Nonlinear morphoelastic energy based theory for stimuli responsive elastic shells

1. Problem Statement

The paper addresses the mechanical modeling of slender, naturally curved active and biological structures (such as shells) undergoing large deformations driven by non-mechanical stimuli (e.g., swelling, growth, or chemical activation).

Existing theories face specific limitations:

• Classical Koiter Shell Theory: While popular, it assumes small strains and decouples stretching and bending. It often relies on the plane-stress hypothesis and fails to capture the intrinsic coupling between geometry, stretching, and bending in the presence of large rotations or incompatible stimuli.
• Previous Morphoelastic Models: Many existing morphoelastic models are derived for flat plates or rely on linearized assumptions. They often neglect the through-the-thickness kinematics or assume specific constitutive behaviors that do not hold for compressible materials or naturally curved geometries.
• The Gap: There is a need for a fully nonlinear, reduced 2D theory that can handle:
  • Naturally curved (non-flat) reference geometries.
  • Large strains and rotations.
  • Incompatible stimuli acting through the thickness.
  • Both incompressible and compressible material behaviors.

2. Methodology

The authors derive a reduced two-dimensional energy functional starting from a fully nonlinear three-dimensional morphoelastic formulation.

A. 3D Framework

• Multiplicative Decomposition: The deformation gradient $\mathbf{F}$ is decomposed into an elastic part $\mathbf{F}^e$ and a growth/stimulus part $\bar{\mathbf{F}}$ (mapping the reference configuration to a stress-free, but potentially incompatible, intermediate configuration).
• Constitutive Models: Two specific hyperelastic materials are examined:
  1. Incompressible Neo-Hookean: Enforced via a Lagrange multiplier (pressure).
  2. Compressible Ciarlet–Geymonat: A model that naturally handles compressibility without requiring a separate pressure variable.
• Stimulus Representation: The non-elastic stimulus is introduced phenomenologically via a target metric $\bar{\mathbf{G}}$ in the intermediate configuration. This metric is allowed to be incompatible (non-Euclidean).

B. Dimensional Reduction (3D to 2D)

The reduction is performed using asymptotic expansion based on the slenderness parameter ($h_0 \ll l_0$, where $h_0$ is thickness and $l_0$ is in-plane length).

• Kinematics: The authors adopt a generalized Kirchhoff-Love type hypothesis but relax the plane-strain assumption. They introduce a thickness function $\phi(\eta^\alpha, Z)$ to describe the deformation through the thickness, allowing for thickness changes.
• Metric Expansion: The covariant and contravariant metrics for the reference, current, and intermediate configurations are expanded in powers of the thickness coordinate $Z$.
• Key Innovation: Unlike linearized theories that neglect $O(Z^2)$ terms in the metric expansion (as they contribute to $O(h^5)$ energy terms), this fully nonlinear derivation shows that $O(Z^2)$ terms contribute to the energy at $O(h^3)$. Therefore, the third fundamental form of the surface must be retained to ensure consistency.
• Invariant Expansion: The invariants of the elastic right Cauchy–Green tensor ($I_1, I_2, I_3$) are expanded in terms of the in-plane and through-the-thickness functions ($\phi$ and $\bar{\phi}$).

C. Derivation of Reduced Energies

The authors derive explicit 2D energy functionals ($\Psi_{2D}$) consisting of stretching ($O(h_0)$) and bending ($O(h_0^3)$) terms.

• Neo-Hookean: The incompressibility constraint is handled variationally, leading to specific relations for the thickness expansion coefficients.
• Ciarlet–Geymonat: The compressible case allows for the determination of thickness coefficients by minimizing the energy, revealing how compressibility affects the coupling between stretching and bending.
• Linearization Check: The authors demonstrate that linearizing their reduced energy does not simply recover the classical morphoelastic Koiter energy. Instead, it retains a stretching-bending coupling term ($w_{gnl}$) that is non-zero for curved shells, a feature absent in classical linear theories.

3. Key Contributions

1. Fully Nonlinear Reduced Theory: Derivation of a consistent 2D energy functional for naturally curved morphoelastic shells that retains geometric and constitutive nonlinearities without ad-hoc kinematic assumptions.
2. Through-the-Thickness Kinematics: Explicit modeling of the thickness deformation ($\phi$) and the stimulus through the thickness ($\bar{\phi}$), allowing for independent control of in-plane metric/curvature changes and thickness changes.
3. Coupling Mechanisms: Identification and mathematical formulation of the intrinsic coupling between stretching and bending in curved shells, which persists even in the limit of small strains if the geometry is curved.
4. Constitutive Versatility: Application of the theory to both incompressible (Neo-Hookean) and compressible (Ciarlet–Geymonat) materials, highlighting how compressibility alters the mechanical response.

4. Results and Applications

The theory is validated and applied to three representative problems:

A. Eversion of a Spherical Cap (Snap-Through)

• Observation: The model predicts the critical curvature stimulus required for snap-through (eversion).
• Compressibility Effect: For incompressible materials, snap-through is robust. However, for compressible materials, the required curvature stimulus increases. Compressibility partially accommodates the geometric frustration, delaying or suppressing the snap-through event.
• Validation: The model recovers the results of Pezzulla et al. [25] for incompressible shells but provides new insights for compressible cases where classical theories might fail.

B. Eversion of a Closed Sphere

• Radial Displacement: The model predicts that a homogeneous curvature stimulus induces a radial contraction or expansion depending on the ratio of bulk to shear modulus ($\lambda$).
  • If $\lambda > 1$ (stiffer bulk), the sphere contracts.
  • If $\lambda < 1$ (highly compressible), the sphere expands.
  • At $\lambda = 1$, the effects cancel out.
• Significance: This highlights that ignoring through-the-thickness behavior or compressibility leads to incorrect predictions of global deformation in closed shells.

C. Budding and Vesiculation (Localized Stimuli)

• Simulation: The model simulates localized curvature and metric growth on a spherical cap to mimic biological budding.
• Morphology:
  • Isotropic curvature stimulus: Produces a symmetric dome.
  • Anisotropic curvature: Produces elongated bulges or ring-like structures (resembling clathrin-coated pits).
  • Metric Stimuli: Adding in-plane area growth (metric stimulus) significantly alters the strain distribution and final shape, creating protrusions or rings depending on the anisotropy ratios.
• Strain Analysis: The model reveals complex strain patterns, including compressive hoop strains outside the stimulated region (potential precursors to fracture) and localized tensile strains driving the protrusion.

5. Significance

• Biological Relevance: The framework provides a rigorous mechanical basis for understanding complex shape transformations in biological systems (e.g., Volvox embryo eversion, cell budding, vesiculation) where large deformations, curvature, and material compressibility are critical.
• Design of Active Materials: The theory offers a predictive tool for designing soft robotic shells and 4D-printed structures that rely on stimuli-responsive actuation, particularly for curved geometries where linear theories are insufficient.
• Theoretical Advancement: By correcting the order of magnitude of neglected terms in the metric expansion, the paper resolves inconsistencies in previous shell theories, ensuring that geometric nonlinearities are correctly captured even for "small strain" problems if the geometry is curved.

In summary, Taffetani and Pezzulla provide a unified, nonlinear framework that bridges the gap between 3D elasticity and 2D shell mechanics for active materials, offering new insights into how compressibility and curvature govern the morphing of biological and synthetic shells.