Theoretical formulation ===================== State variables, state law ---------------------------- The state variables are: total deformation :math:`\mathrm{\epsilon }\left(\mathrm{u}\right)`, plastic deformation :math:`{\mathrm{\epsilon }}^{p}`. Free energy density is written as: .. math:: :label: eq-1 w\ left (\ mathrm {\ epsilon}\ left (\ mathrm {u}\ right), {\ mathrm {\ epsilon}} ^ {p}\ right) =\ frac {1} {2}\ left (\ mathrm {\ epsilon}\ left (\ mathrm {u}\ right) - {\ mathrm {\ epsilon}\ right) - {\ mathrm {\ epsilon}\ right) - {\ mathrm {\ epsilon}} ^ {p}\ right)\ mathrm {:}\ mathrm {A}\ mathrm {A}\ mathrm {:}\ left (\ mathrm {\ epsilon}\ left (\ mathrm {u}\ right) - {\ mathrm {\ epsilon}}} ^ {p}\ right) where :math:`\mathrm{A}` refers to the linear elasticity tensor; it is assumed to be isotropic (2 independent elastic coefficients in 3D). Work hardening is not considered. The elastoplastic three-dimensional law of behavior is then written (state law): .. math:: :label: eq-2 \ mathrm {\ sigma} =\ mathrm {A}\ mathrm {A}\ mathrm {:}\ left (\ mathrm {u}\ right) - {\ mathrm {\ epsilon}}} ^ {p}\ right) - {\ mathrm {\ epsilon}}} ^ {p}\ right) .. _RefNumPara__30728_2770894330: Criterion (load area) smoothed -------------------------------- The criterion delimiting the elastic domain (load surface) is formulated with the 3 tensor invariants: .. math:: :label: eq-3 \ text {f}\ left (\ mathrm {\ sigma}\ right) =\ frac {1} {3} {I} _ {1}\ mathrm {.} \ mathrm {sin} {\ varphi} _ {\ mathit {pp}} _ {\ mathit {pp}}} +\ sqrt {{J} _ {\ varphi} _ {L}\ right) + {a}\ right) + {a} _ {\ mathit {pp}}}} +\ sqrt {{J} _ {tt}}} ^ {2} {\ varphi}\ right) + {a} _ {\ mathit {pp}}} {\ mathit {pp}}}} +\ sqrt {{J} _ {pp}}} +\ sqrt {{J} _ {pp}}} +\ sqrt {{J} _ {pp}}} +\ sqrt {{J} _ {pp}}}} +\ sqrt {{J} _ {it {pp}}} -c\ mathrm {.} \ mathrm {cos} {\ varphi} _ {\ mathit {pp}}\ phantom {\ rule {2em} {0ex}}\ phantom {\ rule {2em} {2em} {0ex}} 0 with :math:`{I}_{1}=\text{tr}\left(\mathrm{\sigma }\right)=3p` the triple of the mean pressure, :math:`{J}_{2}=\frac{1}{2}\left({\mathrm{\sigma }}^{D}\mathrm{:}{\mathrm{\sigma }}^{D}\right)=\frac{1}{3}{\left({\Vert \mathrm{\sigma }\Vert }_{\mathit{VM}}\right)}^{2}`; :math:`{J}_{3}=\text{det}\left({\mathrm{\sigma }}^{D}\right)`, having noted :math:`{\mathrm{\sigma }}^{D}=\mathrm{\sigma }-p\mathrm{Id}` the stress tensor deviator, and the modified Lode angle defined by: :math:`{\theta }_{L}=\frac{1}{3}\mathrm{arcsin}\left(\text{-}\frac{3\sqrt{3}{J}_{3}}{2\sqrt{{J}_{2}^{3}}}\right)\phantom{\rule{2em}{0ex}}\in \phantom{\rule{2em}{0ex}}\left[\text{-}\frac{\pi }{6},\frac{\pi }{6}\right]`, and the three parameters: cohesion :math:`c`, the friction angle :math:`{\varphi }_{\mathit{pp}}`, and the "traction truncation" :math:`{a}_{\mathit{tt}}`, which is used for the approximation hyperbolic used for smoothing at the top of the Mohr-Coulomb pyramid. cf. [:ref:`2 `]. *Note* *1* **:** *According to* [:ref:`2 ` *],* *if* :math:`{a}_{\mathit{tt}}=0` *, criterion (3) finds the top of the pyramid* *of* *Mohr-Coulomb,* *so it is no longer smoothed.