Path-following of growth mechanics in thin biological tissues by efficient 3-DOF and 7-DOF isogeometric shell models with exact tangent operators

We present an isogeometric shell framework for the simulation of growth mechanics in thin biological tissues. Two alternative shell kinematic models are considered: a 7-DOF solid-shell model and a 3-DOF Kirchhoff–Love model. Growth at large strains is modeled via the multiplicative decomposition of the deformation gradient and both compressible and incompressible hyperelastic materials. The 7-DOF model naturally supports fully three-dimensional constitutive laws, while the 3-DOF model is generalized to growth problems through the introduction of tailored plane-stress enforcement strategies, numerical for compressible materials and analytical for incompressible ones. An isogeometric discretization based on cubic C2 NURBS, combined with a patch-wise reduced quadrature, provides high-order accuracy with approximately 2.25 surface integration points per element. A key contribution of this work is the closed-form derivation of all growth-induced tangent operators, enabling an efficient and robust arc-length path-following strategy for both shell theories, particularly suited to highly nonlinear regimes and instabilities. Numerical applications, with a focus on fascinating growth-induced buckling phenomena, demonstrate the robustness and versatility of the proposed simulation tool. The two variants of the 3-DOF model produce identical predictions for typical tissues properties. They remain in excellent agreement with those of the 7-DOF model and of a reference solid model in all the considered benchmarks.