Invariant-based models for incompressible isotropic hyperelasticity are typically formulated as functions of the first and second invariants, $W = W(\bar{I}_1, \bar{I}_2)$. A widely used class of models employs separable representations of the form $W(\bar{I}_1, \bar{I}_2) = W_1(\bar{I}_1) + W_2(\bar{I}_2)$, which enable efficient calibration and straightforward enforcement of modeling constraints. However, this decomposition implicitly restricts the coupling between the invariants and may limit the achievable accuracy for complex material responses. Fully coupled data-driven approaches overcome this limitation but often require nonlinear optimization and large parameter sets. In this contribution, we propose a compact alternative: a bivariate B-spline surface defined directly on the physically admissible invariant domain. By aligning the approximation space with physically realizable states, all model parameters contribute meaningfully to the constitutive response. We utilize homogeneous deformation modes to perform a calibration directly from analytical stress relations, eliminating the need for finite element model updating. Owing to the linear dependence of the spline representation on its coefficients, the resulting parameter identification problem reduces to a constrained linear least-squares problem. This enables fast, robust, and initialization-independent calibration, which makes parameter identification practically instantaneous. The results demonstrate that the proposed model improves accuracy compared to separable approaches while requiring only mild regularization in weakly sampled regions. The combination of computational efficiency and the linear structure of a highly expressive spline surface makes the approach particularly attractive for applications requiring repeated calibration, such as uncertainty quantification or interactive material characterization.

翻译：基于不变量的不可压缩各向同性超弹性模型通常表述为第一和第二不变量的函数，即$W = W(\bar{I}_1, \bar{I}_2)$。广受欢迎的一类模型采用可分离表示形式$W(\bar{I}_1, \bar{I}_2) = W_1(\bar{I}_1) + W_2(\bar{I}_2)$，这种形式能够实现高效标定并直接施加建模约束。然而，该分解隐式地限制了不变量之间的耦合关系，可能制约复杂材料响应的精度。全耦合数据驱动方法虽能克服这一局限，但通常需要非线性优化和大量参数。本文提出一种紧凑替代方案：直接在物理允许的不变量域上定义双变量B样条曲面。通过使逼近空间与物理可实现状态对齐，所有模型参数均能对构型响应产生有意义贡献。我们利用均匀变形模式从解析应力关系直接进行标定，无需有限元模型更新。由于样条表示与其系数呈线性相关，所得参数识别问题简化为约束线性最小二乘问题，从而实现快速、鲁棒且与初始化无关的标定，使参数识别近乎瞬时完成。结果表明，与可分离方法相比，所提模型在仅需对弱采样区域进行轻度正则化处理的情况下提高了精度。计算效率与高表达力样条曲面的线性结构相结合，使得该方法特别适用于需要重复标定的应用场景，如不确定性量化或交互式材料表征。