In the present work, neural networks are applied to formulate parametrized hyperelastic constitutive models. The models fulfill all common mechanical conditions of hyperelasticity by construction. In particular, partially input convex neural network (pICNN) architectures are applied based on feed-forward neural networks. Receiving two different sets of input arguments, pICNNs are convex in one of them, while for the other, they represent arbitrary relationships which are not necessarily convex. In this way, the model can fulfill convexity conditions stemming from mechanical considerations without being too restrictive on the functional relationship in additional parameters, which may not necessarily be convex. Two different models are introduced, where one can represent arbitrary functional relationships in the additional parameters, while the other is monotonic in the additional parameters. As a first proof of concept, the model is calibrated to data generated with two differently parametrized analytical potentials, whereby three different pICNN architectures are investigated. In all cases, the proposed model shows excellent performance.

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

In Elastomeric foam materials find wide applications for their excellent energy absorption properties. The mechanical property of elastomeric foams is highly nonlinear and it is essential to implement mathematical constitutive models capable of accurate representation of the stress-strain responses of foams. A constitutive modeling method of defining hyperfoam strain energy function by a Simplex Strategy is presented in this work. This study will demonstrate that a strain energy function of finite hyperelasticity for compressible media is applicable to describe the elastic properties of open cell soft foams. This strain energy function is implemented in the FE-tool ABAQUS and proposed for high compressible soft foams. To determine this constitutive equation, experimental data from a uniaxial compression test are used. As the parameters in the constitutive equation are linked in a non-linear way, non-linear optimization routines are adopted. Moreover due to the in homogeneities of the deformation field of the uniaxial compression test, the quality function of the optimization routine has to be determined by an FE-tool. The appropriateness of the strain energy function is tested by a complex loading test.

By using the optimized parameters the FE-simulation of this test is in good accordance with the experimental data.

Cet article présente une méthodologie d'identification du comportement de mousses polymères hyperélastiques à partir de mesures de champs par corrélation d'images et de la technique d'identification dite méthode des champs virtuels. Dans un premier temps, les constantes élastiques à faible déformation sont identifiées à partir d'un essai de flexion/cisaillement. Ensuite, une loi hyperélastique de type Ogden est identifiée à partir d'un essai de compression uniaxiale. On observe une forte localisation des déformations en raison de l'effondrement élastique des cellules de la mousse, confirmant qu'une approche basée sur la réponse globale force/déplacement ne permettra pas l'identification d'un comportement matériau (effet de structure). Cette étude préliminaire montre le potentiel de la démarche même s'il reste encore beaucoup à faire pour fiabiliser cette approche.

In the present study, we propose an evaluation of a non-linear homogenization model applied to hyperelastic composites having random microstructure. This modelling approach consists in a 3D implementation of the secondorder method introduced by Ponte Castañeda & Tiberio [1]. We first recall the basic principles of this method. Then, we investigate a composite made up of an hyperelastic matrix reinforced by spherical deformable or rigid particles.Computational issues of the micromechanical model are discussed and some obtained results allowto demonstrate the reinforcement effect of the particles. In order to provide a rigorous evaluation of the methodology, finite elements computations, on an unit cell, are performed and compared to the predictions of the model. Finally, a confrontation with experimental results is provided.

A hyperelastic constitutive law, for use in anatomically accurate finite element models ofliving structures, is suggested for the passive and the active mechanical properties of incompressiblebiological tissues. This law considers the passive and active states as a same hyperelastic continuummedium, and uses an activation function in order to describe the whole contraction phase.The variational and the FE formulations are also presented, and the FE code has been validatedand applied to describe the biomechanical behavior of a thick-walled anisotropic cylinder underdifferent active loading conditions.