Stress-free strains

Fig. 2.14. Energy as a function of shear for a generic crystalline solid. The periodicity of the energy profile reflects the existence of certain lattice invariant shears which correspond to the stress-free strains considered in the method of eigenstrains (adapted from Tadmor et al. (1996)).

In mathematical terms, the implementation of the eigenstrain concept is carried out by representing the geometrical state of interest by some distribution of stress-free strains which can be mapped onto an equivalent set of body forces, the solution for which can be obtained using the relevant Green function. To illustrate the problem with a concrete example, we follow Eshelby in thinking of an elastic inclusion. The key point is that this inclusion is subject to some internal strain e, such as arises from a structural transformation, and for which there is no associated stress. 

Suppose that the piezoelectric thin film is bonded to an isotropic elastic substrate that does not exhibit piezoelectric behavior. If this film-substrate system is exposed to an electric field 8, then the film material will tend to undergo the stress-free strain specified by (3.83). However, the film is constrained from deforming freely by the substrate. As a result, a stress is generated in the film, and this stress may result in a detectable curvature of the substrate. If the X3—axis is normal to the film-substrate interface then the components of mismatch strain in the film are, in reduced notation. 

Upon comparing the two terms on the left side of this equation, it is noted that the first term is essentially an elastic strain times the second term. Because elastic strain is assumed to be very small compared to one, the first term is negligible compared to the second and its contribution is ignored. If the differential equation is expressed in terms of the stress free strain along the film rj t) = (b t) — bo)/bo and it is then linearized for r/(t) 1, it is... 

The general issue of stability of composition of a solid solution is pursued further in the next subsection. Two potentially important physical effects are not taken into account in the discussion of energy variations with composition above. One of these effects arises from the possibility of atomic misfit of one species in the solution with respect to the other. The average unit cell dimension of a solid solution may depend on the composition, so that there is a stress-free volume change (or a more complex stress-free strain, perhaps) with change in concentration. For a spatially nonuniform composition, the associated stress-free strain field will be incompatible, in general, giving rise to a residual stress distribution. 

The elasticity is assumed isotropic in the domain. According to Vegard s law [45], the stress-free strain is isotropic and depends linearly on the composition ... 

Some possible non-DC source processes, such as polymorphic phase transformation, occur throughout a finite volume rather than on a surface. The equivalent force system often can be expressed in terms of the stress-free strain A eij, which is the strain that would occur in the source volume if the tractions on its boundary were held constant by externally applied artificial forces (It might more accurately be called the fixed-stress strain. ). By reasoning that involves a sequence of imaginary cutting, straining, and welding operations (e.g., Aki and Richards 2002, Sect 3.4), the moment tensor of such a volume source is found to be... 

The stress-free strain is not, in general, the strain that actually occurs in a seismic event (Richards and Kim 2005). Because the source is imbedded in the Earth, its deformation is resisted by the stiffness of the surrounding medium, making the tme strain changes smaller than the... 

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 ). 

Obviously, the number of free indices no longer denotes the order of the tensor. Also, the range on the indices no longer denotes the number of spatial dimensions, if the stress and strain tensors are symmetric (they are if no body couples act on an element), then... 

As indicated earlier, protective oxide scales typically have a PBR greater than unity and are, therefore, less dense than the metal from which they have formed. As a result, the formation of protective oxides invariably results in a local volume increase, or a stress-free oxidation strain" . If lateral growth occurs, then compressive stresses can build up, and these are intensified at convex and reduced at concave interfaces by the radial displacement of the scale due to outward cation diffusion (Fig. 7.7) . 

The problem-solving approach that ties the processing variables to products properties includes considering melt orientation, polymer degradation, free volume/molecular packing and relaxation, cooling stresses, and other such factors. The most influential of these four conditions is melt orientation, which can be related to molded-in stress or strain. 

We conclude that high internal stresses are generated by simple shear of a long incompressible rectangular rubber block, if the end surfaces are stress-free. These internal stresses are due to restraints at the bonded plates. One consequence is that a high hydrostatic tension may be set up in the interior of the sheared block. For example, at an imposed shear strain of 3, the negative pressure in the interior is predicted to be about three times the shear modulus p. This is sufficiently high to cause internal fracture in a soft rubbery solid [5]. 

Most mechanical and civil engineering applications involving elastomers use the elastomer in compression and/or shear. In compression, a parameter known as shape factor (S—the ratio of one loaded area to the total force-free area) is required as well as the material modulus to predict the stress versus strain properties. In most cases, elastomer components are bonded to metal-constraining plates, so that the shape factor S remains essentially constant during and after compression. For example, the compression modulus E. for a squat block will be... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

After introduction of cross-links in the strained state, the composite network retracts, upon release, to a stress-free state-of-ease (J9 ) The amount of retraction is determined by the degree of strain during cross-linking and by the ratio >i/v2. The elastic properties relative to the state-of-ease are isotropic for a Gaussian composite network ( 8, 1 9,20). 

It is, of course, necessary to have an accurate measure of d, the stress-free spacing. The strain results can then be converted into stress using a suitable value of the stiffness (e.g., see Ref. 11). 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

T1 under two different compressive stresses. Phase fraction of martensite is proportional to the permanent strain which can be determined by the stress-free specimen length. From Burkart and Read [16],... 

The relationship between stress and strain in a test piece with bonded end pieces is very dependent on the shape factor of the test piece. This is usually defined as the ratio of the loaded cross-sectional area to the total force-free area (Figure 8.15). The larger the shape factor the more stiff the rubber appears and this property is much exploited in the design of rubber springs and mountings. 

Apply several stresses or strains, starting low and working up. If possible, monitor the output signals of the transducer or position sensor while applying test input signals. When the signal is relatively noise free, repeat that step as a complete test. 

Under an applied voltage the stress-free filaments in a bent bimorph will be in two planes, one above and one below the central plane and at equal distances a from it. The centre plane is strain free because the stresses on opposite sides of the joining layer are equal and opposed to one another. The steps in the argument are illustrated in Fig. 6.29 and the stress system is similar to that set up in a heated bimetallic strip, as analysed by S. Timoshenko [20]. 

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