8. Appendix 1: Calculation of logarithmic deformations
8.1. Ratings:
\(\mathrm{A}\) refers to a tensor of order two, and \(\stackrel{̄}{\mathrm{A}}\) a tensor of order four. We adopt Voigt’s notation, (see for example [16]) defined for a tensor of order two:
(8.1)\[\begin{split} \ mathrm {A} =\ left\ {\ begin {array} {c} {c} {A} _ {11}\\ {A} _ {22}\\ {A} _ {33}\\ sqrt {2} {2} {A} {A} {A} _ {12}\\ sqrt {2} {A} _ {23} {A} _ {23}\ end {array} _ {23}\ end {array}\ right\}\end{split}\]
And for a fourth-order tensor:
(8.2)\[\begin{split} \ stackrel {!} {\ mathrm {A}} =\ left\ {\ begin {array} {cccccc} {A} _ {1111} & {A} _ {1122} & {A} _ {1122} & {A} _ {1133} &\ sqrt {2} {A} _ {1123} &\ sqrt {2} {1123} &\ sqrt {2} {1123} &\ sqrt {2} {1123} &\ sqrt {2} {A} _ {1113}\\ {A} _ {2211} _ {2211} & {A} _ {2222} & {A} _ {2233} &\ sqrt {2} {2212} &\ sqrt {2} {2212} &\ sqrt {2} {A} _ {2213}\\ {A} _ {2212} _ {2212} &\ sqrt {2} {A} _ {2212} &\ sqrt {2} {A} _ {2212} & {A} _ {3311} & {A} _ {3322} & {A} _ {3333} &\ sqrt {2} {A} {A} _ {3312} &\ sqrt {2} {A} _ {3323} &\ sqrt {2} {A} {A} _ {A} _ {A} _ {1222} _ {1222} _ {1222} _ {1222} _ {1222} _ {1222} &\ sqrt {2}} {A} _ {1233} & 2 {A} _ {1212} & 2 {A} _ {1223} & 2 {A} _ {1213}\\ sqrt {2} {A} {A} _ {1311} _ {1311} &\ sqrt {2} {1311}} &\ sqrt {2} {A} _ {1333} & 2 {A} _ {1311} & 2 {A} _ {1311} &\ sqrt {2} {1333} & 2 {A} _ {1311} & 2 {A} _ {1311} & 2 {A} _ {1311} & 2 {A} _ {1311} & 2 {A} _ {1311} & 2 {A} _ {1311} & 2 {A} _ {1311}} & 2 {A} _ {1323} & 2 {A} _ {1313}\\ sqrt {2} {A} _ {2311} &\ sqrt { 2} {A} _ {2322} &\ sqrt {2} {A} _ {2333} & 2 {A} _ {2312} & 2 {A} _ {2323} & 2 {A} & 2 {A} & 2 {A} _ {2313} _ {2313}\ end {array}\ right\}\end{split}\]
Where the components relating to the Voigt notation will be designated by a Greek letter:
(8.3)\[ \ parallel {A} _ {\ mathit {ij}}}\ parallel =\ parallel {A} _ {\ alpha}\ parallel\]
We then have the following properties:
(8.4)\[ \ mathrm {A}\ mathrm {:}\ mathrm {B} = {A} _ {\ mathit {ij}} {B} _ {\ mathit {ij}}} = {A}} = {A} _ {\ alpha} _ {\ alpha}} = {A} _ {\ alpha}
\ stackrel {]} {\ mathrm {A}}\ mathrm {A}}\ mathrm {:}\ mathrm {B} = {A} _ {\ mathit {ijkl}} {B} _ {\ mathit {kl}}} = {\ mathit {kl}}} = {A} _ {\ beta}
\ stackrel {] {\ mathrm {A}}\ mathrm {A}}\ mathrm {:}\ stackrel {459} {\ mathrm {B}} = {A} _ {\ mathit {ijkl}}} {B}} {B}} {B}} {\ mathit {ijkl}}} {B}} {B}} {\ mathit {ijkl}} {B}} {B}} {B}} {B}} {B}} {B}} {B} {B}} {B}} {B} {B}} {B}} {B} {B}} {B} {B}} {B}} {B}} {B}} {B}\]
