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In [[mathematics]] the '''cotangent complex''' is roughly a universal linearization of a [[morphism]] of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. [[Luc Illusie]], [[Daniel Quillen]], and M. André independently came up with a definition that works in all cases.
[[Image:Drucker Prager Yield Surface 3Da.png|300px|right|thumb|Figure 1: View of Drucker–Prager yield surface in 3D space of principal stresses for <math>c=2, \phi=-20^\circ</math>]]
==Motivation==
The '''Drucker–Prager yield criterion'''<ref> Drucker, D. C. and Prager, W. (1952). ''Soil mechanics and plastic analysis for limit design''. Quarterly of Applied Mathematics, vol. 10, no. 2, pp. 157–165.</ref> is a pressure-dependent model for determining whether a material has failed or undergone plastic yielding. The criterion was introduced to deal with the plastic deformation of soils. It and its many variants have been applied to rock, concrete, polymers, foams, and other pressure-dependent materials.
Suppose that ''X'' and ''Y'' are [[algebraic variety|algebraic varieties]] and that {{nowrap|''f'' : ''X'' → ''Y''}} is a morphism between them. The cotangent complex of ''f'' is a more universal version of the relative [[Kähler differentials]] Ω<sub>''X''/''Y''</sub>. The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If ''Z'' is another variety, and if {{nowrap|''g'' : ''Y'' → ''Z''}} is another morphism, then there is an exact sequence
In some sense, therefore, relative Kähler differentials are a [[right exact functor]]. (Literally this is not true, however, because the category of algebraic varieties is not an [[abelian category]], and therefore right-exactness is not defined.) In fact, prior to the definition of the cotangent complex, there were several definitions of functors that might extend the sequence further to the left, such as the [[Lichtenbaum–Schlessinger functor]]s ''T''<sup>''i''</sup> and [[imperfection module]]s. Most of these were motivated by [[deformation theory]].
This sequence is exact on the left if the morphism ''f'' is smooth. If Ω admitted a first [[derived functor]], then exactness on the left would imply that the [[connecting homomorphism]] vanished, and this would certainly be true if the first derived functor of ''f'', whatever it was, vanished. Therefore a reasonable speculation is that the first derived functor of a smooth morphism vanishes. Furthermore, when any of the functors which extended the sequence of Kähler differentials were applied to a smooth morphism, they too vanished, which suggested that the cotangent complex of a smooth morphism might be equivalent to the Kähler differentials.
The Drucker–Prager yield criterion has the form
:<math>
\sqrt{J_2} = A + B~I_1
</math>
where <math>I_1</math> is the [[Stress_(physics)#Principal_stresses_and_stress_invariants|first invariant]] of the [[Stress (physics)|Cauchy stress]] and <math>J_2</math> is the [[Stress_(physics)#Invariants_of_the_stress_deviator_tensor|second invariant]] of the [[Stress_(physics)#Stress_deviator_tensor|deviatoric]] part of the [[Stress (physics)|Cauchy stress]]. The constants <math>A, B </math> are determined from experiments.
Another natural exact sequence related to Kähler differentials is the [[conormal exact sequence]]. If ''f'' is a closed immersion with ideal sheaf ''I'', then there is an exact sequence
In terms of the [[von Mises stress|equivalent stress]] (or [[von Mises stress]]) and the [[hydrostatic stress|hydrostatic (or mean) stress]], the Drucker–Prager criterion can be expressed as
This is an extension of the exact sequence above: There is a new term on the left, the conormal sheaf of ''f'', and the relative differentials Ω<sub>''X''/''Y''</sub> have vanished because a closed immersion is [[formally unramified]]. If ''f'' is the inclusion of a smooth subvariety, then this sequence is a short exact sequence.<ref>{{Harvard citations|last = Grothendieck|year = 1967|loc = Proposition 17.2.5|nb = yes}}</ref> This suggests that the cotangent complex of the inclusion of a smooth variety is equivalent to the conormal sheaf shifted by one term.
\sigma_e = a + b~\sigma_m
</math>
where <math>\sigma_e</math> is the equivalent stress, <math>\sigma_m</math> is the hydrostatic stress, and
<math>a,b</math> are material constants. The Drucker–Prager yield criterion expressed in [[Yield_surface#Invariants_used_to_describe_yield_surfaces|Haigh–Westergaard coordinates]] is
:<math>
\tfrac{1}{\sqrt{2}}\rho - \sqrt{3}~B\xi = A
</math>
==Early work on cotangent complexes==
The [[Yield surface#Drucker_-_Prager_yield_surface|Drucker–Prager yield surface]] is a smooth version of the [[Yield surface#Mohr_-_Coulomb_yield_surface|Mohr–Coulomb yield surface]].
