4. Digital integration of behavioral relationships

4.1. Problem reminder

For a given loading increment and a set of given variables (initial field of displacement, constraint and internal variable), we solve the discretized global system (2.2.2.2-1 of [bib3]) which seeks to satisfy the equilibrium equations.

Solving this system gives us \(\mathrm{\Delta u}\), so \(\mathrm{\Delta \epsilon }\). We therefore look locally, at each Gauss point, for the increment of constraint and internal variable corresponding to \(\mathrm{\Delta \epsilon }\) and which satisfy the law of behavior.

The following notations are used: \({A}^{-},A,\mathrm{\Delta A}\) for the quantity evaluated at the known instant t, at the time \(t+\mathrm{\Delta t}\) and its increment, respectively. The equations are discretized implicitly, expressed as a function of variables unknown at time \(t+\mathrm{\Delta t}\).

4.2. Calculation of constraints and internal variables

The elastic prediction of the deviatoric stress is written as:

\({s}^{e}={s}^{-}+\mathrm{2\mu \Delta }\tilde{\epsilon }\) eq 4.2-1

but you can always write \(s\) at the moment + as being:

\(s={s}^{-}+\mathrm{2\mu \Delta }{\tilde{\epsilon }}^{e}\) eq 4.2-2

These two equations allow us to deduce \(s\) as a function of \({s}^{e}\):

\(s={s}^{e}-\mathrm{2\mu \Delta }\tilde{\epsilon }+\mathrm{2\mu \Delta }{\tilde{\epsilon }}^{e}\) eq 4.2-3

\(\text{ou}s={s}^{e}-\mathrm{2\mu \Delta }{\tilde{\epsilon }}^{p}\) eq 4.2-4

Replacing \(\Delta {\tilde{\epsilon }}^{p}\) with its expression according to \({\mathrm{\Delta \epsilon }}_{v}^{p}\), we get:

\(s=\frac{{s}^{e}}{1+\frac{{\mathrm{3\mu \Delta \epsilon }}_{v}^{p}}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}}\) eq 4.2-5

from where,

\(Q=\frac{{Q}^{e}}{1+\frac{{\mathrm{3\mu \Delta \epsilon }}_{v}^{p}}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}}\) eq 4.2-6

Assuming that \({k}_{0}\) is independent of temperature, the incremental writing of \(P\) is written as:

\(P={P}^{-}\text{exp}\left[{k}_{0}{\epsilon }_{v}^{e}-{k}_{0}{\epsilon }_{v}^{{e}^{-}}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[{k}_{0}{\epsilon }_{v}^{e}-{k}_{0}{\epsilon }_{v}^{{e}^{-}}\right]-1)\) eq 4.2-8

\(P={P}^{-}\text{exp}\left[{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e}\right]-1)\) eq 4.2-9

\(\mathrm{\Delta P}={P}^{-}(\text{exp}\left[{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e}\right]-1)+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e}\right]-1)\) eq 4.2.10

Likewise, we can write the expression for \({P}^{e}\) in terms of \({P}^{\text{-}}\):

\({P}^{e}={P}^{-}\text{exp}\left[{k}_{0}{\mathrm{\Delta \epsilon }}_{v}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[{k}_{0}{\mathrm{\Delta \epsilon }}_{v}\right]-1)\) eq 4.2-11

Hence the expression for \(P\) at the moment + is:

\(P={P}^{e}\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]-1)\) eq 4.2-12

In the incremental writing of \({P}_{\text{cr}}\), the coefficient \(k\) does not depend on temperature, so we find the following expression:

\({P}_{\text{cr}}={P}_{\text{cr}0}\text{exp}\left[k({\epsilon }_{v}^{p}-{\epsilon }_{v}^{{p}_{0}})\right]\) eq 4.2-13

\({P}_{\text{cr}}={P}_{\text{cr}}^{-}\text{exp}\left[{\mathrm{k\Delta \epsilon }}_{v}^{p}\right]\) eq 4.2-14