The inverse of a fourth-order tensor with minor symmetries (\({A}_{\mathit{ijrs}}={A}_{\mathit{jirs}}={A}_{\mathit{ijsr}}\)) is written:
(8.5)\[ \ stackrel {Belgium} {\ mathrm {A}}\ mathrm {:} {\ stackrel {3} {\ mathrm {A}}}} ^ {-1} = {\ stackrel {A}} = {\ stackrel {A}}} = {\ stackrel {A}} = {\ stackrel {A}} = {\ stackrel {A}} = {\ stackrel {A}} = {\ stackrel {A}} = {\ stackrel {A}} = {\ stackrel {A}}
{A} _ {\ mathit {ijrs}} {A}} {A} _ {\ mathit {rskl}}} ^ {-1} = {\ mathit {ijkl}}} =\ frac {1} {2}\ left ({\ delta}}\ left ({\ delta}} _ {\ delta}} _ {\ mathit {jl}}} =\ frac {1} {2}}\ left ({\ delta}}}\ left ({\ delta}} _ {\ delta} _ {\ mathit {ik}}} {\ delta}} _ {\ mathit {jl}}} =\ frac {1} {2}}\ left ({\ delta}}}\ left ({\ delta}} _ {\ delta}} _ {\ mathit {jl}}} He}} {\ delta} _ {\ mathit {jk}}\ right)
{A} _ {\ alpha\ gamma} {A} {A} _ {\ gamma\ beta} _ {\ alpha\ beta} = {\ delta} _ {\ delta} _ {\ alpha\ beta} _ {\ alpha\ beta}\]
8.2. Expression of constraints in Lagrangian configuration
The power of inner efforts is written as:
(8.6)\[ {p} _ {\ text {int}} =\ mathrm {T}\ mathrm {T}\ mathrm {:}\ dot {\ mathrm {E}} =\ mathrm {S}\ mathrm {:} {:} {\ stackrel {T}} {\ stackrel {T}} {\ stackrel {e}} {!\]
with \(\stackrel{̄}{\mathrm{P}}=2\frac{\partial \mathrm{E}}{\partial \mathrm{C}}\) which makes it possible to calculate \(\mathrm{S}=\mathrm{T}\mathrm{:}\mathrm{P}\) (or \({S}_{\mathit{ij}}={T}_{\mathit{kl}}\mathrm{:}{P}_{\mathit{klij}}\)). To calculate the Cauchy stress tensor, simply write:
(8.7)\[ \ sigma =\ frac {1} {\ mathit {det}\ mathrm {F}}\ mathrm {F}\ cdot\ mathrm {S}\ cdot {\ mathrm {F}}\ cdot {\ mathrm {F}}} ^ {T}\]
Special case of plane stresses:
In this case, we don’t fully know \(\mathit{det}\mathrm{F}\). In fact, component \(\mathrm{zz}\) of the logarithmic deformation tensor \(\mathrm{E}\) is unknown, as it is dependent on the law of behavior. By limiting ourselves to behaviors such as \(\mathit{det}{\mathrm{F}}^{p}=0\) (plastic incompressibility), we then have \(\mathit{det}\mathrm{F}=\mathit{det}{\mathrm{F}}^{e}\).
Next [16] we can calculate this expression:
(8.8)\[ \ mathit {det} {\ mathrm {F}}} ^ {e}} ^ {e} = {e} = {e} ^ {e}} ^ {e} + {E} _ {\ mathit {yy}} {\ mathit {yy}}} ^ {yy}}} ^ {yy}} ^ {y}} ^ {y}} ^ {e}} ^ {e}}\]
Where \({\mathrm{E}}^{e}\) represents the elastic part of logarithmic deformations, known for any law of elasto-plastic or elasto-viscoplastic behavior by Hooke’s law \({\mathrm{E}}^{e}={\Lambda }^{-1}\cdot \mathrm{T}\).
