Ethan Clarke
Plasticity is a key concept in materials science and engineering that deals with the ability of a solid material to undergo permanent deformation, or flow, under the influence of external forces. The flow rule is a fundamental principle in plasticity theory that relates the plastic strain (change in shape) of a material to the applied stress (external force). It plays a crucial role in understanding and predicting the behaviour of materials under different loading conditions, such as metal plasticity and rock plasticity. The associated flow rule, which assumes equivalence between the plastic potential and yield functions, is commonly accepted for most metals. However, in cases where the plastic potential differs from the yield function, a non-associated flow rule is applied. The flow rule also forms the basis for more advanced theories and models, such as those incorporating hardening laws, pressure sensitivity, and variational constitutive updates, to better describe the complex behaviour of materials under plastic deformation.
| Characteristics | Values |
|---|---|
| Plastic Flow Rule | A relation that encapsulates the assumption that the plastic strain increment and deviatoric stress tensor have the same principal directions |
| Associated Flow Rule | A rule where the plastic potential and yield functions are assumed to be equivalent |
| Non-associated Flow Rule | A rule where the plastic potential and yield functions are assumed to be different |
| Plastic Strain | Induced by the phenomenon of plastic flow |
| Plastic Multiplier | A positive scalar that represents the magnitude of plastic strain |
| Plastic Potential Function | Defines the direction of plastic strain vectors |
| Plastic Deformation | Assumed to occur in the undamaged area of damaged materials |
| Plastic Flow Analysis | A method to predict warpage in injection-molded parts |
| Plastic Viscosity | Can be calculated using the power-law equation |
| Ridging | A phenomenon caused by plastic flow that results in the formation of deep ridges on gear teeth |
| Elastic Limit | Defined by a yield surface that does not depend on plastic strain |
| Prager Consistency Condition | Needed to close the set of constitutive equations and eliminate unknown parameters |
| Deformation Tensor | Can be decomposed into elastic and plastic parts |
| Deformation Gradient Tensor | Can be multiplicatively decomposed into elastic and plastic parts |
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What You'll Learn
The plastic flow rule and evolution equations for internal variables
The plastic flow rule is a fundamental concept in the study of plasticity, particularly in metal plasticity. It encapsulates the relationship between the plastic strain increment and the deviatoric stress tensor, assuming they share the same principal directions. This relationship is expressed as:
> {\displaystyle d{\boldsymbol {\varepsilon }}_{p}=d\lambda \,{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}}
Where dλ is a hardening parameter. This specific form of the flow rule is termed an associated flow rule, and the co-directionality assumption is known as the normality condition.
Rock plasticity theories adopt a similar concept, but with a crucial difference. Due to the pressure-dependence of the yield surface, the assumption of co-directionality needs to be relaxed. Instead, the focus is on the relationship between the plastic strain increment and the normal to the pressure-dependent yield surface:
> {\displaystyle d{\boldsymbol {\varepsilon }}_{p}>0}
This equation signifies that the yield surface remains constant, even with increasing plastic deformation.
The associated flow rule's validity is supported by examining the work done during a cycle of plastic loading-unloading. To complete the set of constitutive equations, the Prager consistency condition is essential:
> {\displaystyle df={\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}:d{\boldsymbol {\sigma }}+{\frac {\partial f}{\partial {\boldsymbol {\varepsilon }}_{p}}}:d{\boldsymbol {\varepsilon }}_{p}=0\,.}
Large deformation flow theories of plasticity often start with two key assumptions: the additive decomposition of the rate of deformation tensor into elastic and plastic parts, or the multiplicative decomposition of the deformation gradient tensor into similar components. The former assumption was initially popular for metal simulations but has been largely replaced by the multiplicative theory.
The evolution equations for internal variables in the context of plastic flow rules are particularly relevant in rate-dependent plasticity. In this scenario, the plastic rate parameter is derived as an empirical function of stress and internal variables. This function is typically an overstress function, as described by Perzyna in 1971. The equation for the overstress function in J2 plasticity flow is provided as a reference:
> {\displaystyle {\begin{aligned}\dot{\varepsilon}_{p}={\frac {\sqrt{3}}{2}}{\frac {\left(\sigma -\sigma _{y}\right)}{\eta }}\end{aligned}}
Where σy is the yield stress under uniaxial tension.
