Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

Nonlinear dynamic equations for isotropic homogeneous hyperelastic materials are considered in the Lagrangian formulation. An explicit criterion of existence of a natural state for a given constitutive law is presented, and is used to derive natural state conditions for some common constitutive relations.For two-dimensional planar motions of Ciarlet–Mooney–Rivlin solids, equivalence transformations are computed that lead to a reduction of the number parameters in the constitutive law. Point symmetries are classified in a general dynamical setting and in traveling wave coordinates. A special value of traveling wave speed is found for which the nonlinear Ciarlet–Mooney–Rivlin equations admit an additional infinite set of point symmetries. A family of essentially two-dimensional traveling wave solutions is derived for that case.     

A Novel Ermakov–Painlevé II System: ‐Dimensional Coupled NLS and Elastodynamic Reductions

Wave packet ansätze are introduced into an ‐dimensional Manakov‐type system and key invariants are isolated. Reduction is made to a novel coupled Ermakov–Painlevé II system and an algorithm presented for the derivation of wave packet representations via the classical Ermakov nonlinear superposition principle. Application of the procedure in the context of certain transverse wave motions in a generalized Mooney–Rivlin hyperelastic material is likewise shown to lead an Ermakov–Painlevé II reduction.     

An operator splitting strategy for fluid–structure interaction problems with thin elastic structures in an incompressible Newtonian flow

We present a computational framework based on the use of the Newton and level set methods to model fluid–structure interaction problems involving elastic membranes freely suspended in an incompressible Newtonian flow. The Mooney–Rivlin constitutive model is used to model the structure. We consider an extension to a more general case of the method described in Laadhari (2017) to model the elasticity of the membrane. We develop a predictor–corrector finite element method where an operator splitting scheme separates different physical phenomena. The method features an affordable computational burden with respect to the fully implicit methods. An exact Newton method is described to solve the problem, and the quadratic convergence is numerically achieved. Sample numerical examples are reported and illustrate the accuracy and robustness of the method.     

Stability of Surface Rayleigh Waves in an Elastic Half‐Space

This paper is concerned with nonlinear analysis for the propagation of Rayleigh surface waves on a homogeneous, elastic half‐space of general anisotropy. We show how to derive an asymptotic equation for the displacement by applying the second‐order elasticity theory. The evolution equation obtained is a nonlocal generalization of Burgers' equation, for which an explicit stability condition is exhibited. Finally, we investigate examples of interest, namely, isotropic materials, Ogden's materials, compressible Mooney–Rivlin materials, compressible neo‐Hookean materials, Simpson–Spector materials, St Venant–Kirchhoff materials, and Hadamard–Green materials.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Incompressible nonlinearly elastic thin membranes

Nonlinearly elastic thin membrane models are derived for hyperelastic incompressible materials using Γ-convergence arguments. We obtain an integral representation of the limit two-dimensional energy owing to a result of singular functionals relaxation due to Ben Belgacem [ESAIM Control Optim. Calc. Var. 5 (2000) 71–85 (electronic)]. To cite this article: K. Trabelsi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Analyzing the harmonic wave propagation in the layered thick tubes

In this study, the propagation of time harmonic waves in prestressed, anisotropic elastic tubes filled with viscous fluid is studied. The fluid is assumed to be incompressible and Newtonian. A two layered hyperelastic anisotropic structural model is used for the compliant arterial wall. The tube is subjected to a static inner pressure P_i and an axial stretch λ. The governing differential equations of tube are obtained in cylindrical coordinates, utilizing the theory of “Superposing small deformations on large initial static deformations”. The analytical solutions of the equations of motion for the fluid have been obtained. Due to variability of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch and material parameters.     

The integration of analysis and testing for the simulation of the response of hyperelastic materials

A family of hyperelastic finite elements capable of modeling arbitrarily large strains for axisymmetric and plane strain analyses has been developed. Constitutive behavior is determined by the selection of a strain energy density function for which user-supplied coefficients are required. Selective reduced integration for the volumetric strain energy terms allows for successful modeling of nearly incompressible materials. Available strain energy density functions are as follows: Mooney-Rivlin, Blatz-Ko, power law, and a nine-term Mooney expansion. The Ogden Strain Energy (OSE) law has also been implemented. The OSE law defines the strain energy relationship entirely in terms of the three principal components of stretch. This differs from the approach of other strain energy formulations, such as the Mooney law in which the strain energy is written as a function of strain invariants. The OSE law as implemented in this formulation is designed to facilitate the user's task of converting physical test data to the numerical (algebraic) form required for input. The family of hyperelastic finite elements has been integrated into ANSYS Revision 4.2 via the user element interface. Numerous verification solutions have been performed. As a representative example, a comparison with a closed-form solution for a Mooney-Rivlin type material is presented. Finally, the difficulties of obtaining test data in the form of user-supplied constants is discussed in the context of the comparison of experimental measurements and analytical simulation of an elastomeric test specimen.     

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.     

