Hooke isotropic

A related measure of the intensity often used for electronic spectroscopy is the oscillator strengdi,/ This is a dimensionless ratio of the transition intensity to tliat expected for an electron bound by Hooke s law forces so as to be an isotropic hanuonic oscillator. It can be related either to the experimental integrated intensity or to the theoretical transition moment integral ... 

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... 

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. 

For an isotropic medium Hooke s law (6.24), taking account of the zero matrix elements, becomes... 

For a sufficiently low stress, the elastic strain magnitude for isotropic bodies is proportional to the magnitude of the stress involved. This relation is expressed by Hooke s law (Fig. 2.1) for tension it is... 

For infinitesimal deformations, we assume that the relation between strain and stress is expressed by Hooke s law the deformation is proportional to the applied force For isotropic bodies, this linear relation... 

Keep in mind that this is for a uniform isotropic material. A lot of materials are not isotropic. Crystalline materials can be weaker or stronger along one crystallographic plane than another depending on the arrangement of atoms, relative strength of the bonds and the presence of defects. The modulus in each of these directions can also be different and we would have to write Hooke s law as (Equation 13-11) ... 

Generalized Stress-Strain Hooke s Law for Isotropic Solids 162... 

GENERALIZED STRESS-STRAIN HOOKE S LAW FOR ISOTROPIC SOLIDS... 

Hooke himself interpreted that dependence as follows if a tensile stress is applied to the ends of a thin rod, then the increment in the rod length Al will be proportional to the force applied. The present-day formulation of Hooke s law was given as early as the 19th century by Cauchy and Poisson and is read as follows if a small deformation occurs in an isotropic body, the stress tensor r is a linear function of the deformation tensor U (and vice versa). 

Assuming all deformations to be small, we can employ Hooke s law (13.16) and the deformation tensor expressed through the displacement vector (13.8). Substitution of (13.16) into (13.21) yields the following form for the equation of motion of a homogeneous isotropic elastic medium ... 

The present hypothesis fully describes the hydrostatic strain/stress state in isotropic samples. Indeed, from the refined parameters e, the macroscopic strain and stress e, x can be calculated and also the intergranular strains and stresses Ae,(g), Ax,(g), both different from zero. Note that nothing was presumed concerning the nature of the crystallite interaction, which can be elastic or plastic. From Equations (112) it is not possible to obtain relations of the type (84) but only of the type (86). For this reason a linear homogenous equation of the Hooke type between the macroscopic stress and strain cannot be established. 

When there is no volume change, as when an elastomer is stretched, Poissons s ratio is 0.5. This value decreases as the Tg of the substance Increases and approaches 0.3 for rigid PVC and ebonite. For simplicity, the polymers can be considered to be isotropic viscoelastic solids with a Poisson ratio of 0.5, and only deformations in tension and shear will be considered. Thus, a shear modulus (G) will usually be used in place of Young s modulus of elasticity (E). Hooke s law for shear is given in Equation 38. E is approximately 2.6 G at temperatures below Tg. 

The development of stresses in the scale is caused by various mechanisms which are briefly considered in the following. The relation between the stress, elastic strain, 8el, within the alumina scale is given by the Hooke s law. The elastic properties of the polycrystalline scale are assumed to be isotropic with E0 as Young s modulus and i as Poisson s ratio. Because of the free surface of the scale, a plane stress state in the scale is supposed with = 0. z is the direction perpendicular to the film plane, and x,v are the in-plane coordinates. The x-component of the stress tensor is then given by... 

Based on Hooke s law, the following relationships have been established for elastic, homogeneous, and isotropic materials under tensile or compressive stresses (see Fig. 4a)... 

The three-dimensional stresses in a flowing, constant-density newtonian fluid have the same form as the three-dimensional stress in a solid body that obeys Hooke s j law (i.e., a perfectly elastic, isotropic solid). 

A perfectly elastic body is one whose stress arises solely in response to the strain from its original state, and if the response is also linear and isotropic, the material obeys Hooke s Law (9) ... 

For materials that are isotropic and under deformations where Hooke s law is valid, the elastic constants and V are related according to the following equations ... 

From Hooke s law, the six independent components of the stress tensor can be expressed as a function of the six components of the strain tensor in a symmetrical matrix of order 6 with 21 modulus components for a general anisotropic sample of material. For an isotropic body, there are only two independent components. The mode of deformation will determine which modulus will be measured. 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

In some texts, the Lame constants, A and fi, are used. These constants are equal to c,2 and c, respectively. In some cases, it is convenient to write Hooke s Law in a form specific for isotropic materials. For example, using the Lame constants. 

The most prevalent and widely developed constitutive connections of polymers between strain and stress are dealt with in linear elasticity by applying the generalized form of Hooke s law which is presented in Chapter 4 for anisotropic solids of different symmetry classes starting with orthotropic solids and progressing up to isotropic solids. Here and in the following chapters we shall develop only the connection for isotropic solids, which is the most useful one and most often is quite sufficient in development of concepts. 

R is of the order of the sample dimensions and fc is the radius of the disclination core, i.e., the distance of molecular order over which Hooke-type elasticity is no longer valid. We must add the disclination core energy to (9.9). This is difficult to calculate exactly, but it does not exceed the energy of the disordered nematic [ k T — Tc)], where is the nematic to isotropic transition temperature. 

Hooke s Law for isotropic materials, in terms of Poisson s ratio, is given in matrix form as ... 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

Elastic Modulao. The mechanical behavior is in general terms concerned with the deformation that occurs under loading. Generalized equations that relate stress to strain are called constitutive relations. The simplest form of such a relation is Hooke s Law which relates the stress s to the strain e for rmiaxial deformation of the ideal elastic isotropic solid ... 

For sufficiently small deformation gradients the coordinates of the stress tensor may be approximated by hnearfunctiorrs of the coordinates of the strain tensor. This geometrical and physical hnearization leads to a generahzation of Hooke s law, well known for isotropic bodies. It takes the form... 

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. 

To develop the generalized Hooke s law for isotropic materials it is convenient to construct equations for the strains Cyy, etc. in terms of the applied stresses o xxj ( yy, etc and so define Young s modulus E and Poisson s ratio v. An applied stress Oxx will produce a strain... 

Note 2.7 (Lode s angle and the response of isotropic solids). If the elastic response of solids is written using Hooke s law as... 

We focus on the case of a drop of liquid L sandwiched between a rigid solid S of low energy (e.g., silanized glass) and a semi-infinite rubber R. We assume the elastomer to be homogeneous and isotropic. This material is soft in the sense that it can easily be deformed. If we apply a stress cr, the resulting deformation e is given by Hooke s law ... 

As discussed in the previous section, strains and stresses are closely linked together through material characteristics and deformation mechanisms. In solid mechanic, it is useful to mathematically express this relation through constitutive laws. Also, as aforementioned, numerical modelling now requires three-dimensional constitutive models. One of the simplest expressions of a constitutive law is the Hooke s law (4) in Voigt notation (Mase and Mase, 1999), for isotropic materials. In this case, assuming the small strain theory, only two... 

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