en>Yobot: WP:CHECKWIKI error fixes using AWB (9475)

2013-09-11T01:52:00Z

WP:CHECKWIKI error fixes using AWB (9475)

New page

[[File:Biegeanimation 2D.gif|300px|thumb|Plastic deformation of a thin metal sheet.]]
'''Flow plasticity''' is a [[solid mechanics]] theory that is used to describe the [[plasticity (physics)|plastic]] behavior of materials.<ref name=lub>{{Citation|last=Lubliner|first=Jacob|year=2008|title=Plasticity Theory|publisher=Courier Dover Publications.}}</ref> Flow plasticity theories are characterized by the assumption that a [[flow rule (plasticity)|flow rule]] exists that can be used to determine the amount of plastic deformation in the material.

In flow plasticity theories it is assumed that the total [[deformation (mechanics)|strain]] in a body can be decomposed additively (or multiplicatively) into an elastic part and a plastic part. The elastic part of the strain can be computed from a [[linear elasticity|linear elastic]] or [[hyperelastic material|hyperelastic]] constitutive model. However, determination of the plastic part of the strain requires a [[flow rule (plasticity)|flow rule]] and a [[hardening model (plasticity)|hardening model]].

== Small deformation theory ==
[[File:Rock plasticity compression plain.svg|thumb|right|300px|Stress-strain curve showing typical plastic behavior of materials in uniaxial compression. The strain can be decomposed into a recoverable elastic strain (<math>\varepsilon_e</math>) and an inelastic strain (<math>\varepsilon_p</math>). The stress at initial yield is <math>\sigma_0</math>. For strain hardening materials (as shown in the figure) the yield stress increases with increasing plastic deformation to a value of <math>\sigma_y</math>.]]
Typical flow plasticity theories (for small deformation perfect plasticity or hardening plasticity) are developed on the basis on the following requirements:
# The material has a linear elastic range.
# The material has an elastic limit defined as the stress at which plastic deformation first takes place, i.e., <math>\sigma = \sigma_0</math>.
# Beyond the elastic limit the stress state always remains on the yield surface, i.e., <math>\sigma = \sigma_y</math>.
# Loading is defined as the situation under which increments of stress are greater than zero, i.e., <math>d\sigma > 0</math>. If loading takes the stress state to the plastic domain then the increment of plastic strain is always greater than zero, i.e., <math>d\varepsilon_p > 0</math>.
# Unloading is defined as the situation under which increments of stress are less than zero, i.e., <math>d\sigma < 0</math>. The material is elastic during unloading and no additional plastic strain is accumulated.
# The total strain is a linear combination of the elastic and plastic parts, i.e., <math>d\varepsilon = d\varepsilon_e + d\varepsilon_p</math>. The plastic part cannot be recovered while the elastic part is fully recoverable.
# The work done of a loading-unloading cycle is positive or zero, i.e., <math>d\sigma\,d\varepsilon = d\sigma\,(d\varepsilon_e + d\varepsilon_p) \ge 0</math>. This is also called the [[Drucker stability]] postulate and eliminates the possibility of strain softening behavior.

The above requirements can be expressed in three dimensions as follows.
* '''Elasticity''' ([[Hooke's law]]). In the linear elastic regime the stresses and strains in the rock are related by
:::<math>
\boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}
</math>
:::where the stiffness matrix <math>\mathsf{C}</math> is constant.
* '''Elastic limit''' ([[Yield surface]]). The elastic limit is defined by a yield surface that does not depend on the plastic strain and has the form
:::<math>
f(\boldsymbol{\sigma}) = 0 \,.
</math>
* '''Beyond the elastic limit'''. For strain hardening rocks, the yield surface evolves with increasing plastic strain and the elastic limit changes. The evolving yield surface has the form
:::<math>
f(\boldsymbol{\sigma}, \boldsymbol{\varepsilon}_p) = 0 \,.
</math>
* '''Loading'''. It is not straightforward to translate the condition <math>d\sigma > 0</math> to three dimensions, particularly for rock plasticity which is dependent not only on the [[deviatoric stress]] but also on the [[mean stress]]. However, during loading <math>f \ge 0</math> and it is assumed that the direction of plastic strain is identical to the [[surface normal|normal]] to the yield surface (<math>\partial f/\partial\boldsymbol{\sigma}</math>) and that <math>d\boldsymbol{\varepsilon}_p:d\boldsymbol{\sigma} \ge 0</math>, i.e.,
:::<math>
d\boldsymbol{\sigma}:\frac{\partial f}{\partial \boldsymbol{\sigma}} \ge 0 \,.
</math>
:::The above equation, when it is equal to zero, indicates a state of '''neutral loading''' where the stress state moves along the yield surface without changing the plastic strain.
* '''Unloading''': A similar argument is made for unloading for which situation <math> f < 0 </math>, the material is in the elastic domain, and
:::<math>
d\boldsymbol{\sigma}:\frac{\partial f}{\partial \boldsymbol{\sigma}} < 0 \,.
</math>
* '''Strain decomposition''': The additive decomposition of the strain into elastic and plastic parts can be written as
:::<math>
d\boldsymbol{\varepsilon} = d\boldsymbol{\varepsilon}_e + d\boldsymbol{\varepsilon}_p \,.
</math>
* '''Stability postulate''': The stability postulate is expressed as
:::<math>
d\boldsymbol{\sigma}:d\boldsymbol{\varepsilon} \ge 0 \,.
</math>