The cotangent complex dates back at least to SGA 6 VIII 2, where [[Pierre Berthelot]] gave a definition when ''f'' is a ''smoothable'' morphism, meaning there is a scheme ''V'' and morphisms {{nowrap|''i'' : ''X'' → ''V''}} and {{nowrap|''h'' : ''V'' → ''Y''}} such that {{nowrap|''f'' {{=}} ''hi''}}, ''i'' is a closed immersion, and ''h'' is a smooth morphism. (For example, all projective morphisms are smoothable, since ''V'' can be taken to be a projective bundle over ''Y''.) In this case, he defines the cotangent complex of ''f'' as an object in the [[derived category]] of [[coherent sheaf|coherent sheaves]] ''X'' as follows:
*<math>L^{X/Y}_0 = i^*\Omega_{V/Y},</math>
*If ''J'' is the ideal of ''X'' in ''V'', then <math>L^{X/Y}_1 = J/J^2 = i^*J</math>,
*<math>L^{X/Y}_i = 0</math> for all other ''i'',
*The differential <math>L^{X/Y}_1 \to L^{X/Y}_0</math> is the pullback along ''i'' of the inclusion of ''J'' in the structure sheaf <math>\mathcal{O}_V</math> of ''V'' followed by the universal derivation <math>d : \mathcal{O}_V \to \Omega_{V/Y}</math>.
*All other differentials are zero.
Berthelot proves that this definition is independent of the choice of ''V''<ref>{{Harvard citations|last = Berthelot|year = 1966|loc = VIII Proposition 2.2|nb = yes}}</ref> and that for a smoothable complete intersection morphism, this complex is perfect.<ref>{{Harvard citations|last = Berthelot|year = 1966|loc = VIII Proposition 2.4|nb = yes}}</ref> Furthermore, he proves that if {{nowrap|''g'' : ''Y'' → ''Z''}} is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an [[exact triangle]]
The correct definition of the cotangent complex begins in the [[homotopic algebra|homotopical setting]]. Quillen and André worked with the [[simplicial set#Simplicial objects|simplicial]] commutative rings, while Illusie worked with simplicial ringed [[topos|topoi]]. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that ''A'' and ''B'' are [[simplicial ring]]s and that ''B'' is an ''A''-algebra. Choose a resolution {{nowrap|''r'' : ''P''<sup>•</sup> → ''B''}} of ''B'' by simplicial free ''A''-algebras. Applying the Kähler differential functor to ''P''<sup>•</sup> produces a simplicial ''B''-module. The total complex of this simplicial object is the '''cotangent complex''' ''L''<sup>''B''/''A''</sup>. The morphism ''r'' induces a morphism from the cotangent complex to Ω<sub>''B''/''A''</sub> called the '''augmentation map'''. In the homotopy category of simplicial ''A''-algebras (or of simplicial ringed topoi), this construction amounts to taking the left derived functor of the Kähler differential functor.
The Drucker–Prager model can be written in terms of the [[stress (physics)#Principal_stresses_and_stress_invariants|principal stresses]] as
:<math>
\sqrt{\cfrac{1}{6}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]} = A + B~(\sigma_1+\sigma_2+\sigma_3) ~.
</math>
If <math>\sigma_t</math> is the yield stress in uniaxial tension, the Drucker–Prager criterion implies
:<math>
\cfrac{1}{\sqrt{3}}~\sigma_t = A + B~\sigma_t ~.
</math>
If <math>\sigma_c</math> is the yield stress in uniaxial compression, the Drucker–Prager criterion implies
:<math>
\cfrac{1}{\sqrt{3}}~\sigma_c = A - B~\sigma_c ~.
</math>
Solving these two equations gives
:<math>
A = \cfrac{2}{\sqrt{3}}~\left(\cfrac{\sigma_c~\sigma_t}{\sigma_c+\sigma_t}\right) ~;~~ B = \cfrac{1}{\sqrt{3}}~\left(\cfrac{\sigma_t-\sigma_c}{\sigma_c+\sigma_t}\right) ~.
