2. Formulation of the model#

The formulation of model GLRC_DAMAGE is presented below, in the formalism of the thermodynamics of irreversible processes.

It should be noted that the use of this model is associated with the use of a plate element. If we choose the family of finite elements DKT (supported modeling: DKTG), we adopt the Love-Kirchhoff theory, that is to say that we do not consider any transverse distortion in the thickness of the plate. Model GLRC_DAMAGE could be usable with Q4G thick plate finite elements, but this extension has not yet been achieved.

The finite element mesh is supposed to be placed on the middle sheet of the slab (at \(z\mathrm{=}0\)).

In order to be able to use the GLRC_DAMAGE behavior model in two different types of analysis, depending on the case, we chose:

for a linear elastic analysis of reinforced concrete plates: to take into account the orthotropy induced by the orthogonal network of steel reinforcements, as well as the flexion-membrane coupling in the case of unequal layers of reinforcements, via a homogenized steel-concrete elastic behavior;
for a nonlinear damagable elastoplastic analysis of reinforced concrete plates: to neglect orthotropy and flexion-membrane coupling in the elasticity phase. This hypothesis makes it possible to simplify the model, by assuming that in the presence of highly non-linear phenomena, orthotropic elasticity becomes negligible, especially during fracture modeling. Moreover, in practice, it is expected that the walls, walls and other structural elements will be reinforced approximately the same between the two main orthogonal directions. This has the effect of reducing the effect of elastic orthotropy. On the other hand, asymmetric reinforcements are very often chosen in order to optimize them according to the direction of loading due to the own weight. This tends to induce a membrane-flexure coupling in elasticity in addition to that in plasticity. However, even if the model neglects asymmetry in elasticity, its influence in elasto-plasticity, the predominant behavior during rupture, can be controlled through threshold functions, which may be asymmetric.

The model is defined by the description of state variables, which represent the mechanical system at each material point on the mean surface of the plate, the surface density of free energy, which includes the form of behavioral relationships and the type of work hardening, the expression of the criteria of plasticity and damage, and the laws of irreversible evolution, deduced from Drücker’s principle of maximum work.

2.1. State variables#

The global state variables are as follows. First of all the global deformation variables:

a membrane deformation tensor: \(\epsilon\) defined in the tangential plane to the plate.
a curvature tensor: \(\kappa\) defined in the tangential plane to the plate.

Then the internal variables:

two damage variables associated with the upper, \({d}_{1}\) and lower, \({d}_{2}\) portions of the plate. They are each capped at one value, \({d}_{1}^{\mathit{max}}\) and \({d}_{2}^{\mathit{max}}\).
two plastic curvature tensors associated with the plasticization of steel beds, upper and lower \({\kappa }_{1}^{p},{\kappa }_{2}^{p}\).
two plastic membrane deformation tensors associated with the plasticization of steel beds, upper and lower, \({\epsilon }_{1}^{p},{\epsilon }_{2}^{p}\).
a tensors of order 2 of kinematic internal variables for hardening \(\alpha\).

2.2. Free energy: linear elastic case#

The free energy surface density is an additive expression of the elastic contributions of membrane and flexure:

(2.24)#\[ {\ Phi} _ {e} ^ {S}\ mathrm {=}\ mathrm {=}\ frac {1} {2}\ epsilon\ mathrm {:}\ mathrm {:}\ epsilon +\ frac {1} {2}\ frac {1} {2}\ frac {1} {2}\ frac {1} {2}\ epsilon +\ frac {1} {2}\ kappa {2}\ kappa {:} {H} _ {f}\ mathrm {:}\ kappa +\ epsilon\ mathrm {:} {H} _ {\ mathit {mf}}\ mathrm {:}\ kappa\]

The tensors \({H}_{m}\), \({H}_{f}\), \({H}_{\mathit{mf}}\) (flexion-membrane coupling in case of unequal layers of reinforcement in the thickness) are described in [§3.1]. In the current state of the model, it is assumed that:

\({H}_{\mathit{mf}}\mathrm{=}0\)

so that the plate is symmetric and that there is no membrane-elastic flexure coupling. In the model, membrane-flexure coupling may appear only as a result of an evolution towards elasto-plasticity (see §2.5).

