Hooke uniaxial

Uniaxial compression of a cylindrical shape is a method of measuring the behavior of a Hookean solid. Hooke s law is represented by the following relationship ... 

Consider a vertically hanging metal rod, to which a load can be apphed (e.g. a steel cable supporting an elevator), as in Figure 10.3. The load exerts a tensHe force over the entire cross-sectional area of the rod, which is said to be under uniaxial stress since only the stress along one of the principal axes is nonzero. The stress is equal to the force divided by the cross-sectional area over which it is distributed. In linear elastic theory, according to Hooke s law, the magnitude of the strain produced in the rod by a small uniform applied stress is directly proportional to the magnitude of the applied stress. Hence ... 

The non-linear response of elastomers to stress can also be handled by abandoning molecular theories and using continuum mechanics. In this approach, the restrictions imposed by Hooke s law are eliminated and the derivation proceeds through the strain energy using something called strain invariants (you don t want to know ). The result, called the Mooney-Rivlin equation, can be written (for uniaxial extension)—Equation 13-60 ... 

The first thing we want to model is a simple linear elastic response, as described by Hooke s law. The equation for simple uniaxial extension is shown in Figure 13-87,... 

In a majority of cases, a body under stress experiences neither pure shear nor pure dilatation. Generally, a mixture of both occurs. Such a situation is exemplified by uniaxial loading which, of course, may be tensile or compressive. Here a test specimen is loaded axially resulting in a change in length, AL. The axial strain, e, is related to the applied stress in an elastic deformation by Hooke s law ... 

These relationships are known as Newton s Law of viscous flow a is termed the fluidity and -q the dynamical shear viscosity. Newton s Law is analogous to Hooke s Law, except shear strain has been replaced by shear strain rate and the shear modulus by shear viscosity. As shown later, this analogy is often very important in solving viscoelastic problems. In uniaxial tension, the viscous equivalent to Hooke s Law would be a=7] ds/dt), where q is the uniaxial viscosity. As v=0.5 for many fluids, this equation can be re-written as <7-=3Tj(de/dO using t7=t /[2(1+v)], the latter equation being the equivalent of the interrelationship between three engineering elastic constants, (fi=E/[2il + v)]). 

For linear elastic materials, Hooke s Law is a constitutive relationship between stress and strain. There have been substantial efforts in identifying similar relationships for plastic solids. In uniaxial tests, the portion of the true stress-true strain curve beyond yielding is often described by... 

Fig, L General view of uniaxial tensile creep machine (Darlington, 1971, unpublished), 1, Lever arm 2, upper hooks 3 linear spring guide 4, tensile extenso-meter 5, lateral extensometer (for thickness strain). 

The law of linear proportionality between uniaxial strain and uniaxial stress discovered by Hooke in 1676 can be generalized to a linear connection relating all nine elements of the strain tensor and all nine elements of the stress tensor, implying the existence of 81 constants of proportionality, or elastic compliances, Sijki, relating generically the strain tensor component sy to the stress tensor component in an expression of the type... 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows ... 

The simplest mechanical test method is tensile testing, where a rectangular or dumbbellshaped specimen is placed between two clamps and then uniaxially drawn with constant speed (64,65). In the case of pure elastic deformation, the stress a and the resulting deformation are proportional to each other. The original dimensions of the test specimen are completely and immediately restored after removal of the stress. The proportionality constant E is called the modulus. It is given by Hooke s law (Eq. 19), where a is the tensile stress (N m ), y the strain, and E Young s modulus (N m ) ... 

The deformation response of a material to a given loading regime is described by generalized equations known as constitutive relations. For uniaxial loading in the limit of small strains, the simplest of these is known as Hooke s Law and linearly relates the stress to the strain ... 

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

In a linear-elastic material under uniaxial loads, stress and strain are related by Hooke s law, a = Ee. In this case, the integral in equation (2.17) can... 

Young s modulus of elasticity quantifies the elasticity of the polymer. Like tensile strength, this is highly relevant in polymer applications involving physical properties of polymers. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke s law holds. This can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material. 

The first one consists of calculating the slope of the stress-strain curve resulting from a uniaxial loading test as described by the one-dimensional (ID) Hooke s law (1) ... 

Generalized Hooke s Law As noted previously, Hooke s law for one dimension or for the condition of uniaxial stress and strain for elastic materials is given by a = E e. Using the principle of superposition, the generalized Hooke s law for a three dimensional state of stress and strain in a homogeneous and isotropic material can be shown to be. 

The latter equation is the uniaxial stress-strain relation for a polymer analogous to Hooke s law for a material that is time independent but is valid only for the case of a constant input of strain. The relaxation test provides the defining equation for the material property identified as the relaxation modulus. More general differential and integral stress-strain relations for an arbitrary loading will be developed in later Chapters. 

A review of the basic definitions of stress and strain was given in Chapter 2. It was noted that a linear elastic solid in uniaxial tension or pure shear obeys Hooke s laws given by,... 

One of the simplest constitutive relations is Hooke s law, which relates the stress a to the strain e for the uniaxial deformation of an ideal elastic isotropic solid. Thus... 

For many materials the relationship between stress and strain can be expressed, at least at low strains, by Hookers law which states that stress is proportional to strain. It must be borne in mind that this is not a fundamental law of nature. It was discovered by empirical observation and certain materials, particularly polymers, tend not to obey it. Hooke s law enables us to define the Young s modulus, of a material which for simple uniaxial extension or compression is given by... 

The situation is not so simple in the case of uniaxial tension where all stresses other than 0 are equal to zero and 82 = 3. The generalized Hooke s law can be expressed as... 

The C constant of the power law can be determined from the continuity of the transition between elastic and plastic deformation. At the border of the uniaxial and biaxial stress state the strain calculated by the Hooke-law and the strain from the flow curve have to be the same ... 

As indicated earlier the different stress-strain curves are not characteristic for particular, chemically defined species of polymers but for the physical state of a polymeric solid. If the environmental parameters are chosen accordingly transitions from one type of behavior (e.g. brittle, curve a) to another (e.g. ductile, curve c) will be observed. These phenomenological aspects of polymer deformation are discussed in detail in [14], [52—53], [55—57], and in the general references of Chapter 1. A decrease of rate of strain or an increase of temperature generally tend to increase the ductility and to shift the type of response from that of curve a) towards that of curves c) and d). At small strains (between zero and about one per cent) the uniaxial stress o and the strain e are linearly related (Hooke s law) ... 

Another important special case of a homogeneous deformation (i.e. y is positionally constant) in which now, however, pressure anisotropies are effective, is the uniaxially stressed cubic crystal. Let us assume this time that there is tensile stress in the x-direction. There it holds for small effects (i.e. Hooke s law fulfilled) that s = P. dcxx/dsxx const = xx/ xx nnd also dsyy/dsxx — ds z/dsxx yy/sxx zz/ xx const. 

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