</math>
Given a commutative square as follows:
=== Uniaxial asymmetry ratio ===
:[[File:Commutative square.svg]]
Different uniaxial yield stresses in tension and in compression are predicted by the Drucker–Prager model. The uniaxial asymmetry ratio for the Drucker–Prager model is
there is a morphism of cotangent complexes {{nowrap|''L''<sup>''B''/''A''</sup> ⊗<sub>''B''</sub> ''D'' → ''L''<sup>''D''/''C''</sup>}} which respects the augmentation maps. This map is constructed by choosing a free simplicial ''C''-algebra resolution of ''D'', say {{nowrap|''s'' : ''Q''<sup>•</sup> → ''D''}}. Because ''P''<sup>•</sup> is a free object, the composite ''hr'' can be lifted to a morphism {{nowrap|''P''<sup>•</sup> → ''Q''<sup>•</sup>}}. Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms {{nowrap|''A'' → ''B'' → ''C''}}, this produces the sequence
:<math>
:<math>L^{B/A} \otimes_B C \to L^{C/A} \to L^{C/B}.</math>
There is a connecting homomorphism <math>L^{C/B} \to (L^{B/A} \otimes_B C)[1]</math> which turns this sequence into an exact triangle.
</math>
The cotangent complex can also be defined in any combinatorial [[model category]] ''M''. Suppose that <math>f\colon A\rightarrow B</math> is a morphism in ''M''. The cotangent complex <math>L^f</math> (or <math>L^{B/A}</math>) is an object in the category of spectra in <math>M_{B//B}</math>. A pair of composable morphisms <math>A\xrightarrow{f} B\xrightarrow{g} C</math> induces an exact triangle in the homotopy category, <math>L^{B/A}\otimes_BC\rightarrow L^{C/A}\rightarrow L^{C/B}\rightarrow (L^{B/A}\otimes_BC)[1]</math>.
=== Expressions in terms of cohesion and friction angle ===
Since the Drucker–Prager [[yield surface]] is a smooth version of the [[Mohr-Coulomb theory|Mohr–Coulomb yield surface]], it is often expressed in terms of the cohesion (<math>c</math>) and the angle of internal friction (<math>\phi</math>) that are used to describe the [[Mohr-Coulomb theory|Mohr–Coulomb yield surface]]. If we assume that the Drucker–Prager yield surface '''circumscribes''' the Mohr–Coulomb yield surface then the expressions for <math>A</math> and <math>B</math> are
:<math>
A = \cfrac{6~c~\cos\phi}{\sqrt{3}(3+\sin\phi)} ~;~~
B = \cfrac{2~\sin\phi}{\sqrt{3}(3+\sin\phi)}
</math>
If the Drucker–Prager yield surface '''inscribes''' the Mohr–Coulomb yield surface then
:<math>
A = \cfrac{6~c~\cos\phi}{\sqrt{3}(3-\sin\phi)} ~;~~
!Derivation of expressions for <math>A,B</math> in terms of <math>c,\phi</math>
|-
|The expression for the [[Mohr-Coulomb theory|Mohr–Coulomb yield criterionddadsa]] in [[Yield_surface#Invariants_used_to_describe_yield_surfaces|Haigh–Westergaard space]] is
:<math>
\left[\sqrt{3}~\sin\left(\theta+\tfrac{\pi}{3}\right) - \sin\phi\cos\left(\theta+\tfrac{\pi}{3}\right)\right]\rho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi
</math>
If we assume that the Drucker–Prager yield surface '''circumscribes''' the Mohr–Coulomb yield surface such that the two surfaces coincide at <math>\theta=\tfrac{\pi}{3}</math>, then at those points the Mohr–Coulomb yield surface can be expressed as
:<math>
\left[\sqrt{3}~\sin\tfrac{2\pi}{3} - \sin\phi\cos\tfrac{2\pi}{3}\right]\rho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi
</math>
or,
:<math>
\tfrac{1}{\sqrt{2}}\rho - \cfrac{2\sin\phi}{3+\sin\phi}\xi = \cfrac{\sqrt{12} c \cos\phi}{3+\sin\phi} \qquad \qquad (1.1)
</math>
==Properties of the cotangent complex==
The Drucker–Prager yield criterion expressed in [[Yield_surface#Invariants_used_to_describe_yield_surfaces|Haigh–Westergaard coordinates]] is
:<math>
\tfrac{1}{\sqrt{2}}\rho - \sqrt{3}~B\xi = A \qquad \qquad (1.2)
</math>
Comparing equations (1.1) and (1.2), we have
:<math>
A = \cfrac{\sqrt{12} c \cos\phi}{3+\sin\phi} = \cfrac{6 c \cos\phi}{\sqrt{3}(3+\sin\phi)} ~;~~ B = \cfrac{2\sin\phi}{\sqrt{3}(3+\sin\phi)}
</math>
These are the expressions for <math>A,B</math> in terms of <math>c,\phi</math>.
