Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

3.11 Critical State Models for Soils

 

Soils consist of a two phase mixture of particles and water.  They exhibit very complex behavior in response to stress, and a number of different constitutive theories are used to model them MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in fact, entire books are devoted to critical state soil models.  Here, we outline a basic soil model known as `Cam-clay,’ developed at Cambridge (UK).   It uses many of the concepts that are used to model plastic deformation of metals, so you will find it helpful to review Sections 3.6 and 3.7 before reading this one.

 

 

3.11.1 Features of the behavior of soils

 

1.      Soils cannot withstand significant tensile stress: we therefore focus on their response to combined pressure and shear loading. 

2.      The behavior of a soil is very sensitive to its water content.   Two types of experiment are conducted on soils: in a `drained’ test, water is allowed to escape from the specimen as it is compressed (so the water pressure is zero); in an `undrained’ test, the volume of the specimen (water + soil particles) is held fixed.   In the latter test, the water pressure can be measured by means of a manometer connected to the pressurized cell.

3.      Under combined pressure and shear loading, soil behaves like a frictional material.  In a drained test, the solid can support shear stresses τ<Mp/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaykW7cqaHepaDcqGH8aapcaWGnbGaam iCaiaac+cadaGcaaqaaiaaiodaaSqabaaaaa@380E@  without excessive deformation, where M is a material property (analogous to friction coefficient).  If the shear stress reaches τ=Mp/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0jabg2da9iaad2eacaWGWbGaai 4lamaakaaabaGaaG4maaWcbeaaaaa@3685@ , the material collapses MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  usually by shearing along one or more discrete shear planes parallel to the maximum resolved shear stress.  If the same test is conducted on an undrained specimen, shear occurs earlier, because the water supports part of the hydrostatic pressure.   In this case shear failure occurs when τ<M(p p w )/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0jabgYda8iaad2eacaGGOaGaam iCaiabgkHiTiaadchadaWgaaWcbaGaam4DaaqabaGccaGGPaGaai4l amaakaaabaGaaG4maaWcbeaaaaa@3AF0@ , where p is the applied hydrostatic pressure, and p w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4Daaqabaaaaa@3285@  is the water pressure.

4.      If subjected to loads below those required to cause catastrophic collapse, soils show a complicated behavior that resembles that of a strain hardening metal MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  except that soils compact in response to combined shear and pressure, whereas metals do not. In addition, the strain hardening occurs only as a result of the compaction: shear strain does not increase the solid’s strength.  For example, the figure shows the response of a soil sample to a test where in which specimen is subjected to a constant pressure, together with a steadily increasing shear stress.   The soil accumulates a permanent shear strain, and also compacts.   The strength of the solid increases up to the limiting shear stress τ=M(p p w )/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0jabg2da9iaad2eacaGGOaGaam iCaiabgkHiTiaadchadaWgaaWcbaGaam4DaaqabaGccaGGPaGaai4l amaakaaabaGaaG4maaWcbeaaaaa@3AF2@ , at which point the compaction reaches a steady state, and the specimen continues to deform at constant shear stress.  

 

 

3.11.2 Constitutive equations for Cam-Clay

 

The constitutive equations for Cam-clay are very similar to the rate independent plastic equations in Section 3.6.  The main concepts are,

1.      Strain rate decomposition into elastic and plastic parts;

2.      Pressure decomposition into contributions from the water pressure (or `pore pressure’) and from the pressure supported by the soil particles.  The pore pressure must be calculated by modeling fluid seepage through the soil.

3.      Elastic stress-strain law, which specifies the  elastic part of the strain in terms of stress;

4.      A yield criterion, which determines the magnitude of the plastic strain rate, given the stresses and the resistance of the material to flow.  Unlike metals, the yield criterion for a soil is a function of the hydrostatic stress, or pressure, in addition to shear stress. The yield criterion is expressed in terms of a State Variable which characterizes the resistance of the material to flow (analogous to yield stress). 

5.      A Plastic flow rule that specifies the ratios of the plastic strain components under multi-axial stress

6.      A hardening law, which specifies how the state variable evolves with plastic straining

7.      The yield surface, flow rule and hardening law also define a critical state criterion for the solid.  The critical state criterion specifies the combination of stresses that lead to unconstrained collapse of the solid.

