Flow plasticity theory: Difference between revisions

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[[File:Biegeanimation 2D.gif|300px|thumb|Plastic deformation of a thin metal sheet.]]

'''Flow plasticity''' is a [[continuumsolid 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 unidirectional loading (for small deformation perfect plasticity or hardening plasticity) are developed on the basis onof 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>.

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# 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 dimensionsdimensional states of stress and multidirectional loading as follows.

* '''Elasticity''' ([[Hooke's law]]). In the linear elastic regime the stresses and strains in the rockmaterial are related by

:::<math>

\boldsymbol{\sigma} = \mathsf{CD}:\boldsymbol{\varepsilon}

</math>

:::where the stiffness matrix <math>\mathsf{CD}</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 rocksmaterials, 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'''. For general states of stress, plastic '''loading''' is indicated if the state of stress is on the yield surface and the stress increment is directed toward the outside of the yield surface; this occurs if the inner product of the stress increment and the outward normal of the yield surface is positive, i.e.,

* '''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>

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* 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 [[Bruce Bilby|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. Kröner,<ref>{{Citation|last=Kröner|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>{{Cite FTP |last=Lee|first=E. H.|year=1969|title=Elastic-Plastic Deformation at Finite Strains|volume=36|issue=1|pages=1–6|url=ftp://melmac.sd.ruhr-uni-bochum.de/kintzel/JoaM_27_04_2008/Lee_69.pdf|doi=10.1115/1.3564580|server=Journal of Applied Mechanics|url-status=dead|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= Journal of Applied Mechanics|volume= 46|issue= 1|pages= 78–82|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 ==

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== See also ==

* [[Plasticity (physics)]]

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[[Category:Continuum mechanics]]