===Flat base change===
On the other hand if the Drucker–Prager surface inscribes the Mohr–Coulomb surface, then matching the two surfaces at <math>\theta=0</math> gives
Suppose that ''B'' and ''C'' are ''A''-algebras such that {{nowrap|Tor<sup>''A''</sup><sub>''q''</sub>(''B'', ''C'') {{=}} 0}} for all {{nowrap|''q'' > 0}}. Then there are quasi-isomorphisms<ref>{{Harvard citations|last = Quillen|year = 1970|loc = Theorem 5.3|nb = yes}}</ref>
:<math>
:<math>L^{B \otimes_A C/C} \cong B \otimes_A L^{C/A},</math>
A = \cfrac{6 c \cos\phi}{\sqrt{3}(3-\sin\phi)} ~;~~ B = \cfrac{2\sin\phi}{\sqrt{3}(3-\sin\phi)}
If ''C'' is a flat ''A''-algebra, then the condition that {{nowrap|Tor<sup>''A''</sup><sub>''q''</sub>(''B'', ''C'')}} vanishes for {{nowrap|''q'' > 0}} is automatic. The first formula then proves that the construction of the cotangent complex is local on the base in the [[flat topology]].
[[Image:MC DP Yield Surface 3Da.png|300px|left|thumb|Comparison of Drucker–Prager and Mohr–Coulomb (inscribed) yield surfaces in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
[[Image:MC DP Yield Surface 3Db.png|300px|none|thumb|Comparison of Drucker–Prager and Mohr–Coulomb (circumscribed) yield surfaces in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
|}
{| border="0"
|-
| valign="bottom"|
[[Image:Drucker Prager Yield Surface 3Db.png|300px|none|thumb|Figure 2: Drucker–Prager yield surface in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
|
|
|valign="bottom"|
[[Image:MC DP Yield Surface sig1sig2.png|300px|none|thumb|Figure 3: Trace of the Drucker–Prager and Mohr–Coulomb yield surfaces in the <math>\sigma_1-\sigma_2</math>-plane for <math>c = 2, \phi = 20^\circ</math>. Yellow = Mohr–Coulomb, Cyan = Drucker–Prager.]]
The Drucker–Prager model has been used to model polymers such as [[polyoxymethylene]] and [[polypropylene]]{{Fact|date=September 2011}}<ref>Abrate, S. (2008). ''Criteria for yielding or failure of cellular materials''. Journal of Sandwich Structures and Materials, vol. 10. pp. 5–51.</ref>. For [[polyoxymethylene]] the yield stress is a linear function of the pressure. However, [[polypropylene]] shows a quadratic pressure-dependence of the yield stress.
*If ''B'' is a [[localization of a ring|localization]] of ''A'', then {{nowrap|''L''<sup>''B''/''A''</sup> {{=}} 0}}.
*If ''f'' is an [[étale morphism]], then {{nowrap|''L''<sup>''B''/''A''</sup> {{=}} 0}}.
*If ''f'' is a [[smooth morphism]], then {{nowrap|''L''<sup>''B''/''A''</sup>}} is quasi-isomorphic to Ω<sub>''B''/''A''</sub>. In particular, it has [[projective dimension]] zero.
*If ''f'' is a [[local complete intersection morphism]], then {{nowrap|''L''<sup>''B''/''A''</sup>}} has projective dimension at most one.
*If ''A'' is Noetherian, {{nowrap|''B'' {{=}} ''A''/''I''}}, and ''I'' is generated by a regular sequence, then <math>I/I^2</math> is a [[projective module]] and ''L''<sup>''B''/''A''</sup> is quasi-isomorphic to <math>I/I^2[1]</math>.
