Hooke shear

The second major assumption is that the material is elastic, meaning that the strains are directly proportional to the stresses applied and when the load is removed the deformation will disappear. In engineering terms the material is assumed to obey Hooke s Law. This assumption is probably a close approximation of the material s actual behavior in direct stress below its proportional limit, particularly in tension, if the fibers are stiff and elastic in the Hookean sense and carry essentially all the stress. This assumption is probably less valid in shear, where the plastic carries a substantial portion of the stress. The plastic may then undergo plastic flow, leading to creep or relaxation of the stresses, especially when the stresses are high. 

A further important property which may be shown by a non-Newtonian fluid is elasticity-which causes the fluid to try to regain its former condition as soon as the stress is removed. Again, the material is showing some of the characteristics of both a solid and a liquid. An ideal (Newtonian) liquid is one in which the stress is proportional to the rate of shear (or rate of strain). On the other hand, for an ideal solid (obeying Hooke s Law) the stress is proportional to the strain. A fluid showing elastic behaviour is termed viscoelastic or elastoviseous. 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

The modulus of elasticity of a material it is the ratio of the stress to the strain produced by the stress in the material. Hooke s law is obeyed by metals but mbber obeys Hooke s law only at small strains in shear. At low strains up to about 15% the stress-strain curve is almost linear, but above 15% the stress and strain are no longer proportional. See Modulus. 

A similar approach can be adopted for the bulk shear modulus. When a small strain is applied to a solid, the latter is stressed, and one can measure the resulting stress. At low deformation, the bulk shear stress, t, is proportional to the strain, r, following Hooke s law ... 

Most whiskers up to 1 micrometer in diameter obey Hooke s law to the point of fracture, regardless of their composition (Evans, 1972). Whiskers of brittle and ductile materials with larger diameters respond to stress differently. Metallic whiskers fail under tension by shear, whereas other compounds fail through fracture. 

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Equation (5.5) is known as Hooke s Law and simply states that in the elastic region, the stress and strain are related through a proportionality constant, E. Note the similarity in form to Newton s Law of Viscosity [Eq. (4.3)], where the shear stress, r, is proportional to the strain rate, y. The primary differences are that we are now describing a solid, not a fluid, the response is to a tensile force, not a shear force, and we do not (yet) consider time dependency in our tensile stress or strain. 

Generalized Hooke s Law. The discussion in the previous section was a simplified one insofar as the relationship between stress and strain was considered in only one direction along the applied stress. In reality, a stress applied to a volnme will have not only the normal forces, or forces perpendicular to the surface to which the force is applied, but also shear stresses in the plane of the surface. Thus there are a total of nine components to the applied stress, one normal and two shear along each of three directions (see Eigure 5.4). Recall from the beginning of Chapter 4 that for shear stresses, the first subscript indicates the direction of the applied force (ontward normal to the surface), and the second subscript indicates the direction of the resnlting stress. Thus, % is the shear stress of x-directed force in the y direction. Since this notation for normal forces is somewhat redundant—that is, the x component of an... 

Recall from Eq. (5.10) that the shear strain can be related to the shear stress through the shear modulus, G, according to Hooke s Law, where we now add subscripts to differentiate the elastic quantities from the viscous quantities ... 

Fig. 6.1. A volume element of a solid subject to shear. The difference between the stresses at the two ends of the volume element is given by Hooke s law as p(d2 /dx2) dx, and by Newton s second law this must be equal to p(d2 /dt2) dx.

In engineering design. Yuung s modulus is used for tension and compression and the rigidity modulus lor shear, as in lorsion springs. According to Hooke s Law, Ihe stress set up within an elastic body is proportional 10 the strain lo which the body is subjected by the applied load. 

An elastic solid has a definite shape. When an external force is applied, the elastic solid instantaneously changes its shape, but it will return instantaneously to its original shape after removal of the force. For ideal elastic solids, Hooke s Law implies that the shear stress (o force per area) is directly proportional to the shear strain (7 Figure H3.2.1A) ... 

At first we have deliberately focused on the applied (technological) importance of the study of melt behavior under extension since the theoretical importance of the analysis of melt extension for polymer physics and mechanics can be regarded as already generally recognized. The scientific success and recognition of melt extension stems, we believe, from several fundamental causes, major of which are as follows. The geometrical pattern of deformation (shear, twisting, tension, etc.) is not very important for mechanics of the usual solid bodies since there is a well-known and multiply verified connection (linear Hooke s mechanics) between the main (if... 

For an ideal solid, Hooke s law holds the stress, cr, applied is proportional to the deformation, e, and the proportionality constant is the modulus of elasticity E, so a = E e. Besides E also other quantities play a role, such as the shear modulus, G, in a shearing deformation or torsion, which is related to E. For the sake of simplicity we shall mainly use as a representative quantity for the elastic stiffness in any geometry of loading. 

Hook F, Ray A, Norden B, Kasemo B (2001) Characterization of PNA and DNA immobilization and subsequent hybridization with DNA using acoustic-shear-wave attenuation measurements. Langmuir 17 8305-8312... 

Hook, F., Rodahl, M., Brzezinski, P., and Kasemo, B. (1998). Energy dissipation kinetics for protein and antibody-antigen adsorption under shear oscillation on a quartz crystal microbalance. Langmuir 14, 729-734. 

Hooke s law for shear stress and shear strain is then (Equation 13-8) ... 

Let s look at this in a little more detail, to make sure that you understand what we mean by strain rate in a shear experiment First, let s go back to Newton s good friend, Hooke. (If you ve read the introduction to this chapter, you know we re being facetious ) We have seen above (Figure 13-11) that for shear the most convenient way to describe the deformation of a solid is in terms of the angle 6 through which a block of the material is deformed (Equation 13-61) ... 

Jnst as stress is proportional to strain in Hooke s law for a solid, the shear stress is proportional to the rate of strain for a fluid i.e., yv. But, for a given shear stress,... 

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