2.3. Free energy: damageable elastoplastic case#

In this case, the elastic orthotropy induced by the reinforcement in the two directions of the plane, as well as the elastic flexion-membrane coupling (for asymmetric reinforcement sheets) are neglected. Steel frames are therefore assimilated to an isotropic elastic membrane, cf. [].

Free energy surface density is an additive expression of elastoplastic membrane, elastoplastic damageable flexure, and kinematic work hardening contributions:

(2.24)#\[ {\ Phi} _ {\ mathit {epd}}} ^ {S} (\ epsilon, {\ epsilon} ^ {p},\ kappa, {\ kappa} ^ {p}, {d} _ {\ mathrm {1,}} _ {\ mathrm {1,}} {s}} {d} _ {d} _ {e, m} ^ {S} (\ epsilon\ mathrm {-} {\ epsilon} ^ {p}) + {\ Phi} _ {\ mathit {ed}, f} ^ {S} (\ kappa\ mathrm {-} {-} {-} {\ kappa} {\ kappa}} {\ kappa}} ^ {p}) + {\ mathrm {1,}} {d} _ {2}) + {\ mathrm {1,}} {d} _ {2}) + {\ mathrm {1,}} {d} _ {2}) + {\ mathrm {1,}} {d} _ {2}) + {\ mathrm {1,}} {d} _ {2}) + {\ mathrm {Phi} _ {p} ^ {S} (\ alpha) +H ({d} _ {j}\ mathrm {-} {d} _ {j}} ^ {\ mathit {max}})\]

with work hardening energy:

(2.24)#\[ {\ Phi} _ {p} ^ {S} (\ alpha) (\ alpha)\ mathrm {=}\ frac {1} {2}\ alpha\ mathrm {:}\ mathrm {:}\ alpha\]

where \(C\) is a Prager kinematic work hardening tensor. In practice, the tensor \(C\) is diagonal, with a coefficient \({C}_{m}\) in membrane and another \({C}_{f}\) in flexure, so we have:

\(C\mathrm{=}(\begin{array}{cccccc}{C}_{m}& 0& 0& 0& 0& 0\\ 0& {C}_{m}& 0& 0& 0& 0\\ 0& 0& {C}_{m}& 0& 0& 0\\ 0& 0& 0& {C}_{f}& 0& 0\\ 0& 0& 0& 0& {C}_{f}& 0\\ 0& 0& 0& 0& 0& {C}_{f}\end{array})\)

In [éq 2.3.1], \(H\) refers to a function indicative of the domain of admissibility of the thermodynamic potential. Concretely, it serves to limit the evolution of damage above \({d}_{j}^{\mathit{max}}\). The \({d}_{j}^{\mathit{max}}\) are identified by [éq 3.2.11].

Membrane and flexure energy densities are given by:

In membrane:

(2.24)#\[ {\ Phi} _ {e, m} ^ {S} (\ epsilon\ mathrm {-} {\ epsilon} ^ {p})\ mathrm {=}\ frac {1} {2} (\ epsilon\ mathrm {2}} (\ epsilon\ mathrm {-} -} {\ epsilon} ^ {p})\ mathrm {:} {H} _ {2} (\ epsilon\ mathrm {-} {-} {\ epsilon} ^ {p})\ mathrm {:} {H} _ {2} (\ epsilon\ mathrm {-} {-} {epsilon} ^ {p})\ mathrm {:} {H} {2}\ frac {1} {:} (\ epsilon\ mathrm {-} {\ epsilon} ^ {p})\]

When flexing:

(2.24)#\[\begin{split} {\ Phi} _ {\ mathit {ed}, f}} ^ {S} (\ kappa\ mathrm {-} {\ kappa} ^ {p}, {d} _ {\ mathrm {1,}} {d}} {d} _ {2})\ mathrm {=} {2})\ mathrm {=}\ frac {{\ lambda} _ {f}} {2}\ mathit {tr} {1,}} {d} {d} {d} {d} _ {2})\ mathrm {=} {2})\ mathrm {=}\ frac {\\ lambda} _ {f}} {2}\ mathit {tr} {1,}} {1,}} {d} {d} {d} kappa\ mathrm {-} {\ kappa} ^ {p})} ^ {p})} ^ {2} {\ xi} _ {f} (\ mathit {tr} (\ kappa\ mathrm {-} {\ kappa} {\ kappa}} ^ {p})} ^ {p})} ^ {p}), {d} _ {2}) + {\ mu} _ {f}\ mathrm {2}) + {\ mu} _ {f}\ mathrm m {\ sum} _ {i\ mathrm {=} {=} 1} 1} ^ {2} {\ tilde {(\ kappa\ mathrm {-} {\ kappa} ^ {p})}} _ {i} ^ {2} ^ {2} {2} {\ xi} _ {2}} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {f} _ {f} ({\ tilde {(\ kappa\ mathrm {-} {\ kappa} ^ {p}) ^ {p}) ^ {p} ^ {2} ^ {2} {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {{i}, {d} _ {\ mathrm {1,}} {d} _ {2})\end{split}\]