==Examples==
== Drucker–Prager model for foams ==
*Let ''X'' be smooth over ''S''. Then the cotangent complex is Ω<sub>''X''/''S''</sub>. In Berthelot's framework, this is clear by taking {{nowrap|''V'' {{=}} ''X''}}. In general, étale locally on ''S'', ''X'' is a finite dimensional affine space and the morphism from ''X'' to ''S'' is projection, so we may reduce to the situation where {{nowrap|''S'' {{=}} Spec ''A''}} and {{nowrap|''X'' {{=}} Spec ''A''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}}. We can take the resolution of {{nowrap|''A''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}} to be the identity map, and then it is clear that the cotangent complex is the same as the Kähler differentials.
For foams, the GAZT model <ref>Gibson, L.J., [[M. F. Ashby|Ashby, M.F.]], Zhang, J. and Triantafilliou, T.C. (1989). ''Failure surfaces for cellular materials under multi-axial loads. I. Modeling''. International Journal of
Mechanical Sciences, vol. 31, no. 9, pp. 635–665.</ref> uses
:<math>
A = \pm \cfrac{\sigma_y}{\sqrt{3}} ~;~~ B = \mp \cfrac{1}{\sqrt{3}}~\left(\cfrac{\rho}{5~\rho_s}\right)
</math>
where <math>\sigma_{y}</math> is a critical stress for failure in tension or compression, <math>\rho</math> is the density of the foam, and <math>\rho_s</math> is the density of the base material.
*Let ''X'' and ''Y'' be smooth over ''S'', and assume that {{nowrap|''i'' : ''X'' → ''Y''}} is a closed embedding. Using the exact triangle corresponding to the morphisms {{nowrap|''X'' → ''Y'' → ''S''}}, we may determine the cotangent complex ''L''<sub>''X''/''Y''</sub>. To do this, note that by the previous example, the cotangent complexes ''L''<sub>''X''/''S''</sub> and ''L''<sub>''Y''/''S''</sub> consist of the Kähler differentials Ω<sub>''X''/''S''</sub> and Ω<sub>''Y''/''S''</sub> in the zeroth degree, respectively, and are zero in all other degrees. The exact triangle implies that ''L''<sub>''X''/''Y''</sub> is nonzero only in the first degree, and in that degree, it is the kernel of the map {{nowrap|''i''<sup>*</sup>Ω<sub>''Y''/''S''</sub> → Ω<sub>''X''/''S''</sub>}}. This kernel is the conormal bundle, and the exact sequence is the conormal exact sequence, so in the first degree, ''L''<sub>''X''/''Y''</sub> is the conormal bundle of ''X'' in ''Y''.
== Extensions of the isotropic Drucker–Prager model ==
The Drucker–Prager criterion can also be expressed in the alternative form
:<math>
J_2 = (A + B~I_1)^2 = a + b~I_1 + c~I_1^2 ~.
</math>
==See also==
=== Deshpande–Fleck yield criterion ===
*[[André–Quillen cohomology]]
The Deshpande–Fleck yield criterion<ref>V. S. Deshpande, and Fleck, N. A. (2001). ''Multi-axial yield behaviour of polymer foams.'' Acta Materialia, vol. 49, no. 10, pp. 1859–1866.</ref> for foams has the form given in above equation. The parameters <math>a, b, c</math> for the Deshpande–Fleck criterion are
:<math>
a = (1 + \beta^2)~\sigma_y^2 ~,~~
b = 0 ~,~~
c = -\cfrac{\beta^2}{3}
</math>
where <math>\beta</math> is a parameter<ref><math>\beta= \alpha/3</math> where <math>\alpha</math> is the
quantity used by Deshpande–Fleck</ref> that determines the shape of the yield surface, and <math>\sigma_y</math> is the yield stress in tension or compression.