* *if* Si*:math:`{a}_{\mathit{tt}}=\frac{1}{2}c\mathrm{.}\mathit{cotan}{\varphi }_{\mathit{pp}}`*, * ** *then at the top of the smoothed criterion, we check:* :math:`{I}_{1}=0` *.* We know that the main constraints can be written according to: .. math:: : label: eq-4 \ {\ begin {array} {c} {\ sigma} {\ sigma} _ {I} _ {1} {1} +\ frac {2\ sqrt {3}}} {3}} {3}\ mathrm {.} \ sqrt {{J} _ {2}}}\ mathrm {.} \ mathrm {sin}\ left ({\ theta} _ {L} _ {L} +\ frac {2\ pi} {3}\ right)\\ {\ sigma} _ {\ mathit {II}}} =\ frac {1}}}} =\ frac {1}} _ {1} +\ frac {2\ sqrt {3}} {3}}\ mathrm {.} \ sqrt {{J} _ {2}}}\ mathrm {.} \ mathrm {sin}\ left ({\ theta} _ {L}\ right)\\ {L}\ right)\\ {\ sigma} _ {\ mathit {III}}} =\ frac {1} {3} {3} {3} {3} {3} {3} {3} {3} {I} _ {1} {1} {I} _ {1}} +\ frac {1} _ {1} +\ frac {1} _ {1} +\ frac {1} _ {1} +\ frac {1} _ {1} +\ frac {1} _ {1} +\ frac { \ sqrt {{J} _ {2}}}\ mathrm {.} \ mathrm {sin}\ left ({\ theta} _ {L} -\ frac {2\ pi} {3}\ right)\ end {array} The :math:`K\left({\theta }_{L}\right)` function in (3) is defined by: .. math:: : label: eq-5 K\ left ({\ theta} _ {L}\ right) =\ {\ begin {\ array} {cc} {\ left (\ mathrm {cos} {\ theta} _ {L} -\ frac {\ sqrt {3}}\ right) =\ {\ sqrt {3}} {\ sqrt {3}} {3}}\ mathrm {3}} {3}\ mathrm {3}} {3}\ mathrm {3}} {3}\ mathrm {3}} {3}\ mathrm {3}} {3}\ mathrm {3}} {3}\ mathrm {3}} {3}\ mathrm {.} \ mathrm {sin} {\ theta} _ {L}\ right)}} ^ {2}}} ^ {2} &\ mathit {si}\ phantom {\ rule {2em} {0ex}}\ left| {\ theta}} _ {L}\\ right} _ {L}\\ right|< {L}\ right|< {L}\ right|< {L}\ right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta} | right|< {\ theta}} | right|<\ left ({\ theta} _ {L}\ right)\ mathrm {.} \ mathrm {sin} 3 {\ theta} _ {L} _ {L} +C\ left ({\ theta} _ {L}\ right)\ mathrm {.} {\ mathrm {sin}} ^ {2} 3 {\ theta} _ {L}\ right)} ^ {2} &\ mathit {si}\ phantom {\ rule {2em} {0ex}}\ left| {\ theta}}}\ left| {\ theta}}}\ left| {\ theta} _ {\ theta} _ {T}\ end {array}}}\ left| {\ theta}}}\ left| {\ theta} _ {\ theta} _ {array} with an additional :math:`{\theta }_{T}\in \phantom{\rule{2em}{0ex}}\left[0,\frac{\pi }{6}\right]` parameter called transition angle and :math:`A\left({\theta }_{L}\right),B\left({\theta }_{L}\right),C\left({\theta }_{L}\right),` functions defined by: .. math:: :label: eq-6 \ {\ begin {array} {c} A\ left ({\ theta} _ {L}\ right) =\ text {-}\ frac {\ sqrt {3}} {3}\ mathrm {sin} {\ sin} {\ sin} {\ varphi} _ {\ varphi}} _ {\ mathit {pp}}\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} -B\ left ({\ theta} _ {L}\ right)\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} -C\ left ({\ theta} _ {L}\ right)\ mathrm {.