A Wrinkling Model for General Hyperelastic Materials based on Tension Field Theory

TL;DR

The paper develops an implicit, isogeometric wrinkling model for general hyperelastic membranes based on tension field theory. Building on Nakashino's linear-elastic TFT, it modifies the deformation gradient to obtain wrinkling strains and stresses, deriving a hyperelastic wrinkling stress $\mathbf{S}'(\mathbf{E})$ that depends on a wrinkling strain term $\mathbf{E}_W$ and a scalar $\gamma(\mathbf{E})$ determined from uniaxial tension constraints via a root-finding problem in the wrinkle orientation $\vartheta$. The approach integrates within a Kirchhoff-Love membrane framework, using Newton–Raphson and dynamic relaxation solvers, and is implemented with Voigt notation for efficient computation of the modified material tensor $\mathcal{C}'$. Validation against isogeometric Kirchhoff-Love shell references across four benchmarks demonstrates accurate mean wrinkling predictions with substantially reduced degrees of freedom, highlighting the method's potential for efficient large-strain wrinkling analysis of hyperelastic membranes. The work also discusses convergence behavior, implementation challenges with tension-field derivatives, and pathways for future enhancements such as tighter coupling of the tension field in the Jacobian and arc-length strategies.

Abstract

Wrinkling is the phenomenon of out-of-plane deformation patterns in thin walled structures, as a result of a local compressive (internal) loads in combination with a large membrane stiffness and a small but non-zero bending stiffness. Numerical modelling typically involves thin shell formulations. As the mesh resolution depends on the wrinkle wave lengths, the analysis can become computationally expensive for shorter ones. Implicitly modeling the wrinkles using a modified kinematic or constitutive relationship based on a taut, slack or wrinkled state derived from a so-called tension field, a simplification is introduced in order to reduce computational efforts. However, this model was restricted to linear elastic material models in previous works. Aiming to develop an implicit isogeometric wrinkling model for large strain and hyperelastic material applications, a modified deformation gradient has been assumed, which can be used for any strain energy density formulation. The model is an extension of a previously published model for linear elastic material behaviour and is generalized to other types of discretisation as well. The extension for hyperelastic materials requires the derivative of the material tensor, which can be computed numerically or derived analytically. The presented model relies on a combination of dynamic relaxation and a Newton-Raphson solver, because of divergence in early Newton-Raphson iterations as a result of a changing tension field, which is not included in the stress tensor variation. Using four benchmarks, the model performance is evaluated. Convergence with the expected order for Newton-Raphson iterations has been observed, provided a fixed tension field. The model accurately approximates the mean surface of a wrinkled membrane with a reduced number of degrees of freedom in comparison to a shell solution.

Figures (14)

• Figure 1: Set-up for the uniaxial tension benchmark problem. In the figure on the left, the filled geometry represents the deformed configuration, and the dashed line indicates the undeformed geometry. The load $p$ indicates a line load acting on the undeformed geometry. The table on the right provides the parameter values for the specific benchmark problem for Neo-Hookean (NH) and Mooney-Rivlin (MR) materials.
• Figure 2: Results for the uniaxial tension benchmark problem from \ref{['fig:benchmarks_UAT_setup']}. The top plots depict the stretch $\lambda$ versus the applied load $p$. The bottom plots present the convergence of the relative residual norm $\Vert \vb*{R}_i \Vert / \Vert \vb*{R}_0 \Vert$ for a load step with $p=1.0\:[\text{MPa}]$, with the triangle indicating second-order convergence. The results are depicted for Neo-Hookean (NH) and Mooney-Rivlin (MR) materials with the parameters from \ref{['fig:benchmarks_UAT_setup']}. The lines indicate the results without a tension field theory (TFT) modification, and the markers indicate the results including the modification proposed in this paper.
• Figure 3: Problem definition for a square membrane with diagonal length $1.2\:[\text{m}]$ subject to pressure $p$. The membrane is modelled using in-plane symmetry boundary conditions on $\Gamma_1$ and $\Gamma_4$. Furthermore, the sides $\Gamma_2$ and $\Gamma_3$ have restricted $z$-displacement. The square membrane has a Neo-Hookean material model with the parameters provided in the table on the right.
• Figure 4: Results for the square membrane subject to a pressure load from \ref{['fig:benchmarks_pillow_setup']}. (\ref{['fig:benchmarks_pillow_deformed']}) represents the deformed shape from a Kirchhoff--Love shell simulation, providing wrinkles. (\ref{['fig:benchmarks_pillow_tensionfield']}) provides the deformed shape from a membrane simulation with the proposed tension field theory modification scheme, together with the tension field. (\ref{['fig:benchmarks_pillow_contours']}) provides the contours of the deformation for both models for different numbers of elements and degrees.
• Figure 5: Problem definition for an annulus with inner radius $R_i$ and outer radius $R_o$ subject to a vertical translation $u_z$ and a rotation $\varTheta$ on the inner boundary $\Gamma_i$ while being fixed on the outer boundary $\Gamma_o$. The annulus has a Neo-Hookean material model with the parameters provided in the table on the right.

Theorems & Definitions (4)

• Remark 1
• Example 1
• Remark 2
• Example 2