Stress Hooke’s law

If one applies tensile stress on a solid, the solid elongates and gets strained. The stress (a) - strain (e) relation is linear for small stresses (Hooke s law) after which nonlinearity appears, in some cases. Finally at a critical stress CTf, depending on the material, amount of disorder and the specimen size etc., the solid breaks into pieces fracture occurs. In the case of brittle solids, the fracture occurs immediately after the Hookean linear region, and consequently the linear elastic theory can be applied to study the essentially nonlinear and irreversible static fracture properties of brittle solids (Lawn and Wilshaw 1975, Thomson 1986, Evans and Zok 1986). 

Thus the strain is a linear function of the applied stress (Hooke s law). The Landau expansion for a paraelectric phase with piezoelectric properties will now contain this new term in addition to a coupling term -XsP, of the same form as in our previous case... 

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. 

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

Dislocations are known to be responsible for die short-term plastic (nonelastic) properties of substances, which represents departure from die elastic behaviour described by Hooke s law. Their concentration determines, in part, not only dris immediate transport of planes of atoms drrough die solid at moderate temperatures, but also plays a decisive role in die behaviour of metals under long-term stress. In processes which occur slowly over a long period of time such as secondaiy creep, die dislocation distribution cannot be considered geometrically fixed widrin a solid because of die applied suess. 

We can now define the elastic moduli. They are defined through Hooke s Law, which is merely a description of the experimental observation that, when strams are small, the strain is very nearly proportional to the stress that is, they are linear-elastic. The nominal tensile strain, for example, is proportional to the tensile stress for simple tension... 

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

The generalized Hooke s law relating stresses to strains can be written in contracted notation as... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

Modulus of elasticity Most materials, including plastics and metals, have deformation proportional to their loads below the proportional limit. Since stress is proportional to load and strain to deformation, this implies that stress is proportional to strain. Hooke s Law, developed in 1676, follows that this straight line (Fig. 2-2) of proportionality is calculated as ... 

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. 

Hooke s Law It is the ratio of normal stress to corresponding strain (straight line) for stresses below the proportional limit of the material. 

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. 

No material is perfectly elastic in the sense of strictly obeying Hooke s law. Polymers, particularly when above their glass transition temperature, are certainly not. For these macromolecular materials there is an element of flow in their response to an applied stress, and the extent of this flow varies with time. Such behaviour, which may be considered to be a hybrid of perfectly elastic response and truly viscous flow, is known as viscoelasticity. 

In order to model viscoelasticity mathematically, a material can be considered as though it were made up of springs, which obey Hooke s law, and dashpots filled with a perfectly Newtonian liquid. Newtonian liquids are those which deform at a rate proportional to the applied stress and inversely proportional to the viscosity, rj, of the liquid. There are then a number of ways of arranging these springs and dashpots and hence of altering the... 

Hint 3 Assume that the fabric is perfectly elastic so that stress is proportional to strain (Hooke s law). 

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-... 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

Most engineering materials, particularly metals, follow Hooke s law by which it is meant that they exhibit a linear relationship between elastic stress and strain. This linear relationship can be expressed as o = E where E is known as the modulus of elasticity. The value of E, which is given by the slope of the stress-strain plot, is a characteristic of the material being considered and changes from material to material. 

This expression indicates that U, the strain energy per unit volume is given by the area under the stress-strain plot over the strain range e = 0 to e = e. If Hooke s law holds, i.e., if a = E e, it follows that... 

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. 

In general, during the initial stages of deformation, a material is deformed elastically. That is to say, any change in shape caused by the applied stress is completely reversible, and the specimen will return to its original shape upon release of the applied stress. During elastic deformation, the stress-strain relationship for a specimen is described by Hooke s law ... 

Hoogsteen hydrogen bonding, 17 609, 610 Hooke s law, 21 719 Hookean region of stress—strain curve, 11 183, 184... 

Hooke s law is the relationship between strain and stress second rank tensors ... 

Systematic measurements of stress and strain can be made and the results plotted as a rheogram. If our material behaves in a simple manner - and it is surprising how many materials do, especially if the strains (or stresses) are not too large - we find a linear dependence of stress on strain and we say our material obeys Hooke s law, i. e. our material is Hookean. This statement implies that the material is isotropic and that the pressure in the material is uniform. This latter point will not worry us if our material is incompressible but can be important if this is not the case. 

Figure 2.1 Stress versus strain for a typical elastomer of vulcanised natural rubber. The straight line shows that a simple Hooke s law response would be observed up to 40% strains...

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence ... 

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