Class Of Hyperelastic Materials Research Articles

A comparison of DAE integrators in the context of benchmark problems for flexible multibody dynamics

In the present work a uniform framework for general flexible multibody dynamics is used to compare state-of-the-art DAE integrators in the context of benchmark problems. The multibody systems considered herein are comprised of rigid bodies, nonlinear beams and shells. The constitutive laws applied in the benchmark problems belong to the class of hyperelastic materials. To numerically integrate the uniform set of DAEs three alternative time-stepping schemes are applied: (i) an energy-momentum consistent method, (ii) a specific variational integrator and (iii) a generalized-a scheme.

On the identifiability of the stored energy function of hyperelastic materials from sensor data at the boundary

This article addresses the inverse problem of reconstructing the stored energy function of a certain class of hyperelastic materials from partial Cauchy data. It is motivated by so-called structural health monitoring systems, whose idea is to disclose defects in elastic, anisotropic structures from data which are recorded at sensors that are applied at the boundary of the object. Damage affects the spatially varying stored energy function and thus its reconstruction reveals defects of the structure. The dynamic behavior of hyperelastic materials is described by Cauchyʼs equation of motion, a second order, non-linear system of partial differential equations, and thus we get an identification problem for this PDE system. We prove conditions under which this identification problem is uniquely solvable and that the solution continuously depends on the measured data as well as on the initial conditions of the structure. An important assumption is that the stored energy function is a conic combination of finitely many given functions. Moreover, we show that in the linear case such a conic decomposition of the elasticity tensor exists if its entries are in a finite dimensional subspace of the space of continuous functions and some spectral conditions are satisfied. In the case of a homogeneous, isotropic medium this decomposition is explicitly known. As a consequence, we get the result that two sensors are sufficient to identify the two independent material parameters, the Lamé coefficients, of such a medium.

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.

Closed form solution for the finite anti-plane shear field for a class of hyperelastic incompressible brittle solids

The equilibrium solution of a damaged zone in finite elasticity is given for a class of hyperelastic materials which does not suffer tension when a critical stretching value is reached. The study is made for a crack in anti-plane shear loading condition. The prescribed loading is that of linearized elastostatics conditions at infinity. The geometry of the damaged zone is found and the stationary propagation is discussed when the inertia terms can be neglected.

On universal relations in orthotropic material subclasses

Universal relations that hold true for specific classes or subclasses of elastic materials form an integral part of the characterisation process for any material that can deform elastically. On the other hand, it has become understood that any class of hyper-elastic materials possessing anisotropic properties more complicated to those of transverse isotropy obeys no universal relations. However, there are met in practice special cases (subclasses) of anisotropic hyper-elastic materials that, under certain circumstances, may still obey a certain number of universal relations. Material characterisation purposes suggest therefore that there is a practical purpose in identifying and classifying different subclasses of anisotropic hyper-elastic materials that may obey any kind of universal relations and, hence, to derive and gradually build up catalogues or tables of different universal relations that such material subclasses obey when subjected to different deformation patterns. This paper heads towards this research direction by initially considering the orthotropic material class which is alternatively identified as the hyper-elastic crystal class of rhombic symmetry [A.E. Green, J.E. Adkins, Large Elastic Deformations, Clarendon Press, Oxford, 1960]. Then, for a certain set of appropriately chosen orthotropic and transverse isotropic hyper-elastic material subclasses, it develops and presents a considerable number of relevant universal relations, most of which are new in the literature.

On compressible materials capable of sustaining axisymmetric shear deformations. III. Helical shear of isotropic hyperelastic materials

A helical shear deformation is a composition of non-universal, axisymmetric, anti-plane shear and rotational shear deformations, shear states that are separately controllable only in special kinds of compressible and incompressible, homogeneous and isotropic hyperelastic materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to determine the anti-plane and rotational shear displacement functions. For compressible materials, however, these shear deformations may not be separately possible in the same specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. We have recently presented simple algebraic conditions necessary and sufficient in order that both anti-plane shear and rotational shear deformations may be separately possible in the same material subclass. In this paper, under the same physical condition that the shear response function be positive, we present an essentially algebraic condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining controllable, helical shear deformations. It is then proved that helical shear deformations are possible in a specified hyperelastic material if and only if that material can separately sustain both axisymmetric, anti-plane shear and rotational shear deformations. The simplicity of the result in applications is illustrated in a few examples.

On compressible materials capable of sustaining axisymmetric shear deformations. Part 2: rotational shear of isotropic hyperelastic materials

It is well known that rotational shear deformations, though not universal, are controllable in specific kinds of compressible and incompressible materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to determine the rotational shear displacement function. For compressible materials, however, rotational shear deformations may not be possible for a specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. Polignone and Horgan have recently derived two conditions in the form of nonlinear ordinary differential equations necessary for hyperelastic materials to sustain rotational shear deformations. In the present paper, under the condition that the shear response function be positive, we present a single, essentially algebraic, condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining rotational shear deformations by application of surface tractions alone. Several examples illustrate the simplicity of the result in applications. We have recently presented similar kinds of results on axisymmetric, anti-plane shear deformations. With the aid of an auxiliary necessary condition on the strain energy function, here we present a simple necessary and sufficient condition for which both anti-plane shear and rotational shear deformations are separately possible in the same material subclass. These algebraic conditions are illustrated in several examples.

On a certain class of elastic materials with nonelliptic energy densities

In this paper we are concerned with the behavior of a certain class of hyperelastic materials with non-elliptic strain energy density functions that satisfy very weak growth conditions at infinity. It is shown, by means of an example, that solutions of boundary-value problems involving these materials do not exist within the usual admissible spaces in view of the severe discontinuities involved (displacement discontinuities).