=== Flow rule ===
In metal plasticity, the assumption that the plastic strain increment and deviatoric stress tensor have the same principal directions is encapsulated in a relation called the [[flow rule]]. Rock plasticity theories also use a similar concept except that the requirement of pressure-dependence of the yield surface requires a relaxation of the above assumption. Instead, it is typically assumed that the plastic strain increment and the normal to the pressure-dependent yield surface have the same direction, i.e.,
:<math>
d\boldsymbol{\varepsilon}_p = d\lambda\,\frac{\partial f}{\partial \boldsymbol{\sigma}}
</math>
where <math>d\lambda > 0</math> is a hardening parameter. This form of the flow rule is called an [[associated flow rule]] and the assumption of co-directionality is called the [[normality condition (plasticity)|normality condition]]. The function <math>f</math> is also called a [[plastic potential]].

The above flow rule is easily justified for perfectly plastic deformations for which <math>d\boldsymbol{\sigma} = 0 </math> when <math>d\boldsymbol{\varepsilon}_p > 0</math>, i.e., the yield surface remains constant under increasing plastic deformation. This implies that the increment of elastic strain is also zero, <math>d\boldsymbol{\varepsilon}_e = 0</math>, because of Hooke's law. Therefore,
:<math>
d\boldsymbol{\sigma}:\frac{\partial f}{\partial \boldsymbol{\sigma}} = 0 \quad \text{and} \quad d\boldsymbol{\sigma}:d\boldsymbol{\varepsilon}_p = 0 \,.
</math>
Hence, both the normal to the yield surface and the plastic strain tensor are perpendicular to the stress tensor and must have the same direction.

For a [[strain hardening|work hardening]] material, the yield surface can expand with increasing stress. We assume Drucker's second stability postulate which states that for an infinitesimal stress cycle this plastic work is positive, i.e.,
:<math>
d\boldsymbol{\sigma}: d\boldsymbol{\varepsilon}_p \ge 0 \,.
</math>
The above quantity is equal to zero for purely elastic cycles. Examination of the work done over a cycle of plastic loading-unloading can be used to justify the validity of the associated flow rule.<ref>Anandarajah (2010).</ref>

=== Consistency condition ===
The [[Prager consistency condition]] is needed to close the set of constitutive equations and to eliminate the unknown parameter <math>d\lambda</math> from the system of equations. The consistency condition states that <math>df = 0 </math> at yield because <math> f(\boldsymbol{\sigma},\boldsymbol{\varepsilon}_p) = 0 </math>, and hence
:<math>
df = \frac{\partial f}{\partial \boldsymbol{\sigma}}:d\boldsymbol{\sigma} + \frac{\partial f}{\partial \boldsymbol{\varepsilon}_p}:d\boldsymbol{\varepsilon}_p = 0 \,.
</math>

== Large deformation theory ==
Large deformation flow theories of plasticity typically start with one of the following assumptions:
* the [[rate of deformation]] tensor can be additively decomposed into an elastic part and a plastic part, or
* the [[deformation gradient]] tensor can be multiplicatively decomposed in an elastic part and a plastic part.

The first assumption was widely used for numerical simulations of metals but has gradually been superseded by the multiplicative theory.

