Finite Elasticity and Viscoelasticity

The following sections are included:

• Preface

• Contents

In this Chapter we introduce basic concepts and establish the main assertions in the tensor theory which are employed in the mechanics of continua. Our exposition of the tensor algebra and analysis is based on the so-called direct tensor notation which allows arbitrary curvilinear coordinate frames to be used. Section 1 is concerned with Eulerian and Lagrangian coordinate frames. In Section 2, basic concepts of the tensor algebra are provided. Section 3 deals with the tensor calculus. We introduce the nabla-operator and covariant derivatives of tensors and prove the Stokes theorem. In Section 4, we discuss corotational derivatives, i.e. time derivatives of objective tensors which are indifferent with respect to superimposed rigid motions. Finally, Section 5 is concerned with scalar and tensor functions of tensors.

This Chapter is concerned with the kinematics and dynamics of continua with finite strains. In Section 1, the main kinematic concepts are introduced: the deformation gradients, the deformation tensors, and the strain tensors. Section 2 provides a classification of forces. We formulate the main principles in the mechanics of continua: the mass conservation law, the principle of linear momentum and the principle of angular momentum, introduce the Cauchy and Piola stress tensors, and derive the motion equations and the boundary conditions in stresses. Finally, in Section 3 basic axioms of the constitutive theory are formulated and discussed.

This Chapter is concerned with constitutive models in the elasticity theory with finite strains. In Section 1, we discuss basic features of elastic media and provide some experimental data. Section 2 is concerned with constitutive equations in finite elasticity. We consider both the Cauchy and Green theories of elastic media, but concentrate on hyperelastic materials. Finally, in Section 3 we formulate boundary-value problems in finite elasticity and discuss the existence and uniqueness of their solutions.

This Chapter is concerned with solutions to several boundary-value problems in the elasticity theory with finite strains. Our objective is to demonstrate techniques employed in solving boundary-value problems in finite elasticity, and to show some physical effects which are typical of large deformations. Section 1 deals with the so-called universal solutions to problems in finite elastostatics. The main attention is focused on the uniaxial and biaxial tension of an incompressible elastic medium, as well as on the correspondence between experimental data and their theoretical prediction. Section 2 is concerned with the simple shear of an elastic medium. It is shown that for large deformation, the simple shear should be accompanied by normal stresses (the Weissenberg effect), and the first difference of the normal stresses is proportional to the square of the shear stress (the Lodge-Meissner formula). In Section 3, we discuss torsion of a circular cylinder made of an incompressible elastic material. We compare the Rivlin solution to this problem with experimental data and demonstrate the Poynting effect: elongation of an elastic bar under the action of torques applied to its ends. Torsion of a circular bar is one of the most interesting (from the applied point of view) problems, and we will return to it in Chapter 7 to demonstrate the separability principle in finite viscoelasticity. Section 4 deals with the discussion of the Saint-Venant principle in finite elasticity. This principle states that the effect of surface load has a local character: far away from the region where surface traction is applied, stresses are determined by the resultant force and the resultant moment, and are practically independent of the load distribution. We provide a detailed proof of the Saint-Venant principle (i) to define correctly the locality property for surface traction, and (ii) to demonstrate the tools employed in the mathematical theory of elasticity. Section 5 is concerned with one-dimensional motions of an elastic medium: elongation and shear. We demonstrate a significant difference between these motions: any classical solution to the dynamic problem for elongation motion exists on a finite interval of time only, and it is transformed into a shock wave independently of the smoothness of initial conditions. This behavior is typical of a number of conservative dynamic systems. Shear motion is an exception from this rule of thumb: its classical solutions may exist on an arbitrary interval of time provided the initial conditions are sufficiently smooth. Finally, Section 6 deals with nonlinear transversal oscillations of a rod. We show that vibrations of an elastic rod are described by the modified Korteweg-de Vries equation, derive the Hirota method for the construction of particular solutions in the form of solitary waves, and discuss some physical properties of n-soliton solutions.

This Chapter is concerned with variational principles in finite elastostatics. In Section 1, we discuss the Lagrange principle and derive the equilibrium equation and the boundary condition in stresses as necessary conditions of minimum for the total energy. We show that the strong ellipticity conditions provide sufficient conditions of the minimum, and establish infinitesimal stability conditions for a hyperelastic medium. Section 2 deals with phase transitions in hyperelastic solids. By using the generalized Lagrange principle, a specific condition on the interface is derived, which is formulated in terms of the Eshelby tensor. For elastic liquids, the interface condition obtained is equivalent to the equality of chemical potentials.

This Chapter is concerned with constitutive models in finite viscoelasticity. In Section 1, we discuss basic physical phenomena typical of viscoelastic media and provide some experimental data. Section 2 deals with differential constitutive models, while in Section 3 we consider integral constitutive models. Finally, in Section 4 we discuss some features of creep kernels and relaxation kernels.

This Chapter is concerned with some boundary problems in finite and infinitesimal viscoelasticity. In Section 1, we consider the Rayleigh problem for a linear non-aging viscoelastic medium and demonstrate that the smoothness of the relaxation kernel affects significantly the propagation of singularities. Section 2 deals with acceleration waves in an aging viscoelastic medium, i.e. solutions of the governing equations which are continuous and have continuous first derivatives, while the second derivatives may have finite jumps of discontinuity on a wave front. It is shown that for shear motions acceleration waves of any amplitude decay in time, whereas for elongation motions acceleration waves with small amplitudes decay exponentially in time, while waves with relatively large amplitudes grow and tend to infinity on finite intervals of time. Section 3 is concerned with radial oscillations of an incompressible viscoelastic thick-walled spherical shell. We demonstrate the effect of initial conditions on the period of natural oscillations for an elastic shell with finite strains, and establish a correspondence between small forced oscillations of viscoelastic shells made of materials with finite and infinite memories. In Section 4, quasi-static torsion of a circular cylinder is studied. We derive relationships which connect torque and compressive force on the one hand, and the twist angle on the other, and validate the obtained formulas by using experimental data for butyl rubber. The obtained results confirm the constitutive model of adaptive links based on the separation principle. Section 5 deals with homogeneous deformations of an incompressible viscoelastic medium loaded on the interval (−∞, 0]. At instant t = 0, external loads are reduced, and the problem consists in the qualitative description of an elastic recoil. The set of homogeneous deformations is divided into classes (elongation motions, shear-free motions, etc.), and we analyze whether the recoil belongs to the same class as the motion under loading. It is shown that for some deformation histories, the body recovers to a state in which it has never been under loading. Section 6 is concerned with continuous growth of a viscoelastic medium with finite strains. The growth means a monotonic increase of the body's mass due to the material influx from the environment. This process is modeled as successive accretion of thin layers on a surface of the growing body. It is shown that final stresses in a growing solid depend on the loading history, the rate of the mass influx, and rheological properties of the medium.

The following sections are included:

• References

• Index