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When a CP model (section 2.8) is used to close a hyper-elastoplastic model, source term splitting is once again used to separate the conservative fluxes from the source terms. The source term part of the equations is

∂Fe ∂t =−F

e_Lp _(3.72)

where Lp, ignoring dependencies on meso-scale variables, is a function of the elastic deformation gradient through its dependence on stress

Lp =Lp(σ) =Lp(Fe) (3.73)

Using a discretization analogous to the source term splitting scheme used in section 3.2,

Fen+1=Fe∗−∆tFe∗Lp(σ∗) (3.74)

where Fe∗ andσ∗ are the elastic deformation and stress predicted by the hyper-elastic model. One may think this a good approximation of the true elastic deformation at the next time step. However, when the stress predicted by the elastic solver is significantly larger than the true yield stress, Lp will be unphysically large in magnitude. Mathematically, this happens because of the exponential

terms with stress appearing as a positive number inside the exponent. These terms appear in various places within the CP model such as the equation for velocity of mobile dislocations (2.160).

Physically speaking, this problem can be explained from a few different perspectives. First, the model itself is designed assuming the true stress is known at any given time. Therefore, a stress that is well outside the yield surface is not even considered a possibility when developing the model. Alternatively, the problem can be viewed as a side effect of combining models on vastly different scales. The macro-scale phenomenon of elasticity occurs on a time scale much larger than the meso-scale phenomenon of dislocation evolution. This issue is seen in the context of a numerical solver by the fact that a time step which is stable in solving the elastic part of the equations is unstable when applied to the plastic part of the equations. After a single step, the elastic solver predicts a stress state well beyond the yield surface. In reality, plasticity would’ve taken effect long before the stress could have ever reach such a fictional state.

One solution to this problem would be too run the entire model on a time step that is within the stable range of the dislocation model, the most restrictive time scale present in the model. The downside of this approach is that running the elastic solver on a time step significantly less than the maximum allowable time step will result in undesired numerical artifacts such as numerical viscosity.

The other solution is to solve (3.72) implicitly. One such implicit discretization of (3.72) is

Fen+1 =Fe∗−∆tFe∗Lp(Fen+1). (3.75)

Here, the only difference is the argument of Lp is replaced with the true state at the next time step as opposed to the elastic-prediction state. This discretization avoids exploring the space of stress states in which the magnitude of Lp is extremely large.

Following the work of [36], (3.75) is solved with the plastic velocity gradient Lp as its unknown. Solving forLp,

Lp = 1

∆t(I−(F

e∗₎−1

Fen+1). (3.76)

The procedure uses a modified Newton method to solve the system of equations for the components of Lp. The initial guess is made by replacing Fen+1 =I

Lp0 = 1

∆t(I −(F

e∗₎−1

Going forward, all matrices should be treated as vectors by placing their components into a single vector one column at a time. LetLpk denote the current guess for Lp. Then the residual of the current guess is

rk=Lpk−Lp(Lpk) (3.78) whereLp can be viewed indirectly as a function of the current guess. The Jacobian is

Jk = dr

k

dLpk =I −

dLp

dLpk. (3.79)

While seeming complex, the Jacobian can be decomposed by chain rule into computationally simpler terms Jk=I−dL p dτ dτ dσpk2 dσpk2 dσ dσ dFe dFe dLp(L pk ). (3.80)

Newton’s method attempts to minimize the residual by picking a new guess as

Lpk+1 =Lpk −(Jk)−1rk. (3.81)

However, this method does not always yield a new residual that is smaller than the previous residual in norm. To resolve this issue, a modified Newton method is employed

Lpk+1=Lpk−αk(Jk)−1rk. (3.82)

Within a step of the modified Newton method,αk _{starts as1. If the resultingr}k+1 _{is less than r}k _in

norm

krk+1_k<krk_{k (3.83)}

then the new guess is accepted and the method moves to the next iteration. If (3.83) doesn’t hold, αk is cut in half and the new Lpk+1 computed by (3.82). This process is repeated until the new residual is less than the previous in norm. The entire loop is terminated when

krk_k

kLpk

When aLp has been settled upon, the new elastic deformation is simply calculated as

CHAPTER 4