Nonlinear morphoelastic energy based theory for stimuli responsive elastic shells

TL;DR

This work develops a nonlinear morphoelastic theory for naturally curved elastic shells by deriving a reduced two-dimensional energy from a fully nonlinear 3D formulation. The authors implement a multiplicative decomposition ${\bf F} = {\bf F}_e \, \bar{\bf F}$ with an intermediate, potentially incompatible metric, and perform a through-thickness expansion that retains curvature-coupled $Z^2$ terms, yielding two 2D energies for incompressible Neo-Hookean and compressible Ciarlet–Geymonat materials. The reduced energies incorporate through-thickness stimuli via functions $\phi$ and $\bar{\phi}$ and depend on geometric nonlinearities, enabling accurate bending–stretching coupling and large deformations; the framework is applied to spherical cap and sphere eversion, illustrating how compressibility and curvature influence snap-through and morphology. The approach provides a rigorous basis for analyzing active and biological shell morphing under non-mechanical stimuli and is poised to advance studies of budding and vesiculation in curved solid-like membranes.

Abstract

Large deformations play a central role in the shape transformations of slender active and biological structures. A classical example is the eversion of the Volvox embryo, which demonstrates the need for shell theories that can describe large strains, rotations, and the presence of incompatible stimuli. In this work, a reduced two-dimensional morphoelastic energy is derived from a fully nonlinear three-dimensional formulation. The resulting model describes the mechanics of naturally curved shells subjected to non-elastic stimuli acting through the thickness, thereby extending previous morphoelastic theories developed for flat plates to curved geometries. Two representative constitutive laws, corresponding to incompressible Neo-Hookean and compressible Ciarlet-Geymonat materials, are examined to highlight the influence of both geometric and constitutive nonlinearities. The theory is applied to the eversion of open and closed spherical shells and to vesiculation processes in biological systems. The results clarify how compressibility, curvature, and through-the-thickness kinematics govern snap-through and global deformation, extending classical morphoelastic shell models. The framework provides a consistent basis for analyzing large deformations in elastic and biological shells driven by non-mechanical stimuli.

Figures (7)

• Figure 1: Multiplicative decomposition of the deformation gradient tensor. The total deformation gradient tensor $\mathbf{F}$ maps the reference configuration $\mathcal{B}_0$ to the current configuration $\mathcal{B}$. The presence of a non-mechanical stimulus is modelled via a deformation gradient tensor $\bar{\mathbf{F}}$ that maps the reference configuration $\mathcal{B}_0$ to the intermediate configuration $\bar{\mathcal{B}}$ that is not necessarily compatible, but where it is still possible to identify the tangent vectors associated with each point. The compatibility is guaranteed by the application of the map given by $\mathbf{F}_e$ from the intermediate to the current configuration from which it is possible to write the energy.
• Figure 2: Snapping of a spherical cap. Natural curvature at snapping, normalized by the theoretical prediction from Pezzulla2018PRL, as a function of the vertical displacement at the free end $w$, normalized by the radius of the shell $R_0$, for $\Theta=\pi/6$ and $\lambda=0.01,0.1,1,10$ when (a) $\rho_1=1$ and $\rho_2=0$ and (b) $\rho_1$ and $\rho_2$ fixed by eqs. \ref{['eq:a']} and \ref{['eq:b']}.
• Figure 3: Dimensionless imposed curvature at snapping, $\kappa_s R_0$ in terms of the rescaled shallowness $\theta\sqrt{R_0/h_0}$. The solid line represents the theoretical prediction from Pezzulla2018PRL, valid for an incompressible Koiter shell model. The colored curves represent the result of our model for different values of $\lambda=0.01,0.1,1,10$ when $\rho_1$ and $\rho_2$ are fixed by eqs. \ref{['eq:a']} and \ref{['eq:b']}.
• Figure 4: Homogeneous dimensionless displacement $w_R/R_0$ as a function of the dimensionless imposed curvature $\kappa R_0$, in the case of plane strain conditions. For $\lambda=100>1$ the shell deflates while for $\lambda=0.1<1$ the shell inflates.
• Figure 5: Morphologies obtained setting $\eta_a=0$ and (a) $\eta_b=0.5$, (b) $\eta_b=1$ and (c) $\eta_b=2$ for $\bar{\kappa}=300$ (panel (a) and (b)) and $\bar{\kappa}=128$ (panel (c)) . Meridional $\epsilon_{11}$ (continuous curves) and azimuthal (dashed curves) $\epsilon_{22}$ strain for a morphoelastic cap with $\eta_a=0$ and (d) $\eta_b=0.5$, (e) $\eta_b=1$ and (f) $\eta_b=2$ for $\bar{\kappa}=300$ (panel (d) and (e)) and $\bar{\kappa}=128$ (panel (f)).