} {\ mathrm {sin}} ^ {2} 3 {\ theta} _ {T} _ {T} +\ mathrm {cos} {\ theta} _ {T}\\ left ({\ theta} _ {L}\ right) =\ frac {\ theta}\ right) =\ frac {\ mathit {sgn}} +\ mathrm {\ theta} _ {L}\ left ({\ theta} _ {L}\ right) =\ frac {\ mathit {sgn}} {\ theta} _ {L}} \ mathrm {sin} 6 {\ theta} _ {T}\ left (\ mathrm {cos} {\ theta} _ {T} -\ frac {\ sqrt {3}}} {3\ mathrm {3}} {3\ mathrm {sin}} {\ mathrm {sin} {\ sin} {\ varphi}} _ {\ mathit {pp}}\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T}}\ right) -6\ mathrm {cos} 6 {\ theta} _ {T}\ left (\ mathit {sgn} {\ sgn} {\ theta}} _ {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} _ {T} +\ frac {\ sqrt {3}} {3\ mathrm {sin} {\ varphi}} _ {\ mathit {pp}}\ mathrm {.} \ mathrm {cos} {\ theta} _ {T}}\ right)} {18 {\ mathrm {cos}} ^ {3} 3 {\ theta} _ {T}}\\ C\ left ({\ theta} _ {theta} _ {L}\ right) =\ frac {-\ mathrm {cos}} 3 {\ theta} _ {T}}\ left (\ mathrm {cos} {\ theta} _ {T} -\ frac {\ sqrt {3}} {3\ mathrm {sin} {\ varphi} _ {\ mathit {pp}} -\ frac {\ sqrt {3}}}\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T}}\ right) -3\ mathit {sgn} {\ theta} _ {L}\ mathrm {sin} 3 {\ theta} _ {T} _ {T}\ left (\ mathit {sgn}} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} _ {T} +\ frac {\ sqrt {3}} {3\ mathrm {sin} {\ varphi}} _ {\ mathit {pp}}\ mathrm {.} \ mathrm {cos} {\ theta} _ {T}}\ right)} {18 {\ mathrm {cos}} ^ {3} 3 {\ theta} _ {T}}}\ end {array} *Note* *2* **:** *According to* [:ref:`2 ` *],* *we should not take a value of* :math:`{\theta }_{T}` *too close to* :math:`\frac{\pi }{6}` *, at the risk of poor conditioning;* *and* *a typical value of* :math:`{\theta }_{T}` *can be taken equal to* :math:`\frac{5\pi }{36}`\ *.* *With this value of* *.* *With this value of* :math:`{\theta }_{T}` *and with* :math:`{a}_{\mathit{tt}}=\frac{1}{20}c\mathrm{.}\mathit{cotan}{\varphi }_{\mathit{pp}}` *, the proposed hyperbolic smoothing leads to an error of at most 0.13% in The plan deviatory from the* *original Mohr-Coulomb pyramid.* The and are a graphical representation of the original Mohr-Coulomb load surface in the principal stress space. The shows the effect of smoothing [:ref:`2 `] on the edges of the Mohr-Coulomb charge surface. .. image:: images/100002000000018A0000011629C869B9FA99F23B.png :width: 3.8146in :height: 2.7319in .. _RefImage_100002000000018A0000011629C869B9FA99F23B.png: Figure 2.2-1: Representation of the Mohr-Coulomb load surface in the three-dimensional space of the main stresses. .. image:: images/10000200000001C7000001BCB4B85E743A07E357.png :width: 2.6098in :height: 2.7319in .. _RefImage_10000200000001C7000001BCB4B85E743A07E357.png: Figure 2.2-2: Representation of the Mohr-Coulomb load surface in plane :math:`\mathrm{\pi }` of the stress deviators (any vector represented in this plane corresponds to a deviatoric