These are discussed in more detail below.

 

 Strain rate decomposition

 

We assume small strains, so shape changes are characterized by ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaqadaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyA aaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaki abgUcaRiabgkGi2kaadwhadaWgaaWcbaGaamOAaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaai aac+cacaaIYaaaaa@4880@ . The strain is decomposed into elastic and plastic parts as

ε ij = ε ij e + ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aa0baaSqaaiaadMgacaWGQb aabaaaaOGaeyypa0JaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGa amyzaaaakiabgUcaRiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaai aadchaaaaaaa@3FBD@

 Pressure Decomposition

 

Assume that the soil is subjected to a stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@ .  The pressure is p= σ kk /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH9aqpcqGHsislcqaHdpWCda WgaaWcbaGaam4AaiaadUgaaeqaaOGaai4laiaaiodaaaa@3899@  (note the negative sign).  In general, part of this pressure is supported by the water in the soil; while the rest is supported by the soil particles themselves.  The pressure is decomposed into two parts

p= p w + p s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH9aqpcaWGWbWaaSbaaSqaai aadEhaaeqaaOGaey4kaSIaamiCamaaBaaaleaacaWGZbaabeaaaaa@3785@

where p s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4Caaqabaaaaa@3281@  is the contribution to the pressure from the soil particles and p w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4Daaqabaaaaa@3285@  is the contribution from the water.

 

When using the constitutive equation in a boundary value problem, the water pressure must be calculated as a separate problem, in addition to solving the usual mechanical field equations.  Here, we outline briefly a  simple approximate description of fluid seepage through a soil.  More general treatments are also available, which include nonlinear versions of the flow law, finite strain effects, as well as the effects of fluid absorption by the soil particles to form a gel, the tendency of soil to absorb fluid due to capillarity, and the effects of partial soil saturation.

 

1.      Fluid seepage through the soil is driven by gravity and by fluid pressure variations.  The driving force is quantified by the piezometric head, defined as

ϕ=z+ p w ρ w g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjabg2da9iaadQhacqGHRaWkda WcaaqaaiaadchadaWgaaWcbaGaam4DaaqabaaakeaacqaHbpGCdaWg aaWcbaGaam4DaaqabaGccaWGNbaaaaaa@3B2C@

where z is the height above some arbitrary datum, p w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4Daaqabaaaaa@3285@  is the fluid pressure (compressive pressure is positive); ρ w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaWG3baabeaaaa a@3350@  is the fluid density; and g is the acceleration due to gravity.

2.      The volume of material flowing through unit area of solid in the x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaaa@327F@  direction per unit time obeys Darcy’s law

q i =k ϕ x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadghadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcqGHsislcaWGRbWaaSaaaeaacqGHciITcqaHvpGzaeaacqGH ciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaaaaa@3C20@

where k is a material parameter, known as the permeability of the medium.

3.      The fluid itself may be compressible, with bulk modulus K w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaam4Daaqabaaaaa@3260@

4.      The fluid can be absorbed in cavities in the soil.   The volume fraction of cavities n is defined as

n= d V c dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gacqGH9aqpdaWcaaqaaiaadsgaca WGwbWaaSbaaSqaaiaadogaaeqaaaGcbaGaamizaiaadAfaaaaaaa@3717@

where d V c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGwbWaaSbaaSqaaiaadogaae qaaaaa@3340@  is the cavity volume in a volume of soil (including both cavities and soil particles) dV.

5.      At time t=0 the solid starts with some cavity volume fraction n 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3241@ ; this volume fraction evolves as the solid is deformed.  Usually the dominant contribution to the cavity volume change occurs as a result of plastic compaction of the soil (more sophisticated treatments include an elastic contribution).  The cavity volume fraction after the solid is subjected to an infinitesimal plastic strain ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadchaaaaaaa@350E@  is

n= n 0 + ε kk p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gacqGH9aqpcaWGUbWaaSbaaSqaai aaicdaaeqaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadUgacaWGRbaa baGaamiCaaaaaaa@39CF@

6.      At time t=0, a (possibly zero) fluid pressure p w0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4Daiaaicdaae qaaaaa@333F@  acts on the solid; and for t>0 the values of either fluid pressure, or volumetric flow rate must be specified on the boundary of the solid.

