3. Digital integration

3.1. Discretized equations

« Integrating the law of behavior consists in calculating the internal variables and the stress with a given deformation history (mechanical). Given the additive nature of the constraint (), it is possible to proceed independently for each branch. The tangent operator of the system will also be the sum of the tangent operators specific to each branch.

The time discretization of behavioral equations is based on an implicit Euler schema (including for the viscoelastic branch), that is to say that the various variables of the problem are expressed at the final instant of the time step in question. We note \({Q}_{n}\) the value of a quantity \(Q\) at the start of the time step, \(\mathrm{\Delta }Q\) its increment during the time step and (simply) \(Q\) its value at the end of the time step. The mechanical state at the start of time step \(({\epsilon }_{n},{{\epsilon }^{p}}_{n},{{\epsilon }^{v}}_{n})\) is assumed to be known as well as the deformation increment \(\mathrm{\Delta }\epsilon\) (and therefore also the deformation \(\epsilon\)). It is then a question of calculating the increments of internal variables \(\mathrm{\Delta }{\epsilon }^{p}\) and \(\mathrm{\Delta }{\epsilon }^{v}\) as well as the constraint at the end of the time step \(\sigma\).

Note: Since the law is written in relation to true thermodynamic variables, it does not take into account an initial state under constraints defined by the keyword factor ETAT_INIT. An initial non-zero state must therefore be defined by the internal variables of deformation.

3.2. Elastoplastic branch

The discretization of the stress-strain relationship () is written as follows:

(3.1)\[ {\ mathrm {\ sigma}} ^ {p} =3 {K} ^ {p} =3 {K} ^ {p} =3 {K} ^ {p}} =3 {K} ^ {p}} =3 {K}} ^ {p}} _ {n} _ {n} -\ mathrm {\ Delta} {\ mathrm {\ epsilon}} - {\ epsilon}} ^ {p}}) S+2 {p}} _ {n} _ {n} -\ mathrm {\ Delta} {\ mathrm {\ epsilon}} ^ {p}) S+2 {p}} thrm {\ mu}} ^ {p}\ mathit {dev} (\ mathrm {\ epsilon} - {{\ mathrm {\ epsilon}}} ^ {p}} _ {n} -\ mathrm {\ Delta} {\ delta} {\ mathrm {\ epsilon}}} ^ {p}} _ {n} -\ mathrm {\ Delta} {\ Delta} {\ delta} {\ mathrm}} ^ {p})\]

According to the flow equation (), it can be seen that the trace of the plastic deformation increment is zero, so that the same is true for the plastic deformation at any time (isochoric plastic deformation). The relationship () is simplified and it is possible to reveal an elastic stress (or elastic test stress) noted \({\sigma }^{e}\) and an elastic flow (test) direction noted \({N}^{e}\), which are known:

(3.2)\[ {\ mathrm {\ sigma}} ^ {p} = {\ mathrm {\ sigma}}} ^ {e} -2 {\ mathrm {\ mu}} ^ {p}\ mathrm {\ Delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {\} {\ delta} {}\ mathit {sph} (\ mathrm {\ epsilon}) S+2 {\ mathrm {\ mu}} ^ {p}\ mathit {dev} (\ mathrm {\ epsilon} - {{\ mathrm {\ epsilon}} - {\ epsilon}}} ^ {epsilon}}} ^ {e}}} ^ {e}}} ^ {e}} ^ {e}} =\ mathit {dev}} =\ mathit {dev}} ({\ mathrm {\ sigma}}} ^ {e}) -C {\ mathrm {\ epsilon}} _ {n} ^ {p}\]

If the elastic test remains within the elasticity range, the plastic deformation increment is zero (elastic evolution):

(3.3)\[ {\ Green {N} ^ {e}\ Green}\ Green} _ {\ mathit {eq}}\ the {\ mathrm {\ sigma}} ^ {c}\ text {}\ Rightarrow\ text {}\ mathrm {\ delta} {\ mathrm {\ Delta}} {\ delta} {\ delta}} {\ mathrm {\ Delta}} {\ delta} {\ delta} {\ mathrm} {\ Delta} {\ delta} {\ mathrm} {\ Delta} {\ delta} {\ mathrm} {\ Delta} {\ delta} {\ mathrm} {\ Delta} {\ delta} {\ mathrm} {\ Delta} {\ delta} {\ mathrm} {\ Delta} {\ delta} {\ mathrm} {} = {\ mathrm {\ sigma}}} ^ {e}\]