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The plastic flow rule in metal plasticity
Plasticity is a critical aspect of materials science, and understanding how materials respond to stress beyond their elastic limit is essential for engineering and design applications. The plastic flow rule is a fundamental concept in metal plasticity, describing the relationship between plastic strain increments and the deviatoric stress tensor. This rule is encapsulated in an equation known as the flow rule, which assumes that these quantities have the same principal directions.
The flow rule equation, often referred to as Eq. (4.86), includes a positive scalar called the plastic multiplier (p) and the plastic potential function (g). The plastic multiplier represents the magnitude of plastic strain, while the plastic potential function defines the direction of plastic strain vectors. This equation is fundamental to understanding the plastic flow rule and its applications in metal plasticity.
It's important to distinguish between associated and non-associated flow rules. An associated flow rule assumes that the plastic potential function is the same as the yield function, resulting in co-directional plastic strain increments and the normal to the pressure-dependent yield surface. This assumption is known as the normality condition. On the other hand, a non-associated flow rule assumes that the plastic potential function differs from the yield function, leading to distinct directions for the plastic strain increments.
The plastic flow rule is not limited to a single plastic mechanism. Eq. (4.86) indicates that a single mechanism is considered, but in practice, multiple plastic mechanisms may be relevant, each associated with a specific plastic potential function. This flexibility allows for a more comprehensive modelling of the mechanical behaviour of materials, particularly geomaterials, where non-associated flow rules may offer a more accurate representation.
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The plastic flow rule in rock plasticity
Plasticity refers to the irreversible deformation of a material without fracture. In the context of rock plasticity, it is important to distinguish it from metal plasticity, where the size of a dislocation is sub-grain size, whereas in soil, plasticity arises from the relative movement of microscopic grains. Rock mass plasticity specifically refers to the study of rock responses to loads beyond the elastic limit.
The associated flow rule is expressed mathematically as:
> d{\boldsymbol {\varepsilon }}_{p}=d\lambda \,{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}}
Where dλ is a hardening parameter. The yield surface remains constant even as plastic deformation increases.
The validity of the associated flow rule can be justified by examining the work done over a cycle of plastic loading and unloading. The Prager consistency condition is necessary to close the set of constitutive equations and eliminate unknown parameters.
In rock plasticity, the plastic potential function may differ from the yield function, leading to a nonassociated flow rule. This type of flow rule assumes that the principal axes of the plastic strain tensor do not align with those of the stress tensor. Nonassociated flow rules can be more suitable for modelling the mechanical behaviour of geomaterials accurately.
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The associated flow rule
The validity of the associated flow rule can be justified through the examination of work done over a plastic loading-unloading cycle. The Prager consistency condition plays a vital role in closing the set of constitutive equations and eliminating unknown parameters. The associated flow rule also serves as a foundation for developing phenomenological plastic flow rules, which are valuable for simulating the behaviour of elastoplastic responses in rate-independent polycrystals.
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The non-associated flow rule
The associated flow rule is a commonly accepted law in the theory of plastic deformation of most metals, assuming equivalency between the plastic potential and yield functions. However, this rule has limitations, especially in predicting formability in complex stamping processes and spring-back in general. This has led to the proposal of alternate material models, such as the non-associated flow rule, which is the focus of this discussion.
One of the key advantages of the non-associated flow rule is its ability to account for pressure-sensitive mechanisms. For example, Spitzig and Richmond (1983) observed a small pressure-sensitivity in the yield criterion for aluminium and steel that was not accompanied by the expected plastic dilatancy as predicted by the associated flow rule. By adopting a non-associated flow rule, Lee (1988) successfully developed a self-consistent material model that explained the Spitzig and Richmond experiments.
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Frequently asked questions
In metal plasticity, the flow rule encapsulates the assumption that the plastic strain increment and deviatoric stress tensor share the same principal directions.
The associated flow rule assumes equivalency between the plastic potential and yield functions. It is commonly accepted as a valid law in the theory of plastic deformation of most metals.
The non-associated flow rule is when the plastic potential is assumed to differ from the yield function. In this case, the plastic potential function usually has a mathematical expression similar to the yield function, but with differences in coefficients related to material properties.
The plastic flow rule relates the direction of plastic strain rate to the direction of strain rate. It also yields the reduction of the shear modulus after plastic yielding.
The non-associated flow rule proposed by Spitzig and Richmond is significant because it fully accounts for the strength differential effect (SDE). This is important as the SDE is sometimes described through kinematic hardening models using only pressure-insensitive yield criteria.