7.      Finally, the rate of change of fluid pressure follows from conservation of fluid volume as

n K w p w t = q i x i + dn dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamOBaaqaaiaadUeadaWgaa WcbaGaam4DaaqabaaaaOWaaSaaaeaacqGHciITcaWGWbWaaSbaaSqa aiaadEhaaeqaaaGcbaGaeyOaIyRaamiDaaaacqGH9aqpdaWcaaqaai abgkGi2kaadghadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG 4bWaaSbaaSqaaiaadMgaaeqaaaaakiabgUcaRmaalaaabaGaamizai aad6gaaeaacaWGKbGaamiDaaaaaaa@4636@

 

 Elastic constitutive equations

 

The elastic strains are related to the stresses using the standard linear elastic stress-strain law.   The elastic strain is related to stress by

σ ij = C ijkl ε kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadMgacaWGQbGaam4Aaiaa dYgaaeqaaOGaeqyTdu2aa0baaSqaaiaadUgacaWGSbaabaGaamyzaa aaaaa@3E9F@

where C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@351A@  are the components of the elastic compliance tensor.  For the special case of an isotropic material with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3132@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@

σ ij = E 1+ν { ε ij + ν 12ν ε kk δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4kaSIaeqyV d4gaamaacmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaaki abgUcaRmaalaaabaGaeqyVd4gabaGaaGymaiabgkHiTiaaikdacqaH 9oGBaaGaeqyTdu2aaSbaaSqaaiaadUgacaWGRbaabeaakiabes7aKn aaBaaaleaacaWGPbGaamOAaaqabaaakiaawUhacaGL9baaaaa@4D98@

 

 Yield criterion and critical state surface

 

The yield criterion specifies the stresses that are required to cause plastic flow in the soil.  The concept is identical to the yield criterion used in metal plasticity, except that, unlike metals, hydrostatic pressure can cause yield in a soil.  The yield criterion is

f( σ ij )= ( p s a 1 ) 2 + ( σ e Ma ) 2 1=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH9aqpdaqadaqaamaalaaabaGa amiCamaaBaaaleaacaWGZbaabeaaaOqaaiaadggaaaGaeyOeI0IaaG ymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaa bmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaamyzaaqabaaakeaaca WGnbGaamyyaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGc cqGHsislcaaIXaGaeyypa0JaaGimaaaa@4A36@

where

1.        p s = σ kk /3 p w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4CaaqabaGccq GH9aqpcqGHsislcqaHdpWCdaWgaaWcbaGaam4AaiaadUgaaeqaaOGa ai4laiaaiodacqGHsislcaWGWbWaaSbaaSqaaiaadEhaaeqaaaaa@3CD1@  is the pressure exerted by the stresses on the soil skeleton,

2.        σ e = 3 S ij S ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGLbaabeaaki abg2da9maakaaabaGaaG4maiaadofadaWgaaWcbaGaamyAaiaadQga aeqaaOGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGVaGaaG OmaaWcbeaaaaa@3C6E@  where S ij = σ ij 1 3 σ kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiab gkHiTmaalaaabaGaaGymaaqaaiaaiodaaaGaeq4Wdm3aaSbaaSqaai aadUgacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqa baaaaa@422B@  is the Von-Mises effective stress;

3.       M is a material property, whose physical significance was described in Sect 3.10.1.  Usually M<1.

4.       a is a state variable that quantifies the current yield strength of the soil. At time t=0, the soil has some finite strength a= a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH9aqpcaWGHbWaaSbaaSqaai aaicdaaeqaaaaa@3420@ , which subsequently evolves with plastic straining, as described below.