Propagation of a plane wave to a materially uniform but inhomogeneous body

We produce the equations of small deformations superimposed upon large for materially uniform but inhomogeneous bodies and specialize to an isotropic material and to a homogeneous finite elastic deformation. By assuming the small deformation to be a plane wave, a set of equations for the amplitude of the wave is produced which is accompanied by an additional set of conditions. By requiring a non-trivial solution for the amplitude, we obtain the secular equation and from it a set of necessary and sufficient conditions for having a real wave speed. The second set of conditions that have to be satisfied is due to the materials inhomogeneity. Essentially, the present analysis enhances the approach of Hayes and Rivlin for materially uniform but inhomogeneous bodies. The outcome is that for such bodies the restrictions on the constitutive law for having real wave speeds for an isotropic material subjected to a pure homogeneous deformation involves the field of the inhomogeneity as well.     

Numerical evaluation of the temperature field of steady-state rolling tires

Rubber is the main component of pneumatic tires. The tire heating is caused by the hysteresis effects due to the deformation of the rubber during operation. Tire temperatures can depend on many factors, including tire geometry, inflation pressure, vehicle load and speed, road type and temperature and environmental conditions. The focus of this study is to develop a finite element approach to computationally evaluate the temperature field of a steady-state rolling tire. For simplicity, the tire is assumed to be composed of rubber and body-ply. The nonlinear mechanical behavior of the rubber is characterized by a Mooney–Rivlin model while the body-ply is assumed to be linear elastic material. The coupled effects of the inflation pressure and vehicle loading are investigated. The influences of body-ply stiffness are studied as well. The simulation results show that loading is the main factor to determine the temperature field. The stiffer body-ply causes less deformation of rubber and consequently decreases the temperature.     

Conmutación temporal entre técnicas numéricas implícita y explícita para simulación de estructuras no lineales inestables

Simulation problems involving non-linear materials imply in numerous cases divergence of the implicit method which use return mapping algorithms for modelling of the nonlinear response. A switching implicit-explicit numerical technique in the context of Finite Element Methods is presented in this paper. Implicit/explicit mesh partitions are not considered whatsoever. Formulation for application to nonlinear hyperelastic materials and nonlinear elastic-plastic materials is provided. Furthermore, the response of the solid subjected to large deformations is presented and is embedded in the proposed technique. Numerical tests for nonlinear problems (geometric and/or material) showed the accurateness of the technique.     

Extension of the Arruda-Boyce model to the modelling of the curing process of polymers

A phenomenologically motivated finite strain general framework to simulate the curing of polymer have been developed and discussed in our recently published papers [2,4]. The Arruda-Boyce model is a classical hyperelastic model for polymeric materials. This contribution presents an extension of the Arruda-Boyce model towards modelling the curing process of polymers following our previous framework. In this paper, we will show how to model the elastic behaviour and shrinkage effects of the polymer curing process in the isothermal case using the Arruda-Boyce model. Several numerical examples have been demonstrated to verify our newly proposed modified approach in case of curing process. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material models in principal stretches for elastodynamics

In the present work we deal with the conserving integration of elastic bodies undergoing finite deformations. In particular, we make use of constitutive laws formulated in terms of principal stretches. Most material models for hyperelastic isotropic materials are described in terms of principal stretches (Simo and Taylor [1]), like the Neo–Hooke material which is a special case of the Ogden material, or in invariants. The main advantage of principal stretches is the fact that they can be measured directly, which means that the numerical results can be compared easily with experimental ones, see for example, Ogden [2]. Moreover, it is advantageous to describe viscoelastic material behaviour (e.g. rubberlike materials) in terms of principal stretches. Concerning the discretization in space we apply the enhanced assumed strain (EAS) method, see Simo and Armero [3]. For the discretization in time we aim at numerical integrators which inherit fundamental conservation laws from the underlying continuous system. In particular, we propose an energy and momentum conserving time–stepping scheme which relies on the notion of a discrete gradient (or derivative) in the sense of Gonzalez [4]. The proposed approach starts from our previous developments in [5]. Numerical examples demonstrate the advantageous properties of the present formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Monotonicity-preserving interproximation of B–H-curves

B–H-curves are used for modelling ferromagnetic materials in connection with electromagnetic field computations. They are needed for the numerical simulation of devices such as transformers or magnetic valves. Starting from real-life measurement data, we present an approximation technique which is based on the use of spline functions and a data-dependent smoothing functional. It preserves physical properties, such as monotonicity, and is robust with respect to noise in the measurements.     

Homogenization of Linear Elastic Properties of Silicon Nitride

The effective linear elastic properties of silicon nitride (Si₃N₄) are estimated based on first–, third–, and fifth–order bounds of the strain energy density. This specific type of material is a mixture of two linear elastic materials with different material symmetries. The β-Si₃N₄ grains have a hexagonal symmetry with significant amount of anisotropy, whereas the glassy phase is approximately isotropic. The results are as follows: i) The fifth–order upper and lower bounds are almost identical. Therefore, these bounds are sufficient for estimating the effective elastic properties. ii) For fixed elastic constants of the hexagonal β-Si₃N₄ grains, the effective properties of Si₃N₄ are determined as a function of properties of the glassy phase and its volume fraction. The corresponding diagrams allow for the inverse identification of the elastic properties of the glassy phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)