Scalar strain rate

The scalar strain rate is defined as the octahedral strain rate, which for small strains is given by... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

Direct numerical simulation studies of turbulent mixing (e.g., Ashurst et al. 1987) have shown that the fluctuating scalar gradient is nearly always aligned with the eigenvector of the most compressive strain rate. For a fully developed scalar spectrum, the vortexstretching term can be expressed as... 

For fast equilibrium chemistry (Section 5.4), an equilibrium assumption allowed us to write the concentration of all chemical species in terms of the mixture-fraction vector c(x, t) = ceq( (x, 0). For a turbulent flow, it is important to note that the local micromixing rate (i.e., the instantaneous scalar dissipation rate) is a random variable. Thus, while the chemistry may be fast relative to the mean micromixing rate, at some points in a turbulent flow the instantaneous micromixing rate may be fast compared with the chemistry. This is made all the more important by the fact that fast reactions often take place in thin reaction-diffusion zones whose size may be smaller than the Kolmogorov scale. Hence, the local strain rate (micromixing rate) seen by the reaction surface may be as high as the local Kolmogorov-scale strain rate. 

In words, (ef (t) is the mean scalar dissipation rate for all particles with the same strain-rate history. The reader should appreciate that two fluid particles can have different values of ) (f), even though they share the same value of a> (t). 

Ashurst, W. T., A. R. Kerstein, R. M. Kerr, and C. H. Gibson (1987). Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence. The Physics of Fluids 30, 2343-2353. 

Systems approach borrowed from the optimization and control communities can be used to achieve various other tasks of interest in multiscale simulation. For example, Hurst and Wen (2005) have recently considered shear viscosity as a scalar input/output map from shear stress to shear strain rate, and estimated the viscosity from the frequency response of the system by performing short, non-equilibrium MD. Multiscale model reduction, along with optimal control and design strategies, offers substantial promise for engineering systems. Intensive work on this topic is therefore expected in the near future. 

For gaseous flames, the LES/FMDF can be implemented via two combustion models (1) a finite-rate, reduced-chemistry model for nonequilibrium flames and (2) a near-equilibrium model employing detailed kinetics. In (1), a system of nonlinear ordinary differential equations (ODEs) is solved together with the FMDF equation for all the scalars (mass fractions and enthalpy). Finite-rate chemistry effects are explicitly and exactly" included in this procedure since the chemistry is closed in the formulation. In (2). the LES/FMDF is employed in conjunction with the equilibrium fuel-oxidation model. This model is enacted via fiamelet simulations, which consider a laminar counterflow (opposed jet) flame configuration. At low strain rates, the flame is usually close to equilibrium. Thus, the thermochemical variables are determined completely by the mixture fraction variable. A fiamelet library is coupled with the LES/FMDF solver in which transport of the mixture fraction is considered. It is useful to emphasize here that the PDF of the mixture fraction is not assumed a priori (as done in almost all other flamelet-based models), but is calculated explicitly via the FMDF. The LES/FMDF/flamelet solver is computationally less expensive than that described in (1) thus, it can be used for more complex flow configurations. 

Newton s law of motion for liquids describes a linear relationship between the deformation of a fluid and the corresponding stress, as indicated in Equation 22.16, where the constant of proportionality is the Newtonian viscosity of the fluid. The generalized Newtonian fluid (GNF) refers to a family of equations having the structure of Equation 22.16 but written in tensorial form, in which the term corresponding to viscosity can be written as a function of scalar invariants of the stress tensor (x) or the strain rate tensor (y). For the GNF, no elastic effects are taken into account [12, 33] ... 

The scalar product of a stress tensor and a strain rate tensor T E is determined by equation (5.31). Then (5.184) takes the form... 

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

It is important to note that in the expressions in eqs. (3.32)-(3.36) the shear rate is to be considered as a scalar-average deviatoric rate, which is related to a reference tensile equivalent strain rate by... 

The range of material behavior considered next is broadened significantly by appeal to the notion of a plastic rate equation as a model for any possible physical mechanism of deformation that may be operative. The ideas will be developed for general states of stress, but will be applied primarily for the case of thin films in equi-biaxial tension. Constitutive relationships that serve as models for inelastic response of materials for a wide variety of physical mechanisms of deformation have been compiled by Frost and Ashby (1982). These constitutive equations are represented as scalar equations expressing the inelastic equivalent strain rate /3e in terms of the effective stress (Tm/ /3 and temperature T. These strain rate and stress measures are denoted by 7 and as by Frost and Ashby (1982), and the rate equations representing models of material behavior all take the form... 

Equations (12.17) and (12.18) can be used in three-dimensional stress analyses on the basis that they give scalar rates of plastic strain, which can be converted to tensor strain rate components via the use of a flow rule. The shear stress r is defined as the octahedral shear stress in terms of the principal stresses... 

Equation (3.26) gives a relaxation modulus that is independent of the applied strain magnitude, and will obey the homogeneity requirement of linearity for all scalars, with any arbitrary strain input. Equation (3.26) is not linear however as norms are not superposable except in the most trivial examples. The hypothetical material represented by Equation (3.26) is therefore non-linear, but for many types of tests used for material characterization it could not be distinguished from a linear viscoelastic material. In fact, the parameters n and P appearing in this constitutive equation can be adjusted so that the time derivative of the stress for a constant strain rate input is proportional to the relaxation modulus a commonly cited property of a linear viscoelastic material [12,37]. Careful examination of the stress output to various strain inputs confirms the non-linear nature of this equation and indicate it is within the range of this simple equation to describe the onedimensional response of solid propellants at small strains. To demonstrate this ability, the stress output for a variety of strain inputs have been determined for different values of n and p. 

GNF-based constitutive equations differ in the specific form that the shear rate dependence of the viscosity, t](y), is expressed, but they all share the requirement that the non-Newtonian viscosity t](y) be a function of only the three scalar invariants of the rate of strain tensor. Since polymer melts are essentially incompressible, the first invariant, Iy = 0, and for steady shear flows since v = /(x2), and v2 V j 0 the third invariant,... 

In the last chapter we discussed the relation between stress and strain (or instead rate-of-strain) in one dimension by treating the viscoelastic quantities as scalars. When the applied strain or rate-of-strain is large, the nonlinear response of the polymeric liquid involves more than one dimension. In addition, a rheological process always involves a three-dimensional deformation. In this chapter, we discuss how to express stress and strain in three-dimensional space. This is not only important in the study of polymer rheological properties in terms of continuum mechanics " but is also essential in the polymer viscoelastic theories and simulations studied in the later chapters, into which the chain dynamic models are incorporated. 

Here, is any scalar field quantity such as a Cartesian component of stress or strain, and d is the increment of crack advance in the xi direction. Finally, Hutchinson (1974) derived the following path-independent integral that can be use to determine the crack tip energy release rate Gtip during steady crack growth. 

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