Flow plasticity theory: Difference between revisions

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== 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 of 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 material 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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=== 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>{{CitationCite FTP |last=Lee|first= E. H. |year=1969|title=Elastic-Plastic Deformation at Finite Strains|journal= Journal of Applied Mechanics|volume= 36|issue=1|pages= 11–6|url=ftp://melmac.sd.ruhr-uni-bochum.de/kintzel/JoaM_27_04_2008/Lee_69.pdf|doi=10.1115/1.3564580|bibcode 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

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\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|issue= 1|pages= 78.78–82|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)

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== See also ==

* [[Plasticity (physics)]]

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[[Category:Continuum mechanics]]