Hooke’s Law of Elasticity

To better understand the nature and features of these vibrations, bonds can be considered as springs. Given this analogy, the behaviour of these molecular springs approximately follows Hooke s law of elasticity. In physics, Hooke s law relates the strain on a body (spring) to the force (load or mass) causing the strain . In essence, molecular bonds follow this linear relationship, where the... 

Three equations are basic to viscoelasticity (1) Newton s law of viscosity, a = ijy, (2) Hooke s law of elasticity. Equation 1.15, and (3) Newton s second law of motion, F = ma, where m is the mass and a is the acceleration. One can combine the three equations to obtain a basic differential equation. In linear viscoelasticity, the conditions are such that the contributions of the viscous, elastic, and the inertial elements are additive. The Maxwell model is ... 

Since the stress has units of force/area and the strain is dimensionless, the modulus has units of force/area. Equation (7.98) is Hooke s law of elasticity and it is valid for all solids at sufficiently small strains. 

Combining equation (1.28) with the Hooke s law of elasticity and Newton s law of viscosity, one can obtain ... 

The stress field of a screw dislocation is pure shear. As indicated earlier, high strains exist in the core region and, therefore, Hooke s Law of elasticity does not apply and so will not be considered. The dislocation line is parallel to the z axis there are no displacements in the x and y directions and the other stress components are zero ... 

Here, represents the Cauchy stress tensor, p is the mass density, and ft and m, are the body forces and displacements in the i direction within a bounded domain Q. The two dots over the displacements indicate second derivative in time. The indices i and j in the subscripts represent the Cartesian coordinates x, y, and z. When a subscript follows a comma, this indicates a partial derivative in space with respect to the corresponding index. For the special case of elastic isotropic solids, the stress tensor can be expressed in terms of strains following Hooke s law of elasticity, and the strains, in turn, can be expressed in terms of displacements. The resulting expression for the stress tensor is... 

The next great scientist to address these problems was Hooke (Figure 13-4), a cantankerous and brilliant contemporary of Newton s (there is a long list of inventions and discoveries that can be attached to Hooke s name). So, the two laws we will be concerned with, Hooke s law for elastic solids and Newton s law for fluids, both had their genesis in the same time period, the latter part of the 17th century. But let s stick to solids, for now. 

Before plastic flow occurs during loading of a specimen, only elastic deformation takes place. With Hooke s law those elastic deformations can be described unequivocally by stresses and flow conditions can be derived. According to the theory of elasticity, a stress pattern causes a distortion pattern. The correlation of both by Hooke s law is the so-called material law . 

For a perfectly elastic material that obeys Hooke s law, the elastic modulus (G) is defined as the ratio of the stress to the strain. As previously described, the strain... 

On the basis of these observations, Maxwell suggested to combine Hooke s law (for elastic bodies) and Newton s law (for viscous fluids) additively into a single rheological equation of state, which has the following form in the onedimensional case ... 

According to Hooke s law, the elastic deformation of solid leads to the accumulation of elastic energy in it. The density of this energy is given by... 

The development of stresses in the scale is caused by various mechanisms which are briefly considered in the following. The relation between the stress, elastic strain, 8el, within the alumina scale is given by the Hooke s law. The elastic properties of the polycrystalline scale are assumed to be isotropic with E0 as Young s modulus and i as Poisson s ratio. Because of the free surface of the scale, a plane stress state in the scale is supposed with = 0. z is the direction perpendicular to the film plane, and x,v are the in-plane coordinates. The x-component of the stress tensor is then given by... 

Thus, the conditions of the experiment described above are adequately expressed by Hooke s Law for elastic materials. For materials under tension, strain (e) is proportional to applied stress a. 

Now suppose the parallelepiped is subjected to equal normal pressure (compressive stress, a) in such a way that its shape remains unchanged but the volume, V, is changed by the amount AV (Figure 13.11). Deformation of this type is called pure dilatation, and Hooke s law for elastic dilatation is written as... 

Hook s law of energy elastic bodies (elastic deformation) a = ... 

We first develop the generalized Hooke s law of energy elasticity as the linear connection between stress and strain in tensorial form and then proceed to consider the most relevant special forms for cases of high material symmetry. 

Hooke, Robert (1635-1703) English physicist, chemist, and architect. One of the most brilliant scientists of his age and one of the most quarrelsome. He formulated Hooke s law of the extension and compression of elastic bodies and effectively invented the quadrant, the microscope, and the first Gregorian telescope. He was curator of experiments at the Royal Society (1662) and later its secretary. 

It can be seen that the shear stress is directly proportional to the rate of change of shear strain with time. This formulation brings out the analogy between Hooke s law for elastic solids and Newton s law for viscous liquids. In the former... 

Hooke s law for elastic solids Also back in the 17th century, Robert Hooke developed his True Theory of Elasticity . In this he proposed that the power of any spring is in the same... 

A special case is the linearly elastic solid, which obeys Hooke s law of proportionality between stress and strain. For the Hookean solid the shear modulus is a material constant, which may depend on pressure and temperature but is independent of the magnitude of the shear strain. Most elastic solids obey Hooke s law for small shear strains, but become non-Hookean at large strains. Energy expended to deform an elastic solid is conserved, and may be recovered when the solid returns to its original shape upon gradual removal of the stress. The energy w stored per unit volume is... 

In a direct analogy to the tensile case, a Hooke s law of shear may be used to define the elastic shear modulus ... 

The states of stress and strain in a deformed crystal being idealized as a continuum are characterized by symmetric second-rank tensors and Cjj, respectively, each comprising six independent components. Hooke s law of linear elasticity for the most general anisotropic solid expresses each component of the stress tensor linearly in terms of all components of the strain tensor in the form... 

The perturbed stress field that arises as a result of the nonuniform composition can be found by straightforward application of the stress equilibrium equations and Hooke s law of linear response for an isotropic elastic material, subject to the constraints that the perturbation alters neither the mean extensional strain in any direction in the a z—plane nor the zero net force per wavelength in the y—direction. The mean normal stress implied by these constraints is... 

As indicated in Sec. 5.7.1, viscosity is a property typical of the liqnid state describing its flow due to externally applied stresses. In normal discnssion, it is the shear viscosity that is of concern. Analogously to Hooke s law for elasticity, an externally applied shear stress and the developed shear strain rate are related through Newton s law of viscosity ... 

Because of their complex structure the mechanical behavior of polymeric materials is not well described by the classical constitutive equations Hooke s law (for elastic solids) or Newton s law (for viscous liquids). Polymeric materials are said to be viscoelastic inasmuch as they exhibit both viscous and elastic responses. This viscoelastic behavior has played a key role in the development of the understanding of polymer structure. Viscoelasticity is also important in the understanding of various measuring devices needed for rheometric measurements. In the fluid dynamics of polymeric liquids, viscoelasticity also plays a crucial role. " Also in the polymer-processing industry it is necessary to include the role of viscoelastic behavior in careful analysis and design. Finally there are important connections between viscoelasticity and flow birefringence. ... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

---