Hooke’s law equation

However, Hooke s law, Equation (2.1), can be derived from Equation (2.5) ... 

The elastic component is dominant in solids, hence their mechanical properties may be described by Hooke s law (Equation 14.1), which states that the applied stress (.s) is proportional to the resultant strain (y) but is independent of the rate of this strain (dy/dt). 

The applications of Hooke s law [equations (2-14) and (2-18)] discussed above have assumed that the volume of the material is invariant with strain during a tensile deformation. However, because the pressure is not zero, this may not be the case, and the strains in each direction must be known to account for this. By measuring the actual transverse (yyy) and longitudinal (y ) strains, one can define the ratio of these two strains as a material property. This is called Poisson s ratio ju, and is defined as ... 

The heart of force detection in an AFM apparatus is a scanning cantilever microfabricated with a tip for force sensing (Figure 6.29). Forces generated by surface-to-tip interactions, Fst> lead to vertical tip displacements, Az, that are related to each other in a quasi-static way and to a hrst approximation by a form of the Hooke s law equation ... 

The moduli of elasticity determined by stress / strain measurements are generally much lower than the lattice moduli of the same polymers (Table 11-3). The difference is to be found in the effects of entropy elasticity and viscoelasticity. Since the majority of the polymer chains in such polymer samples do not lie in the stress direction, deformation can also occur by conformational changes. In addition, polymer chains may irreversibly slide past each other. Consequently, E moduli obtained from stress/strain measurements do not provide a measure of the energy elasticity. Such E moduli are no more than proportionality constants in the Hooke s law equation. The proportionality limit for polymers is about 0.l%-0.2% of the... 

The two extreme cases of mechanical behavior can be reproduced very well by mechanical models. A compressed Hookean spring can serve as a model for the energy-elastic body under load (Figure 11-11). On releasing the load, the compressed spring immediately returns to its original position. The relationship between the shear stress (021) = Oe, the shear modulus Ge, and the elastic deformation ye is given by Hooke s law [Equation (11-1)] ... 

The rates of deformation dyjdt are additive in these processes. By combining the expressions for the rates of deformation according to Hooke s law [Equation (11-50)] and according to Newton s law [Equation (11-51)], we obtain the following for the total rate of deformation ... 

Hooke s law, equation (2.20), can also be written in differential form ... 

In a more complete, tensor form, one would write compliance matrix) to generalize Hooke s law (Equation (1.9b)). Indeed, reducing y and Voigt notation in mathematics, one... 

From the Hooke s law equation, would you expect the C-X bonds of common haloalkanes (X = Cl, Br, I) to have IR bands at higher or lower wavenumbers than are typical for bonds between carbon and lighter elements (e.g., oxygen) ... 

Using Hooke s law (Equation 5), > and defining the mode deformation potentials for a given stress, dj, bj, and Cj, we can obtain an alternative form of Equation 8 that relates the mode frequency shift with the stress tensor [14]. 

For bonded structures exhibiting bulk linear-elastic behaviour, i.e. away from the crack tip regions, they obey Hooke s Law. Equation 7.3 may be expressed [1-4] as ... 

For most metallic materials, elastic deformation persists only to strains of about 0.005. As the material is deformed beyond this point, the stress is no longer proportional to strain (Hooke s law. Equation 6.5, ceases to be valid), and permanent, nonrecoverable, or plastic deformation occurs. Figure 6.10a plots schematically the tensile stress-strain behavior into the plastic region for a typical metal. The transition from elastic to plastic is a gradual one for most metals some curvature results at the onset of plastic deformation, which increases more rapidly with rising stress. 

For tensile and compressive loading, the slope of the linear elastic region of the stress-strain curve is the modulus of elasticity ( ), per Hooke s law (Equation 6.5). For a material that exhibits nonlinear elastic behavior, tangent and secant moduli are used. 

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