Cauchy-Green Strain Tensor Research Articles

Finite-amplitude inhomogeneous waves in Mooney–Rivlin viscoelastic solids

New exact solutions are exhibited within the framework of finite viscoelasticity. More precisely, the solutions correspond to finite-amplitude, transverse, linearly polarized, inhomogeneous motions superposed upon a finite homogeneous static deformation. The viscoelastic body is composed of a Mooney–Rivlin viscoelastic solid, whose constitutive equation consists in the sum of an elastic part (Mooney–Rivlin hyperelastic model) and a viscous part (Newtonian viscous fluid model). The analysis shows that the results are similar to those obtained for the purely elastic case; inter alia, the normals to the planes of constant phase and to the planes of constant amplitude must be orthogonal and conjugate with respect to the B -ellipsoid, where B is the left Cauchy–Green strain tensor associated with the initial large static deformation. However, when the constitutive equation is specialized either to the case of a neo-Hookean viscoelastic solid or to the case of a Newtonian viscous fluid, a greater variety of solutions arises, with no counterpart in the purely elastic case. These solutions include travelling inhomogeneous finite-amplitude damped waves and standing damped waves.

Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity

Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. It is a direct generalization of the classical Hooke's law for isotropic infinitesimal elasticity by replacing the Cauchy stress tensor and the infinitesmal strain tensor with the foregoing stress and strain tensors. A simple, straightforward proof is presented to show that Hencky's elasticity model is exactly a hyperelasticity model, derivable from a quadratic potential function of the Hencky strain tensor. Generally, Hill's isotropic linear hyperelastic relation between any given Doyle-Ericksen or Seth-Hill strain tensor and its work-conjugate stress tensor is studied. A straightforward, explicit expression of this general relation is derived in terms of the Kirchhoff stress and left Cauchy-Green strain tensors. Certain remarkable properties of Hencky's model are indicated from both theorectical and experimental points of view.

A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I1 of the Cauchy–Green strain tensor, is a simple logarithmic function of I1 and involves just two material parameters, the shear modulus μ and a parameter Jm which measures a limiting value for I1−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Pade approximant for this function. The constants μ and Jm in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function.

Anisotropic membrane wrinkling: theory and analysis

A complete theory of wrinkling of anisotropic elastic membranes is presented based on the notion of relaxed energy function. The theory is obtained as a particular case of a general theory of saturated elasticity. It is shown that when any given hyperelastic constitutive law is supplemented with a saturation condition, the right Cauchy–Green strain tensor abides by an additive decomposition law governed by a normality rule. For the case of wrinkling, these results are implemented into a numerical procedure, and examples are presented.

New descriptions of plane strain

This paper describes some new descriptions of plane strain, based on a frame attached to the soil rather than an external coordinate frame. The new descriptions are not limited to small strain processes. The paper starts with physical interpretations of the deformation gradient and right Cauchy–Green strain tensors. A new set of ‘zensor’ quantities are then derived, based on a conceptual frame that is attached to the soil and deforms with it. This parallels the physical reality of strain-gauge measurements. It is then shown that invariants in plane strain are intimately associated with three specific linear operators on zensors, here called the scale, relativistic and arc operators. This leads to identification of three independent in-plane strain-increment invariants, not just two, and three independent work-conjugate in-plane effective stress invariants. Fixed-frame zensor transformation laws in plane strain are then derived. As well as the familiar concept of change of frame, these also allow change of reference state to be handled. Finally, some advantages of the new approach to continuum mechanics are discussed.

New descriptions of plane strain

This paper describes some new descriptions of plane strain, based on a frame attached to the soil rather than an external coordinate frame. The new descriptions are not limited to small strain processes. The paper starts with physical interpretations of the deformation gradient and right Cauchy–Green strain tensors. A new set of ‘zensor’ quantities are then derived, based on a conceptual frame that is attached to the soil and deforms with it. This parallels the physical reality of strain-gauge measurements. It is then shown that invariants in plane strain are intimately associated with three specific linear operators on zensors, here called the scale, relativistic and arc operators. This leads to identification of three independent in-plane strain-increment invariants, not just two, and three independent work-conjugate in-plane effective stress invariants. Fixed-frame zensor transformation laws in plane strain are then derived. As well as the familiar concept of change of frame, these also allow change of reference state to be handled. Finally, some advantages of the new approach to continuum mechanics are discussed.

