The state of strain

For convenience, these nine quantities are regrouped and denoted as follows  

The first three quantities en, C22 and C33 correspond to the fractional expansions or contractions along the 1,2 and 3 axes of an infinitesimal element at X - the normal strains. The second three quantities C23, 3i and C12 correspond to the components of shear strain in the 23, 31 and 12 planes respectively. The last three quantities wi, m2 and mj, do not correspond to a deformation of the element at X, but are the components of its rotation as a rigid body. 

2mi =9i —62 does not correspond to a deformation of ABCD but to twice the angle through which AC has been rotated. 

Therefore, the deformation is defined by the first six quantities 611, 22, 33, 23, 3i 12 that are called the components of strain. It is important to note that engineering strains have been defined, fn Chapter 3, we take a more general approach and examine a number of strain-related tensor quantities in Section 3.1.5. For the purposes of this chapter, which concerns small strains, we define the strain tensor, as 

In elementary elasticity, which is concerned with the elastic behaviour of isotropic materials, it is usual to consider two types of strain only. First, there is extensional 

For deformation of a material, which involves extensions and shears superimposed in quite general directions, we require a more general starting point to define extensional and shear strains, i.e. components of extensional and shear strain, analogous to the components of tensile and shear stress. 

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The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

To investigate the effect of the porous layer on the state of strain in the epitaxial film, the radius of curvature Rc of the GaN/SiC and GaN/PSC samples using XRD was measured [55,56]. The measurements showed that GaN films grown on nonporous SiC experienced biaxial tensile stress, which resulted from film/substrate mismatches. In contrast, the films grown on PSC were compressed. The Rc measurements also revealed that with increase of the thickness of the PSC layer the value of the compressive stress increased. 

The other extreme of behavior involves the "phantom chain" approximation. Here, it is assumed that the individual chains and crosslink points may pass through one another as if they had no material existence that is, they may act like phantom chains. In this approximation, the mean position of crosslink points in the deformed network is consistent with the affine transformation, but fluctuations of the crosslink points are allowed about their mean positions and these fluctuations are not affected by the state of strain in the network. Under these conditions, the distribution function characterizing the position of crosslink points in the deformed network cannot be simply related to the corresponding distribution function in the undeformed network via an affine transformation. In this approximation, the crosslink points are able to readjust, moving through one another, to attain the state of lowest free energy subject to the deformed dimensions of the network. 

The phenomenological theory, as its name implies, concerns itself only with the observed behavior of elastomers. It is not based on considerations of the molecular structure of the polymer. The central problem here is to find an expression for the elastic energy stored in the system, analogous to the free energy expression in the statistical theory [equation (6-72)]. Consider again the deformation of our unit cube in Figure 6-3. In order to arrive at the state of strain, a certain amount of work must be done which is stored in the body as strain energy ... 

Koh et al. [6] have rigorously modeled the electromechanics of this interaction for the simplified case of uniform biaxial stretching of an incompressible polymer film including many important effects such as the nonlinear stiffness behavior of the polymer film and the variation in breakdown field with the state of strain. With regard to the latter effect, Pelrine et al. [5] showed the dramatic effect of prestrain on the performance of dielectric elastomers (specifically silicones and acrylics) as actuators. We would expect the same breakdown enhancement effects to be involved with regard to power generation. There are many additional effects that may be important, such as electrical and mechanical loss mechanisms, interaction with the environment or circuits, frequency, and temperature-dependent effects on material parameters. The analysis by Koh provides the state equations... 

It is shown in the appendix that the state of stress within a homogeneously stressed material can be represented by a symmetric second-rank tensor [cr,y] with six independent components. For a cube with faces perpendicular to OT1T2T3 the component cr represents the outwardly directed force per unit area applied in the direction parallel to Ox,- on the face perpendicular to Ox,. It thus represents a normal stress. The component a,y for i j represents the force per unit area applied in the Ox, direction to the face perpendicular to Ox,-, so that it represents a shear stress. It is also shown in the appendix that the state of strain in the material can be... 

These transformation equations show that as the axes are rotated, the magnitude of the strain components will change, i.e., the state of strain at a point depends on the reference axes. Consider the example of a transformation, depicted in Fig. 2.21, in which a square (bold) is elongated by a strain e along the x axis and is contracted by e along the x axis. The problem is to determine the deformation of the (bold) diamond shape that is inscribed at 45 in the square. From the rotation of the axes, one finds that... 

Most of the experimental results obtained confirm the stress or strain profile along the fibre length in agreement with Cox approach (Figure 7). However, interfacial failure at the fibre ends modifies the strain profile and thus can be directly evidenced by Raman spectrometry. Moreover, the state of strain (strain mapping) of fibres into real composite due to residual thermal stresses can also be determined. 

