I have developed a VUMAT material model for Johnson-Cook plasticity. Now, I have to add Johnson-Cook damage/failure model to my code.
Johnson-Cook Damage model :
D = ∑ εp(eff)/εf
Fracture strain:
εf = [D1 + D2exp(D3σ*)][1 + D4(εp*)][1+ D5T*]
How can I calculate equivalent plastic strain increment, εp(eff)? Though I have already calculated equivalent deviatoric plastic strain increment in the VUMAT.
And how can I update the new stress after every time step?
The JC model is supposed to provide you with the effective plastic strain rate. Its implementation within a explicit FE solver is quite straight forward. You should use the following steps:
STEP 0: Get the stress tensor s, the damage variable D, the temperature T and the equivalent plastic strain ep,eq at the beginning of the time increment
STEP 1: Calculate the equivalent stress seq at beginning of the time increment with the yield criterion of your choice (e.g. von Mises, Tresca...)
seq=f(s)
STEP 2: Calculate the plastic flow direction n according to:
nij=dseq / dsij
STEP 3: Calculate the equivalent plastic strain rate dep,eq/dt from the JC constitutive equation
dep,eq/dt = f(seq, T, ep,eq)
STEP 4: Calculate the plastic strain rate tensor
dep,ij/dt=dep,eq/dt*nij
STEP 5: Calculate the stress rate tensor
dsij/dt=(1-D)*Cijkl *(dekl/dt -dep,kl/dt)
where C is the fourth rank stiffness tensor. For the specific case of isotropic elasticity, the above relation becomes:
dsij/dt=(1-D)*L *(dekk/dt -dep,kk/dt)*1ij+2*(1-D)*G *(deij/dt -dep,ij/dt)
where L and G are the Lamé coefficients. For the even more specific case of isotropic elasticity and purely deviatoric plasticity, the stress rate is given by:
dsij/dt=(1-D)*L dekk/dt*1ij+2*(1-D)*G *(deij/dt -dep,ij/dt)
STEP 6: Calculate the plastic strain tensor and equivalent plastic strain at the end of the time increment
ep,ij=ep,ij+dep,ij/dt*dt
ep,eq=ep,eq+dep,eq/dt*dt
STEP 7: Calculate the stress tensor at the end of the time increment
sij=sij+dsij/dt*dt
STEP 8: Calculate the damage indicator at the end of the time increment
D=ep,eq/ef
Hello
You can read all principal strains in your VUMAT then you have to calculate the Equivalent plastic strain using the formula regarding your constitutive equation.
I hope it helps
The JC model is supposed to provide you with the effective plastic strain rate. Its implementation within a explicit FE solver is quite straight forward. You should use the following steps:
STEP 0: Get the stress tensor s, the damage variable D, the temperature T and the equivalent plastic strain ep,eq at the beginning of the time increment
STEP 1: Calculate the equivalent stress seq at beginning of the time increment with the yield criterion of your choice (e.g. von Mises, Tresca...)
seq=f(s)
STEP 2: Calculate the plastic flow direction n according to:
nij=dseq / dsij
STEP 3: Calculate the equivalent plastic strain rate dep,eq/dt from the JC constitutive equation
dep,eq/dt = f(seq, T, ep,eq)
STEP 4: Calculate the plastic strain rate tensor
dep,ij/dt=dep,eq/dt*nij
STEP 5: Calculate the stress rate tensor
dsij/dt=(1-D)*Cijkl *(dekl/dt -dep,kl/dt)
where C is the fourth rank stiffness tensor. For the specific case of isotropic elasticity, the above relation becomes:
dsij/dt=(1-D)*L *(dekk/dt -dep,kk/dt)*1ij+2*(1-D)*G *(deij/dt -dep,ij/dt)
where L and G are the Lamé coefficients. For the even more specific case of isotropic elasticity and purely deviatoric plasticity, the stress rate is given by:
dsij/dt=(1-D)*L dekk/dt*1ij+2*(1-D)*G *(deij/dt -dep,ij/dt)
STEP 6: Calculate the plastic strain tensor and equivalent plastic strain at the end of the time increment
ep,ij=ep,ij+dep,ij/dt*dt
ep,eq=ep,eq+dep,eq/dt*dt
STEP 7: Calculate the stress tensor at the end of the time increment
sij=sij+dsij/dt*dt
STEP 8: Calculate the damage indicator at the end of the time increment
D=ep,eq/ef
plastic strain ( and damage) is local, stress tensor resulting from it is global, assuming the increment as linear (this presumes convexity) actual plastic strain+increment (epi+deltepi) result in a damage value (Di), that have to be in equilibrium for the global problem resulting in using them, iteration over i can converge using a criteria based on final damage value (convergence is not insured if non convex)
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