Semi-linear Material Research Articles

Thermohyperelastic constitutive response for semilinear materials with application to deformation of a rotating thin disk

Constitutive modeling of soft elastomers is a well researched subject due to its numerous engineering applications. This study derives a thermodynamically consistent constitutive model for semilinear thermoelastic solids for which the strain energy density depends on the deformation gradient through the Biot stretch tensor. The Frechet derivatives, Coleman-Noll procedure, and product decomposition-based technique are employed to formulate the constitutive model. The combination of the derived model and thermomechanical field equations is specialized for application to problem of radial deformation of a rotating thin disk. The formulated model provides simple closed-form expressions for simulating thermomechanical responses of solids. Under suitable conditions, the derived three-configuration-based constitutive model reduces to the two-configuration-based constitutive equations. Finally, and among other things, the numerical solutions illustrate the effects of angular speed of rotation on thermal stress and displacement fields at points within the radially deforming rotating disk.

Nonlinear electroelastic theory for dielectric semilinear materials

Modeling of constitutive theory of smart materials is an intensely researched topic in mechanics of solids. This is due to its complexity and widespread applications. This article presents general constitutive theory for dielectric semilinear materials and specializes the theory for applications to problems of extension-inflation coupling and radial deformation of electro-sensitive tubes based on prototype energy function of John’s semilinear material. During extension-inflation coupling of an incompressible tube, it is obtained that a lower pressure is required to maintain a prescribed azimuthal stretch when an electric field is applied. Finally, and among other things, the numerical solutions of the considered problems coincide with those predicted by the Dorfmann and Ogden theory.

Rational design of hyperelastic semi-linear material and its application to elastic wave control

Incremental elastic wave in a hyperelastic semi-linear (SL) material can be purposely controlled by pre-deformation with help of hyperelastic transformation (HT) theory. However, this material model, assuming a linear stress-strain relation for large deformation, has neither been realized with naturally occurring materials nor through microstructure design. By balancing the stiffness of axial and rotational springs in a hexagonal mass-spring lattice, we propose a lattice prototype to realize the SL material. The performance of the model is systematically verified from static strain energy to wave property on different finitely deformed configurations. Based on this SL lattice, an elastic wave cloak is designed and validated through numerical simulation. This work provides the first two-dimensional (2D) microstructure model for realizing SL material and paves the way for elastic wave control with HT method.

On pure bending in non-linear elasticity: A circular closed-form 2D solution for semi-linear orthotropic material

Large-deformation pure bending of a straight beam of uniform rectangular cross-section in plane strain is re-visited and generalised to an orthotropic linear-elastic material and the case of plane stress. The beam is bent into a completely shearless state by means of normal surface traction of a specific profile. For a linear-elastic material expressed in terms of Biot strain and stress tensors (orthotropic generalisation of the so-called semi-linear material) the solution still exists in closed form and is expressed in terms of the applied cross-sectional traction. As such, it provides a convenient benchmark reference for numerical methods in non-linear elasticity.In agreement with the isotropic case, through-the-thickness stresses develop in the form a hyperbolic cosine, but now with the magnitude actually depending on a measure of orthotropy in the material, while the longitudinal stresses change over the thickness in the form of a hyperbolic sine. In general, the unstrained axis is displaced from the centroidal axis.For an ideal material with infinite yield strength, the theoretical maximum of the magnitude of the longitudinal traction is limited to the amount of the geometric mean of the effective Young's moduli in the longitudinal and the through-the-thickness directions while, for a brittle material, bending strength reduces by approximately 715 of the square of the ratio between the magnitude of the applied traction and this mean value.

