Hook law

The constitutive law solid models. The classical Hooke law... 

We introduce the most basic aspects of elasticity. We begin with Hookes law the change in length of a strut is proportional to the applied force, or 6L = FL/EA. Note that this is a linear relationship. Restated in a normalized way, a = Ee, where a is the stress (Pa or N/m ), E is Young s modulus (Pa) a property of the material, and e is the strain (6LjL) a dimensionless quantity. 

For a simple system, such as a rod under compression, one can define the stiffness, or the spring constant. If we examine Hookes law, we get 6L = L SFjEA, where A is the cross sectional area, and 6F is the applied load. The spring constant k is defined as dF/dL, or k = EA/L. The basic physics equation F = kx is just a statement of this. For many degree of freedom systems there will be multiple spring constants, each connected to a modal shape. 

The hysteresis property of any polyurethane is composed of two components, namely the spring and the viscous component. The viscous component is responsible for the absorption of the force and the conversion of the energy into heat. Any deflections must be within the limits where Hookes Law is obeyed (i.e., only 1 to 10%). The ratio between the forced and natural frequencies needs to be determined. The natural frequency is a function of the static deflection of the system. The damping ratio of the polyurethane also must be known. This can vary from 0.05 for highly resilient materials to 0.15 for low-resiliency materials. To obtain damping, the forced-to-natural damping frequency ratio must be greater than 1.4. 

The value of S-EWetching in the force field equation (see equation (5.1)) for a structure is given by the sum of appropriate expressions for E for every pair of bonded atoms in the structure. For example, using the Hooke law model, for a molecule consisting of three atoms bonded a-b-c the expression would be ... 

Experimentally, we can introduce a built-in strain in an epitaxial layer by growing it on a lattice mismatched substrate. As long as the mismatched epitaxial layer is below the critical thickness, the produced strain is uniform and no dislocations are induced. As a result, the in-plane lattice constant of the epitaxial layer is fitted to that of the substrate, and the out-of plane lattice constant is adjusted to a new lattice constant according to the Hook law. Then, the subband structure is modified by introducing a built-in strain, and the strain has a dramatic influence on the electronic properties of the system. Theoretically, we can easily include the strain effect in the k.p theory. 

Steepness of a potential energy function, for instance, spring strength in Hookes law. Empirical force constants are different from spectroscopic force constants (see molecular mechanics, force field). 

Figure 2 shows the principle of electrochemical Kosters laser interferometry for the determination of changes of surface energy by the resulting deformation of an elastic plate. The height AZc of the center of the plate with respect to a plane at a given radius yields Ag from the appropriate form of Hookes law... 

In most cases the variation in surface stress has been determined indirectly by measuring the potential dependence of the strain (i.e., electrode deformation) and then obtaining the variation in stress from the appropriate form of Hookes law (-> bending beam or cantilever beam method, bending plate methods e.g., the measurement of the deformation of the electrode with the help of a —r Kosters laser interferometer). 

For quantitative analysis, Inglis considered a uniformly stressed two-dimensional solid like a thin plate, containing an elliptic hole representing the crack (see Fig. 3.4). Let the lengths of the semi-major and -minor axes of the ellipse to be 21 and 2b repectively, and a denote the external (say tensile) stress applied on the sample along the y-direction. We assume the (linear) Hooke law to hold everywhere in the plate and that the boundary surface of the elliptic hole, represented by the equation... 

I. In the case of elastic behavior stress and strain are proportional to each other, i.e. there is a linear dependence between y and x, described by the Hooke law, namely... 

We will now examine the simplest rheological model, the Hooke law for elasticity and Newton law for viscosity. In Hooke s law the tensor of strain Un is a linear fimction of the stress tensor of S, i.e. the deformation is proportional the acting forces. If the inertial stress, the elastic or Hooke s stress and the viscous stress are additive, we can write... 

We have illustrated two types of vibration, one in which the atoms vibrate together in an equal distance (shown as a longitudinal vibration) and the other where the atoms vibrate an unequal distance, relative to one another (here shown as a transverse vibration). Note however, that the atoms vibrate in phase, regardless of the direction and the distance. We can write equations using these terms by considering u as a coordinate and p as a displacement, in terms of the total force required. Ft, and the individual forces on each atom, Fi, as a form of Hookes Law ... 

The Maxwell model. One of the first attempts to explain the mechanical behavior of matmals such as pitch and tar was made by James Clark Maxwell. He argued that when a material can undergo viscous flow and also respond elastically to a stress it should be described by a combination of both the Newton and Hooke laws. This assumes that both contributions to the strain are additive so that e= e ias, + e jsc-Expressing this as the dilfeimtial equation leads to the equation of motion of a Maxwell unit... 

Infrared Absorption Spectra. Following West and Edwards (63), Bauer and Magat (64) examined the electrostatic effect of the solvent on the stretching frequency of a polar bond. Mathematically, their approach is simple to the restoring force acting on the bond in the gas phase (Hooke law type) the treatment incorporates the perturbing force created by the R.F. on the polar bond. This contribution is proportional to the solvent R.F. Let vq be the frequency of the vibrator in the gas phase, i>2, the frequency in the solvent S and Aj the difference... 

Unfortunately, this wonderful two-electron system is (at least partially) non-physical. It represents a strange helium atom, in which the two electrons (with their distance denoted by ri2) interact through the Coulombic potential, but each is attracted to the nucleus by a harmonic spring [i.e., satisfying the Hooke law (of equilibrium length 0 and force constant k, with electron-nucleus distances denoted by r and r2) see Fig. 4.25]. 

The mathematics of spring deformation, as described by Hookes law, is used to model the elastic forces between these spheres. This treatment requires that only the motions of nuclei are studied and that the electrons are not examined explicitly electrons are assumed to find an optimal distribution around the nuclei. 

The Hooke law states that the strain of a body is proportional to the applied stress. The law is applicable at the macroscopic level of strength of a soUd (see Table 1.1, the characteristic length is greater than 0.01 m). Practically, the Hooke law holds within the elastic range of a material for relative small values of strain e (e 1). 

Cauchy generalized the Hooke law to a three-dimensional elastic body and stated that the six components of stress are hnearly related to the six components of strain. The law can be written in a tensor form as... 

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