Strains second order

At higher strains second-order terms cannot be neglected. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

Stress and strain are both second-order tensors. 

The stiffness and compliances in stress-strain and strain-stress relations are fourth-order tensors because they relate two second-order tensors ... 

Contracted notation is a rearrangement of terms such that the number of indices is reduced although their range increases. For second-order tensors, the number of indices is reduced from 2 to 1 and the range increased from 3 to 9. The stresses and strains, for example, are contracted as in Table A-1. Similarly, the fourth-order tensors for stiffnesses and compliances in Equations (A.42) and (A.43) have 2 instead of 4 free indices with a new range of 9. The number of components remains 81 (3 = 9 ). 

It is worth to be noted that these definitions of first- and second-order distortions according to Warren-Averbach are model-free. From a linear or a quadratic increase of peak breadths it can neither be concluded in reverse that strain broadening, nor that paracrystalline disorder were detected. 

The second-order fluctuating rate-of-strain tensor is real and symmetric. Thus, its three eigenvalues are real and, due to continuity, sum to zero. The latter implies that one eigenvalue (a) is always positive, and one eigenvalue (y) is always negative. In the turbulence literature (Pope 2000), y is referred to as the most compressive strain rate. 

Tensile Modulus. Tensile samples were cut from the 0.125 in. plates of the compositions according to Standard ASTM D638-68, into the dogbone shape. Samples were tested on an Instron table model TM-S 1130 with environmental chamber. Samples were tested at temperatures of -30°C, 0°C. 22°C, 50°C, 80°C, 100°C and 130°C. Samples were held at test temperature for 20 minutes, clamped into the Instron grips and tested at a strain rate of 0.02 in./min. until failure. The elastic modulus was determined by ASTM D638-68. Second order polynomial equations were fitted to the data to obtain the elastic modulus as a function of temperature for each of the compositions. 

It should be noted that the theory described above is strictly vahd only close to Tc for an ideal crystal of infinite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Flere the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc. The reason for this might also be technical It is extremely difficult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat flow that can only occur when the temperature gradient is different from zero. 

The elastic excitation mode (strain mode) is the soft mode in many of the second-order CJTE transitions. 

Figure 4.25 Comparison of the different order parameters for the 151 K second-order transition of PrAlOj. (After Sturge et al., 1975.) Unbroken line is the smooth curve through the internal displacement order parameter, cos 2(, from ESR measurements. Black circles represent the electronic order parameter from optical absorption studies. Squares represent the reduced macroscopic strain from elastic neutron scattering.

Reconstructive phase transitions occur when major changes are made in the topology, i.e. when the bond graph is reorganized. The transitions usually observed in structures with lattice-induced strain are displacive and often second order (no latent heat). Reconstructive transitions arise when two quite different structures with the same composition have similar free energies. Unlike the displacive transitions they involve the dissolution of one structure and the recrystallization of a quite different structure. These phase transitions possess a latent heat and often display hysteresis. 

Spectrum simulation treated the hyperfine interactions by second order perturbation theory and there were distributions in D and E/D, because strain in these parameters dominated the spectra. Spectral features grow in up to 1 equivalent of added Mn(II) at geff = 15.4, 5.3, 3.0 and 2.0 (Bi 1B) and a broad signal with a... 

Usually the analysis is limited to the first order in the strain (harmonic approximation) and second order terms (anharmonic coupling) are neglected. This second order magnetoelasticity has not been analyzed for the compounds under consideration. 

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... 

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