We are going to demonstrate this relationship. We recall the definition of logarithmic deformations:
\[\]
: label: eq-50
{E} _ {mathit {ij}} =frac {1} {2}}sum _ {k=mathrm {1.3}}mathrm {log} ({lambda} ^ {(k)}) {N} ^ {(k)}) {N}}) {N} ^ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}
Which leads to this expression of the determinant:
(8.9)\[ \ mathit {det} {\ mathrm {F}}} ^ {e} =\ sqrt {\ mathit {det} {\ mathrm {F}} ^ {T}\ mathrm {F}}} =\ sqrt {{\ lambda}}} =\ sqrt {{\ lambda}}} =\ sqrt {{\ lambda}}} =\ sqrt {{\ lambda}} _ {3}}} =\ sqrt {\ lambda}} _ {3}}\]
\({\lambda }_{i}\) being the eigenvalues of \({\mathrm{F}}^{T}\cdot \mathrm{F}\). So:
\[\]
: label: eq-52
mathrm {log} (mathit {det} {mathrm {F}}} ^ {e}) =frac {1} {2}left{mathrm {log} ({lambda} _ {1}) +mathrm {lambda} _ {1}) +mathrm {log} ({lambda} _ {1}) +mathrm {log} ({lambda} _ {1}) +mathrm {log} ({lambda} _ {1}) +mathrm {log} ({lambda} _ {1}) +mathrm {log} ({lambda} _ {1}) +mathrm {log} ({lambda} _ {1}) + 3})right}
Applying the exponential function, we get the result ().
8.3. Expression of the tangent operator in Lagrangian configuration
By deriving the expression \(\mathrm{S}=\mathrm{T}\mathrm{:}\mathrm{P}\) in relation to time:
(8.10)\[ \ dot {\ mathrm {S}}} =\ dot {\ mathrm {T}}\ mathrm {T}}\ mathrm {T}} =\ mathrm {T}\ mathrm {:}\ mathrm {:}\ dot {\ mathrm {T}}}\ dot {\ mathrm {T}}} =\ left (\ frac {\ mathrm {T}}\ mathrm {T}}\ mathrm {T}}\ mathrm {T}}\ mathrm {T}}\ mathrm {T}\ mathrm {:}:}\ dot {\ stackrel {:}}\ dot {\ stackrel {! {T}} {\ partial\ mathrm {E}}\ mathrm {:}}\ dot {\ mathrm {E}}\ right)\ mathrm {:}\ stackrel {tante} {\ mathrm {P}}} +\ mathrm {P}}} +\ mathrm {:}\ mathrm {:}\ left (\ frac {\ partial\ stackrel {279} {\ stackrel {!} {\ stackrel {!} {\ stackrel {!} {\ stackrel {]} {\ mathrm {P}}} +\ mathrm {P}}}\ dot {\ mathrm {:}\ right)\ mathrm {{P}}} {\ partial\ mathrm {C}}\ mathrm {:}\ dot {\ mathrm {C}}\ right) =\ left [\ frac {\ partial\ mathrm {T}}} {\ partial\ mathrm {T}}} {\ partial\ mathrm {E}} {\ partial\ mathrm {E}} {\ partial\ mathrm {E}} {\ partial\ mathrm {E}}} {\ partial\ mathrm {E}}\ mathrm {C}}\ mathrm {:}\ dot {\ mathrm {C}}\ right)\ right]\ mathrm {:}\ stackrel {tante} {\ mathrm {P}} {\ mathrm {P}}} +\ mathrm {P}} +\ mathrm {P}} +\ mathrm {P}} :}\ left (\ frac {\ partial\ stackrel {ounty}} {\ mathrm {P}}}} {\ partial\ mathrm {C}}\ mathrm {:}:}\ dot {\ mathrm {C}}\ right)\]
Either:
(8.11)\[ \ dot {\ mathrm {S}} =\ left ({\ stackrel {E}} {\ mathrm {P}}}} ^ {T}\ mathrm {:} {\ stackrel {tante} {\ stackrel {tante}} {\ mathrm {E}}} {\ mathrm {E}}} {\ mathrm {E}}}} {\ mathrm {E}}}} {\ mathrm {E}}}} ^ {p}\ mathrm {}}} ^ {p}\ mathrm {:}\ stackrel {P}} {\ mathrm {P}}} +\ mathrm {P}} T}\ mathrm {:}\ stackrel {tante} {\ stackrel {tante} {\ mathrm {L}}}\ right)\ mathrm {:}\ frac {1} {2} {2}\ dot {\ mathrm {C}}\]