\({\mathrm{\Delta P}}_{\text{cr}}={P}_{\text{cr}}^{-}\left[\text{exp}({\mathrm{k\Delta \epsilon }}_{v}^{p})-1\right]\) eq 4.2-15

Summary:

\(f({s}^{e},{P}^{e},{P}_{\text{cr}}^{-})\le 0\) in this case \({\mathrm{\Delta P}}_{\text{cr}}=0\) or \(s={s}^{-}+\mathrm{\Delta s}={s}^{e}\)

\(P={P}^{e}\)

\(f({s}^{e},{P}^{e},{P}_{\text{cr}}^{-})>0\) in this case \({\mathrm{\Delta P}}_{\text{cr}}>0\), \(\Delta {\tilde{\epsilon }}^{p}\ne 0\) and \({\mathrm{\Delta \epsilon }}_{v}^{p}\ne 0\)

Be \(s={s}^{e}-\mathrm{2\mu \Delta }{\tilde{\epsilon }}^{p}\)

\(P={P}^{e}\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]-1)\)

\({P}_{\text{cr}}={P}_{\text{cr}}^{-}\text{exp}\left[{\mathrm{k\Delta \epsilon }}_{v}^{p}\right]\)

Note:

The main unknown is \({\mathrm{\Delta \epsilon }}_{v}^{p}\) .

4.3. Calculating the unknown \({\mathrm{\Delta \epsilon }}_{v}^{p}\)

By entering into the criterion the expressions of \(P\) and \(Q\) as a function of \({P}^{e}\) and \({Q}^{e}\) and using the equation [éq 4.2-6]:

\({Q}_{e}^{2}=-{\left[1+\frac{{\mathrm{3\mu \Delta \epsilon }}_{v}^{p}}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}\right]}^{2}{M}^{2}(P-{P}_{\text{trac}})(P-{P}_{\text{trac}}-{\mathrm{2P}}_{\text{cr}})\) eq 4.3-1

\(\begin{array}{}{Q}_{e}^{2}=-{M}^{2}{\left[1+\frac{{\mathrm{3\mu \Delta \epsilon }}_{v}^{p}}{{M}^{2}({P}_{e}\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]-1)-{P}_{\text{trac}}-{P}_{\text{cr}}^{-}\text{exp}\left[{\mathrm{k\Delta \epsilon }}_{v}^{p}\right])}\right]}^{2}\\ ({P}_{e}\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]-1)-{P}_{\text{trac}})\\ ({P}_{e}\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]-1)-{P}_{\text{trac}}-{\mathrm{2P}}_{\text{cr}}^{-}\text{exp}\left[{\mathrm{k\Delta \epsilon }}_{v}^{p}\right])\end{array}\) eq 4.3-2

In the following sub-paragraph we determine limits to this function that facilitate the resolution of equation [éq 4.3-2] with, for example, the string method or the Newton method.

4.4. Determining the limits of the function

We’re asking \({\mathrm{\Delta \epsilon }}_{v}^{p}=x\) the unknown of the problem.

So we have:

\(P(x)={P}^{e}\text{exp}(-{k}_{0}x)+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}(-{k}_{0}x)-1)\) eq 4.4-1

\({P}_{\text{cr}}(x)={P}_{\text{cr}}^{-}\text{exp}(\text{kx})\) eq 4.4-2

\(\Lambda (x)=\frac{x}{{\mathrm{2M}}^{2}(P(x)-{P}_{\text{trac}}-{P}_{\text{cr}}(x))}\) eq 4.4-3

\(Q(x)=\frac{{Q}^{e}}{1+6\mu \Lambda (x)}\) eq 4.4-4

\(f(x)={Q}^{2}(x)+{M}^{2}{(P(x)-{P}_{\text{trac}})}^{2}-{\mathrm{2M}}^{2}(P(x)-{P}_{\text{trac}}){P}_{\text{cr}}(x)=0\) eq 4.4-5

At the point \(x=0;P(0)={P}^{e};{P}_{\text{cr}}(0)={P}_{\text{cr}}^{-};\lambda (0)=0;Q(0)={Q}^{e}\) eq 4.4-6