Equilibrium stability of nonlinear elastic sphere with distributed dislocations

An exact formulation and a solution of the stability problem for a three-dimensional elastic body containing distributed dislocations are given. The buckling phenomenon for a hollow nonlinear elastic sphere of the semi-linear (harmonic) material with edge dislocations is studied. The study is carried out within the framework of the continuum theory of continuously distributed dislocations using the bifurcation method of buckling analysis. The bifurcation method is to find the equilibrium positions of an elastic body, which differ little from the subcritical (unperturbed) state. The perturbed equilibrium state is described by a linearized boundary value problem. By solving a homogeneous linear boundary value problem, the minimum critical value of the external pressure at which the sphere loses stability is found. The influence of dislocations on the buckling of both thin and thick spherical shells is analyzed.

The study of nonlinear deformation of a plane with an elliptical inclusion for harmonic materials

Analytical methods are used to study nonlinear deformation of a plane with an elliptical inclusion. The elastic properties of a material of the plane and inclusion are described with a semi-linear material. The external load is constant nominal (Piola) stresses at infinity. At the inclusion boundary, the conditions of the continuity for stresses and displacements are satisfied. Semi-linear material belongs to the class of harmonic, the methods of the theory of functions of a complex variable are applicable to solving nonlinear plane problems. Stresses and displacements are expressed in terms of two analytical functions of a complex variable, determined by the boundary conditions on the inclusion contour. It is assumed that the stress state of an inclusion is uniform (the tensor of nominal stresses is constant). This hypothesis made it possible to reduce the difficult nonlinear problem of conjugation of two elastic bodies to the solution of two more simpler problems for a plane with an elliptical hole. The validity of this hypothesis is justified by the fact that the constructed solution exactly satisfies all the equations and boundary conditions of the problem. The same hypothesis was used earlier by other authors to solve linear and nonlinear problems of an elliptical inclusion. In the article, a comparative analysis of the stresses and strains is carried out for two models of harmonic materials — semi-linear and John’s. Various variants of values of elasticity parameters of the inclusion and matrix have been considered.

In-Plane Semi-Linear Cloaks with Arbitrary Shape

Soft materials with semi-linear strain energy function can be used as smart transformation media to manipulate elastic waves via finite pre-deformation. However, the intrinsic constraints involved in such materials limit the shapes of transformation devices to very simple cases. In this work, combining theoretical and numerical analyses, we report an approach of achieving the in-plane elastodynamic cloak with arbitrary shape. We demonstrate that with the appropriate out-of-plane stretch applied on the semi-linear material, cloaking effect can be achieved for both P- and SV-waves in the symmetric plane of a 3D domain, and the performance of the cloak with arbitrary cross section can be guaranteed for relatively small in-plane rotation. In addition, we propose an empirical formula to predict the deformation limit of the cloaks with semi-linear materials. This work may stimulate the experimental research on soft-matter-based transformation devices. Potential applications can be anticipated in nondestructive testing, structure impact protection, biomedical imaging and soft robotics.

Hyperelastic antiplane ground cloaking.

Hyperelastic materials possess the appealing property that they may be employed as elastic wave manipulation devices and cloaks by imposing pre-deformation. They provide an alternative to microstructured metamaterials and can be used in a reconfigurable manner. Previous studies indicate that exact elastodynamic invariance to pre-deformation holds only for neo-Hookean solids in the antiplane wave scenario and the semi-linear material in the in-plane compressional/shear wave context. Furthermore, although ground cloaks have been considered in the acoustic context they have not yet been discussed for elastodynamics, either by employing microstructured cloaks or hyperelastic cloaks. This work therefore aims at exploring the possibility of employing a range of hyperelastic materials for use as antiplane ground cloaks (AGCs). The use of the popular incompressible Arruda-Boyce and Mooney-Rivlin nonlinear materials is explored. The scattering problem associated with the AGC is simulated via finite element analysis where the cloaked region is formed by an indentation of the surface. Results demonstrate that the neo-Hookean medium can be used to generate a perfect hyperelastic AGC as should be expected. Furthermore, although the AGC performance of the Mooney-Rivlin material is not particularly satisfactory, it is shown that the Arruda-Boyce medium is an excellent candidate material for this purpose.

Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material

Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material

Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material

Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (проект № 16-31-00065).

Longitudinal elastic wave control by pre-deforming semi-linear materials.

An incremental wave superimposed on a pre-deformed hyper-elastic material perceives an elastic media with the instantaneous modulus of the current material. This offers a new route with a broadband feature to control elastic waves by purposely creating finite deformation field. This study proves that the governing equation of a semi-linear material under a symmetric pre-deformation condition maintains the form invariance for longitudinal wave, so the longitudinal wave control can be made by transformation method without the constraint condition on principle stretches; however, this is not the case for shear waves. Therefore, pre-deforming a semi-linear material provides a potential method for treating longitudinal and shear waves differently. Examples with elastic wave control and band structure shift through pre-deforming a semi-linear material are provided to illustrate this finding. Finally, a one-dimensional spring lattice is proposed to mimic a semi-linear material, and the dispersion relation for longitudinal waves in a sandwich structure with such spring lattice is shown to be invariant during elongation, confirming the result found based on a homogeneous semi-linear material. These results may stimulate researches on designing new hyper-elastic microstructures as well as designing new devices based on pre-deformed hyper-elastic materials.

Soft phononic crystals with deformation-independent band gaps

Soft phononic crystals have the advantages over their stiff counterparts of being flexible and reconfigurable. Normally, the band gaps of soft phononic crystals will be modified after deformation due to both geometric and constitutive nonlinearity. Indeed these are important properties that can be exploited to tune the dynamic properties of the material. However, in some instances, it may be that one wishes to deform the medium while retaining the band gap structure. A special class of soft phononic crystals is described here with band gaps that are independent or almost-independent of the imposed mechanical deformation, which enables the design of phononic crystals with robust performance. This remarkable behaviour originates from transformation elasticity theory, which leaves the wave equation and the eigenfrequencies invariant after deformation. The necessary condition to achieve such a property is that the Lagrangian elasticity tensor of the hyperelastic material should be constant, i.e. independent of deformation. It is demonstrated that incompressible neo-Hookean materials exhibit such a unique property. Semilinear materials also possess this property under special loading conditions. Phononic crystals composed of these two materials are studied theoretically and the predictions of invariance, or the manner in which the response deviates from invariance, are confirmed via numerical simulation.

Large bending strains in an orthotropic beam with a preliminarily stretched or compressed layer: Exact solution

The model of an orthotropic nonlinear elastic compressible material generalizing the known model of an isotropic semilinear (harmonic) material and admitting a number of exact solutions at large strains is proposed. For this model of the material, an exact solution of the plane problem on large bending strains of a compound rectangular beam consisting of two layers, one of which is preliminarily deformed, is obtained. When formulating and solving the problem, the theory of superimposed large strains is used. On the basis of the calculations carried out, the effect of anisotropy on a strong bending of the compound beam with a preliminarily deformed layer is analyzed.

Nonlinear isotropic elastic reduced Cosserat continuum as a possible model for geomedium and geomaterials. Spherical prestressed state in the semilinear material

We suggest a nonlinear elastic reduced Cosserat continuum as a possible model for geomedium and geomaterials and also for a soil or rock with heterogeneities. If a medium has a block structure, or if it contains heterogeneities that may have their proper rotational dynamics, the presence of rotational degrees of freedom may influence wave propagation and stability of the medium. In the reduced Cosserat model, translations and rotations are independent, the medium resists to the rotation of each “particle” relatively to the background continuum, but it does not resist to the gradient of rotation. We consider a nonlinear spherical stress state in an isotropic elastic reduced Cosserat material. We write down small deviations from this nonlinear equilibrium. They coincide in form with the equations of the linear elastic isotropic reduced Cosserat continuum. Depending on the level of the stress and on the type of elastic energy, the equilibrium can be stable or unstable. In the domain of stability, shear–rotational wave demonstrates dispersion. There is a resonant frequency corresponding to the independent rotational oscillations. The bulk plane shear–rotational wave has a forbidden band of frequencies. In this zone, the shear–rotational wave localises near heterogeneities or external sources. We show that for a semilinear medium (medium with large deformations but linear dependence of stress on the strain tensor), strong compression leads to the material instability caused by shear perturbations, and strong tension for some class of parameters yields in instability caused by rotational perturbations.