=== Kinematics of multiplicative plasticity ===
The concept of multiplicative decomposition of the deformation gradient into elastic and plastic parts was first proposed independently by B. A. Bilby,<ref>{{Citation|last1=Bilby|first1=B. A.|last2=Bullough|first2=R.|last3=Smith|first3= E.|year=1955|title= Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry|journal= [[Proceedings of the Royal Society A]]|volume= 231|pages= 263–273.|issue=1185|bibcode=1955RSPSA.231..263B|doi=10.1098/rspa.1955.0171}}</ref> E. Kroner,<ref>{{Citation|last=Kroner|first=E.|title=Kontinuumstheorie der versetzungen und eigenspannungen|journal=Erg. Angew. Math.|volume= 5 |year=1958|pages=1–179.}}</ref> in the context of [[crystal plasticity]] and extended to continuum plasticity by Erasmus Lee.<ref>{{Citation|last=Lee|first= E. H. |year=1969|title=Elastic-Plastic Deformation at Finite Strains|journal= Journal of Applied Mechanics|volume= 36|pages= 1|url=ftp://melmac.sd.ruhr-uni-bochum.de/kintzel/JoaM_27_04_2008/Lee_69.pdf|doi=10.1115/1.3564580|bibcode = 1969JAM....36....1L }}</ref> The decomposition assumes that the total deformation gradient ('''''F''''') can be decomposed as:
:<math>
\boldsymbol{F} = \boldsymbol{F}^e\cdot\boldsymbol{F}^p
</math>
where '''''F'''''e is the elastic (recoverable) part and '''''F'''''p is the plastic (unrecoverable) part of the deformation. The [[finite strain theory#Time-derivative of the deformation gradient|spatial velocity gradient]] is given by
:<math>
\begin{align}
\boldsymbol{l} & = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}
= \left(\dot{\boldsymbol{F}}^e\cdot\boldsymbol{F}^p + \boldsymbol{F}^e\cdot\dot{\boldsymbol{F}}^p\right)\cdot
\left[(\boldsymbol{F}^p)^{-1}\cdot(\boldsymbol{F}^e)^{-1}\right] \\
& = \dot{\boldsymbol{F}}^e\cdot(\boldsymbol{F}^e)^{-1} + \boldsymbol{F}^e\cdot[\dot{\boldsymbol{F}}^p\cdot
(\boldsymbol{F}^p)^{-1}]\cdot(\boldsymbol{F}^e)^{-1} \,.
\end{align}
</math>
where a superposed dot indicates a time derivative. We can write the above as
:<math>
\boldsymbol{l} = \boldsymbol{l}^e + \boldsymbol{F}^e\cdot\boldsymbol{L}^p\cdot(\boldsymbol{F}^e)^{-1} \,.
</math>
The quantity
:<math>
\boldsymbol{L}^p := \dot{\boldsymbol{F}}^p\cdot
(\boldsymbol{F}^p)^{-1}
</math>
is called a '''plastic velocity gradient''' and is defined in an intermediate ([[compatibility (mechanics)|incompatible]]) stress-free configuration. The symmetric part ('''''D'''''p) of '''''L'''''p is called the '''plastic rate of deformation ''' while the skew-symmetric part ('''''W'''''p) is called the '''plastic spin''':
:<math>
\boldsymbol{D}^p = \tfrac{1}{2}[\boldsymbol{L}^p +(\boldsymbol{L}^p)^T] ~,~~
\boldsymbol{W}^p = \tfrac{1}{2}[\boldsymbol{L}^p -(\boldsymbol{L}^p)^T] \,.
</math>
Typically, the plastic spin is ignored in most descriptions of finite plasticity.

=== Elastic regime ===
The elastic behavior in the finite strain regime is typically described by a [[hyperelastic material]] model. The elastic strain can be measured using an elastic right [[Cauchy-Green deformation tensor]] defined as:
:<math>
\boldsymbol{C}^e := (\boldsymbol{F}^e)^T\cdot\boldsymbol{F}^e \,.
</math>
The [[logarithmic strain|logarithmic]] or [[logarithmic strain|Hencky strain]] tensor may then be defined as
:<math>
\boldsymbol{E}^e := \tfrac{1}{2}\ln\boldsymbol{C}^e \,.
</math>
The symmetrized [[Mandel stress]] tensor is a convenient stress measure for finite plasticity and is defined as
:<math>
\boldsymbol{M} := \tfrac{1}{2}(\boldsymbol{C}^e\cdot\boldsymbol{S} + \boldsymbol{S}\cdot\boldsymbol{C}^e)
</math>
where '''''S''''' is the [[stress measures|second Piola-Kirchhoff stress]]. A possible hyperelastic model in terms of the logarithmic strain is <ref>{{Citation|last=Anand|first= L.|year=1979|title= On H. Hencky's approximate strain-energy function for moderate deformations|journal= ASME Journal of Applied Mechanics|volume= 46|pages= 78.|bibcode=1979JAM....46...78A|doi=10.1115/1.3424532}}</ref>
:<math>
\boldsymbol{M} = \frac{\partial W}{\partial \boldsymbol{E}^e} = J\,\frac{dU}{dJ} + 2\mu\,\text{dev}(\boldsymbol{E}^e)
</math>
where ''W'' is a strain energy density function, ''J'' = det('''''F'''''), ''μ'' is a modulus, and "dev" indicates the deviatoric part of a tensor.

=== Flow rule ===
Application of the [[Clausius-Duhem inequality]] leads, in the absence of a plastic spin, to the finite strain flow rule
:<math>
\boldsymbol{D}^p = \dot{\lambda}\,\frac{\partial f}{\partial \boldsymbol{M}} \,.
</math>

=== Loading-unloading conditions ===
The loading-unloading conditions can be shown to be equivalent to the [[Karush-Kuhn-Tucker conditions]]
:<math>
\dot{\lambda} \ge 0 ~,~~ f \le 0~,~~ \dot{\lambda}\,f = 0 \,.
</math>

=== Consistency condition ===
The consistency condition is identical to that for the small strain case,
:<math>
\dot{\lambda}\,\dot{f} = 0 \,.
</math>

== References ==
{{Reflist}}

== See also ==
* [[Plasticity (physics)]]

[[Category:Continuum mechanics]]
[[Category:Solid mechanics]]

Coupling coefficient of resonators - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (9475)