Ideal elastic material

For an ideally elastic material, the stress is linearly related to the strain by... 

For ideal elastic materials 8 = 0, whereas for purely viscous fluids 8 = %/2 rad. 

An ideal elastic material is one which exhibits no time effects. When a stress o is applied the body deforms immediately to a strain t. (These terms were defined broadly in Section 1.8). The sample recovers its original dimensions completely and instantanously when the stress is removed. Further, the strain is always proportional to the stress and is independent of the rate at which the body is deformed ... 

Fig. 1. Crack opening and tensile stress near the crack tip in an ideal elastic material...

From the foregoing, one infers that the state of an ideal elastic material under small deformations is totally defined once two of the four parameters E, G, K, and v are known. The relations between the elastic parameters can be obtained from the expressions indicated above. The pertinent relations are given in Table 4.1 (5). 

The complex relationship between the configurational distortion produced by a perturbation field in polymers and the Brownian motion that relaxes that distortion make it difficult to establish stress-strain relationships. In fact, the stress at a point in the system depends not only on the actual deformation at that point but also on the previous history of deformation of the material. As a consequence the relaxation between the stress and strain or rate of strain cannot be expressed by material constants such as G or /, as occurs in ideal elastic materials, but rather by time-dependent material functions, G t) and J t). It has been argued that the dynamics of incompressible liquids may be characterized by a function of the evolution of the strain tensor from the beginning up to the present time. According to this criterion, the stress tensor would be given by (3,4)... 

For an Ideally elastic material the ratio between stress and the accompan3dng strain is called the modulus G (units N m ). The elastic counterpart of (3.6.2] reads... 

The net work of compaction is a function of material properties. For example, the energy needed to deform an ideal elastic material will be completely recovered during the decompression phase and there will be zero net work compact. On the other hand, for... 

The ideal elastie response is typified by the stress-strain behavior of a spring. A spring has a constant modulus that is independent of the strain rate or the speed of testing stress is a funetion of strain only. For the pure Hookean spring the inertial effects are neglected. For the ideal elastic material, the mechanical response is deseribed by Hooke s law ... 

As Equation 14.7 shows, the Maxwell element is merely a linear combination of the behavior of an ideally elastic material and pure viscous flow. Now let us examine the response of the Maxwell element to two typical experiments used to monitor the viscoelastic behavior of polymer. 

In fact, concrete is not an ideal elastic material. Under constant stress, strain will increase as the time increases, which is known as creep effect. Creep is not only connected with load time T, but also connected with concrete s age t. The creep degree is usually be expressed as a variable by the formula as follow (Bofang Zhu. 1999) ... 

Polymeric liquids have a microstructure that is like springs representing the linear chain. Restoration of these springs to their equilibrium state is through the elastic energy that is stored during the deformation process. But polymeric fluids are not ideal elastic materials, and they also have a dissipative reaction to deformation, which is the viscous dissipation. For small deformations, the response of the system is linear, meaning that the response is additive effect of sum of two small deformations is equal to the sum of the two individual responses. 

An ideal elastic material where stress is directly proportional to strain. 

Criteria for understanding and characterizing the mechanical patency of an implant, (a) An ideally elastic material with indefinite fatigue life, and (b) and (c) visco-elastic materials with limited fatigue life. 

The stress a in an ideally elastic materials is uniquely related to a specific strain y, a = Gy (where G is the shear modulus), whereas the stress in a Newtonian fluid is uniquely related to the rate of strain, y = dy/dt,... 

A sinusoidal stress applied to an ideal elastic material produces a sinusoidal strain proportional to the stress amplitude and in phase with it. For ideal viscous materials the stress and strain are out of phase by 90°. Figure 15 gives an example of a stress-strain diagram for a sinusoidal stress applied to a real material. The amplitude of the deformation (strain) in response to the stress is proportional to that of the stress, but lags behind the strain curve by some angle 5 between 0 and 90°, depending on the elastic/viscous characteristic of the material. This behavior is usually analyzed by the use of complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, x and y > respectively, are... 

Keeping in mind that our aim is to describe the mechanical behaviour of a piezoelectric element we start by using the model of an ideal elastic material. The basic property of this model is that the Cauchy stress tensor at an arbitrary material poirrt at a certain moment depends only on the deformation gradierrt at this same poirrt at the same moment. This implies that a rigid body motion cannot produce ary stresses. It is also presupposed that the elastic properties of the rtraterial are irrde-pendent of time. Therefore, the stresses are independent of the strairring rate as well as of previous treatment, in short of the history of the rrraterial. 

Loss modulus represents the loss of energy as heat during the deformation of materials. When loss modulus becomes low, damping loss factor decreases, which means the material is closer to ideal elastic material. The loss modulus of maleicing PLA/PBAT blends is higher than that of both PLA/PBAT blends wifliout maleicing and plain PLA, which means that better impact resistance of modified PLA foams can be achieved. 

Figure 5.10. DMA sinusoidal stress-strain response curves for a typical viscoelastic material illustrating the phase shift between strain and stress (Sperling 2006) (a) vectorial representation (b) hysteresis effect for a viscoelastic material—area within the ellipse represents viscous heat loss (c) an ideal elastic material shows no heat loss. (Reproduced with permission of JohnWiley Sons.)...

Polymeric materials often tend to behave as springs (that is, they have a tendency to retract on stretching), thus displaying some degree of elasticity. An ideal elastic material responds instantaneously to application or removal of stress, with the strain (y) being proportional to the stress (Hookean), independently of the strain rate [Eq. (33)]. Here, the constant G is the modulus of the elastic material. 

A few results could be found in the literature for the relation between F/N and the radius of a hemispherical or spherical slider. When this radius R was [20] 0.285, 1-33, and 10.0 cm, respectively, F/N was 0.4, 0.46, and 0.57- The slider was of glass, the polymer was Nylon-66, and N was constant at 10 dynes. From Hertz s theory for ideally elastic materials, the product wd is independent of R (w is proportional to the cubic root of R and d is inversely proportional to this root). It is seen that F/N also was little affected by the radius when R rose in the ratio of 35 to 1, F/N increased only in the ratio 1.4 to 1. Presumably it was not quite constant because Nylon-66 was not a Hookean solid. The change of the track width with R also seems to confirm the poor applicability of Hertz s equation to nylon this width increased less steeply than did the cubic root of R. 

Silicon is a nearly ideal elastic material, so its intrinsic quality factor is very high. However, this quality factor is only realized if the device is operated in high vacuum, where Q may be 10,000 or more. Otherwise the damping effect of a surrounding medium reduces Q. In air, it is... 

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