Simon Wiesheier

Simon Wiesheier, Paul Steinmann, Miguel Angel Moreno-Mateos

This work revisits the recently proposed data-adaptive viscoelasticity (DAVIS) framework, a spline-based formulation of finite viscoelasticity within the generalized standard materials setting. DAVIS enables a data-driven representation of equilibrium and non-equilibrium constitutive functions while retaining thermodynamic consistency and supporting parameter identification via finite element model updating. The present contribution focuses on improving the robustness and identifiability of non-equilibrium branches in generalized Maxwell-type models. To this end, two extensions of the original formulation are introduced. First, the spline representation is reformulated in terms of curvature-based variables, which is especially convenient to enforce monotonicity and convexity constraints by construction through a smooth parameter mapping. Second, the adaptation of interpolation domains is decoupled from the inner parameter identification by means of a staggered, block-alternating strategy: spline coefficients are optimized for fixed domain endpoints, while the endpoints are updated in an outer loop based on smooth statistics of sampled invariants. This separation alleviates an inherent scaling ambiguity between interpolation domains and spline coefficients that can impair conditioning in viscoelastic inverse problems. The underlying constitutive model remains the finite strain viscoelasticity framework of Reese and Govindjee. The proposed identification strategy is assessed for homogeneous uniaxial loading-unloading tests, which facilitates the study of identifiability and robustness of non-equilibrium branches.

Miguel Angel Moreno-Mateos, Simon Wiesheier, Paul Steinmann et al.

Highly compressible solids, such as foams, exhibit complex responses, including pronounced tension-compression asymmetry. Capturing such behaviors within unified hyperelastic frameworks remains challenging. Invariant-based hyperelastic models are commonly identified from standard tests such as homogeneous uniaxial tension/compression and simple shear, implicitly assuming a unique energy representation. Here we show that this assumption is fundamentally violated and that, oftentimes, the choice of which term should prevail is just a matter of taste. Using spline-based strain-energy density functions as a data-adaptive tool and stress-strain experimental data for elastomeric foams, we expose this non-uniqueness, often hidden in low-parameter formulations. Our framework captures the volumetric deformation of ultra-light foams used in racing shoes using homogeneous experimental data from tension, compression, and shear. We formulate an overly rich ansatz of separable and non-separable energies in the ($\bar{I}_1$, $\bar{I}_2$, $J$) space à la Money-Rivlin. These constructs, defined by multiplicative decompositions, resemble classical invariant-based models while generalizing them to a data-driven spline representation. This serves two purposes: (i) to capture the response under complex volumetric deformation modes and (ii) to allow non-uniqueness in the identification problem to emerge naturally. We find that a coupling term between isochoric and volumetric deformation, such as $Ψ(\bar{I}_1,J)$ or $Ψ(\bar{I}_2,J)$, is essential and that additional coupling terms help but are not fully necessary; rather, they pronounce the non-uniqueness. As a consequence, different models may be indistinguishable on available data. Importantly, these challenges are not specific to splines but extend to traditional and neural network-based models.

Simon Wiesheier, Miguel Angel Moreno-Mateos, Paul Steinmann

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.