Deviatoric Stress Calculation

With the stress being separated into deviatoric and pressure (hydrostatic) stress (Stresses in Solids), it is the deviatoric stress that is responsible for the plastic deformation of the material. The hydrostatic stress will either shrink or expand the volume uniformly, that is, with proportional change in shape. The determination of the deviatoric stress tensor and whether the material will plastically deform requires a number of steps.

Perform an Elastic Calculation

The deviatoric stress is time integrated from the previous known value using the strain rate to compute an elastic trial stress:(1)
s i j e l ( t + Δ t ) = s i j ( t ) + s ˙ i j r Δ t + 2 G ( ε ˙ i j 1 3 ε ˙ k k δ i j ) Δ t
G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raaaa@36C2@
Shear modulus

This relationship is Hooke's Law, where the strain rate is multiplied by time to give strain.

Compute von Mises Equivalent Stress and Current Yield Stress

Depending on the type of material being modeled, the method by which yielding or failure is determined will vary. The following explanation relates to an elastoplastic material (LAW2).

The von Mises equivalent stress relates a three dimensional state of stress back to a simple case of uniaxial tension where material properties for yield and plasticity are well known and easily computed.

The von Mises stress, which is strain rate dependent, is calculated using the equation:(2)
σ v m e = 3 2 s i j e l s i j e l
The flow stress is calculated from the previous plastic strain:(3)
σ y ( t ) = a + b ε p n ( t )

For material laws 3, 4, 10, 21, 22, 23 and 36, Equation 3 is modified according to the different modeling of the material curves.

Plasticity Check

The state of the deformation must be checked.(4)
σ v m e σ y 0
If this equation is satisfied, the state of stress is elastic. Otherwise, the flow stress has been exceeded and a plasticity rule must be used (Figure 1).

Figure 1. Plasticity Check

The plasticity algorithm used is due to Mendelson. 1

Compute Hardening Parameter

The hardening parameter is defined as the slope of the strain-hardening part of the stress-strain curve:(5)
H = d σ y d ε p
This is used to compute the plastic strain at time t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG0baaaa@39D0@ :(6)
ε ˙ p Δ t = σ v m σ y 3 G + H
This plastic strain is time integrated to determine the plastic strain at time t + Δ t :(7)
ε p ( t + Δ t ) = ε p ( t ) + ε ˙ p Δ t
The new flow stress is found using:(8)
σ y ( t + Δ t ) = a + b ε p n ( t + Δ t )

Radial Return

There are many possible methods for obtaining s i j p a from the trial stress. The most popular method involves a simple projection to the nearest point on the flow surface, which results in the radial return method.

The radial return calculation is given in Equation 9. Figure 2 is a graphic representation of radial return.(9)
s i j p a = σ y σ v m s i j e l

Figure 2. Radial Return
1 Mendelson A., “Plasticity: Theory and Application”, MacMillan Co., New York, 1968.