“Drucker–Prager yield criterion”的意思、由来-开放百科全书

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 first invariant of the Cauchy stress and

is the second invariant of the deviatoric part of the Cauchy stress. The constants

are determined from experiments.

In terms of the equivalent stress (or von Mises stress) and the hydrostatic (or mean) stress, the Drucker–Prager criterion can be expressed as

are material constants. The Drucker–Prager yield criterion expressed in Haigh–Westergaard coordinates is

The Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface.

Expressions for A and B

is the yield stress in uniaxial tension, the Drucker–Prager criterion implies

is the yield stress in uniaxial compression, the Drucker–Prager criterion implies

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.^[2] 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 middle circumscribes the Mohr–Coulomb yield surface then

If the Drucker–Prager yield surface inscribes the Mohr–Coulomb yield surface then

Drucker–Prager model for polymers

The Drucker–Prager model has been used to model polymers such as polyoxymethylene and polypropylene{{Citation needed|date=September 2011}}.^[3] For polyoxymethylene the yield stress is a linear function of the pressure. However, polypropylene shows a quadratic pressure-dependence of the yield stress.

Drucker–Prager model for foams

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

Deshpande–Fleck yield criterion or isotropic foam yield criterion

The Deshpande–Fleck yield criterion^[5] for foams has the form given in above equation. The parameters

for the Deshpande–Fleck criterion are

where

is a parameter^[6] 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.^[7] This yield criterion is an extension of the generalized Hill yield criterion and has the form

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 ^[8] 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 ^[9] which has the form

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

See also

References

1. ^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.
2. ^https://www.onepetro.org/conference-paper/SPE-20405-MS
3. ^Abrate, S. (2008). Criteria for yielding or failure of cellular materials. Journal of Sandwich Structures and Materials, vol. 10. pp. 5–51.
4. ^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 ofMechanical Sciences, vol. 31, no. 9, pp. 635–665.
5. ^V. S. Deshpande, and Fleck, N. A. (2001). Multi-axial yield behaviour of polymer foams. Acta Materialia, vol. 49, no. 10, pp. 1859–1866.
6. ^

where

is the quantity used by Deshpande–Fleck
7. ^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
8. ^Drucker, D. C. (1949) Relations of experiments to mathematical theories of plasticity, Journal of Applied Mechanics, vol. 16, pp. 349–357.
9. ^{{citation|last1=Cazacu|first1=O.|last2=Barlat|first2=F.|year=2001|title=Generalization of Drucker's yield criterion to orthotropy|journal=Mathematics & Mechanics of Solids|volume=6|issue=6|pages=613–630}}.