Macroscopic stresses

FIGURE 21.2 Macroscopic stress-strain (S-S) curve of natural rubber (NR) vulcanizate. 

G. Therefore, in a polycrystal, the macroscopic stress needed general plastic deformation is ... 

A close inspection of the right-hand side of eq. (2.34) reveals that it is simply equal to twice the excess free energy of the considered chain in a flowing system compared with that in a system at rest. By summing up the contributions of all chains per unit of volume, one obtains for the first invariant of the macroscopic stress tensor ... 

As a consequence, the given proof for the validity of the stress-optical law remains formally true. The same holds for the relation between the diagonal components of the macroscopic stress tensor and the stored free energy per unit of volume. In fact, it does not make any difference, whether this energy is thought to be built up of the contributions of all complete chains or all subchains contained in the unit of volume. Only one statement will be revized, viz. that with respect to the coil expansion of the entire chain. A detailed discussion of this point will be given in Section 3.3. 

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...

Fig. 4 In order to obtain wrinkles that are stable in the absence of macroscopic stresses, a macroscopic substrate of large thickness has to be modified in stressed state as explained below. While the substrate does not react to compressive strains by wrinkling (1), the thin membrane that was created in the stressed state wrinkles when the substrate is relaxed (2). While the system is still under tension microscopically, no macroscopic stress is necessary to maintain the wrinkle structure and long-term stable wrinkles can be created...

Having constructed the microscopic mesh, we specify the microscopic problem based on the macroscopic nodal displacements. The displacements of the elemental boundaries are given by the macroscopic solution (although the internal microscopic scale displacements are not necessarily affine). The microscopic problem is to find node positions and segment lengths such that the boundary nodes are as specified by the macroscopic displacements and the internal nodes experience no net force. The boundary nodes have displacement specified and are subjected to a non-zero net force. The next step in the solution process is to convert those forces into the macroscopic stress tensor. 

In our current model, the stress is assumed to be constant over the element (i.e., the stress is treated as piecewise constant on the macroscopic scale). We therefore assign the macroscopic stress a value equal to the spatial average of the microscopic stress ... 

For simplicity, let us consider perfectly drained conditions (p = 0) and start from an equilibrium between solid and solute (xjrc — xj/x). The equilibrium is disturbed by application of a constant macroscopic stress X = <5 ( > 0). 

The title of the book, Optical Rheometry of Complex Fluids, refers to the strong connection of the experimental methods that are presented to the field of rheology. Rheology refers to the study of deformation and orientation as a result of fluid flow, and one principal aim of this discipline is the development of constitutive equations that relate the macroscopic stress and velocity gradient tensors. A successful constitutive equation, however, will recognize the particular microstructure of a complex fluid, and it is here that optical methods have proven to be very important. The emphasis in this book is on the use of in situ measurements where the dynamics and structure are measured in the presence of an external field. In this manner, the connection between the microstructural response and macroscopic observables, such as stress and fluid motion can be effectively established. Although many of the examples used in the book involve the application of flow, the use of these techniques is appropriate whenever an external field is applied. For that reason, examples are also included for the case of electric and magnetic fields. 

Hence, the macroscopic deformation A plays the role of an external driving force. The macroscopic stress (a) can be defined as the average local stress over the solid portion Vs of the unit cell... 

Among possible macroscopic dynamical quantities of interest, the most useful is the macroscopic stress,... 

To conclude this subsection, we expose an interesting paradox arising from the time dependence of the particle configuration. As discussed in Section III, Frankel and Acrivos (1967) developed a time-independent lubrication model for treating concentrated suspensions. Their result, given by Eq. (3.7), predicts singular behavior of the shear viscosity in the maximum concentration limit where the spheres touch. Within the spatially periodic framework, the instantaneous macroscopic stress tensor may be calculated for the lubrication limit, e - 0. The symmetric portion of its deviatoric component takes the form (Zuzovsky et al, 1983)... 

The previous paragraph has made it clear that if there are elastic fibers and a constant macroscopic stress is applied, the longitudinal creep rate will eventually fall to zero. With constant transverse stresses applied as well, the process of transient creep will be much more complicated than that associated with Eqns. (27) and (28). However, it can be deduced that the longitudinal creep rate will still fall to zero eventually. Furthermore, any transverse steady creep rate must occur in a plane strain mode. During such steady creep, the fiber does not deform further because the stress in the fiber is constant. In addition, any debonding which might tend to occur would have achieved a steady level because the stresses are fixed. 

The shear stress transmitted to a fiber is limited to the shear strength r. As a result, the formula given in Eqn. (63) is valid only up to a composite macroscopic stress of... 

The specimen is composed of a mixture of matrix, unbroken fibers, and broken fibers. The volume fraction of intact fibers is given by Eqn. (57) with L = Ls, the specimen length. To the neglect of transients, the macroscopic stress supported by these intact fibers is given by Eqn. (60). The strain will now exceed the level of Eqn. (61) associated with the ultimate strength of the fiber bundle. Therefore, the stress supported by the intact fibers will be less than ac, which is the ultimate strength of the fiber bundle without matrix. The applied stress exceeds composite material to creep. 

The modulus and yield kinetic parameters of the block polymer B can be related to those of the homopolymer in terms of a microcomposite model in which the silicone domains are assumed capable of bearing no shear load. Following Nielsen (10) we successfully applied the Halpin-Tsai equations to calculate the ratio of moduli for the two materials. This ratio of 2 is the same as the ratio of the apparent activation volumes. Our interpretation is that the silicone microdomains introduce shear stress concentrations on the micro scale that cause the polycarbonate block continuum to yield at a macroscopic stress that is half as large as that for the homopolymer. The fact that the activation energies are the same however indicates that aside from this geometric effect the rubber domains have little influence on the yield mechanism. 

A detailed atomistic approach was used to investigate the molecular segment kinematics of a glassy, atactic polypropylene system dilated by 30%. ° The microstructural stress—dilation response consists of smooth, reversible portions bounded by sudden, irreversible stress jumps. But compared to the micro-structural stress—strain curve of the shear simulation, the overall trend more closely resembles macroscopic stress—strain curves. The peak negative pressure was in the neighborhood of 12% dilatation, with a corresponding secondary maximum in the von Mises shear stress. The peak negative pressure was re-... 

The macroscopic stress cr is this force times the number of interparticle bonds that cross a unit area of the sample this latter factor should scale as 0 /a (Russel et al. 1989). As long as the local applied force increases with increased strain, cr increases with increasing strain, and the gel maintains its mechanical stability. But once the strain reaches the point -that the slope W of the potential is a maximum (see Fig. 7-23), any further strain produces a decreasing force, and the interparticle structure breaks apart. This corresponds to the point of yield. Thus, the yield strain yy is given by the condition that the second derivative W of W D) is zero that is, W" Dy) = 0, where Dy = lyyU + (yy + 1)Dq is the value of D for which W" = 0. Very roughly, we might expect that W is a maximum (W — 0) when separation D = Dy % on the order of twice Dq, the value of D at static equilibrium. This would imply that the yield strain yy is roughly hence, for particles 100 nm in... 

To find the stress tensor in the same system we place Equation (87) into Equation (73a) and one obtains an expression similar to Equation (84b). To obtain the macroscopic stress we must integrate this expression over the Euler space. The integral acts only on the single-crystal stiffness tensor elements Equation (74a) and can be calculated analytically. The macroscopic stress is ... 

Here and Sy are the trace of s and, respectively, the macroscopic stress along y ... 

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. 

The elements of the macroscopic stress tensor Sj have exactly the same expressions only the matrix g must be replaced by g defined as follows ... 

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