with \(\stackrel{̄}{\stackrel{̄}{\mathrm{L}}}=4\frac{{\partial }^{2}\mathrm{E}}{\partial \mathrm{C}\partial \mathrm{C}}\) and \({\stackrel{̄}{\mathrm{E}}}^{p}=\frac{\partial \mathrm{T}}{\partial \mathrm{E}}\). This defines the tangent operator \({\stackrel{̄}{\stackrel{̄}{\mathrm{C}}}}^{\mathit{ep}}=\left({\stackrel{̄}{\mathrm{P}}}^{T}\cdot {\stackrel{̄}{\mathrm{E}}}^{p}\cdot \stackrel{̄}{\mathrm{P}}+\mathrm{T}\mathrm{:}\stackrel{̄}{\stackrel{̄}{\mathrm{L}}}\right)\) that checks for \(\dot{\mathrm{S}}={\stackrel{̄}{\stackrel{̄}{\mathrm{C}}}}^{\mathit{ep}}\mathrm{:}\frac{1}{2}\dot{\mathrm{C}}\). Or, depending on the Green-Lagrange \(\Delta =\frac{1}{2}\left(\mathrm{C}-{\mathrm{I}}_{d}\right)\) deformations:
(8.12)\[ \ frac {\ partial\ mathrm {S}} {\ partial\ Delta}} =\ frac {\ partial\ mathrm {S}} {\ partial\ mathrm {C}}\ mathrm {:}\ frac {\ partial\ mathrm {S}}\ frac {\ partial\ mathrm {C}} {\ partial\ mathrm {C}} {\ partial\ mathrm {S}} {\ partial\ mathrm {S}} {\ partial\ mathrm {S}} {\ partial\ mathrm {S}} {\ partial\ mathrm {S}} {\ partial\ mathrm {S}} m {C}} = {\ stackrel {tante} {\ stackrel {tante} {\ mathrm {C}}}}}} ^ {\ mathit {ep}}\]
The expression of this tangent operator as well as the stress tensor, both in Lagrangian configuration, allow, for the calculation of internal forces, to use a variational formulation in the initial configuration, as in [R5.03.20] for example. We write the balance in variational form on the initial configuration:
(8.13)\[ \ delta {W} _ {\ text {int}}\ cdot\ delta\ mathrm {v} + {\ mathit {SW}}} _ {\ text {ext}}}\ cdot\ delta\ delta\ delta\ mathrm {v} =0
\ forall\ delta\ mathrm {v}\]
Under the assumption that the load does not depend on the geometric transformation, the virtual work of external forces is written as a linear form:
(8.14)\[ \ delta {W} _ {\ text {ext}}}\ cdot\ delta\ mathrm {v} =\ underset {{\ omega} _ {0}} {\ rho} _ {0} {F} _ {0} {F} _ {i} {F} _ {i}\ delta {v} _ {v} _ {i} d {\ omega} _ {0}} +\ underset {{\ partial}} _ {\ partial} _ {F} {F} {F} {F} {F} {F} {F} {F} {F} {\ Omega} _ {0}} {\ int} {T} _ {i} _ {i} ^ {d}\ delta {v} _ {i} {\ mathit {dS}}} _ {o}\]
With \(F\) the volume loading and \({\mathrm{T}}^{d}\) a surface loading exerted on the \({\partial }_{F}{\Omega }_{0}\) edge. Again, we choose the initial configuration as the reference configuration, to express the work of internal efforts [R5,03,20]. [R7.02.03]:
(8.15)\[ {\ mathit {SW}} _ {\ text {int}}}\ cdot\ delta\ mathrm {v} =-\ underset {{\ Omega} _ {0}} {\ int} {F} _ {\ mathit {ik}} _ {\ mathit {ik}}} {\ mathit {ik}}} {S} _ {S} _ {l}}\ delta {v} _ {i, l} d {\ omega} {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _\]
With a view to resolving by a Newton method, it is important to also express the second variation of the virtual work of internal forces, namely, the geometric rigidity:
(8.16)\[ {d} ^ {2} {W} _ {\ text {int}}\ cdot\ text {int}}\ cdot\ delta\ mathrm {v} =-\ underset {{\ Omega} _ {0} _ {0}} {0}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}}\ delta {u} _ {i, k} _ {i, k} _ {i, k} _ {i, l} d {\ omega} _ {0}\]
And the elastic stiffness:
(8.17)\[ -\ underset {{\ Omega} _ {0}} {\ int}} {\ int}\ delta {u} _ {i, q} {F} _ {\ mathit {ip}} {\ left (\ frac {\ partial\ mathrm {S}}} {\ partial\ mathrm {S}}} {S}} {S}} {\ partial\ mathrm {S}}} {S}} {S}}\ delta {v}} {\ partial\ delta}\ right)} _ {\ mathit {jk}}\ mathrm {S}} {S}}\ delta {v} _ {j, l} d {\ omega} _ {0}\]
8.4. Effective calculation of logarithmic deformations
Logarithmic deformations are defined by:
(8.18)\[ \ mathrm {E} =\ frac {1} {2}\ mathrm {log} (\ mathrm {C}) =\ frac {1} {2}\ mathrm {log} ({\ mathrm {F}}} {2}\ mathrm {F}} ({\ mathrm {F}})\]
Strictly speaking, the metric tensor should be added in the case of an initial configuration defined in a space different from Euclidean space (case of shells for example). To simplify the writings, we will use the case of an initial Euclidean configuration, the components of the vectors and tensors being written in a 3D orthonormal coordinate system. The 2D restriction is immediate. The calculation of the logarithmic deformation can only be done in the natural coordinate system. It is therefore necessary to determine the three eigenvalues \({\lambda }^{(i)}\) and eigenvectors \({N}^{(i)}\) solutions of the problem to the following eigenvalues:
\[\]
: label: eq-62
mathrm {C} {N} ^ {(i)} = {lambda} ^ {(i)} {N} ^ {(i)} ^ {(i)}
We can then calculate the three values in the « clean » space:
(8.19)\[ {e} ^ {(i)} =\ frac {1} {2}\ mathrm {log} ({\ lambda} ^ {(i)})\]
The logarithmic deformations are then carried into the original space by:
(8.20)\[ {E} _ {\ mathit {ij}} =\ frac {1} {2}}\ sum _ {k=\ mathrm {1.3}}\ mathrm {log} ({\ lambda} ^ {(k)}) {N} ^ {(k)}) {N}}) {N} ^ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}) {N} _ {(k)}\]
Because the logarithmic function is an isotropic function of the tensor \(\mathrm{C}\) [16]. For post-processing (see § 3.3), i.e. the calculation of the stress tensor and the tangent operator, you have to calculate the quantities \(\stackrel{̄}{\mathrm{P}}=2\frac{\partial \mathrm{E}}{\partial \mathrm{C}}\):
(8.21)\[ \ stackrel {tante} {\ mathrm {P}}} =\ sum _ {i=\ mathrm {1.3}}\ frac {1} {2} {d} ^ {(i)} {N}} {N} ^ {(i)} {(i)}\ otimes {N}}}\ otimes {N} {(i)}}\ otimes {M} {2} {(i)} {d} ^ {(i)}} {N} ^ {(i)}} {N} ^ {(i)}} {N} ^ {(i)}} +\ sum {i)} +\ sum {i)} +\ sum {i)} +\ sum {i)} +\ sum {i=\ mathrm {1,3}}\ sum _ {j\ne i} ^ {3} {3} {\ theta} _ {\ mathit {ij}} {N} ^ {(i)}\ otimes {N} ^ {(j)} ^ {(j)}}\ otimes {(j)}}\ otimes {(j)}}\ otimes {N} ^ {(j)}}\]