==Notes==
== Anisotropic Drucker–Prager yield criterion ==
{{reflist}}
An anisotropic form of the Drucker–Prager yield criterion is the Liu–Huang–Stout yield criterion <ref>Liu, C., Huang, Y., and Stout, M. G. (1997). ''On the asymmetric yield surface of plastically orthotropic materials: A phenomenological study.'' Acta Materialia, vol. 45, no. 6, pp. 2397–2406</ref>. This yield criterion is an extension of the [[Hill yield criteria|generalized Hill yield criterion]] and has the form
:<math>
\begin{align}
f := & \sqrt{F(\sigma_{22}-\sigma_{33})^2+G(\sigma_{33}-\sigma_{11})^2+H(\sigma_{11}-\sigma_{22})^2
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''')
K = \cfrac{\sigma_{3c}-\sigma_{3t}}{2\sigma_{3c}\sigma_{3t}}
| pages = xii+700
\end{align}
| nopp = true
</math>
}}
where
*{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1967__32_ | year=1967 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=32 | pages=5–361 | doi=10.1007/BF02732123}}
:<math>
*{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | title=Catégories cofibrées additives et complexe cotangent relatif | publisher=[[Springer-Verlag]] | location=Berlin, New York | language=French | series=Lecture Notes in Mathematics '''79''' | isbn=978-3-540-04248-8 | date=01/07/1969 }}
*{{Citation | last1=Lichtenbaum | last2=Schlessinger | title=The cotangent complex of a morphism | journal=Transactions of the American Mathematical society | issue=128 | year=1967 | pages=41–70}}
</math>
*{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=On the (co-)homology of commutative rings | series=Proc. Symp. Pure Mat. | volume=XVII | publisher=[[American Mathematical Society]] | year=1970}}
and <math>\sigma_{ic}, i=1,2,3</math> are the uniaxial yield stresses in '''compression''' in the three principal directions of anisotropy, <math>\sigma_{it}, i=1,2,3</math> are the uniaxial yield stresses in '''tension''', and <math>\sigma_{23}^y, \sigma_{31}^y, \sigma_{12}^y</math> are the yield stresses in pure shear. It has been assumed in the above that the quantities <math>\sigma_{1c},\sigma_{2c},\sigma_{3c}</math> are positive and <math>\sigma_{1t},\sigma_{2t},\sigma_{3t}</math> are negative.
{{DEFAULTSORT:Cotangent Complex}}
== The Drucker yield criterion ==
[[Category:Algebraic geometry]]
The Drucker–Prager criterion should not be confused with the earlier Drucker criterion <ref> Drucker, D. C. (1949) '' Relations of experiments to mathematical theories of plasticity'', Journal of Applied Mechanics, vol. 16, pp. 349–357.</ref> which is independent of the pressure (<math>I_1</math>). The Drucker yield criterion has the form
[[Category:Category theory]]
:<math>
[[Category:Homotopy theory]]
f := J_2^3 - \alpha~J_3^2 - k^2 \le 0
</math>
where <math>J_2</math> is the second invariant of the deviatoric stress, <math>J_3</math> is the third invariant of the deviatoric stress, <math>\alpha</math> is a constant that lies between -27/8 and 9/4 (for the yield surface to be convex), <math>k</math> is a constant that varies with the value of <math>\alpha</math>. For <math>\alpha=0</math>, <math>k^2 = \cfrac{\sigma_y^6}{27}</math> where <math>\sigma_y</math> is the yield stress in uniaxial tension.
== Anisotropic Drucker Criterion ==
An anisotropic version of the Drucker yield criterion is the Cazacu–Barlat (CZ) yield criterion <ref>Cazacu, O. and Barlat, F. (2001). ''Generalization of Drucker's yield criterion to orthotropy.'' Mathematics and Mechanics of Solids, vol. 6, no. 6, pp. 613–630.</ref> which has the form
:<math>
f := (J_2^0)^3 - \alpha~(J_3^0)^2 - k^2 \le 0
</math>
where <math>J_2^0, J_3^0</math> are generalized forms of the deviatoric stress and are defined as
=== Cazacu–Barlat yield criterion for plane stress ===
For thin sheet metals, the state of stress can be approximated as [[plane stress]]. In that case the Cazacu–Barlat yield criterion reduces to its two-dimensional version with
The Drucker–Prager yield criterion[1] is a pressure-dependent model for determining whether a material has failed or undergone plastic yielding. The criterion was introduced to deal with the plastic deformation of soils. It and its many variants have been applied to rock, concrete, polymers, foams, and other pressure-dependent materials.