 

The yield criterion is sketched in principal stress space on the figure MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it resembles a football (if you are American MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  or a rugby ball to the rest of the world) with its axis parallel to the line σ 1 = σ 2 = σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaaabeaaki abg2da9iabeo8aZnaaBaaaleaacaaIYaaabeaakiabg2da9iabeo8a ZnaaBaaaleaacaaIZaaabeaaaaa@3A89@ .  The shape of the football depends on M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for M=1 it is a sphere; for M<1 it is stretched parallel to its axis.  The size of the yield locus is determined by a.

 

The yield criterion, together with the hardening law, also define a critical state surface, which determines the stress where unrestricted shear deformation can occur at constant shear stress (i.e. with zero hardening).  If the stresses lie inside the critical state surface (this is known as the `wet’ side of critical state), the material shows stable strain hardening behavior.  If the stresses lie outside the critical state surface (known as the `dry’ side of the critical state), the material softens with plastic straining, and so violates the Drucker stability condition.  Under these conditions the material is unstable, and plastic strain tends to localize into shear bands.

 

The critical state surface for Camclay is

g( σ ij )= σ e M p s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH9aqpcqaHdpWCdaWgaaWcbaGa amyzaaqabaGccqGHsislcaWGnbGaamiCamaaBaaaleaacaWGZbaabe aaaaa@3E44@

The material is stable for g<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacqGH8aapcaaIWaaaaa@3312@  and unstable for g>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacqGH+aGpcaaIWaaaaa@3316@ .  The critical state surface is sketched in the figure: it is a cone, which cuts through the fattest part of the yield surface.

 

 Flow law

 

The flow law specifies the plastic strain components under a multiaxial state of stress.  Like metal plasticity, the Cam-clay model bases the flow law on the yield criterion, so that

d ε ij p =dλa f σ ij =dλ( 2 3 [ p s a 1 ] δ ij +3 S ij M 2 a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaeyypa0JaamizaiabeU7aSjaadggadaWc aaqaaiabgkGi2kaadAgaaeaacqGHciITcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaaaakiabg2da9iaadsgacqaH7oaBdaqadaqaaiab gkHiTmaalaaabaGaaGOmaaqaaiaaiodaaaWaamWaaeaadaWcaaqaai aadchadaWgaaWcbaGaam4CaaqabaaakeaacaWGHbaaaiabgkHiTiaa igdaaiaawUfacaGLDbaacqaH0oazdaWgaaWcbaGaamyAaiaadQgaae qaaOGaey4kaSIaaG4mamaalaaabaGaam4uamaaBaaaleaacaWGPbGa amOAaaqabaaakeaacaWGnbWaaWbaaSqabeaacaaIYaaaaOGaamyyaa aaaiaawIcacaGLPaaaaaa@5B79@

where dλ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBaaa@3305@  is a dimensionless constant that depends on the increment of stress applied to the solid, and is proportional to the plastic strain magnitude. The procedure to calculate dλ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBaaa@3305@  is discussed in more detail below: if the stress state lies inside the critical state surface  ( g( σ ij )<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH8aapcaaIWaaaaa@3841@  ), then dλ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBaaa@3305@  can be expressed in terms of the stress increment d σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaHdpWCdaWgaaWcbaGaamyAai aadQgaaeqaaaaa@351D@  and the fluid pressure increment d p w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGWbWaaSbaaSqaaiaadEhaae qaaaaa@336E@  applied to the solid as

dλ={ 0f( σ ij )<0 1 3a [ p s a 1 ]( d σ kk +3d p w )+ 3 2 S ij d S ij ( Ma ) 2 2c p s a [ p s a 1 ] f( σ ij )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBcqGH9aqpdaGabaqaau aabeqaceaaaeaacaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caWGMbGaaiikaiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaGccaGGPaGaeyipaWJaaGimaaqaamaalaaa baWaaaWaaeaacqGHsisldaWcaaqaaiaaigdaaeaacaaIZaGaamyyaa aadaWadaqaamaalaaabaGaamiCamaaBaaaleaacaWGZbaabeaaaOqa aiaadggaaaGaeyOeI0IaaGymaaGaay5waiaaw2faamaabmaabaGaam izaiabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccqGHRaWkcaaI ZaGaamizaiaadchadaWgaaWcbaGaam4DaaqabaaakiaawIcacaGLPa aacqGHRaWkdaWcaaqaaiaaiodaaeaacaaIYaaaamaalaaabaGaam4u amaaBaaaleaacaWGPbGaamOAaaqabaGccaWGKbGaam4uamaaBaaale aacaWGPbGaamOAaaqabaaakeaadaqadaqaaiaad2eacaWGHbaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaOGaayzkJiaawQYiaa qaaiaaikdacaWGJbWaaSaaaeaacaWGWbWaaSbaaSqaaiaadohaaeqa aaGcbaGaamyyaaaadaWadaqaamaalaaabaGaamiCamaaBaaaleaaca WGZbaabeaaaOqaaiaadggaaaGaeyOeI0IaaGymaaGaay5waiaaw2fa aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaadAgacaGGOaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaakiaacMcacqGH9aqpcaaIWaaaaaGa ay5Eaaaaaa@507E@