stress). .. image:: images/1000000000000108000000F4F8C4825CC122FB21.png :width: 2.2in :height: 2.0335in .. _RefImage_1000000000000108000000F4F8C4825CC122FB21.png: Figure 2.2-3: Representation of the Mohr-Coulomb load surface smoothed in plane :math:`\pi` of the stress deviators, for the **friction angle** :math:`{\varphi }_{\mathit{pp}}=25°` **and two values of** :math:`{\theta }_{T}` (taken from [:ref:`3 `]). An enrichment of the criterion () is provided by adding a term for hardening to cohesion, in the following form: .. math:: :label: eq-7 -c\ mathrm {.} \ mathrm {cos} {\ varphi} _ {\ mathit {pp}\ mathrm {.}} \ left (1+ {r} _ {\ mathit {eg}}}\ cdot {\ vert {\ epsilon} ^ {p}\ green} _ {\ mathit {VM}}}\ right) where the parameter :math:`{r}_{\mathit{eg}}` (expressed without units) makes it possible to introduce linear cohesion work hardening. .. _RefNumPara__31986_2770894330: Plastic flow potential -------------------------------- The plastic flow of this law is not standard (non-associated law), and therefore not oriented according to the normal to the load surface. The authors [:ref:`2 `] propose an expression of plastic flow using another potential with the same analytical form as that of the criterion (), by involving the dilatance angle :math:`\psi`: .. math:: :label: eq-8 \ text {g}\ left (\ mathrm {\ sigma}\ right) =\ frac {1} {3} {I} _ {1}\ mathrm {.} \ mathrm {sin}\ psi +\ sqrt {{J} _ {2}\ mathrm {.} {K} _ {g}\ left ({\ theta}} _ {L}\ right) + {a} _ {\ mathit {tt}} ^ {2} {\ mathrm {tan}}} ^ {2} {2} {\ varphi}} _ {\ varphi} _ {\ mathit {pp}}\ mathrm {.}} {\ mathrm {cos}} ^ {2}\ psi} -c\ mathrm {.} \ mathrm {cos}\ psi When :math:`\psi ={\varphi }_{\mathit{pp}}`, the law of plastic flow becomes associated. Function :math:`{K}_{g}\left({\theta }_{L}\right)` in (7) is defined by: .. math:: :label: eq-9 {K} _ {g}\ left ({\ theta} _ {L}\ right) =\ {\ begin {array} {cc} {\ left (\ mathrm {cos} {\ theta} {\ theta} _ {L} -\ frac {\ theta} _ {L}} -\ frac {\ sqrt {3}} {L} -\ frac {\ sqrt {3}}} {3}\ mathrm {sin}\ left (\ mathrm {cos} {\ theta} {\ theta} _ {L} -\ frac {\ sqrt {3}}} {3}\ mathrm {sin}\ psi\ mathrm {.} \ mathrm {sin} {\ theta} _ {L}\ right)}} ^ {2}} &\ mathit {si}\ phantom {\ rule {2em} {0ex}}\ left| {\ theta}} _ {theta} _ {L}\\ right|< {L}\\ right|< {L}\ right|< {\ theta} | right|< {\ theta} | right|< {\ theta} _ {\ theta} _ {T}\\ left ({A} _ {\ text {g}}}\ left ({\ theta}}\ left ({\ theta}}} _ {L}\ right) + {B} _ {\ text {g}}\ left ({\ theta} _ {L}\ right)\ mathrm {.} \ mathrm {sin} 3 {\ theta} _ {L} _ {L} + {L}} + {L}\ right)\ mathrm {.}\ left ({\ theta} _ {L}\ right)\ mathrm {.