Without question the results summarized here afford just a first glimpse of a rich field in which the magnetism of epitaxial films responds in an interesting and sensitive manner to the epitaxial constraint. The actual state of strain in this limit depends on both the film thickness and the growth conditions. In turn the magnetic state must depend on the state of strain and other factors that may influence, for example, the magnetic domain structure, in addition to the natural variables of field and temperature. An eventual complete description must include the statistical behavior of the spin-slip system. 

As the thickness of a film is reduced, the pseudomorphism is improved and the state of strain generally becomes more uniform. At the same time, however, the definition of magnetic phase structure may be complicated by boundary effects, and the sharp symmetry distinctions among alternative phases is blurred. For a recent review of thin-film magnetism see the article by Falicov et al. (1990), which deals mainly with transition... 

Third, and finally, it has been established that the lanthanide magnetism is in general remarkably robust, and in particular is insensitive to the interfaces, even in crystals only a few atomic layers thick. A reservation of critical importance in this regard is the central role of the state of strain in the description of the magnetic behavior. Specifically it has been established for Dy that epitaxial strains 2% are sufficient to double the Curie temperature or completely suppress the ferromagnetic phase. The twin assets of robustness and strain sensitivity make these materials at one time both ideal systems with which to explore epitaxial effects, and attractive models with which new states of magnetic order may be designed and synthesized. 

Figure 1. Entanglement network cross-linked while strained in simple extension showing the unstretched state,X., the state of strain, X, and the equilibrium state of ease, X. ...

Moreover, the critical current density depends strongly on the state of strain in Nb3Sn, in particular close to Bc2- Figure 4.2-15 shows the strain sensitiv-... 

Quantitative evaluation of the stress-strain characteristics of the rubber network then involves calculation of the configurational entropy of the whole assembly of chains as a function of the state of strain. This calculation is considered in two stages calculation of the entropy of a single chain and calculation of the change in entropy of a network of chains as a function of strain. 

Perhaps the most important feature of the BKZ model, which distinguishes it from all the models discussed so far, is the choice of strain measure. Hitherto, all the materials have been assumed to be solids, in that they have an initial undeformed, stress-free state that acts as a reference relative to which all strained states are measured. In BKZ theories there is no such special state, and the material therefore may be classed as a flmd. At any present time t, the state of strain is measured relative to the state at previous times t. This is done by adopting as the strain measure the quantity A(f)/A(T). Reflecting on the remarks in the paragraph above now suggests the importance of the quantity [A (r)/A (r)] — [A(T)/A(f)] in a theory in which the stress depends on the strain history. The BKZ form given by Zapas and Craft [21] in uniaxial stretching is... 

By syaaetry the two coaponents of displaceaents in any plain section of bodies of revolution along their axis of syaaetry define coapletely the state of strain and, therefore the state of stress. So, first of all, we look for the relation between displaceaents and strain. Froa Zienkiewicz [61 we have ... 

The same equation was derived by Gibbs [6] using the internal energy. This equation clearly applies to equilibrium between any two phases. For the special cases where a = y, e.g. both phases are fluids, Equation (6.33) reduces to the case obtained for planar interfaces, i.e. the chemical potentials are equal. Both y and a appear in Equation (6.33). Cahn [7] has pointed out that those processes which change the surface (interface) area cause y to be introduced and those which change the state of strain (pressure) cause cr to be introduced. 

These formulas, like the corresponding ones in Chapter 5, imply small deformations and are approximations based on simplified representations of the states of strain. Corrections may be made for edge effects and bulging of the sides of shear sandwich samples of appreciable thickness, and for a more complete description of the state of strain in a twisted rod of rectangular cross-section. ... 

The state of strain in large deformations is commonly described either by the principal extension ratios, Xi, X2, X3, deflned in the notation of Chapter 1 as X,- = 1 + Uilxi, with the coordinate axes suitably oriented, or by three strain invariants whose values are independent of the coordinate system. In simple extension, Xi = 1 + e, where e is the (practical) tensile strain U jx cf. equation 8 above), not to be confused with the e in equations 3,4 and 6. Most of this section is concerned with simple extension. 

This result, however, is not yet complete. For an incompressible solid like a rubber, superposition of a hydrostatic pressure onto the other applied external forces leaves the shape of the sample and thus the state of strain unchanged. We can account for this arbitrariness by including the undetermined hydrostatic pressure, denoted p, as a further component in the equation and rewrite it as... 

The Boltzmann superposition principle represents the stress as a result of changes in the state of strain at previous times. In the linear theory valid for small strains, these can be represented by the linear strain tensor. In Lodge s equation the changes in the latter are substituted by changes in the time dependent Finger tensor... 

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