Here x =x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaamaabaGaamiEaaGaayzkJiaawQYiai abg2da9iaadIhaaaa@3538@  for x>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhacqGH+aGpcaaIWaaaaa@3327@  and x =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaamaabaGaamiEaaGaayzkJiaawQYiai abg2da9iaaicdaaaa@34F5@  for x<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhacqGH8aapcaaIWaaaaa@3323@ , and c is a material property governing the hardening rate, as defined below.  These expressions are valid only if the stress lies inside the critical state surface.  If the stress lies on or outside the critical state surface g( σ ij )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGHLjYScaaIWaaaaa@3903@  then dλ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBaaa@3305@  cannot be determined from the stress increment. 

 

 Hardening law

 

A soil becomes stronger if it is compacted to crush the soil particles together.   This is described in the constitutive law by making the state variable a evolve with plastic straining in some appropriate way.  A simple hardening law that captures the main features of experiments is

da=cad ε kk p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGHbGaeyypa0JaeyOeI0Iaam 4yaiaadggacaWGKbGaeqyTdu2aa0baaSqaaiaadUgacaWGRbaabaGa amiCaaaaaaa@3B8A@

where c is a dimensionless material property, which determines the hardening rate.  Notice that, in this law, hardening occurs only as a result of compaction, and not as a result of shear deformation.

 

 Calculating the plastic stress-strain relation

 

When using the constitutive equation, the formulas outlined in the preceding sections must be combined to predict the plastic strain d ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaaaa@35F7@  resulting from an increment in stress d σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaHdpWCdaWgaaWcbaGaamyAai aadQgaaeqaaaaa@351D@  and fluid pressure increment d p w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGWbWaaSbaaSqaaiaadEhaae qaaaaa@336E@ .  This is done as follows:

1.      Check the yield criterion.  If f<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacqGH8aapcaaIWaaaaa@3311@ , the plastic strain is zero d ε ij p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaeyypa0JaaGimaaaa@37C1@ .

2.      Check to see whether the stresses lie inside the critical state surface.  If g( σ ij )<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH8aapcaaIWaaaaa@3841@  the material behaves like a stable, strain hardening plastic solid, and the plastic strain increment can be calculated by following steps 3-5 below.

3.      Check for elastic unloading.   The solid will unload elastically, with d ε ij p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaeyypa0JaaGimaaaa@37C1@ , if the stress increment brings the stress below yield. This is the case whenever

f σ ij d σ ij <0( 2 3a [ p s a 1 ]( d σ kk +3d p w )+3 S ij d S ij ( Ma ) 2 )<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamOzaaqaaiabgk Gi2kabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaamizaiab eo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH8aapcaaIWaGaey O0H49aaeWaaeaacqGHsisldaWcaaqaaiaaikdaaeaacaaIZaGaamyy aaaadaWadaqaamaalaaabaGaamiCamaaBaaaleaacaWGZbaabeaaaO qaaiaadggaaaGaeyOeI0IaaGymaaGaay5waiaaw2faamaabmaabaGa amizaiabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccqGHRaWkca aIZaGaamizaiaadchadaWgaaWcbaGaam4DaaqabaaakiaawIcacaGL PaaacqGHRaWkcaaIZaWaaSaaaeaacaWGtbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaadsgacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaa aOqaamaabmaabaGaamytaiaadggaaiaawIcacaGLPaaadaahaaWcbe qaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaeyipaWJaaGimaaaa@65D1@

4.      If plastic strain does occur, the yield criterion must be satisfied throughout plastic straining.  This requires that

df= f σ ij d σ ij + f a da=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGMbGaeyypa0ZaaSaaaeaacq GHciITcaWGMbaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadMgacaWG QbaabeaaaaGccaWGKbGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabe aakiabgUcaRmaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadgga aaGaamizaiaadggacqGH9aqpcaaIWaaaaa@48BC@

It is straightforward to show that

f a = 2 p s a 2 ( p s a 1 )2 σ e 2 M 2 a 3 = 2 p s a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamOzaaqaaiabgk Gi2kaadggaaaGaeyypa0ZaaSaaaeaacqGHsislcaaIYaGaamiCamaa BaaaleaacaWGZbaabeaaaOqaaiaadggadaahaaWcbeqaaiaaikdaaa aaaOWaaeWaaeaadaWcaaqaaiaadchadaWgaaWcbaGaam4Caaqabaaa keaacaWGHbaaaiabgkHiTiaaigdaaiaawIcacaGLPaaacqGHsislca aIYaWaaSaaaeaacqaHdpWCdaqhaaWcbaGaamyzaaqaaiaaikdaaaaa keaacaWGnbWaaWbaaSqabeaacaaIYaaaaOGaamyyamaaCaaaleqaba GaaG4maaaaaaGccqGH9aqpcqGHsisldaWcaaqaaiaaikdacaWGWbWa aSbaaSqaaiaadohaaeqaaaGcbaGaamyyamaaCaaaleqabaGaaGOmaa aaaaaaaa@51C3@

5.      The hardening law and plastic flow rule give

da=cad ε kk p =cadλ f σ kk =2cdλ[ p s a 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGHbGaeyypa0JaeyOeI0Iaam 4yaiaadggacaWGKbGaeqyTdu2aa0baaSqaaiaadUgacaWGRbaabaGa amiCaaaakiabg2da9iabgkHiTiaadogacaWGHbGaamizaiabeU7aSn aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kabeo8aZnaaBaaaleaa caWGRbGaam4AaaqabaaaaOGaeyypa0JaaGOmaiaadogacaWGKbGaeq 4UdW2aamWaaeaadaWcaaqaaiaadchadaWgaaWcbaGaam4Caaqabaaa keaacaWGHbaaaiabgkHiTiaaigdaaiaawUfacaGLDbaaaaa@558C@

6.      Finally, combining 3-5 leads to

dλ= ( 1 3a [ p s a 1 ]( d σ kk +3d p w )+ 3 2 S ij d S ij ( Ma ) 2 ) 2c p s a [ p s a 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBcqGH9aqpdaWcaaqaam aabmaabaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4maiaadggaaaWa amWaaeaadaWcaaqaaiaadchadaWgaaWcbaGaam4Caaqabaaakeaaca WGHbaaaiabgkHiTiaaigdaaiaawUfacaGLDbaadaqadaqaaiaadsga cqaHdpWCdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaey4kaSIaaG4mai aadsgacaWGWbWaaSbaaSqaaiaadEhaaeqaaaGccaGLOaGaayzkaaGa ey4kaSYaaSaaaeaacaaIZaaabaGaaGOmaaaadaWcaaqaaiaadofada WgaaWcbaGaamyAaiaadQgaaeqaaOGaamizaiaadofadaWgaaWcbaGa amyAaiaadQgaaeqaaaGcbaWaaeWaaeaacaWGnbGaamyyaaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaaaeaa caaIYaGaam4yamaalaaabaGaamiCamaaBaaaleaacaWGZbaabeaaaO qaaiaadggaaaWaamWaaeaadaWcaaqaaiaadchadaWgaaWcbaGaam4C aaqabaaakeaacaWGHbaaaiabgkHiTiaaigdaaiaawUfacaGLDbaaaa aaaa@638A@

 

 