\(f(0)={Q}^{{e}^{2}}+{M}^{2}({P}^{e}-{P}_{\text{trac}})({P}^{e}-{P}_{\text{trac}}-{\mathrm{2P}}_{\text{cr}}^{-})\) eq 4.4-7

\(f(0)>0\)

At the point:

\(P-{P}_{\text{trac}}={P}_{\text{cr}};\Lambda ({x}_{b})=\infty ;Q({x}_{b})=0\text{et}f({x}_{b})=-{M}^{2}{(P-{P}_{\text{trac}})}^{2}\) eq 4.4-8

\(f({x}_{b})<0\)

In \(x=0\); \(f(0)>0\) and in \(x={x}_{b}\); \(f({x}_{b})<0\)

We are looking for x between 0 and \({x}_{b}\); to determine it, we write:

\(P({x}_{b})-{P}_{\text{trac}}={P}_{\text{cr}}({x}_{b})\)

\(\iff {P}^{e}\text{exp}(-{k}_{0}{x}_{b})+\frac{{K}_{\text{cam}}}{{k}_{0}}\text{exp}(-{k}_{0}{x}_{b})-{P}_{\text{cr}}^{-}\text{exp}({\text{kx}}_{b})=\frac{{K}_{\text{cam}}}{{k}_{0}}+{P}_{\text{trac}}\) eq 4.4-9

It’s a non-linear equation in \({x}_{b}\), we do a limited expansion of order 1 to deduce the expression for \({x}_{b}\):

If \({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}}=0\); \({x}_{b}=0\) and \({\mathrm{\Delta \epsilon }}_{v}^{p}=0\)

If \({k}_{0}{P}^{e}+{K}_{\text{cam}}+{\text{kP}}_{\text{cr}}^{-}\ne 0\); \({x}_{b}=(\frac{{P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}}}{{k}_{0}{P}^{e}+{K}_{\text{cam}}+{\text{kP}}_{\text{cr}}^{-}})\)

Otherwise we do a limited development of order 2 and we find;

\(({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}})-({k}_{0}{P}^{e}+{K}_{\text{cam}}+{\text{kP}}_{\text{cr}}^{-}){x}_{b}+\frac{1}{2}({k}_{0}{P}^{e}+{K}_{\text{cam}}-{\text{kP}}_{\text{cr}}^{-}){x}_{b}^{2}=0\)

Like \({k}_{0}{P}^{e}+{K}_{\text{cam}}+{\text{kP}}_{\text{cr}}^{-}=0\) then \({k}_{0}{P}^{e}+{K}_{\text{cam}}-{\text{kP}}_{\text{cr}}^{-}\ne 0\)

And we solve

\(({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}})+\frac{1}{2}({k}_{0}{P}^{e}+{K}_{\text{cam}}-{\text{kP}}_{\text{cr}}^{-}){x}_{b}^{2}=0\)

If \({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}}=0\); \({x}_{b}=0\) and \({\mathrm{\Delta \epsilon }}_{v}^{p}=0\)

Otherwise \({x}_{b}=\pm \sqrt{\frac{-2({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}})}{({k}_{0}{P}^{e}+{K}_{\text{cam}}-{\text{kP}}_{\text{cr}}^{-})}}\)

If \(\sigma\) <0 we choose an approximate value for \({x}_{b}\) to be \({x}_{b}=\frac{1}{{k}_{0}+k}\text{Log}(\frac{\mid {P}^{e}-{P}_{\text{trac}}\mid }{{P}_{\text{cr}}^{-}})\)

Otherwise we have the choice between two values of \({x}_{b}\);

We do the following test:

If \(({P}^{e}-{P}_{\text{trac}}>{P}_{\text{cr}}^{-})\) then \({x}_{b}=\sqrt{\frac{-2({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}})}{({k}_{0}{P}^{e}+{K}_{\text{cam}}-{\text{kP}}_{\text{cr}}^{-})}}\); the solution would be positive; x>0

