Linear, isotropic, elastic solid

A model substance that represents closely the deformation behaviour for small strains of a wide range of materials is the linear isotropic, elastic solid. As the name impUes, the substance is isotropic, satisfies Hooke s law for aU strains and shows perfect elasticity. 

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have... 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

As usual, we have invoked the summation convention and in addition have assumed that the material properties are homogeneous. For an isotropic linear elastic solid, the constitutive equation relating the stresses and strains is given by... 

Note that these equations are a special case of the equilibrium equations revealed in eqn (2.53) in the constitutive context of an isotropic linear elastic solid. 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

Isotropic Elasticity and Nervier Equations Use the constitutive equation for an isotropic linear elastic solid given in eqn (2.54) in conjunction with the equilibrium equation of eqn (2.84), derive the Navier equations in both direct and indicial notation. Fourier transform these equations and verify eqn (2.88). 

For an isotropic linear elastic solid, write the effective body forces associated with this eigenstrain. (b) Use the effective body force derived in part (a) in conjunction with eqn (2.96) to obtain an integral representation for the displacements due to the spherical inclusion. 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

In the present setting p is the mass density while the subscript i identifies a particular Cartesian component of the displacement field. In this equation recall that Cijki is the elastic modulus tensor which in the case of an isotropic linear elastic solid is given by Ciju = SijSki + ii(5ikSji + SuSjk). Following our earlier footsteps from chap. 2 this leads in turn to the Navier equations (see eqn (2.55))... 

These are most easily represented by the equation E = E + iE". where E is the ratio between (the amplitude of the in-phase stress component strain, a/e) and E" is the loss modulus (the amplitude of the out-of-phase component. strain amplitude). Similarly for (7 and K and the ratio between the Young s modulus E and the shear modulus C includes Poisson s ratio u, for an isotropic linear elastic solid with a uniaxial stress. (Poisson s ratio is more correctly defined as minus the ratio of the perpendicular. strain to the plane strain, or for one orthogonal direction 22 which equals the. 3.3 strain if the sample is... 

As discussed in section 6.2.2, the values of Young s modulus for isotropic glassy and semicrystalline polymers are typically two orders of magnitude lower than those of metals. These materials can be either brittle, leading to fracture at strains of a few per cent, or ductile, leading to large but non-recoverable deformation (see chapter 8). In contrast, for rubbers. Young s moduli are typically of order 1 MPa for small strains (fig. 6.6 shows that the load-extension curve is non-linear) and elastic, i.e. recoverable, extensions up to about 1000% are often possible. This shows that the fundamental mechanism for the elastic behaviour of rubbers must be quite different from that for metals and other types of solids. 

The mechanical behavior of metals in service can often be assumed to be that of a linear, isotropic, and elastic solid. Thus, design analysis can be based on classical strength of materials theory extensively reviewed in textbooks and literature. Practically, results may be used in the form of standard formulae, or design charts for a selected class of applications. Such uses are most appropriate to components of simple geometric shapes for which standard solutions exist, or for more shapes that are complex where they can possibly be used for initial approximate design calculations. 

The first stage in the calculation is to choose a constitutive equation that relates the applied stresses to the resulting strains. For an elastic material, the behavior is described by two independent elastic constants, such as the shear modulus G and the bulk modulus K. The constitutive equation for an isotropic linear elastic solid has the form (28)... 

The mechanical behavior of metals in service can often be assumed to be that of a linear, isotropic, and elastic solid. Thus, design analysis can be based on classical strength of materials theory extensively reviewed in textbooks and literature. Such uses are most appropriate to... 

Most of the experimental work on linear viscoelastic behaviour is confined to a single mode of deformation, usually corresponding to a measurement of the Young s modulus or the shear modulus. Our initial discussion of linear viscoelasticity will therefore be confined to the one-dimensional situation, recognising that greater complexity will be required to describe the viscoelastic behaviour fully. For the simplest case of an isotropic polymer at least two of the modes of deformation defining two of the quantities E, G and K for an elastic solid must be examined, if the behaviour is to be completely specified. 

In section 2.9 the theorem will be applied to complex stress systems in the linear, isotropic solid to obtain relations between the various elastic constants of such a material. 

The linear isotropic approximation is widely used in calculations on the elastic behaviour of solids and the equations [2.14] are often expressed in terms of two material parameters known as the Lame elastic constants. 

The coefficients of elasticity describe strains caused by stress. The two moduli of linear elasticity for isotropic solids are well known ... 

The most prevalent and widely developed constitutive connections of polymers between strain and stress are dealt with in linear elasticity by applying the generalized form of Hooke s law which is presented in Chapter 4 for anisotropic solids of different symmetry classes starting with orthotropic solids and progressing up to isotropic solids. Here and in the following chapters we shall develop only the connection for isotropic solids, which is the most useful one and most often is quite sufficient in development of concepts. 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

The diffusion of positrons in solid is limited by the scattering of positrons by phonons [92]. Under the simplest assumptions (an isotropic phonon spectrum with a linear dispersion curve, elastic scattering), this gives the following expression for the mean free path of the positrons ... 

The shear stress r and shear strain y are related through equation [2.13] when Hooke s law is obeyed. Therefore, for an isotropic, linear solid in which the elastic strains are very small ... 

More generally, a Newtonian viscous liquid may be defined as one in which the components of stress at a given point in the liquid, at a given time, are linear functions of the first spatial derivatives of the velocity components at the same point and at the same instant. The general equations for Newtonian liquid flow may then be obtained from the elastic equations of an isotropic solid by replacing the strain components with the rate of strain components, remembering that, in a liquid, there is a hydrostatic stress —p, where p is the pressure, superposed on the viscous stresses. Equations [2.10] and [2.26] then become ... 

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