If the stress state is at yield f( σ ij )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH9aqpcaaIWaaaaa@3842@  and also lies on the critical state g( σ ij )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH9aqpcaaIWaaaaa@3843@  the material behaves like a perfectly plastic solid (with constant flow stress).  In this case,

1.      The solid unloads elastically, with d ε ij p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaeyypa0JaaGimaaaa@37C1@  if

f σ ij d σ ij <0( 2 3a [ p s a 1 ]( d σ kk +3d p w )+3 S ij d S ij ( Ma ) 2 )<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamOzaaqaaiabgk Gi2kabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaamizaiab eo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH8aapcaaIWaGaey O0H49aaeWaaeaacqGHsisldaWcaaqaaiaaikdaaeaacaaIZaGaamyy aaaadaWadaqaamaalaaabaGaamiCamaaBaaaleaacaWGZbaabeaaaO qaaiaadggaaaGaeyOeI0IaaGymaaGaay5waiaaw2faamaabmaabaGa amizaiabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccqGHRaWkca aIZaGaamizaiaadchadaWgaaWcbaGaam4DaaqabaaakiaawIcacaGL PaaacqGHRaWkcaaIZaWaaSaaaeaacaWGtbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaadsgacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaa aOqaamaabmaabaGaamytaiaadggaaiaawIcacaGLPaaadaahaaWcbe qaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaeyipaWJaaGimaaaa@65D1@

2.      If the solid deforms plastically, the stress state must satisfy

f σ ij d σ ij =0( 2 3a [ p s a 1 ]( d σ kk +3d p w )+3 S ij d S ij ( Ma ) 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamOzaaqaaiabgk Gi2kabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaamizaiab eo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaaIWaGaey O0H49aaeWaaeaacqGHsisldaWcaaqaaiaaikdaaeaacaaIZaGaamyy aaaadaWadaqaamaalaaabaGaamiCamaaBaaaleaacaWGZbaabeaaaO qaaiaadggaaaGaeyOeI0IaaGymaaGaay5waiaaw2faamaabmaabaGa amizaiabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccqGHRaWkca aIZaGaamizaiaadchadaWgaaWcbaGaam4DaaqabaaakiaawIcacaGL PaaacqGHRaWkcaaIZaWaaSaaaeaacaWGtbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaadsgacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaa aOqaamaabmaabaGaamytaiaadggaaiaawIcacaGLPaaadaahaaWcbe qaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaaaa@65D5@

In this case the plastic strain cannot be determined from the stress increment: any dλ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH7oaBcqGH+aGpcaaIWaaaaa@34C7@  is admissible.  In a situation where the total strain of the solid is prescribed, the plastic strain increment can be determined by first solving for the elastic strain increment, and subtracting it from the total strain.

 

If the stress lies outside the critical state g( σ ij )>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH+aGpcaaIWaaaaa@3845@  and is at yield f( σ ij )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaacMcacqGH9aqpcaaIWaaaaa@3842@ , the material softens.  In this case it is impossible to distinguish unambiguously between elastic unloading and plastic flow accompanied by strain softening.  The deformation in this regime usually consists of intense plastic shearing along one or more discrete shear bands, while the rest of the material unloads elastically.   Boundary value problems with material behavior in the unstable regime are generally ill-posed and cannot be solved uniquely. However, attempting to load a soil past the critical state usually results in catastrophic collapse (such as a landslide), so detailed solutions to boundary value problems in this regime are not of much practical interest.  The critical state surface can be used as a failure criterion to avoid collapse.

 

 

3.11.3 Application of the critical state equations to simple 2D loading

 

The constitutive equations for soils are complicated, and a simple 2D example helps to interpret them. To this end, consider a solid subjected to a 2D stress state of the form σ ij =p δ ij +q( δ i1 δ j2 + δ i2 δ j1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqGHsislcaWGWbGaeqiTdq2aaSbaaSqaaiaadMga caWGQbaabeaakiabgUcaRiaadghacaGGOaGaeqiTdq2aaSbaaSqaai aadMgacaaIXaaabeaakiabes7aKnaaBaaaleaacaWGQbGaaGOmaaqa baGccqGHRaWkcqaH0oazdaWgaaWcbaGaamyAaiaaikdaaeqaaOGaeq iTdq2aaSbaaSqaaiaadQgacaaIXaaabeaakiaacMcaaaa@4D05@ , as illustrated in the picture. Assume that the specimen is drained, so that the water pressure p w =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4DaaqabaGccq GH9aqpcaaIWaaaaa@344F@ .  In addition, assume that at time t=0 the solid has strength a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaGimaaqabaaaaa@3234@

