Hookean body

Hence, we arrive at the conclusion that only in the limit a - 0 the Hookean body is the ideal energy-elastic one (r = 0) and the uniform deformation of a real system is accompanied by thermal effects. Equation (19) shows also that the dependence of the parameter q (as well as to) on strain is a hyperbolic one and a, the phenomenological coefficient of thermal expansion in the unstrained state, is determined solely by the heat to work and the internal energy to work ratios. From Eqs. (17) and (18), we derive the internal energy of Hookean body... 

It can be easily demonstrated that for a Hookean body a thermomechanical inversion of the internal energy (AU = 0) must occur at the deformation... 

For most solids, one can neglect the difference between Pp f (ap f/3 for an isotropic body) and the coefficient of thermal expansion at constant P is usually used. Therefore, we may use P and a without subscripts. Assuming that E and p are independent of temperature and ignoring the change in lateral dimensions during defonnation (i.e. we take the Poisson s ratio p = 0, because this simplification gives effects of only the second order of smallness), one can arrive at relations similar to Eqs. (17)—(21). To do this, it is necessary to replace in Eq. (16) the volume deformation e by e, the modulus K by E and a by p (see Fig. 1). For the simple deformation of a Hookean body the characteristic parameter r is also inversely dependent on strain, viz. r = 2PT/e and sinv = —2PT. It is interesting to note that... 

According to the theory of elasticity, the pure shear and torsion are accompanied only by a change in shape of an elastic body but its volume remains unchanged. The equation of state of the Hookean body for pure shear can be represented in the following form 7,10)... 

Figure 8-9 (A) Stress-Time and (B) Strain-Time Curves of a Hookean Body...

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

In order to derive some simple linear viscoelastic models, it is necessary to introduce the mechanical equivalents for a Newtonian and a Hookean body. 

Fig. 9 Steel spring as the mechanical model for an ideal Hookean body the length of the spring increases proportionally to the force applied, which is here represented by a weight that stretches the spring.

The most popular dynamic test procedure for viscoelastic behavior is the application of an oscillatory stress of small amplitude. This shear stress applied produces a corresponding strain in the material. If the material were an ideal Hookean body, the shear stress and shear strain rate waves would be in phase (Fig. 14A), whereas for an ideal Newtonian sample, there would be a phase shift of 90° (Fig. 14B), because for Newtonian bodies the shear strain is at a maximum, when a maximum of stress is present. The shear strain, when assuming an oscillating sine fimction, is at a maximum in the middle of the slope, because there is the steepest increase in shear strain due to the change in direction. For a typical viscoelastic material, the phase shift will have a value between >0° and <90° (Fig. 14C). 

In the case of an ideal Hookean body with = 0°, the loss compliance is zero and / relates to the elastic energy, which has been stored in the material. No... 

Fig. 14 Oscillating shear stress and resulting shear strain Hookean body (A) Newtonian body (B) viscoelastic material (C).

Hookean body. All constants a and b except Aq and bo are zero. Equation 3.65 becomes... 

Weakly cross-linked polymers are rubberlike above their glass transition temperatures. Such rubbers simultaneously exhibit characteristic properties of solids, liquids, and gases. Like solids, they have dimensional stability and behave as Hookean bodies for small deformations. On the other hand, they possess similar expansion coefficients and moduli of elasticity as liquids. Just as the pressures of compressed gases increase with increasing temperature, the stresses for rubbers also increase (Figure 11-3). In contrast, the stresses increase with decreasing temperature below the glass transition temperature. 

As indicated, the inability to slip makes ceramics more difficult to deform. However, since ceramics behave like a Hookean body until fracture, the known stress-strain relations in elastic deformation can be applied. Assuming that the force, P, is acting normally on a small area, AA, of a ceramic test specimen ... 

Hooke number Hooke-Zahl Hookean bodies Hookesche Kdrper hoop n Reifen, Band, Ring hoop vb... 

Due to their infinitely large size, which confers insolubility and nonfusibility to them, polymer networks cannot be characterized by the traditional methods used for linear and soluble polymers. It is mainly through the study of their mechanical properties (experiments of traction and compression) that one can attain a better knowledge of the structure of networks. As for other solid materials, polymer networks behave like Hookean bodies (within the limit of moderate deformations) that is the deformation is directly proportional to the applied stress. At this stage, it is necessary to make a distinction between rigid networks, made up of crystallized... 

I o denotes the maximum amplitude of this stress and co denotes its frequency. For purely elastic Hookean bodies with no energy dissipated, the strain is written as... 

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