Flow rules

In a given motion, a particular material particle will experience a strain history The stress rate relation (5.4) and flow rule (5.11), together with suitable initial conditions, may be integrated to obtain the eorresponding stress history for the particle. Conversely, using (5.16) instead of (5.4), may be obtained from by an analogous ealeulation. As before, may be represented by a continuous curve, parametrized by time, in six-dimensional symmetric stress spaee. 

Arthington AH, Bunn SE, LeRoy Poff N et al (2006) The challenge of providing environmental flow rules to sustain river ecosystems. Ecol Appl 16 1311-1318... 

Table 2. Maximum disposable tunnel water flow rules far the Gotthard Rase Tunnel (Switzerland),...

In pressure-swing processes, it Is quite common to use a fraction of the less-adsorbed gas product as a low-pressure purge gas, as shown In Figure 2. Often the purge flow is in the opposite direction from the feed flow. Rules for the minimum fraction of less-adsorbed gas product for displacing... 

Eq. (8.24), and the modified Mohr-Coulomb yield (or failure) criterion, Eq. (8.27). It should be noted that other yield criteria, such as the von Mises criterion, are used to model the flow of bulk solids in hoppers, and more conditions may need to be imposed, such as the Levy flow rule, in order to close the system of equations [Cleaver and Nedderman, 1993],... 

The constitutive model used to describe large plastic deformations of glassy polymers involves a separate formulation for temperatures above and below the glass transition Tg, since the underlying deformation mechanisms are different. In either regime, the formulation is based on the decomposition of the rate of deformation into an elastic part and a plastic part Z)P so that 0 = 0° + D. By assuming an isotropic yield stress, the isochoric plastic strain rate is given by the flow rule... 

Normally, the permeability of a porous medium can be related to other porous medium properties through a model of the porous medium structure. Pragmatically, the complexity of the flow rules out any rigorous analytical attempts to resolve the problem. Ideally, one would like to use heuristic arguments to derive an expression for k or F in terms of universal constants and easily measurable properties of the porous material and the flowing fluid. Such attempts have been made by many researchers in the past. A large collection of these studies has been... 

We use a non-associated plastic flow rule in order to better describe the transition from plastic contractance to dilatance. Based on the previous work by Pietruszczak et al. (1988), the following function is adopted ... 

For this part of the model the flow rule is non-associated... 

The incremental strain-stress relation is obtained following the standard procedure in which the consistency condition, the flow rule and the hardening rule are used. This relation reads... 

Material Flow Rule Cutting processes of metallic materials involve highly complex interactions between plastic material behavior, strain rate, temperature, and material microstructure. Even in conventional cutting, very high equivalent strain rates of approximately 10 s are... 

Associated flow rule used to obey a metal plasticity rule for rock like material. Also, sweeping assumptions made to homogenization of fractures due to demand of the continuous damage modelling. 4GPa is applied as input for shear modulus for a rook with 52 GPa elastic modulus and 0.33 Poisson s ratio. 

The condition of the associated-flow rule is depicted as a vector def parallel to the outward normal to the yield surface where a — Y. 

It is informative to note that when the deformation resistance of a solid becomes significantly rate-dependent, but a tensile stress-strain relation F(eP,eP) at different strain rates is still available for different cases of monotonic straining at different strain rates eP, without interruptions, holdings, or reversals in the deformation, the associated-flow-rule relations of eqs. (3.28a-f) can still serve as a useful guide. However, dealing with more complex resistances and paths of... 

Gurson, A. L. (1977) Continuum theory of ductile rupture by void nucleation and growth part I - yield criteria and flow rules for porous ductile media, J. Eng. Mater. Technol., 99, 1-15. 

When determining the radial displacements in the plastic zone, a plastic potential needs to be specified in advance. However, different-form plastic potentials have significant influences on dilatant plastic deformations (Zienkiewicz et al. 1975). In this study the dilatant plastic deformations are assumed to be related to stress levels. A non-linear non-associated flow rule is employed (Clausen Damkilde 2008) ... 

In continuum mechanics, constitutive modeling of materials follows certain steps, including deformation response, stress response, as well as other particular steps based on materials studied, such as structural relaxation for polymers and a plastic flow mle, and a hardening rule for materials with plastic deformation. In the following, we will present the deformation response, structural relaxation, stress response, and flow rule for the thermosetting SMP programmed by cold-compression programming. 

More recently, Nguyen et al. [20] further extended the viscous flow rule to a structure dependent glass transition region by introducing the Active temperature 7 into the temperature dependence ... 

Equation (5.3) represents the magnitude of the amorphous inelastic strain rate, while the direction of the amorphous inelastic flow rate,, is governed by the deviatoric driving stress, s y. The following flow rule is proposed for the inelastic deformation in the amorphous phase [88] ... 

The well-established elastic-predictor/plastic-corrector return mapping algorithm can be utilized to obtain the inelastic responses of the microscale amorphous and crystalline phases. Here, we only outline the steps to be used. A detailed description of this solution algorithm can be foimd in References [103] to [105]. The return mapping technique is capable of handling both associative and nonassociative flow rules with variant tangent stiffnesses and results in a consistent solution approach [105]. It is noted that this algorithm is applicable to the material, intermediate, or spatial formulations. 

An example of a material model based on the physics of material behavior is classical metals plasticity theory. This theory, often referred to as /2-flow theory, is based on a Mises yield surface with an associated flow rule, followed by rate-independent isotropic hardening (Khan and Huang 1995). Physically, plastic flow in metals is a result of dislocation motion, a mechanism known to be driven by shear stresses and to be insensitive to hydrostatic pressure. 

The yielding and plastic flow of the material is captured using the tensorial flow rule (Bergstrom et al. 2002b, Bergstrom, Rimnac, and Kurtz 2003) ... 

Onsager showed theoretically that the coefficients Ly must be symmetrical. To ensure the non-nej tivity of the dissipation, it suffices to require Ly to be definite positive, other than being symmetiicaL The off-diagonal coefficients allow to account for cross-couphngs. This formulation seems to be better suited to moderately non-hnear problems. For example, it cannot lead to the classical plastic flow rule in solids. 

Note that the associative flow rule renders the tangent matrix Dgp symmetric. This... 

However, there are two effective stresses a and a", which is confusing. The situation will be optimum if we can assume either b = b, hence a = a", or matrix incompressibility which implies b = b = 1 and that a = a" = cr + pi. The last case is of particular importance and corresponds to the majority of cases in soils.The above flow rule is known as associative since the strain rate is normal to the yield surface, with the advantage that the nonnegativity of the dissipation is always satisfied. Geomateiials exhibit complex volumetric behaviours and sometimes call for non associative flow rules ... 

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