A Geometric Theory of Surface Elasticity and Anelasticity

TL;DR

This work develops a unified geometric theory of elasticity and anelasticity for bodies with elastic material surfaces by embedding both bulk and surface physics in a Riemannian framework. Surface mechanics are derived consistently from the ambient material metric and the surface metric, with stresses obtained as derivatives of energy w.r.t. the respective metrics, and eigenstrains incorporated via material metrics. The Lagrange–d’Alembert principle yields coupled bulk and surface balance laws, including a generalized Laplace-type equation for surface traction, residual stresses, and a complete nonlinear spherical cavity example that demonstrates stiffening/softening due to surface and fluid eigenstrains. The approach generalizes to anisotropic surfaces, arbitrary geometries, and time-dependent processes (growth/remodeling), offering a principled foundation for surface effects in soft solids and composites. Overall, the paper provides a rigorous, invariant framework that integrates surface elasticity and anelasticity within differential geometry, eliminating ad hoc assumptions and enabling systematic constitutive modeling.

Abstract

In this paper we formulate a geometric theory of elasticity and anelasticity for bodies containing material surfaces with their own elastic energies and distributed surface eigenstrains. Bulk elasticity is written in the language of Riemannian geometry, and the framework is extended to material surfaces by using the differential geometry of hypersurfaces in Riemannian manifolds. Within this setting, surface kinematics, surface strain measures, surface material metric, and the induced second fundamental form follow naturally from the embedding of the material surface in the material manifold. The classical theory of surface elasticity of Gurtin and Murdoch (1975) is revisited and reformulated in this geometric framework, and then extended to anelastic bodies with anelastic material surfaces. Constitutive equations for isotropic and anisotropic material surfaces are formulated systematically, and bulk and surface anelasticity are introduced by replacing the elastic metrics with their anelastic counterparts. The balance laws are derived variationally using the Lagrange-d'Alembert principle. These include the bulk balance of linear momentum together with the surface balance of linear momentum, whose normal component gives a generalized Laplace's law. As an application, we obtain the complete solution for a spherical incompressible isotropic solid ball containing a cavity filled with a compressible hyperelastic fluid, where the cavity boundary is an anelastic material surface with distributed surface eigenstrains. The analytical and numerical results quantify the effects of surface and fluid eigenstrains on the pressure-stretch response and residual stress.

Figures (9)

• Figure 1: The commutative diagram of projection and surface projection maps.
• Figure 2: Schematic representation of a body $\mathcal{B}$ containing $m$ material surfaces. The material surfaces form an abstract $2$-manifold $\mathsf{S}=\bigsqcup_{i=1}^{m}\mathsf{S}_i$, and their inclusion in the body manifold is denoted by $\mathtt{S}=\iota_{\mathsf{S}}(\mathsf{S})\subset \mathcal{B}$, where $\iota_{\mathsf{S}}:\mathsf{S}\hookrightarrow\mathcal{B}$ is the inclusion map. The bulk region is $\mathring{\mathcal{B}}=\mathcal{B}\setminus\mathtt{S}$, with connected components $\mathring{\mathcal{B}}=\bigsqcup_{i=1}^{m+1}\mathcal{B}_i$. For $i=1,\cdots,m$, the boundary of the $i$th inclusion is $\partial\mathcal{B}_i=\mathtt{S}_i$, whereas $\partial\mathcal{B}_{m+1}=\partial_o\mathcal{B}\,\sqcup\,\bigsqcup_{i=1}^{m}(-\mathtt{S}_i)$, where $-\mathtt{S}_i$ denotes the orientation-reversed surface. A material surface $\mathtt{S}_i$ moves under the deformation $\varphi_t$ to the deformed surface $\mathtt{s}_i=\varphi_t(\mathtt{S}_i)$ in the ambient space.
• Figure 3: Motion of a material surface $\mathsf{S}$, which is an abstract $2$-manifold. Its inclusion in the material manifold $\mathtt{S}=\iota_{\mathsf{S}}\mathsf{S}\subset\mathcal{B}$ is the undeformed material surface. For a fixed $X\in \mathtt{S}$, $\varphi(X,t)$ is a curve in the ambient space. Tangent to this curve at $x=\varphi(X,t)$ is the material velocity $\mathbf{V}(X,t)$, which has parallel $\mathbf{V}_\parallel(X,t)$ and normal $\mathbf{V}_\perp(X,t)$ components with respect to the deformed material surface $\mathtt{s}$.
• Figure 4: The schematic relationship between tangential and surface deformation gradients and the tangents of inclusion maps.
• Figure 5: The solid red curve shows the normalized equilibrium cavity radius $x^*=r_i^*/R_i$ as a function of $\Omega_s$ for $\alpha=3.0$, $\xi=1.0$, and $\eta=2\xi$. The dotted curve is the zero-stress radius $e^{-\Omega_s}R_i/R_i=e^{-\Omega_s}$. When $\Omega_s=0$ and $\hat{p}_o=0$, one has $x=1$ as an equilibrium solution. For $\Omega_s>0$ the material surface prefers the smaller natural radius $e^{-\Omega_s}R_i$ while the surrounding bulk is stress-free at $r(R)=R$. The relaxed radius results from balancing these effects and is always larger than the zero-stress radius.

Theorems & Definitions (27)

• Lemma 2.1
• proof
• Definition 2.2: Projection Map
• Definition 2.3: Surface Projection Map
• Proposition 2.4
• Remark 2.5
• Remark 2.6
• Definition 3.1
• Definition 3.2
• Remark 3.3