Plasticity flow rule

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... 

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 ... 

Not all products adhere to this rule, but the product designer must understand the flow of plastic within the mold. The plastic pressure is highest near the gate, and, due to the restriction in the space between cavity and core, the injection pressure drops as the cavity space is filled with plastic farther away from the gate. This means that, by the time the plastic reaches an area where more plastic is required (such as in a thickening of the product), the pressure available to fill that area is low, and it will be more difficult to fill the cavity and to pack out the product to specification. 

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. 

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. 

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. 

Plastic flow is governed by a flow potential function G([Pg.101]

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... 

The flow rule determines the direction of plastic straining. The flow mle is termed associative if the plastic strain occurs in a direction normal to the yield surface. The plasticity model is termed associated plasticity in the case of associative flow mle. In the Dmcker-Prager plasticity model, the associated plasticity is estabhshed by setting ip = i/r, in which case there will be a volumetric expansion of the material... 

Based on the mathematical theory of plasticity, the plastic deformation behavior of the material can be described by the three components of the rate-independent plasticity model, namely yield criterion, flow rule, and hardening rule. The yield criterion determines the stress level at which yielding is initiated. This is represented by the equivalent stress Ueq, which is a function of the individual stress vector components a. Plastic strain is developed in the metal parts when the equivalent stress is equal to a material yield parameter ay finally, the flow rule determines the direction of plastic straining ... 

The material form of the tangent constitutive tensor is calculated taking the time derivative of Equation (17), considering the flow rule of Equation (20) and the plastic consistency parameter of Equation (22) as Oiler et al. (1996b)... 

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 ... 

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 ... 

The flow rule (2.302) implies that the direction of the plastic strain increment deP is normal to the surface g = constant, and coincides with the stress a. For isotropic materials this can be described as follows. We introduce the unit tensors (see Sect. 2.8.3) as... 

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. ... 

Since the plastic compliance tensor of (2.312), determined by the flow rule, is represented by a product of two second-order tensors, the determinant is identically zero (detC = 0, if we set the second-ordCT tensors as vectors as mentioned in (2.310)). Since it is not possible to obtain the inverse of Cp directly, we use the properties of the elastic compliance C, which has the inverse, along with the direct sum of the strain increment given by (2.293). That is. 

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 ... 

Yield criteria, the subject of sections 3.3.1 to 3.3.3, describe the transition between elastic and plastic behaviour for arbitrary stress states. Next, we will study flow rules that can be used to calculate how the material deforms. 

As we saw in the previous sections, yield criteria can be used to assert for any stress state whether a material yields. How the material deforms plastically is not governed by a yield criterion. The plastic deformation itself is described using flow rules. We will discuss them rather briefly here, a more... 

Furthermore, the relation between stresses and plastic strain rates must be unique. From this, it can be seen that the yield surface must be strictly convex and continuously differentiable to allow the formulation of a flow rule. The Tresca yield criterion is not continuously differentiable (there is no unique normal vector at its corners), and on the surfaces, different stress states fulfil equation (3.42) for a given Therefore, a flow rule cannot be derived using this criterion. 

A concept that has been central to the development of relationships between plastic strain rate and current state of stress, which are the flow rules of plasticity, underlies the postulate of a maximum plastic resistance. This postulate can be stated in the following way. Consider an elastic-plastic material under circumstances in which the state of stress aij satisfies the yield condition = 0. In geometrical terms, is a point on the... 

The components of the viscoplastic strain rate are calculated using the above-mentioned plastic flow rule for rate-sensitive material ... 

From the flow rule of normality principle, the following relationship exists between the plastic strain increment and the plastic stress increment ... 

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],... 

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... 

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