Otherwise, a plastic correction must be carried out (plastic evolution). The flow equation () and the coherence equation () are then discretized as follows:

(3.4)\[ {\ mathrm {\ Delta}\ mathrm {\ epsilon}}} ^ {epsilon}}} ^ {p} =\ frac {\ delta}\ mathrm {\ delta}\ mathrm {\ kappa}}} {{\ mathrm {\ epsilon}}} ^ {p}} =\ frac {\ delta}}\ frac {\ mathrm {\ kappa}}} {\ mathrm {\ kappa}}} {{\ mathrm {\ epsilon}}} ^ {p}} =\ frac {\ delta}\ mathrm {\ kappa}}} {\ mathrm {\ kappa}}} {{\ mathrm {\ epsilon}}} {\ mathrm {\ epsilon}} ^ {p} +C)\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} ^ {p}\ right]\ text {;} {\ green {N} ^ {e} - (2 {\ mathrm {\ delta}} - (2 {\ mathrm {\ mu}}}}} {\ mu}}}} ^ {mu}}}} ^ {mu}}}} ^ {mu}}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}}} _ {\ mathit {eq}} = {\ mathrm {\ sigma}} ^ {c}}\]

After a few algebraic manipulations, we deduce:

(3.5)\[ N= {\ mathrm {\ sigma}} ^ {c}\ frac {{N}} ^ {e}} {{\ green {N} ^ {e}\ green} _ {\ mathit {eq}}}}\ text {;}}\ text {;}\ text {;}}\ text {;}}\ text {;}}\ text {;}}\ text {;};}\ mathrm {\ Delta} {\ delta} {\ mathrm {\ epsilon}}} ^ {p} =\ frac {{N} {eq}}}}}\ text {; eq}}}}\ text {;};}\ text {;};}\ text {;};}\ mathrm {\ Delta} {\ delta} {\ mathrm {N} {2 {\ mathrm {\ mu}}} ^ {p} +C}\ text {;} {\ mathrm {\ sigma}} ^ {p} = {\ mathrm {\ sigma}}} ^ {e}} ^ {e} -2 {\ mathrm {\ mu}} -2 {\ mathrm {\ mu}} -2 {\ mathrm {\ mu}}} ^ {p}} ^ {p}} ^ {p}\]

The tangent operator of the elastoplastic branch, such as \(\delta {\sigma }^{p}={H}^{p}\mathrm{:}\delta \epsilon\), is now denoted \({H}^{p}\). In the case of an elastic evolution, we simply have according to ():

(3.6)\[ {H} ^ {p} =3 {K} ^ {p} S\ otimes S+2 {\ mathrm {\ mu}} ^ {p} {P} ^ {\ mathit {dev}}\ text {;} {dev}}\ text {;} {;} {P}} ^ {\ mathit {P}} ^ {\ mathit {dev}}}\ equiv {I} _ {4} -S\ otimes S\]

where we introduced \({I}_{4}\) the fourth-order identity tensor and \({P}^{\mathit{dev}}\) the spotlight on the space of deviators. And in the case of plastic evolution, we proceed by deriving the equations in (). After some calculations, we get:

(3.7)\[ {H} ^ {p} =3 {K} ^ {p} S\ otimes S+\ p} S\ otimes S+\ left (\ frac {2 {\ mathrm {\ mu}} {\ mathrm {\ mu}}}} ^ {mu}}}} ^ {mu}}}} ^ {mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {\ mu}} {} ^ {e}\ Green} _ {\ mathit {eq}}}}} 2 {\ mathrm {\ mu}} ^ {p} +C\ right) {P} ^ {\ mathit {dev}}} -3 {\ mathrm {\ dev}}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}} -3 {\ mathrm {\ dev}}} -3 {\ m {\ mu}} ^ {p} +C}\ right)\ right)\ left (\ frac {{\ mathrm {\ sigma}} ^ {c}} {{\ green {N} ^ {e}\ green}\ green} _ {\ mathit {eq}}\ right)\ left (\ frac {{N} ^ {e}}} {{\ green {N} ^ {e}\ green} _ {\ mathit {eq}}} _ {\ mathit {eq}}}}\ right)\ right)\ left (\ frac {{N} ^ {e}}\ green} _ {\ mathit {eq}}}\ otimes\ frac {{N} ^ {e}} {{\ green {N} ^ {e}\ green}} _ {\ mathit {eq}}}}\ right)\]

