Stress/Strain Invariants
In FLAC3D, assuming sij is the deviatoric part of the stress tensor σij, and eij is the deviatoric part of the strain tensor εij, the stress/strain invariants in FLAC3D are defined and summarized in Table 1.
For example, in a triaxial compression test, the cell pressure in the xy-plane is 100, the vertical compression pressure in the z-direction is 400, the stress tensor in the sample is thus
σij={σxx,σyy,σzz,σxy,σyz,σxz}={−100,−100,−400,0,0,0}
According to equations in Table 1, we have
I1=−600,I2=9e4,I3=−4e6σv=−600,σm=−200,eij={sxx,syy,szz,sxy,syz,sxz}={100,100,−200,0,0,0},J1=0,J2=3e4,J3=−2e6,θσ=π/6,σ1=−400,σ2=σ3=−100,σeq=q=300,σoct=100√2,τmax=100,σnorm=300√2, andσtm=300√2.
Stress Invariant | Definition | Strain Invariant | Definition |
---|---|---|---|
1st stress invariant | I1=σkk | 1st strain invariant | I′1=εkk |
2nd stress invariant | I2=(σxxσyy+σyyσzz+σzzσxx)−(σxyσyz+σyzσzx+σzxσxy) | 2nd strain invariant | I′2=(εxxεyy+εyyεzz+εzzεxx)−(εxyεyz+εyzεzx+εzxεxy) |
3rd stress invariant | I3=|σij| | 3rd strain invariant | I′3=|εij| |
volumetric stress | σv=σkk | volumetric strain | εv=εkk |
mean stress | σm=σkk/3 | mean strain | εm=εkk/3 |
1st deviatoric stress invariant | J1≡0 | 1st deviatoric strain invariant | J′1≡0 |
2nd deviatoric stress invariant | J2=(sijsij)/2 | 2nd deviatoric strain invariant | J′2=(eijeij)/2 |
3rd deviatoric stress invariant | J3=(sijsikski)/3 | 3rd deviatoric strain invariant | J′3=(eijeikeki)/3 |
stress Loge’s angle | θσ=13arcsin(−3√32J3J1.52) | strain Loge’s angle | θε=13arcsin(−3√32J′3J′1.52) |
minimum principal stress | σ1=2√3√J2sin(θσ−23π)+σm | minimum principal strain | ε1=2√3√J′2sin(θε−23π)+εm |
intermediate principal stress | σ2=2√3√J2sin(θσ)+σm | intermediate principal strain | ε2=2√3√J′2sin(θε)+εm |
maximum principal stress | σ3=2√3√J2sin(θσ+23π)+σm | maximum principal strain | ε3=2√3√J′2sin(θε+23π)+εm |
von Mises equivalent stress | σeq=√3J2 | von Mises equivalent strain | εeq=√43J′2 |
equivalent deviatoric stress | q=√3J2 | equivalent (engineering) shear strain | γ=√4J′2 |
octahedral stress | σoct=√23J2 | octahedral strain | εoct=√83J′2 |
maximum shear stress | τmax=(σ3−σ1)/2 | maximum shear strain | γmax=ε3−ϵ1 |
stress norm | σnorm=√σijσij | strain norm | εnorm=√εijεij |
stress total measure | σtm=√σ21+σ22+σ23=√I21/3+2J2 | strain total measure | εtm=√ε21+ε22+ε23=√I′21/3+2J′2 |
Was this helpful? ... | FLAC3D © 2019, Itasca | Updated: Feb 25, 2024 |