State of strain

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

The basis for the determination of an upper bound on the apparent Young s modulus is the principle of minimum potential energy which can be stated as Let the displacements be specified over the surface of the body except where the corresponding traction is 2ero. Let e, Tjy, be any compatible state of strain that satisfies the specified displacement boundary conditions, l.e., an admissible-strain tieldr Let U be the strain energy of the strain state TetcTby use of the stress-strain relations... 

The active molecule must be regarded as in a state of strain or as a distorted molecule and the ease of distortion or the energy of activation will be dependent on the molecular structure. The following values for the energies of activation as calculated from the temperature coefficients indicate clearly the effect of substitution in a simple molecule on the energy of activation. 

Flory.P.J. Elasticity of polymer networks cross-linked in state of strain. Trans. Faraday Soc. 56,722-743 (1960). 

Kramer,0.,Ty,V, Ferry, J.D. Entanglement coupling in linear polymers demonstrated by networks crosslinked in states of strain. Proc. Natl. Acad. Sci. 69,2216-2218 (1972). 

Kramer, O., Carpenter, R. L., Ty, V., Ferry, J. D. Entanglement networks crosslinked in states of strain, paper presented at Society of Rheology Meeting, Montreal, October 1973. Part of this work is described in Entanglement networks of 1,2-poly butadiene crosslinked in states of strain. I. Cross-linking at 0° C. Macromolecules 7,79-84 (1974), by the same authors. 

A defining characteristic of a solid is the ability to resist shear. Therefore, stress is an additional feature which has to be taken into account when the physical chemistry of solids is at issue. Gibbs treated the thermodynamics of stressed solids a century ago in his classic work Equilibrium of Heterogeneous Substances under the title The Conditions of Internal and External Equilibrium for Solids in Contact with Fluids with Regard to all Possible States of Strain of the Solid . We have already mentioned in the introduction that stress is an unavoidable result of chemical processes in solids. Let us therefore briefly discuss the basic concepts of the thermodynamics of stressed solids. 

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. 

The heterogeneous character of solid surfaces is a matter of the greatest importance to Chemistry, as it is on the exceptional state of strain in certain atoms in the surface that the catalytic properties of surfaces usually depend. This subject will be dealt with in Chapter VII in this chapter, those properties which can be averaged over considerable areas of the solid surface, such as their power of being wetted by liquids, will be considered. The average properties of solids are often of very great industrial importance. 

A plane strain state is defined as a state of strain where the components of the vector displacement take the form... 

Figure 18.2 (a) A state of strain resulting from the stress state in Figure 18.1 in absence of self-diffusion. The strain rate inside the inclusion is uniform, and it is uniform again at points remote from the inclusion, with different principal values, (b) An arbitrary but convenient system for designating points around the periphery. 

As mentioned previously, the boundary condition in DD and FE are different. Periodic boundary condition is used in DD analysis to take into account the periodicity of single crystals whereas confined boundary condition is used in the FE analysis to achieve the uniaxial state of strain. In order for the boundary conditions in FE and DD to be consistent, periodic FE boundary condition is implemented as well. This implementation of periodic FE boundary condition yields a relaxed state of stress with low peak pressure when compared to the experiment as illustrated in Fig. 9(a). Furthermore, both shear and longitudinal waves are generated which is discordant with plane wave characteristics as shown in Fig 9(b). Fig 10 shows the deformed shape when confined and periodic boundary conditions are used. In the confined case there is no distortion in the RVE. However, for the periodic case, considerable... 

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 increase in the TD density in the films grown on relatively thick (6-8 pm) PSC is most probably caused by a specific plastic relaxation process, occurring as a reaction to a particular state of strain that appears in these epitaxial films. This can be stated on the basis of strain inversion in the films grown on PSC, as well as on the increase in compressive stress with the thickness of the PSC layer increasing. These effects show that apart from the stress caused by the GaN/SiC lattice mismatches, an additional built-in stress arises in the films. Obviously, the additional stress is caused by the presence of (0001) PDs, because one can expect that a part of GaN film within the faulted region may have altered its mechanical properties as compared with unfaulted material [72]. Then the increase in dislocation density in GaN grown on relatively thick PSC can be explained by a plastic relaxation process, which relieves the built-in stress and occurs because this internal stress/(0001) PD density reaches a certain critical value. 

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... 

Table 10.3 Current state of strain development of C thermocellum and T. saccharolyticum.

The current state of strain development for ethanol production via C. thermocellum and T. saccharolyticum is summarized in Table 10.3 with reference to key performance metrics. It may be noted that solubilization data and fermentation of high substrate concentrations have been summarized in Table 10.1. Volumetric productivities (g ethanol 1 h ) calculated from the data in Table 10.3 include 0.78gl h for T. saccharolyticum fermentation of mixed cellodextrins and 0.20 gl h for C. thermocellum fermentation of Avicel. Similarly to the wild type, C. thermocellum mutant strains still secrete amino acids into the culture medium, providing a target for further increasing ethanol yields and titers. 

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