And quantity \(\mathrm{T}\mathrm{:}\stackrel{̄}{\stackrel{̄}{\mathrm{L}}}\):
(8.22)\[\begin{split} \ begin {array} {c}\ mathrm {T}\ mathrm {T}\ mathrm {:}\ stackrel {tante} {\ mathrm {L}}}} =\ sum _ {i} ^ {i} ^ {3} ^ {3} ^ {3}\ 3}\ frac {:}\ frac {1} {4} {4} {f} {f}} {(i)} {\ zeta} ^ {(\ mathit {ii}}) ^ {(\ mathit {ii}) ^ {3} ^ {3}\ frac {1} {4} {4} {f} {(i)} ^ {(i)} ^ {(\ mathit {ii}) ^ {3} ^ {3}\ frac {1} {4} {thrm {M}} ^ {(\ mathit {ii})}}\ otimes {\ mathrm {M}}} ^ {(\ mathit {ii})}} +\ sum _ {i} ^ {3}\ sum _ {j\ne i}} {j\ne i}} ^ {3}\ sum _ {k\ne i}} ^ {3} 2\ eta {\ zeta} ^ {3}\ sum _ {\ zeta} ^ {3}\ sum _ {j\ne i} ^ {3}\ sum _ {j\ne i} ^ {3} 2\ eta {\ zeta} ^ {3}\ sum _ {j\ne i} ^ {3}\ sum _ {j\ne i} ^ {3} 2\ eta {\ zeta} ^ {3}\ sum it {ij})} {\ mathrm {M}}} ^ {{(\ mathit {ik})}\ otimes {\ mathrm {M}} ^ {(\ mathit {ij})}\\ +\ sum _ {i}}} {i}} ^ {3}\ sum _ {3}\ sum _ {j\ne i})}\ otimes {\ mathrm {M}}} ^ {(\ mathit {ij})}\\ +\ sum _ {i} ^ {3}\ sum _ {i} ^ {3}\ sum _ {i} ^ {3}\ sum _ {j\ne i}} 2 {\ xi} ^ {i}} ^ {(\ mathit {ij})} ({\ mathrm {M}}} ^ {(\ mathit {ij})}\ otimes {\ mathrm {M}} ^ {(\ mathit {jj})}}}}} + {\ mathrm {M}}}} ^ {\ mathrm {M}})} + {\ mathrm {jj}}}} + {\ mathrm {M}})} + {\ mathrm {M}})} + {\ mathrm {M}})} + {\ mathrm {M}})} + {\ mathrm {M}})} + {\ mathrm {M}})} + {\ mathrm {M}})} + {\ mathrm {M}}} ij})}) + {\ zeta} ^ {(\ zeta} mathit {jj})} {\ mathrm {M}}} ^ {(\ mathit {ij})}\ otimes {\ mathrm {M}}} ^ {(\ mathit {ij})}}} ^ {(\ mathit {ij})}}]\ end {array}\end{split}\]
With \({d}^{(i)}=\frac{1}{{\lambda }^{(i)}}\), \({f}^{(i)}=\frac{-2}{{({\lambda }^{(i)})}^{2}}\), \({\zeta }^{(\mathit{ij})}=\mathrm{T}\mathrm{:}{N}^{(i)}\otimes {N}^{(j)}\), \({\mathrm{M}}_{\mathit{ab}}^{(\mathit{ij})}={N}_{a}^{(i)}{N}_{b}^{(j)}+{N}_{a}^{(j)}{N}_{b}^{(i)}\). \({\theta }^{(\mathrm{ij})}\), \({\xi }^{(\mathrm{ij})}\), and \({\eta }^{(\mathrm{ij})}\) are defined by:
(8.23)\[ {\ theta} ^ {(\ mathrm {ij})}} =\ frac {({e} ^ {(i)}} - {e} ^ {(j)})} {({\ lambda}} ^ {\ lambda} ^ {\ lambda} ^ {(j)})} {(lambda} ^ {(j)})} {
{\ xi} ^ {(\ mathrm {ij})}} =\ frac {({\ theta} ^ {(\ mathrm {ij})}} -\ frac {1} {2} {d} {d} ^ {(j)})}} {(j)})} {({\ lambda} ^ {(j)})})} {(\ lambda} ^ {(j)})}} {(\ lambda} ^ {(j)})}
\ eta =\ sum _ {i} ^ {3}\ sum _ {j\ne i} ^ {3}\ sum _ {k\ne i, k\ne j} ^ {3}\ frac {{e} ^ {(i)} ^ {(i)}}} {2 ({\ lambda} ^ {(i)}} {2 (\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda} ^ {(i)}) ({\ lambda})} - {\ lambda} ^ {(k)})}\]
(8.24)\[ {\ theta} ^ {(\ mathrm {ij})}} = {\ theta} ^ {(\ mathrm {ji})}} =\ frac {1} {2} {2} {d} {d} ^ {(j)}
\ textrm {for}
n=k, m\ in\ {i, j\}
{\ theta} ^ {(\ mathrm {mn})}} =\ frac {({e} ^ {(m)}} - {e} ^ {(n)})} {(\ lambda})} {(\ lambda} ^ {\ lambda} ^ {(n)})} {(lambda} ^ {(n)})} {\]
(8.25)\[ {\ theta} ^ {(\ mathrm {ij})}} =\ frac {1} {2} {d} ^ {(j)} ^ {(j)}\]