Linear elastic solid

Fig. 8.1. Stress-strain behaviour for a linear elastic solid. The axes are calibrated for a material such as steel.

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

Using the negative sign convention, the equation for this model can be written by simply combining the rheological equation for a Hookean linear elastic solid... 

The macroscopic behavior of a saturated porous material undergoing a dissolution of its linear elastic solid matrix is therefore described by the classical Biot s theory, where the poroelastic properties now depend on the morphological parameter . Formally, plays the role of a damage parameter accounting for the dissolution. 

Currently, mathematical tools are available only for the modeling of cohesive or bridging zones for cracks in linear-elastic solids, although the closure pressure function p(u) or p(u,t) can itself be nonlinear. We first review some basic approaches for the modeling of cohesive zones, beginning with time-independent bridging and then discuss the relationship between cohesive zones and crack growth at elevated temperature primarily based on some recent or just-completed studies.29,30,32,33... 

In principle, then, the surface work which determines the fracture stress of the body can be calculated from the physical properties of the material. In practice this is not easy, since the energy density distribution can only be calculated exactly for linear elastic solids, for which 1 and Eq. (5) reverts to the Griffith theory. 

The porous medium is compressible and behaves as a linear elastic solid ... 

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))... 

Using identical methods one can write the stress-strain equation for an orthotropic linear elastic solid in terms of the principal values of stress and strain as... 

Rigid Solid (Euclidian) Linear Elastic Solid (Hookean) Nonlinear Elastic Solid (Non-Hookean) Visco-Elastic Fluids and Solids (Non-Linear) Nonlinear Viscous Fluid (Non-Newtonian) Linear Viscous Fluid (Newtonian) Inviscid Fluid (Pascalian)... 

Electromigration is the displacement of atoms in a conductor due to an electric current. A metal, such as aluminum, consists of positively charged ions, and Z conduction electrons per ion (Z = 3 for aluminum). To simplify the discussions, we will consider an idealized homogenous conductor, rather than the more realistic polycrystalline description. Our mixture is then made of two components, an electron gas and an ionic body, which we will describe as a linearly elastic solid. 

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... 

Although bone is a viscoelastic material, at the quasi-static strain rates in mechanical testing and even at the ultrasonic frequencies used experimentally, it is a reasonable first approximation to model cortical bone as an anisotropic, linear elastic solid with Hooke s law as the appropriate constitutive equation. Tensor notation for the equation is written as ... 

In section 6.2 the formalism of the elastic behaviour of an ideal linear elastic solid for small strains was considered. Rubbers may, however, be reversibly extended by hundreds of per cent, implying that a different approach is required. The previous ideas suggest a possible plausible generalisation, as follows. 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

For a highly oriented sample at 0 = 0° c/c shear makes a very small contribution, because only a few crystallites experience a sufficiently large shear stress. The observed linear recoverable and incompressible behaviour, with Vu = 0-50, corresponded to that of a linear elastic solid with an extremely high bulk modulus, and was thought to be associated with the non-crystalline regions. 

Another model describing the mechanical properties of materials originated in the seventeenth century is Hooke s law for a linear elastic solid [1],... 

Merodio, J. and Ogden, R. W. (2005). Mechanical response of fiber-reinforced incompressible non-linearly elastic solids. International Journal of Norv-Linear Mechanics 40, pp. 213-227. 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows ... 

As discussed in Chapter 1, fracture mechanics relies on the assumption that the materials tested behave as linearly elastic solids. Structural adhesive bonds fail by crack initiation and propagation in the adhesive layer. Assuming that the structural adhesive is brittle, then it would seem appropriate... 

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 nonlineaxity of a stress-strain trace is caused by the material properties, a transition from elastic deformation to plastic deformation, and/or a large change in the configuration of the specimen [57]. The same applies to a stress intensity/displacement (SD) trace. The stress intensity in the case of a cracked body is quantified by the stress intensity factor for Mode I (ifj). The SD trace is linear for linear, elastic solids (before gross plastic deformation occurs). When a cracked solid contains compressive stresses arormd microspheres (practically in the form of residual stresses), as illustrated in Figure 3.22, also the compressive stresses affect the SD trace and then manifest as nonlinearity. The reason is that, as the applied load increases, the compressive stresses around microspheres in the vicinity... 

In several of the following chapters we consider the behaviour of solid polymers subject to large deformations and show also that in general these materials are viscoelastic, which means that stress (or strain) varies with time. As a starting point, however, we need to consider a polymer as a linear elastic solid when a load is applied the deformation is instantaneous, after which it remains constant until the load is removed, when the recovery is instantaneous and complete linearity means that stress and strain are always proportional to one another. 

Hooke s law describes the behaviour of a linear elastic solid and Newton s law that of a linear viscous liquid. A simple constitutive relation for the behaviour of a linear viscoelastic solid is obtained by combining these two laws ... 

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