On the influence of the gravity force on the stability of an inhomogeneous layer under biaxial extension-compression

In the present paper, in the framework of the three-dimensional nonlinear theory of elasticity, we study the stability of a heavy layer under biaxial extension-compression. The elastic properties of the layer are assumed to be inhomogeneous along thickness and are described by a semilinear material model. We study the stability by using the bifurcation approach. By solving the linearized equilibrium equations, we obtain the critical curves and the stability domain in the plane of the loading parameters, for which we take the material elongation ratios along the coordinate axes lying in the layer plane. We analyze the influence of the layer thickness, specific weight, and material parameters on buckling. In particular, we find that, when studying stability, it is expedient to take the gravity force into account only if the layer rigidity decreases with increasing depth.

Analysis of a stress singularity in a non-linear Flamant problem for certain models of a material

A generalized plane problem in the non-linear theory of elasticity is considered for a half-plane loaded on the boundary with a concentrated external force (the non-linear Flamant problem). The properties of the material of the half-plane are described by different (known) models, and each model of the non-linearly elastic material generates its own specific boundary-value problem. Analytical solutions of the problems are obtained for two models of an incompressible material: the neo-Hookean model and the Bartenev–Khazanovich model, and a model of a compressible semi-linear (harmonic) material. The dependence of the stress state as a whole on the adopted model of the material and the effect of the model of the material on the form of the stress singularity in the neighbourhood of a pole are investigated.

The non-linear theory of the pure bending of prismatic elastic solids

The problem of the bending of a prismatic elastic solid by finite torques under large deformation conditions is considered. Using the semi-inverse method, the initial three-dimensional boundary-value problem of the non-linear theory of elasticity is reduced to a two-dimensional non-linear boundary-value problem for a region in the form of the cross-section of the beam. Two formulations of the problem are given in the cross-section: in terms of the displacements and of the stresses. Stress functions are introduced and a variational formulation of the two-dimensional problem is obtained, based on the supplementary energy principle. An approximate solution of the problem of the strong bending of a beam of rectangular cross-section is found for a semi-linear material and for a Bartenev-Khazanovich material using the Ritz method.

An energy function for transversely-isotropic elastic material and the Ponyting Effect

On the basis of the semi-linear material of John, invoking the theory of homogenization for heterogeneous media and the theory of invariants for isotropic scalar functions, an energy function is built for a transversely-isotropic medium in finite elastic deformation. The Ponyting Effect, for material in simple shear, is reviewed for this case of transversal isotropy. It is shown that this effect is apprehended by the constructed energy function.

On interacting longitudinal-shear waves in large elastic deformation of a composite cylindrical laminate

We consider a large deformation of a cylindrical layer. Through the process of homogenization and the theory of effective moduli this heterogeneous medium is converted to a homogeneous but anisotropic medium. On the basis of John’s semilinear material, an energy function is identified and equations of motion obtained. Using the method of characteristics the speeds of propagating linear longitudinal and non-linear shear cylindrical waves are obtained: these “p-wave” and “s-wave” do interact, contrary to the known view, of the small deformation theory.

Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block

Under consideration is the problem of flexure of compressible nonlinearly elastic rectangular blocks. The discussion is confined to deformations describing the bending of a rectangular block into a sector of a circular cylindrical tube. The predictions based upon the well-known semilinear material model are investigated. Addressed, in particular, are some problems concerning the existence, uniqueness and stability of solutions to specific boundary value problems.