The tangent operator is not continuous at the transition between the elastic and plastic regimes. In principle, knowledge of the evolution regime and therefore the choice of the tangent operator depend on whether or not inequality \(\Vert {N}^{e}\Vert \le {\sigma }^{c}\) is achieved. However, in the prediction phase and if the previous step corresponded to a plastic evolution, we may find ourselves in the case where \(\Vert {N}^{e}\Vert \approx {\sigma }^{c}\), with the precision of resolution, while we expect a new time step in the plastic regime. This is why we choose to « expand » the plastic regime a bit for the choice of the tangent operator in prediction: if \(\Vert {N}^{e}\Vert >{\sigma }^{c}(1-\text{prec})\), with prec set to \({10}^{-6}\), then we choose the plastic tangent operator.

3.3. Viscoelastic branch

The temporal discretization of the behavior equations () and () leads respectively to:

(3.8)\[ {\ mathrm {\ sigma}} ^ {v} = {C} ^ {v} = {C} ^ {v}\ mathrm {\ epsilon}} ^ {v}) = {C} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}\ mathrm {\ epsilon} - {\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}}} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}) =3 {K} ^ {v}}\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}} on}} ^ {v}) S+2 {\ mathrm {\ mu}}} ^ {v}} ^ {v}\ mathit {dev} (\ mathrm {\ epsilon}}} ^ {v}} ^ {v})\ text {v})\ text {})\ text {}};\ text {}};\ text {}};\ text {} 3 {K}} ^ {v} =\ frac {{E} ^ {v}}} {1-2 {\ mathron}} {1-2 {\ mathrm}} {v}} {1-2 {\ mathrm}} {m {\nu}} ^ {v}}\ text {}}\ text {}} 2 {\ mathrm {\ mu}}} ^ {v} =\ frac {{E} ^ {v}}} {1+ {\ mathrm {\nu}} {\ mathrm {\nu}} {\\]

(3.9)\[ \ frac {{\ mathrm {\ Delta}\ mathrm {\ epsilon}}} ^ {v}} {\ mathrm {\ Delta} t} =\ frac {1} {{\ mathrm {\ Delta}} {\ mathrm {\ eta}}} ^ {\ eta}}} ^ {eta}}} ^ {eta}}} ^ {eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {\ eta}}} ^ {eta}}} ^ {eta}}} {\ mathrm {\ gamma} -1} -1} V\ mathrm {:} {\ mathrm {\ sigma}} ^ {v}\]

The instantaneous constraint \({\sigma }^{i}\), which is a known quantity, is introduced:

(3.10)\[ {\ mathrm {\ sigma}} ^ {i} = {C} ^ {v} = {C} ^ {v}\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}} _ {n} ^ {v}) = {n} ^ {v}) = {\ mathrm {\ sigma}}} ^ {v} = {\ mathrm {\ sigma}}} ^ {v} = {\ mathrm {\ sigma}}} ^ {v}} ^ {i})} - {C} ^ {v}\ mathrm {:}\ mathrm {:}\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} ^ {v}\]

Substituting () into (), we get an equation for \({\sigma }^{v}\):

(3.11)\[ \ left ({I} _ {4} +\ frac {\ mathrm {\ sigma}} +\ frac {\ mathrm {\ delta}}} {\ mathrm {\ gamma}}} {\ green {\ mathrm {\ sigma}}}} {\ mathrm {\ sigma}}}} {\ mathrm {\ sigma}}}} {\ mathrm {\ gamma}}}} {\ green {\ mathrm {\ gamma}}}} {\ green {\ mathrm {\ gamma}}}} {\ green {\ mathrm {\ sigma}}}} {\ mathrm {\ sigma}}}} {\ mathrm {\ sigma}}} {\ mathrm {\ sigma}}}} {\ mathrm {\ sigma}} {:} V\ right)\ mathrm {::} {\ mathrm {\ sigma}} ^ {v} = {\ mathrm {\ sigma}}} ^ {i}\]

The method for solving this tensor equation is based on the isotropic nature of the fourth-order tensors that appear there. We start by introducing the unknown \(x\ge 0\) such as:

\[\]

: label: eq-21

{green {mathrm {sigma}}} ^ {v}}green} _ {v} =x {green {mathrm {sigma}} ^ {i}\ green} _ {v}

By separating the parts spherical and deviatoric of (), we then obtain respectively:

(3.12)\[ \ mathit {sph} ({\ mathrm {\ sigma}}} ^ {v}) =\ frac {\ mathit {sph} ({\ mathrm {\ sigma}} ^ {i})} {1+ {b} _ {1} _ {1} {1} {1} {x} {x} ^ {\ mathrm} -1}}}\ text {;}}\ mathit {dev})} {\ mathrm {\ mathrm {\ gamma} -1}}}\ text {;}}\ mathit {dev} ({\ mathrm} {\ sigma}} ^ {v}) =\ frac {\ mathit {dev} ({\ mathrm {\ sigma}} ^ {i})} {1+ {b} _ {2} {2} {x} {x} ^ {\ mathrm {\ gamma} -1}}}\]

where the coefficients were introduced:

(3.13)\[ {b} _ {0} =\ frac {\ mathrm {\ delta} t} {{\ eta} ^ {\ gamma}} {\ green {\ sigma} ^ {i}\ green} _ {v} ^ {\ gamma -1} ^ {\ gamma -1}}\ text {\ gamma -1}}\ text {\ gamma -1}\ text {\ gamma -1}\ text {;} {b} _ {1}} = 3 {K} ^ {v} (1-2 {\nu} ^ {d}) {v} _ {d}) {b} _ {d}) {b} _ {0}}\ text {;} {b} _ {2} =2 {\ mu} ^ {v} (1+ {\nu} ^ {d}) {b} _ {0}\]

Knowledge of the unknown \(x\) thus makes it possible to calculate the entire stress tensor \({\sigma }^{v}\). By substituting the latter with its expression according to \(x\) in (), it comes:

(3.14)\[ \ sum _ {k=1} ^ {2}\ frac {{a} _ {k}} {{(1+ {b}} _ {k} {x} ^ {\ mathrm {\ gamma} -1})}}} ^ {2}})} ^ {2}} = {x}} ^ {2}} = {x} ^ {2}}\ text {;} {a} _ {1} = (1-2 {\ mathrm {\nu} -1})} ^ {u} -1})} ^ {d}}) {\ left (\ frac {\ mathit {sph}} ({\ mathrm {\ sigma}} ^ {i})} {{\ green {\ mathrm {\ sigma}} ^ {i}\ green}\ green} _ {v} {sph}} (green} _ {sph}} (green} _ {sph}}\ green} _ {sph}}\ green} _ {sph}} {green} _ {sph}}\ green} _ {sph}} green} _ {sph}}\ green} _ {sph}} _ {sph} (green} _ {sph}} _ {sph}} green} _ {sph}} green} _ {sph}} green} _ {sph}} green} _ {sph}} _ d}) {\ left (\ frac {\ mathit {dev}} ({\ mathrm {\ sigma}} ^ {i})} {{\ green {\ mathrm {\ sigma}}} ^ {i}}\ sigma}}} ^ {sigma}}}} ^ {sigma}}}} ^ {sigma}}} ^ {2}}\]

This is a scalar equation in \(x\) that has a unique solution. It is solved by a Newton method. Like \(n\le 1\) and therefore \(\gamma \ge 1\), we can determine boundaries:

(3.15)\[ \ sqrt {\ sum _ {k=1} ^ {2}\ frac {{a} _ {k}} {{(1+ {b} _ {k})}} ^ {2}}} ^ {2}}}}\ le x\ le 1\]

We proceed with the iterations of Newton’s method until we obtain a framework for solution \({x}_{m}\le x\le {x}_{p}\) such as:

(3.16)\[ {x} _ {p} - {x} _ {m}\ the\ text {RESI\ _ INTE\ _ RELA}\]

The quality of solution \({\sigma }^{v}\) is thus controlled in relation to \(\Vert {\sigma }^{i}\Vert\).

The tangent operator \({H}^{v}\) such as \(\delta {\sigma }^{v}={H}^{v}\mathrm{:}\delta \epsilon\) is obtained by differentiating the preceding equations in succession. The expression for this is given directly:

\[\]

: label: eq-27

{H} ^ {v} =frac {3 {K} ^ {v}} {1+ {b}}} {1+ {b} _ {x} ^ {mathrm {gamma} -1}} Sotimes S+frac {frac {2}frac {2}frac {frac {2}frac {2}frac {2}frac {2}frac {2}frac {2}frac {2}frac {2}frac {frac {2}}frac {2}frac {frac {2}}frac {2} {gamma} -1}}} {mathrm {gamma} -1}}} {H} ^ {mathrm {} -1}} {P} ^ {mathit {dev}}} - {x}}} - {x} ^ {mathrm {gamma} -1} {times {B} _ {0} - (mathrm {gamma} -1) {mathrm {gamma} -1} {b} _ {0} - (mathrm {gamma} -1} - (mathrm {gamma} -1) - (mathrm {gamma} -1) {x} ^ {x} ^ {mathrm {gamma} -1} {b} _ {0} hotimes X

where the following various intermediate quantities were introduced:

\({A}_{1}=2(1-2{\mathrm{\nu }}^{d})\left[\left(\frac{3{K}^{v}\mathit{sph}({\mathrm{\sigma }}^{i})}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}S\right)-{\left(\frac{\mathit{sph}({\mathrm{\sigma }}^{i})}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)}^{2}\left(\frac{{C}^{v}\mathrm{:}V\mathrm{:}{\mathrm{\sigma }}^{i}}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)\right]\)
\({A}_{2}=2(1+{\mathrm{\nu }}^{d})\left[\left(\frac{2{\mathrm{\mu }}^{v}\mathit{dev}({\mathrm{\sigma }}^{i})}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)-{\left(\frac{\mathit{dev}({\mathrm{\sigma }}^{i})}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)}^{2}\left(\frac{{C}^{v}\mathrm{:}V\mathrm{:}{\mathrm{\sigma }}^{i}}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)\right]\)
\({B}_{0}=\frac{(\mathrm{\gamma }-1)\mathrm{\Delta }t}{{\mathrm{\eta }}^{\mathrm{\gamma }}}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}^{\mathrm{\gamma }-1}\left(\frac{{C}^{v}\mathrm{:}V\mathrm{:}{\mathrm{\sigma }}^{i}}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)\text{;}{B}_{1}=3{K}^{v}(1-2{\mathrm{\nu }}^{d}){B}_{0}\text{;}{B}_{2}=2{\mathrm{\mu }}^{v}(1+{\mathrm{\nu }}^{d}){B}_{0}\)
\(X={\left[\sum _{k=1}^{2}\frac{2{a}_{k}(1+{b}_{k}\mathrm{\gamma }{x}^{\mathrm{\gamma }-1})}{{(1+{b}_{k}{x}^{\mathrm{\gamma }-1})}^{3}}\right]}^{-1}\left[\sum _{k=1}^{2}\frac{(1+{b}_{k}{x}^{\mathrm{\gamma }-1}){A}_{k}-2{a}_{k}{x}^{\mathrm{\gamma }-1}{B}_{k}}{{(1+{b}_{k}{x}^{\mathrm{\gamma }-1})}^{3}}\right]\)
\(h=\frac{3{K}^{v}(1-2{\mathrm{\nu }}^{d})}{{(1+{b}_{1}{x}^{\mathrm{\gamma }-1})}^{2}}\left(\frac{\mathit{sph}({\mathrm{\sigma }}^{i})}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}S\right)+\frac{2{\mathrm{\mu }}^{v}(1+{\mathrm{\nu }}^{d})}{{(1+{b}_{2}{x}^{\mathrm{\gamma }-1})}^{2}}\left(\frac{\mathit{dev}({\mathrm{\sigma }}^{i})}{{\Vert {\mathrm{\sigma }}^{i}\Vert }_{v}}\right)\)

3.4. Post-processing variables

As mentioned above, the cumulative plastic deformation and the cumulative viscous deformation are calculated at the end of the behavior integration phase. After discretization in time, equation () thus leads to:

(3.17)\[ {\ mathrm {\ Delta}\ mathrm {\ epsilon}}} _ {\ mathit {cum}} ^ {p} =\ sqrt {\ frac {2} {3} {\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ epsilon}}\ mathrm {\ epsilon}}} ^ {p}\ mathrm {\ epsilon}} ^ {p}\ mathrm {:} {\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm on}} ^ {p}}\ text {;} {\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ epsilon}} _ {\ mathit {cum}}} ^ {v} =\ sqrt {\ frac {2} {2} {2} {3} {3} {3} {3} {3} {3} {\ 3} {\ mathrm {3} {3} {\ mathrm {\ Delta}}\ mathrm {\ Delta}} _ {\ epsilon}} ^ {v}\ mathrm {:} {\ mathrm {2} {2} {3} {3} {3} {3} {3} {\ mathrm} {\ Delta}}\ mathrm {\ Delta}}} _ m {\ Delta}\ mathrm {\ epsilon}}} ^ {v}}\]