where is the equivalent stress, is the hydrostatic stress, and
are material constants. The Drucker–Prager yield criterion expressed in Haigh–Westergaard coordinates is
The Drucker–Prager model can be written in terms of the principal stresses as
If is the yield stress in uniaxial tension, the Drucker–Prager criterion implies
If is the yield stress in uniaxial compression, the Drucker–Prager criterion implies
Solving these two equations gives
Uniaxial asymmetry ratio
Different uniaxial yield stresses in tension and in compression are predicted by the Drucker–Prager model. The uniaxial asymmetry ratio for the Drucker–Prager model is
Expressions in terms of cohesion and friction angle
Since the Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface, it is often expressed in terms of the cohesion () and the angle of internal friction () that are used to describe the Mohr–Coulomb yield surface. If we assume that the Drucker–Prager yield surface circumscribes the Mohr–Coulomb yield surface then the expressions for and are
If the Drucker–Prager yield surface inscribes the Mohr–Coulomb yield surface then
If we assume that the Drucker–Prager yield surface circumscribes the Mohr–Coulomb yield surface such that the two surfaces coincide at , then at those points the Mohr–Coulomb yield surface can be expressed as
where is a critical stress for failure in tension or compression, is the density of the foam, and is the density of the base material.
Extensions of the isotropic Drucker–Prager model
The Drucker–Prager criterion can also be expressed in the alternative form
Deshpande–Fleck yield criterion
The Deshpande–Fleck yield criterion[4] for foams has the form given in above equation. The parameters for the Deshpande–Fleck criterion are
where is a parameter[5] that determines the shape of the yield surface, and is the yield stress in tension or compression.
Anisotropic Drucker–Prager yield criterion
An anisotropic form of the Drucker–Prager yield criterion is the Liu–Huang–Stout yield criterion [6]. This yield criterion is an extension of the generalized Hill yield criterion and has the form
The coefficients are
where
and are the uniaxial yield stresses in compression in the three principal directions of anisotropy, are the uniaxial yield stresses in tension, and are the yield stresses in pure shear. It has been assumed in the above that the quantities are positive and are negative.
The Drucker yield criterion
The Drucker–Prager criterion should not be confused with the earlier Drucker criterion [7] which is independent of the pressure (). The Drucker yield criterion has the form
where is the second invariant of the deviatoric stress, is the third invariant of the deviatoric stress, is a constant that lies between -27/8 and 9/4 (for the yield surface to be convex), is a constant that varies with the value of . For , where is the yield stress in uniaxial tension.
Anisotropic Drucker Criterion
An anisotropic version of the Drucker yield criterion is the Cazacu–Barlat (CZ) yield criterion [8] which has the form
where are generalized forms of the deviatoric stress and are defined as
Cazacu–Barlat yield criterion for plane stress
For thin sheet metals, the state of stress can be approximated as plane stress. In that case the Cazacu–Barlat yield criterion reduces to its two-dimensional version with
For thin sheets of metals and alloys, the parameters of the Cazacu–Barlat yield criterion are
Table 1. Cazacu–Barlat yield criterion parameters for sheet metals and alloys
Material
6016-T4 Aluminum Alloy
0.815
0.815
0.334
0.42
0.04
-1.205
-0.958
0.306
0.153
-0.02
1.4
2090-T3 Aluminum Alloy
1.05
0.823
0.586
0.96
1.44
0.061
-1.302
-0.281
-0.375
0.445
1.285
References
↑ Drucker, D. C. and Prager, W. (1952). Soil mechanics and plastic analysis for limit design. Quarterly of Applied Mathematics, vol. 10, no. 2, pp. 157–165.
↑Abrate, S. (2008). Criteria for yielding or failure of cellular materials. Journal of Sandwich Structures and Materials, vol. 10. pp. 5–51.
↑Gibson, L.J., Ashby, M.F., Zhang, J. and Triantafilliou, T.C. (1989). Failure surfaces for cellular materials under multi-axial loads. I. Modeling. International Journal of
Mechanical Sciences, vol. 31, no. 9, pp. 635–665.
↑V. S. Deshpande, and Fleck, N. A. (2001). Multi-axial yield behaviour of polymer foams. Acta Materialia, vol. 49, no. 10, pp. 1859–1866.
↑Liu, C., Huang, Y., and Stout, M. G. (1997). On the asymmetric yield surface of plastically orthotropic materials: A phenomenological study. Acta Materialia, vol. 45, no. 6, pp. 2397–2406
↑ Drucker, D. C. (1949) Relations of experiments to mathematical theories of plasticity, Journal of Applied Mechanics, vol. 16, pp. 349–357.
↑Cazacu, O. and Barlat, F. (2001). Generalization of Drucker's yield criterion to orthotropy. Mathematics and Mechanics of Solids, vol. 6, no. 6, pp. 613–630.