} {\ mathrm {sin}} ^ {2} 3 {\ theta} _ {L}\ right)} ^ {2} &\ mathit {si}\ phantom {\ rule {2em} {0ex}}\ left| {\ theta}}}\ left| {\ theta}}}\ left| {\ theta} _ {\ theta} _ {T}\ end {array}}}\ left| {\ theta}}}\ left| {\ theta} _ {\ theta} _ {array} functions :math:`{A}_{\text{g}}\left({\theta }_{L}\right),{B}_{\text{g}}\left({\theta }_{L}\right),{C}_{\text{g}}\left({\theta }_{L}\right),` being defined by: .. math:: :label: eq-10 \ {\ begin {array} {c} {A} _ {\ text {g}}}\ left ({\ theta} _ {L}\ right) =\ text {-}\ frac {\ sqrt {3}} {3}} {3} {3}\ mathrm {g}} {3}\ mathrm {sin}\ psi\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} - {T} - {B} _ {\ text {g}}\ left ({\ theta} _ {L}\ right)\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} _ {T} - {C} _ {\ text {g}}\ left ({\ theta} _ {L}\ right)\ mathrm {.} {\ mathrm {sin}} ^ {2} 3 {\ theta} _ {T} _ {T} +\ mathrm {cos} {\ theta} _ {\ text {g}} _ {\ text {g}}}\ left ({\ theta}}}\ left ({\ theta}}}\ left ({\ theta}}\ left ({\ theta}}\ left ({\ theta}})\ left ({\ theta}}\ left ({\ theta}}\ left ({\ theta}})\ left ({\ theta}}\ left ({\ theta}}\ left ({\ theta}})\ left ({\ theta}}\ left ({\ theta}}\ left ({\ theta}})\ left} \ mathrm {sin} 6 {\ theta} _ {T}\ left (\ mathrm {cos} {\ theta} _ {T} -\ frac {\ sqrt {3}}} {3\ mathrm {3}} {3\ mathrm {sin}\ left (\ mathrm {sin}\ left) \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T}}\ right) -6\ mathrm {cos} 6 {\ theta} _ {T}\ left (\ mathit {sgn} {\ sgn} {\ theta}} _ {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} _ {T} +\ frac {\ sqrt {3}} {3\ mathrm {sin}\ psi\ mathrm {.} \ mathrm {cos} {\ theta} _ {T}}\ right)} {18 {\ mathrm {cos}} ^ {3} 3 {\ theta} _ {T}}\\ {C} _ {\ C} _ {\ text {g}} _ {g}}\ right)}\ left ({\ theta}}}\ right) =\ frac {-\ mathrm {cos}}}\\ {C} _ {C} _ _ {C} _ _ {C} _ _ {C} _ _ {C} _ {C} _ _ {C} _ {T}\ left (\ mathrm {cos} {\ theta} {\ theta} _ {T} -\ frac {\ sqrt {3}}} {3\ mathrm {sin}\ psi\ mathrm {.} \ mathit {sgn} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T}}\ right) -3\ mathit {sgn} {\ theta} _ {L}\ mathrm {sin} 3 {\ theta} _ {T} _ {T}\ left (\ mathit {sgn}} {\ theta} _ {L\ mathrm {.}} \ mathrm {sin} {\ theta} _ {T} _ {T} +\ frac {\ sqrt {3}} {3\ mathrm {sin}\ psi\ mathrm {.} \ mathrm {cos} {\ theta} _ {T}}\ right)} {18 {\ mathrm {cos}} ^ {3} 3 {\ theta} _ {T}}}\ end {array} When the stress state is on load surface :math:`\text{f}\left(\mathrm{\sigma }\right)=0`, the plastic deformation rate is normal to potential :math:`\text{g}\left(\mathrm{\sigma }\right)` and is written with the plastic multiplier :math:`d\lambda`: .. math:: :label: eq-11 d {\ mathrm {\ epsilon}}} ^ {p} =d\ lambda\ mathrm {.} \ frac {\ partial\ text {g}} {\ partial\ mathrm {\ sigma}} =d\ lambda\ mathrm {.