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=4e6
σv=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=1002,
τmax=100,
σnorm=3002, and
σtm=3002.

Table 1: Stress/Strain Invariants
Stress Invariant Definition Strain Invariant Definition
1st stress invariant I1=σkk 1st strain invariant I1=εkk
2nd stress invariant I2=(σxxσyy+σyyσzz+σzzσxx)(σxyσyz+σyzσzx+σzxσxy) 2nd strain invariant I2=(εxxεyy+εyyεzz+εzzεxx)(εxyεyz+εyzεzx+εzxεxy)
3rd stress invariant I3=|σij| 3rd strain invariant I3=|εij|
volumetric stress σv=σkk volumetric strain εv=εkk
mean stress σm=σkk/3 mean strain εm=εkk/3
1st deviatoric stress invariant J10 1st deviatoric strain invariant J10
2nd deviatoric stress invariant J2=(sijsij)/2 2nd deviatoric strain invariant J2=(eijeij)/2
3rd deviatoric stress invariant J3=(sijsikski)/3 3rd deviatoric strain invariant J3=(eijeikeki)/3
stress Loge’s angle θσ=13arcsin(332J3J1.52) strain Loge’s angle θε=13arcsin(332J3J1.52)
minimum principal stress σ1=23J2sin(θσ23π)+σm minimum principal strain ε1=23J2sin(θε23π)+εm
intermediate principal stress σ2=23J2sin(θσ)+σm intermediate principal strain ε2=23J2sin(θε)+εm
maximum principal stress σ3=23J2sin(θσ+23π)+σm maximum principal strain ε3=23J2sin(θε+23π)+εm
von Mises equivalent stress σeq=3J2 von Mises equivalent strain εeq=43J2
equivalent deviatoric stress q=3J2 equivalent (engineering) shear strain γ=4J2
octahedral stress σoct=23J2 octahedral strain εoct=83J2
maximum shear stress τmax=(σ3σ1)/2 maximum shear strain γmax=ε3ϵ1
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=I21/3+2J2