Finite-amplitude inhomogeneous plane waves in a deformed Mooney-Rivlin material

The propagation of finite-amplitude linearly-polarized inhomogeneous transverse plane waves is considered for a Mooney–Rivlin material maintained in a state of finite static homogeneous deformation. It is shown that such waves are possible provided that the directions of the normal to the planes of constant phase and of the normal to the planes of constant amplitude are orthogonal and conjugate with respect to the B-ellipsoid, where Bis the left Cauchy–Green strain tensor corresponding to the initial deformation. For these waves, it is found that even though the system is nonlinear, results on energy flux are nevertheless identical with corresponding results in the classical linearized elasticity theory. Byproducts of the results are new exact static solutions for the Mooney–Rivlin material.

Thermoelastic Shear of Rubberlike Solid Cylindrical Tubes

The condition of incompressibility, which is acceptable in the theory of large homothermal, elastic deformation of materials such as rubbers, is questionably so when the deformation gradient tensor is distortional, i.e., not spherical, and accompanied by large temperature variations. A form of the two-parameter Mooney-Rivlin model, modified by Chadwick to take account of compressibility and successfully matched by him to experiment by suitable choice of the material constants, is described as an example of a more general theory of rubberlike materials. This specific model is used to analyze by numerical methods the boundary value problem of azimuthal shear of cylindrical tubes, which are subject to a temperature gradient in the radial direction. It is found that the physical component B <12> of the left Cauchy-Green strain tensor, which corresponds to a constant S<12> component of the distortional stress tensor additional to the pressure, differs markedly from the value obtained from the incompressible theory when the temperature gradient is high and the inner radius of the tube is large in comparison to its thickness. The difference is even more marked if the tube of compressible material is allowed to expand in thickness by a small percentage.

Finite element analysis of nonlinear orthotropic hyperelastic membranes

A finite element method is presented for geometrically and materially nonlinear orthotropic hyperelastic membranes. The constitutive relations are formulated in terms of the invariants of the 2D right Cauchy-Green strain tensor and the resulting system of nonlinear equations solved using a Newton-Raphson approach. Both axisymmetric and nonaxisymmetric versions of the method are developed and validated. Example problems are solved for isotropic and orthotropic membranes, and the effect of various parameters investigated. Finally, convergence studies are performed for various degrees of anisotropy.

Compatibility conditions for the Cauchy-Green strain fields: Solutions for the plane case

This paper considers the issues related to uniqueness and existence of a finite deformation generated by prescribed right or left Cauchy-Green strain tensor field in the plane. First, the questions of uniqueness and existence to a pre-assigned right strain field C are discussed. It is shown that the existence condition, in the context of continuum mechanics, are naturally posed using the field corresponding to the square root of C instead of C, the latter a classical approach. Then, the corresponding questions for the left strain field are considered, which is more involved. The analysis of uniqueness gives rise to an appropriate classification of the deformation fields. The question of existence is discussed and a complete solution is presented. In both the right and left cases, we stress the techniques for obtaining the corresponding deformation fields.

The nonlinear elastic behavior of polydisperse hexagonal foams and concentrated emulsions

A complete constitutive theory that describes the nonlinear elastic behavior of foams and concentrated emulsions with two‐dimensional, polydisperse hexagonal structure is derived. The individual bubbles are irregular hexagons that can vary widely in size and shape. The straight thin films and threefold symmetric Plateau borders exhibited by perfectly ordered monodisperse systems are retained by polydisperse hexagonal structures. The constitutive theory is expressed as explicit relations between the Cauchy stress and the Cauchy–Green strain tensors, or in terms of a strain energy function. The important material characteristics are surface tension, number density of bubbles, and if the dispersed phase is gas, an equation of state. The specific spatial arrangement and size distribution of the bubbles does not affect the elastic response. The volume fraction of the phases only influences the stress through the dispersed phase pressure. Weaire, Kermode, and Fu have conjectured that the elastic response associated with monodisperse structure provides an upper bound for all two‐dimensional fluid–fluid networks; we extend this conjecture to polydisperse hexagonal structure, not just monodisperse structure.

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ3 normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.

A second order fluid of the differential type

In this paper a new constitutive equation is proposed for viscoelastic fluids. It is based on the time derivatives of the left Cauchy-Green strain tensor and is thus believed to characterize a new material. From this constitutive equation, a second-order approximation is derived and it is shown that this second order fluid has a bounded and unique solution to the initial value problem of cessation of steady shear flow. This stability is in direct contrast with the presently well known second order models. Further, the natural time of the fluid is positive, making the theory consistent with the intuitive notion that phenomena in dissipative materials should depend on the past rather than the future. Finally, a test is proposed to distinguish the present second-order model from the earlier version.