The associated-flow rule

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 

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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. 

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

In what concerns incompressibility, several researchers stated after measuring volume variation of different polymers, that there is negligible volume variation after yielding and concluded that the normality rule, typical of associated plasticity, does not hold in case of the pressure dependent yield surfaces of polymers (Whitney and Andrews, 1967). This conclusion was confirmed by Spitzig and Richmond (1979) who showed that the associated flow rule based on a pressure sensitive yield surface leads to predictions... 

By considering the rigid-plastic constitutive equations derived from the associated flow rule (Equation 8.2) under plane strain conditions ... 

Since ao is arbitrary, the above condition is satisfied if g = / (the associated flow rule) and / is convex (Fig. 2.20). This gives a strong restriction for elasto-plastic materials, especially for granular media, since most of the experimental data show that if we apply the associated flow rule with a yield function, such as the Coulomb or Drucker-Prager type, the dilatancy (i.e., the volume change due to shearing) is over-estimated. ... 

Let us apply the associated flow rule (g = /) and assume that / is an w-th order homogeneous function, so that the work-hardening rule (2.320) can be written as... 

A normality rule is assumed on the yield locus f p, q) = 0 (namely, the associated flow rule) as shown in Fig. 6.5, so that we have... 

In this case, the associated flow rule (7.55) takes the form... 

It follows from Eq. (3) that v can be expressed, in the subset Bt of tangential components, as the associated flow rule... 

In 3D, the stress dilatancy equation is not sufficient to determine the strain increment tensor and the flow rule becomes necessary. In this case, the stress-dilatancy equation is considered as a constraint condition for strain increments. Kanatani [2] proposed a modified associated flow rule having a constraint condition on the deformation, and he states that the differentiation in the associated flow rule is to be made in keeping the constraint force constant. The constraint force is a force to make work with the deformation, which must disappear by the given constraint condition. Thus, as is seen in Eq.(3l)> de = 0 is the constaint condition and p" is the corresponding constraint force, and by the condition of stress path, p is kept constant in the considered decomposition, as is shown in Eq.(l9) ... 

The plastic dissipation occurs along thin interface planes which are velocity discontinuity lines. Both tangential and normal velocity discontinuity occur for shear zones corresponding to Coulomb yield condition. Only normal velocity discontinuity occurs for planes where tension failure condition is satisfied. Using the associated flow rule (3), the rate of dissipation D per unit area of discontinuity plane can be expressed as follows, cf.Izbicki and Mr6zr3l,... 

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. 

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) ... 

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. 

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 ... 

The angle of internal friction q> can also be a function of the internal state variable for hardening hypothesis. The initial angle of internal friction is given by (Po- The flow rule is given by a general non-associated flow rule g f, with the plastic potential given by... 

By knowledge of the total strain the only remaining unknown variable is eP. In other words, the decomposition of g into an elastic and plastic part must be determined, see Figure 4. For this reason Drucker s postulate can be used, which yields, inter alia, an associated flow rule as follows ... 

The yield criteria and associated flow rules follow the modified versions of the Tresca and von Mises criteria that were proposed by Whimey and Andrews (1967), Raghava et al. (1973) and Caddell et al. (1974). 

Since both the original and modified Cam clay models employ an associated flow rule, the plastic potential function g is the same as the yield function /. We use (6.88) and (6.98), and obtain the hardening coefficient h as follows ... 

The Levy-Mises equations define one of a number of possible flow rules that can be derived via an argument that depends upon a concept known as the plastic potential. This idea has been discussed by Hill [ 15]. It is assumed that the components of the plastic strain increment tensor are proportional to the partial derivatives of the plastic potential, which is a scalar function of stress. The flow rule can thus be generated by this differentiation process. We may choose to assume, for a particular form of yield criterion, that the plastic potential has the same functional form as the yield criterion then, the derived flow rule is described as being associated with the yield criterion (or as an associative flow rule). However, this assumption is not obligatory and when it is not true we will be applying a yield criterion together with a non-associated flow rule. This is discussed further by de Souza Neto etal. [19],... 

The principle of maximum plastic work for granular materials is explained by using newly proposed decompositions of stress and strain increment tensors. In forming the decompositions both the condition of stress path and the stress-dilatancy equation are taken into account. The 3D stress-dilatancy equation in a tensorial form, which is a natural extension of the form in 2D, is proposed. The application of the modified associated flow rule for obtaining the strain increment tensor in 3D is explained by virtue of the proposed decompositions. 

Consequently, the modified associated flow rule is expressed as... 

It is further assumed that the material is isotropic and initially homogeneous. The limit analysis method is based upon the idea of perfectly plastic behaviour of solids in which the strain rate tensor CiJ s related to the stress tensor S iJ by the potential flow rule associated with the yield condition F(6ij) = 0... 

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