} {\ mathrm {n}} _ {\ text {g}} where the normal tensor of order two outside the surface is: .. math:: :label: eq-12 {\ mathrm {n}} _ {\ text {g}} =\ frac {\ partial\ text {g}\ left (\ sigma\ right)} {\ partial {I} _ {1}\ left (\ sigma\ right)}\ left (\ sigma\ right)}\ cdot\ mathrm {right)}\ cdot\ mathrm {Id}} +\ left (\ sigma\ right)}\ cdot\ mathrm {Id} +\ left (\ sigma\ right)}\ cdot\ mathrm {Id} +\ left (\ sigma\ right)}\ cdot\ mathrm {Id} +\ left (\ sigma\ right)} {\ partial {right)} {\ partial {right)} J} _ {2}\ left (\ sigma\ right)}} +\ frac {\ partial\ text {g}\ left (\ sigma\ right)} {\ partial {\ theta} _ {L}}\ cdot\ frac {\ dot\ frac {\ frac {\ frac {\ frac {\ partial {\ theta}} _ {L}}} {\ partial\ theta}}} {\ partial\ theta}}} {\ partial {\ theta}}} {\ partial\ theta}} {\ left (\ sigma\ right)}\ right) {\ sigma\ right)}\ right) {\ sigma\ frac {\ frac {\ partial {\ theta}} _ {L}}}\ right) {\ sigma\ right)}\ right) {\ sigma} ^ {D} +\ frac {\ partial\ text {g}\ left (\ sigma\ right)} {\ partial {\ theta} _ {L}}\ cdot\ frac {\ partial {\ theta} _ {L} _ {L}}} {L}}} {\ partial {J}}}} {\ partial {J} _ {L}}} {\ partial {\ theta} _ {L} _ {L}}} {L}}} {\ partial {\ theta} _ {L}}} {\ partial {\ theta} _ {L}}} {L}}} {\ partial {\ theta} _ {L}}} {\ partial {\ theta} _ {L}}} {L}}} {\ partial {\ theta} _ {L}}} {sigma\ right)} {\ partial\ sigma} The only true thermodynamic internal variable is the equivalent plastic deformation :math:`{\Vert {\epsilon }^{p}\Vert }_{\mathit{VM}}`. We naturally have :math:`\frac{\partial \text{g}\left(\sigma \right)}{\partial {I}_{1}\left(\sigma \right)}=\frac{1}{3}\cdot \mathrm{sin}\psi` and we remind you that: :math:`\frac{\partial {J}_{2}\left(\sigma \right)}{\partial \sigma }={\sigma }^{D}`. *Note* *3* **:** *L* *e flow potential chosen, in the same shape as the criterion, therefore the* *flow* *tensor* *of flow* :math:`{\mathrm{n}}_{\text{g}}` *, is not affected by the possible introduction of linear cohesion work hardening, cf.* * *()* () *\*.* *In fact, the evolution of cohesion does not modify the direction normal to the faces of the pyramid* :math:`\text{g}\left(\mathrm{\sigma }\right)=0` *.* Energy dissipation by plastic flow --------------------------------------------- The mechanical power density dissipated by the plasticity mechanism is: .. math:: :label: eq-13 D=\ mathrm {\ sigma}\ mathrm {:} {\ dot {\ mathrm {\ epsilon}}}} ^ {p} Mechanical energy is dissipated by accumulation during the loading path. Behavioral relationship parameters ----------------------------------------- The parameters of the behavioral relationship are: .. csv-table:: "YoungModulus", "Young's Modulus (positive)", ":math:`E`" "PoissonRatio", "Poisson Ratio", ":math:`\nu`" "Cohesion", "Cohesion (positive or zero)", ":math:`c`" "FrictionAngle", "Friction angle (supplied in°)", ":math:`{\varphi }_{\mathit{pp}}`" "dilatancyAngle", "Expansion angle (supplied in°)", ":math:`\psi`" "TransitionAngle", "Transition Lode Angle (supplied in°, less than 30°)", ":math:`{\theta }_{T}`" "TensionCutoff", "Traction truncation (positive or zero)", ":math:`{a}_{\mathit{tt}}`" "HardeningCoef", "Cohesion hardening (positive or zero)", ":math:`{r}_{\mathit{eg}}`"