclusion (denoted by the subscript "$I$") and matrix (denoted by the subscript "$M$") are indicated. The inclusion has the three semi-axes $A_j$ in the reference configuration, $\lambda^*A_j$ in the transformed configuration, and $a_j$ in the deformed configuration aligned with the right-handed Cartesian coordinate system ($\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$). The red region in the transformed configuration denotes the incompatibility present between the inclusion and matrix. The mappings $\boldsymbol{\chi}^e$, and $\boldsymbol{\chi}$ and their gradients $\mathbf{F}^*$, $\mathbf{F}^e$, and $\mathbf{F}$ represent the transformation, elastic part of the deformation, and total deformation respectively.
• Figure 2: (a) A visualization of the difference between a confocal set of ellipses, characterized by the aspect ratio function $\hat{\Phi}^C(\Lambda)$, and the generalized set of ellipses, characterized by the aspect ratio function $\hat{\Phi}^G(\Lambda)$ with $\Phi_\infty=2$. The gray region represents an undeformed inclusion with $\Phi_0=5$. (b) The aspect ratio functions $\hat{\Phi}^C(\Lambda)$ and $\hat{\Phi}^G(\Lambda)$ versus $\Lambda$ of (a).
• Figure 3: The Deformed aspect ratio of the inclusion $\varphi_0$ (a) and volume averaged dimensionless inclusion pressure $\bar{p}_I/\mu$ (b) vs. the volume growth ratio ($J^*_I-1$) of an incompressible and , neo-Hookean, isotropically growing prolate spheroidal inclusion-matrix system with undeformed aspect ratio $\Phi_0 = 5$. The dashed grey lines in (a) and (b) denote the isomorphic aspect ratio of the inclusion-matrix system $\varphi_\propto = 1.788$ and the dimensionless isomorphic inclusion pressure of the inclusion-matrix system $p_\propto/\mu= 2.802$ respectively (as derived in Section \ref{['sec:4.3']}). The numerical labels in (a) denote the volume growth ratios of the deformation magnitude plots in Fig. \ref{['fig:5_fullfield']}.
• Figure 4: Full field plots of the normalized displacement magnitude $|\mathbf{u}|/A_1$ in the normalized reference coordinates $\mathbf{X}/A_1$ of an incompressible, neo-Hookean, isotropically growing prolate spheroidal inclusion-matrix system with undeformed aspect ratio $\Phi_0 = 5$. The volume growth ratios in each set of subplots are are (a) $J^*_I = 1.01$, (b) $J^*_I = 10$, (c) $J^*_I = 100$, and (d) $J^*_I = 26424$. Note that the colorbar scale and zoom level vary between each set of plots.
• Figure 5: The deformed aspect ratio of the inclusion $\varphi_0$ (a) and volume averaged dimensionless inclusion pressure $\bar{p}_I/\mu$ (b) vs. the volume growth ratio ($J^*_I-1$) of an incompressible and , neo-Hookean, isotropically growing prolate spheroidal inclusion-matrix system with undeformed aspect ratio $\Phi_0 = 1/5$. The dashed grey lines in (a) and (b) denote the isomorphic aspect ratio of the inclusion-matrix system $\varphi_\propto = 0.471$ and the dimensionless isomorphic inclusion pressure of the inclusion-matrix system $p_\propto/\mu= 3.045$ respectively (as derived in Section \ref{['sec:4.3']}). The numerical labels in (a) denote the volume growth ratios of the deformation magnitude plots in Fig. \ref{['fig:0.2_fullfield']}.