 

For this loading, the yield surface can be plotted in 2D, as a graph of the critical combinations of p and q that cause yield, as shown in the picture below.  The yield surface is an ellipse, with semi-axes a and Ma.  The critical state surface is a straight line with slope M.


We can now examine the behavior of the solid as it is loaded.  Consider first the response to a constant pressure on the `wet’ side of critical state p> a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH+aGpcaWGHbWaaSbaaSqaai aaicdaaeqaaaaa@3431@ , together with a steadily increasing shear stress q.  In this case

1.      The solid first reaches yield when ( p/ a 0 1 ) 2 +3 ( q/M a 0 ) 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiCaiaac+cacaWGHbWaaS baaSqaaiaaicdaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgUcaRiaaiodadaqadaqaaiaadghaca GGVaGaamytaiaadggadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIXaaaaa@4237@

2.      If the shear stress is raised beyond yield, the solid will deform plastically.  Since the flow law is derived from the yield criterion, the plastic strain direction is normal to the yield surface.  That is to say, if the solid experiences a plastic shear strain dγ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaHZoWzaaa@32F8@  and volumetric strain dv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2baaaa@324C@ , the vector ( dv,dγ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bGaaiilaiaadsgacqaHZo Wzaaa@358C@  ) is normal to the yield surface, as shown in the picture.

3.      On the `wet’ side of critical state, the volumetric plastic strain component dv is always compressive.  This means the solid compacts, and its strength increases (recall that da=cadv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGHbGaeyypa0JaeyOeI0Iaam 4yaiaadggacaWGKbGaamODaaaa@37DC@  and dv<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bGaeyipaWJaaGimaaaa@340A@  during compaction). 

4.      As the yield surface expands, the volumetric strain component associated with an increase in shear stress dq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGXbaaaa@3247@  decreases (remember that we assume a constant pressure).    The hardening rate therefore decreases with strain, until the stress reaches the critical state.  At this point dv=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bGaeyypa0JaaGimaaaa@340C@ , so there is no further hardening.

Next, consider behavior on the `dry’ side of critical state. In this case

1.      The solid first reaches yield when ( p/ a 0 1 ) 2 +3 ( q/M a 0 ) 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiCaiaac+cacaWGHbWaaS baaSqaaiaaicdaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgUcaRiaaiodadaqadaqaaiaadghaca GGVaGaamytaiaadggadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIXaaaaa@4237@

2.      As before, the direction of the plastic strain is normal to the yield surface.

3.      Notice that on the `dry’ side of critical state, the volumetric plastic strain component dv is always dilatational.  This means its strength decreases with plastic straining, as shown in the figure (recall that da=cadv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGHbGaeyypa0JaeyOeI0Iaam 4yaiaadggacaWGKbGaamODaaaa@37DC@  and dv>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bGaeyOpa4JaaGimaaaa@340E@  during dilation). 

4.      The yield surface contracts during plastic straining, and this process continues until the stress reaches the critical state.  At this point the solid continues to deform at a constant shear stress.

 

 

 

3.11.4 Typical values of material properties for soils

 

Soil properties are highly variable, and for accurate predictions you will need to measure directly the properties of the soil you are intending to model.  In addition, soil models that are used in practice are somewhat more sophisticated than the simplified version given here. As a rough guide, material properites estimated from data in D.M. Wood, “Soil Behavior and Critical State Soil Mechanics,” Cambridge University Press, Cambridge, 1990 are listed in the table below.

 

 

Bulk modulus

 K

Water bulk modulus K w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaam4Daaqabaaaaa@3260@

Hardening rate

 c

Critical state constant

M

Initial strength

  a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaGimaaqabaaaaa@3234@

2GPa

2.2GPa

5

0.8

0.2MPa