If \(({P}^{e}-{P}_{\text{trac}}<{P}_{\text{cr}}^{-})\) then \({x}_{b}=-\sqrt{\frac{-2({P}^{e}-{P}_{\text{cr}}^{-}-{P}_{\text{trac}})}{({k}_{0}{P}^{e}+{K}_{\text{cam}}-{\text{kP}}_{\text{cr}}^{-})}}\); the solution would be negative; x<0

4.5. Special case of the critical point

Figure 4.5-a: Mechanical state around the critical point

If at the moment \({t}^{-}\) we reach critical condition, then \({P}_{\text{cr}}^{+}={P}_{\text{cr}}^{-},{\mathrm{\Delta \epsilon }}_{v}^{p}=0\) and \({Q}^{-}={\text{MP}}^{-}\). If \(f=\mathrm{0,}\dot{f}=0\), then the point \((P,Q)\) at the time \({t}^{+}\) moves on the initial ellipse (cf. [Figure 4.5-a]). We immediately deduce from the elastic law and from the condition \({\mathrm{\Delta \epsilon }}_{v}^{p}=0\):

\(\mathrm{\Delta P}={k}_{0}\Delta {\epsilon }_{v}{P}^{-}\) eq 4.5-1

The criterion being verified at time \({t}^{+}\), we have using [éq 4.5-1]:

\({Q}^{+2}={M}^{2}{P}^{+}({\mathrm{2P}}_{\text{cr}}^{-}-{P}^{+})={M}^{2}({P}^{-}+\mathrm{\Delta P})({P}^{-}-\mathrm{\Delta P})={M}^{2}{P}^{-(2)}(1-{k}_{0}^{2}{\mathrm{\Delta \epsilon }}_{v}^{2})={Q}^{-(2)}(1-{k}_{0}^{2}{\mathrm{\Delta \epsilon }}_{v}^{2})\) eq 4.5-2

On the other hand, the constraint deviator can be written as:

\(s={s}^{e}-\mathrm{2\mu }\Delta {\tilde{\epsilon }}^{p}={s}^{e}-\mathrm{2\mu }\Lambda \frac{\partial f}{\partial s}={s}^{e}-\mathrm{6\mu }\Lambda s\) eq 4.5-3

From this we deduce:

\(1+6\mu \Lambda =\frac{{Q}^{e}}{Q}\), eq 4.5-4

and:

\(s=\frac{{Q}^{-}\sqrt{(1-{k}_{0}^{2}{\mathrm{\Delta \epsilon }}_{v}^{2}}}{{Q}^{e}}{s}^{e}\) eq 4.5-5

4.6. Summary

The discretization of the equations and the law of behavior in an implicit way leads to the resolution of the equation [éq 4.3-2].

If \({P}^{-}\ne {P}_{\text{cr}}^{-}\), then we solve the equation [éq 4.3-2] whose unknown is \({\mathrm{\Delta \epsilon }}_{v}^{p}\).

We then deduce:

\(\begin{array}{}{P}_{\text{cr}}={P}_{{\text{cr}}^{-}}\text{exp}({\mathrm{k\Delta \epsilon }}_{v}^{p}),\\ P={P}^{e}\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}\left[-{k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{p}\right]-1),\\ \text{puis}s=\frac{{s}^{e}}{1+\frac{{\mathrm{3\mu \Delta \epsilon }}_{v}^{p}}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}}\end{array}\) eq 4.6-1

Finally, we deduce:

\(\Delta {\tilde{\epsilon }}^{p}=\frac{3}{2}\frac{{\mathrm{\Delta \epsilon }}_{v}^{p}}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}s\) eq 4.6-2

At the critical point:

\({\mathrm{\Delta \epsilon }}_{v}^{p}=\mathrm{0,}{P}_{\text{cr}}={P}_{\text{cr}}^{-}\) eq 4.6-3

At this point, there is no change in work hardening; on the other hand, the stress state can continue to evolve either in contraction or in expansion (the tangent to the criterion is horizontal). The new stress state moves over the load surface of the previous state.