Elastic body, ideal

With respect to the harmonic content of the strain signal, a correction method was therefore developed for torque harmonics, based on observations made when testing an ideal elastic body, for instance, the calibration spring. T(n(x>/ (a) data are corrected according to... 

For each stress component a there exists a corresponding strain component y. Even for an ideally elastic body, however, a pure tension does not produce a pure yss strain y components exist which constnct the body in the y and z directions,... 

Fracture mechanics approach. Fracture mechanics provides the basis for many modern fatigue crack-growth studies. AK is the stress intensity range (kjnilx-kjnm) where K is the magnitude of the mathematically ideal crack-tip stress field in a homogeneous linear-elastic body and is a function of applied load and crack geometry. 

Although a compliance is the inverse of a modulus for an ideal elastic body this is not true for viscoelastic materials. That is,... 

If the solid does not shows time-dependent behavior, that is, it deforms instantaneously, one has an ideal elastic body or a Hookean solid. The symbol E for the modulus is used when the applied strain is extension or compression, while the symbol G is used when the modulus is determined using shear strain. The conduct of experiment such that a linear relationship is obtained between stress and strain should be noted. In addition, for an ideal Hookean solid, the deformation is instantaneous. In contrast, all real materials are either viscoplastic or viscoelastic in nature and, in particular, the latter exhibit time-dependent deformations. The rheological behavior of many foods may be described as viscoplastic and the applicable equations are discussed in Chapter 2. 

Most real bodies are viscoelastic and obey laws (349) and (353) only under certain conditions. Hence, the concept of the stress decay time or the relaxation time x is introduced to characterize the stress-strain state of real bodies. For absolutely elastic bodies, x —> 0, whereas, for ideally viscous bodies, x > oo. Real viscous, anomalous viscous, and viscoelastic media are described in the interval 0 < x < oo. 

An ideal elastic body (also called Hooke s body) is defined as a material that deforms reversibly and for which the strain is proportional to the stress, with recovery to the original volume and shape occurring immediately upon release of the stress. In a Hooke body, stress is directly proportional to strain, as illustrated in Fig. 3. The relationship is known as Hooke s law, and the behavior is referred to as Hookean behavior. 

An ideal elastic material is one which exhibits no time effects. When a stress is applied the body deforms immediately, and it recovers its original dimensions completely and instantaneously when the stress is removed. When the ideal elastic body is subjected to tensile (or compressive) stress, the... 

A compliance is the inverse of a modulus for an ideal elastic body, but this is not true for viscoelastic materials. Consider, for example, two experiments carried out with identical samples of a viscoelastic material. In experiment (a) the sample is subjected to a tensile stress tensile strain at t is j, and the creep compliance measured at that time is Di(t) = edai. In experiment (b) a sample is stressed to a level cr2 such that strain i is achieved immediately. The stress is then gradually decreased so that the strain remains at i for time t (i.e., the sample does not creep further). Let the stress on the material at time t be ay, the corresponding relaxation modulus will be E2(t) = In measurements of this type, it can be expected that cr2>cri>cr3 and (f) HD t). 

In a realistic situation the adhesive filament will not act as a perfect elastic body uniformly stressed up to fracture. Uneven stress distributions and plastic yielding would be expected to increase the energy dissipation observed beyond that calculated for the ideal elastic model. It will be very interesting to see whether in the future auxetic materials can be developed to an extent that they can be used as coatings for such porous substrates. Even greater increases in fracture energy can then be anticipated. 

In terms of rheology ceramic bodies hold a special position between ideal elastic and ideal plastic bodies, as they exhibit Bingham behaviour. Plotted on a shear stress/shearing speed graph, ceramic plastic bodies start to deform only after having reached a certain shear stress tq, the so-called yield point. 

General Point Defects, Lattice atoms can oscillate thermally about their ideal positions. This oscillation can be conceived in terms of the oscillation of an elastic body with the energy hv. Such elastic bodies are called phonons. Electrons and holes are especially important with nonmetallic semiconducting solids. A semiconductor is considered to be perfect when it has an empty semiconducting band. An isolated electron in a perfect solid will, of course, produce a defect. Holes are quantum states in a normally filled semiconducting band. They behave in an electric field like a positive charge. 

An ideal-elastic or energy-elastic body deforms under the influence of a force by a definite amount which does not depend on the duration of the force. For comparison purposes, reference is made, not to the force, but to the force per unit surface area, i.e., the stress. The deformation may be a stretching, shearing, turning, compression, or bending (see Table 11-1). 

In contrast to ideal entropy-elastic bodies, real entropy-elastic bodies have an energy-elastic component. The force Fe resulting from this component is given for a uniaxial deformation by... 

With dynamic measurements, the sample is subjected to a periodic stress. In the simplest case, the stress is applied sinusoidally to an ideal energy-elastic body. The applied stress o t) then alters with time t and the angular frequency CO according to... 

General Point Defects. Lattice atoms can oscillate thermally about their ideal positions. This oscillation can be conceived in terms of the oscillation of an elastic body with the energy hv. Such elastic bodies are called phonons. Electrons and holes are especially important with non-... 

A basic measure of hardness, of course, is the modulus of the material. While the modulus of an ideal elastic body remains independent of the conditions of measurement, the modulus of a real viscoelastic body varies as a function of time and temperature. The modulus decrease with increasing temperature was illustrated in Figures 6.26 and 6.28. Through the application of the concepts embodied in the time-temperature superposition principle, the modulus would be expected to increase with increasing strain rate. 

Such behavior is approximated over limited ranges. The energy input to deform an ideal elastic body is entirely recoverable. In other words, the mechanical energy is conserved. The energy input to deform a Newtonian fluid is completely dissipated into heat. None of the deformation is recoverable. Most materials are measurably... 

The phase of the obtained stress response then shifts by an angle d from the phase of the applied strain. For an ideal elastic body (i.e., spring), the phase between the stress and strain is in phase, namely, = 0° for a Newtonian fluid, <5 = 90° and for a viscoelastic body, d is between 0° and 90° (Fig. 3). By... 

Fig. 3 Relationship between sinusoidal force and deformation for an ideal elastic body, Newtonian liquid, and viscoelastic body.

The theoretical yield strain of was calculated for a ccMlete-ly elastic body deformed by tensile stress in the idealized manner described in the text. We found an average elongation of 24.7 - 0.5 for ten D lon 6/6 diMbell tensile samples equilibrated at 50 relative humidity for three months. When a sample behaves in a conqpletely elastic manner, the load at tensile failure is a measure of the force required to break a specific number of bonds. This number of bonds can be estimated from the original crossectional area and the density. Therefore, the tensile stress, based on the load and this area, has physical meaning only when completely elastic behavior is observed. 

If an ideal elastic body is subjected to a shear stress Oq for a period of time and then the stress is removed, it will follow the strain pattern shown in Figure 2(a) in this case the shear strain Sq = J(To, where J is the compliance. If, on the other hand, the material is viscoelastic the strain will in general be an increasing function of time until the stress is removed, as shown in Figure 2(b). A creep compliance can be defined as... 

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