Strain/stress macroscopic

Macroscopic and Intergranular StrainjStress. Similar to the case of texture we can call 8, g) = Oq, (P2) the strain/stress orientation distribution functions (SODF) (a,- is still a placeholder). In contrast to the texture case, the average of the SODF over all variables is not unity but is the macroscopic strain/stress ... 

Fig. 3 A typical situation in which wrinkles occur in the presence of a macroscopic stress is schematically depicted A thin sheet is exposed to a uniaxial macroscopic deformation (1). As a consequence, the sheet is compressed in the direction perpendicular to the elongation axis and the reacts by a buckling instability. Wrinkles are formed, which however relax as the macroscopic strain is released (2), unless plastic deformations occur in the macroscopically stressed state...

In our previous paper (2), we proposed a possible mechanism to interpret the stress-strain behavior of gradient polymers. We perceived the gradient polymer as consisting of infinite number of layers of varying compositions. Upon deformation, the macroscopic strain is the same for the entire sample. Because of the fact that the moduli of the various... 

Low-symmetry crystais posses some interesting, difficuit to assimiiate properties reiat-ing to stress waves and eiasticity. Such properties reiate to what is caiied internai strain , by which an externai stress produces strains in the unit ceii of the crystai different from the macroscopic strain. This can occur if, for exampie, a iattice has at each iattice point a group of atoms (the basis) with symmetry different from that of the iattice. 

The stress state, where the stress can be both applied and residual, and the associated strain influence many different material properties, which is especially important in engineering and technological applications. The residual stress and strain can be advantageous or, on the contrary, can provoke a faster failure of machine parts or other manufactured materials. There are different methods to determine the strain and stress in materials mechanical, acoustical, optical and the diffraction of X-ray and neutrons. The diffraction method is applicable for crystalline materials and is based on the measurements of the elastic strain effects on the diffraction lines. There are two kinds of such effects, a peak shift and a peak broadening. The strain modifies the interplanar distances d. In a polycrystalline specimen a peak shift is produced if the average of the interplanar distance modifications on the crystallites in reflection is different from zero. If the dispersion of interplanar distance modifications is different from zero, then a peak broadening occurs. The effect of the strain on the peak breadth is described in Chapter 13. Here we deal only with the peak shift effect caused by the macroscopic, or Type I strain/stress. There is a substantial amount of literature on this subject. The comprehensive... 

The SODF is itself a macroscopic quantity, as is the following difference, called intergranular strain/stress ... 

To calculate the macroscopic strains and stresses. Equations (112) are averaged over the Euler space. The average acts only on the matrix P and, presuming isotropic polycrystals, one obtains ... 

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. 

Arguments for recent developments of the spherical harmonics approach for the analysis of the macroscopic strain and stress by diffraction were presented in Section 12.2.3. Resuming, the classical models describing the intergranular strains and stresses are too rough and in many cases cannot explain the strongly nonlinear dependence of the diffraction peak shift on sin even if the texture is accounted for. A possible solution to this problem is to renounce to any physical model to describe the crystallite interactions and to find the strain/ stress orientation distribution functions SODF by inverting the measured strain pole distributions ( h(y)). The SODF fully describe the strain and stress state of the sample. 

In this product the strain tensor components in the crystallite reference system are used for the SODFs. With this choice the calculation of the macroscopic strains and stresses e, and x,- requires only the harmonic coefficients of /=0 and 1 = 2 (see Section 12.2.6.3). Similar to the ODF [Equation (23)], the WSODFs are expanded in a series of generalized spherical harmonics ... 

The complicated morphology of crystalline polymer solids and the coexistence of crystalline and amorphous phases make the stress and strain fields extremely nonhomogeneous and anisotropic. The actual local strain in the amorphous component is usually greater and that in the crystalline component is smaller than the macroscopic strain. In the composite structure, the crystal lamellae and taut tie molecules act as force transmitters, and the amorphous layers are the main contributors to the strain. Hence in a very rough approximation, the Lennard-Jones or Morse type force field between adjacent macro-molecular chain sections (6, 7) describes fairly well the initial reversible stress-strain relation of a spherulitic polymer solid almost up to the yield point, i.e. up to a true strain of about 10%. 

In the case that the macroscopical uniform stress o, 0°, 0° is acting at far field of the crack homogenization model, the macroscopical uniform strain °, °, ° is generated. The relation between the strain and the stress is expressed by using compliance A,ju] as... 

Actually, the data to be described in this paper are not primarily designed to monitor surface damage directly rather the influence that this damage has on the static friction of monofilament contacts. The acquisition of these data are an intrinsic part of an attempt to model the mechanical properties of non-woven monofilament assemblies. An assembly of this type accommodates a macroscopic strain by two means which involve internal motion in an element of monofilament which is strained locally by at least two contacts with adjacent monofilaments. The element may distort between fixed contact points. At high strains, however, the stresses on the contact points induce relative motion between monofilaments the critical stress is produced by the friction at the point contact. 

Fig. 9.28 Model results for the normalized equivalent macroscopic flow stress 5e/ro as a function of the equivalent macroscopic strain Se for several deformation histories of tension, plane-strain compression, uniaxial compression, and simple shear compared with experimental results of plane-strain compression (o) and uniaxial compression ( ), where To = 7.8 MPa is the plastic-shear resistance of HDPE (from Lee et al. (1993a) courtesy of Elsevier).

Therefore, surface tension takes the same value as surface energy only when the atom arrangement at the surface is independent of external stress in a reasonable period of time and this conclusion is satisfied when the atom mobility is high enough. The relationship between surface tension and surface energy is determined by the atomic arrangement at the surface and this can be understood by introducing macroscopic strains to a body. 

One sees in Eqs. (6.48) and (6.51) that the strain rate is linearly proportional to the stress and inversely proportional to the grain size. In Eq. (6.48), the expression is given in terms of shear strain and macroscopic shear stress. The above expressions explain why large-grained materials are preferential for creep applications at high temperatures. 

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