Macroscopic Strain and Stress

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 

1 Elastic Strain and Stress in a Crystallite - Mathematical Background 

By definition it is a symmetric second-rank tensor. The stress tensor ffy, i,j= 1,3), is also a symmetric second-rank tensor defined as follows (Landau and Lifchitz ) the element Oy is the i component of the force acting on the unit area normal to the axis x. The symmetry of the stress tensor is imposed by the condition of mechanical equilibrium. 

In different reference systems the strain and stress tensors have different components, the transformation being easily derived starting from the definitions. Let us consider, for example, the sample reference system (y ) and denote by Latin letters etm and stm the components of the strain and stress tensors in this system. If the transformation of the sample reference system (y,) into the crystal reference system (x ) is given by Equation (1) then the transformations of the strain tensors are the following  

Similar transformations occur between t,/ and The d-spacing variation caused by a strain along the lattice vector H hk[) is observable in a diffraction experiment  

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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 ... 

These formulae can be easily implemented in Rietveld codes with e, refinable parameters. In fact they were already implemented in GSAS (profile 5) but the derivation presented in the GSAS manual is different, the concrete connection of the refined parameters with the macroscopic hydrostatic strain and stress not being revealed. 

As known, the macroscopic load-deformation or stress-strain and stress relaxation characteristics of any polymeric material, during and following deformation, are a consequence of the mean molecular motions of its chains ( ). These properties are additionally affected In the case of cartilage by the following (1) many of the tissue s macromolecules are associated In a variety of fibrous and other arrays (7 ) (2) the tissue Is both... 

In a uniaxial tension test to determine the elastic modulus of the composite material, E, the stress and strain states will be assumed to be macroscopically uniform in consonance with the basic presumption that the composite material is macroscopically Isotropic and homogene-ous. However, on a microscopic scSeTBotFTfhe sfre and strain states will be nonuniform. In the uniaxial tension test,... 

Use the bounding techniques of elasticity to determine upper and lower bounds on the shear modulus, G, of a dispersion-stiffened composite materietl. Express the results In terms of the shear moduli of the constituents (G for the matrix and G for the dispersed particles) and their respective volume fractions (V and V,j). The representative volume element of the composite material should be subjected to a macroscopically uniform shear stress t which results in a macroscopically uniform shear strain y. 

The macroscopic upper yield stress is lineary related to Tg — T for a given strain rate, and can be adjusted with Argon s, Bowden s, and Kitagawa s models. The influence of strain rate is well represented by the phenomenological Eyring s equation. 

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...

On the basis of what has been discussed, we are in the position to provide a unified understanding and approach to the composite elastic modulus, yield stress, and stress-strain curve of polymers dispersed with particles in uniaxial compression. The interaction between filler particles is treated by a mean field analysis, and the system as a whole is macroscopically homogeneous. Effective Young s modulus (JE0) of the composite is given by [44]... 

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. 

In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio X of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n, of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n 5 por polymer SI). 

Since then, TEM has been used to study dislocation microstructures in a wide range of naturally and experimentally deformed minerals and rocks. In general, the aim of the experimental studies is to determine the deformation mechanisms by relating the evolution of the observed mi-crostructures to the macroscopic deformational behavior observed under varying conditions of temperature, confining pressure, chemical environment, strain-rate, stress, and total strain, and then to use this knowledge to interpret the microstructures observed in naturally deformed specimens and hence to determine their deformational history. 

The previous sections have been mostly concerned with the dislocations and microstructures observed in single crystals deformed to various strains under known experimental conditions. In some minerals, notably quartz and olivine, the macroscopic deformational behavior, as revealed by the creep and stress-strain curves, can be understood in terms of the micro-structural evolution during deformation and, furthermore, certain quantifiable characteristics of the microstructure correlate with the imposed... 

The behavior of the material under high stresses and strains in the microregion at the crack tip should also reflect specific features determined in macroscopic experiments. In thermoplastic materials the dependence on strain and time is of prime importance. These questions will be addressed in the following section. 

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