where the Lamé parameters for bending \({\lambda }_{f}\) and \({\mu }_{f}\) are introduced:

\(\begin{array}{c}{\lambda }_{f}\mathrm{=}\frac{{h}^{3}}{12}\lambda \\ {\mu }_{f}\mathrm{=}\frac{{h}^{3}}{12}\mu \end{array}\)

\(h\) being the thickness of the plate and \(\lambda\), \(\mu\) being the Lamé coefficients of the homogenized material.

\({\tilde{(\kappa \mathrm{-}{\kappa }^{p})}}_{i}\) refers to the \(i\) th eigenvalue of \(\kappa -{\kappa }^{p}\). Hereinafter, we will note \({\kappa }^{e}\) the elastic curvature variation tensor defined according to the hypothesis of partitioning flexural deformations by: \({\kappa }^{e}\mathrm{=}\kappa \mathrm{-}{\kappa }^{p}\).

And finally, we define the characteristic function of flexural damage \({\xi }_{f}\):

(2.24)#\[ {\ xi} _ {f} (x, {d} _ {\ mathrm {1,}} {d} _ {2})\ mathrm {=}\ frac {1+\ gamma {d} _ {1} _ {1}}} {1}} {1}} {\ mathrm {1,}} {\ mathrm {1,}} {1}} {1}} {1}} {1}} {1+ {d}} {1}} {1+ {d} _ {2}} {1+ {d}} _ {2}} {1+ {d} _ {2}} {1+ {d}} _ {2}} {1+ {d} _ {2}} {1+ {d} _ {2}} {1+ {d} _ {2}} H (\ mathrm {-} x)\]

In this expression \(H\) is the Heaviside function and \(\gamma\) is a damage parameter between 0 and 1. This function \({\xi }_{f}\) characterizes the weakening of stiffness by damage. It is decreasing for \({d}_{1}\), \({d}_{2}\) positives. It is convex (thanks to the choice of \(\gamma\), identified by the procedure described in [§3.2.2]) to ensure the stability of the reinforced concrete « material » of the slab.

2.4. Law of damageable elastoplastic behavior#

The law of damageable elastoplastic behavior (state law) provides dual variables: membrane forces, bending moments, which are second order tensors defined on the tangential plane of the plate, damage forces and irreversible work-hardening tensors. They are written:

Membrane stress:

(2.24)#\[ N\ mathrm {=}\ frac {\ mathrm {\ partial} {\ partial} {\ partial} {\ partial} {\ partial}\ epsilon}\ mathrm {\ partial}\ epsilon}\ mathrm {=} {partial} {\ partial} {\ partial} {\ partial} {\ partial}}\ epsilon} {\ partial}\ epsilon}\ partial}\ epsilon}\ partial}\ epsilon}\ partial}\ epsilon}\ partial}\ epsilon}\ partial}\ epsilon}\ partial} {\ partial}\ epsilon}\ partial}\ epsilon}\ partial} {\ partial}\ epsilon}\ partial}\ epsilon} {\ partial}\ epsilon}\ partial} {\ partial}})\]

Bending moment:

(2.24)#\[ M\ mathrm {=}\ frac {\ mathrm {\ partial} {\ partial} {\ partial} {\ partial}} {\ mathrm {\ partial}\ kappa}\ mathrm {=} {partial}\ kappa}}\ mathrm {=} {\ partial} {\ partial}}\ mathrm {\ partial}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}}\ kappa}} {\ mathrm {1,}} {d} _ {2})\ mathrm {:} (\ kappa\ mathrm {-} {\ kappa}} ^ {p})\]

The membrane elasticity tensor \({H}_{m}\) is given in [§3.1], while \({H}_{f}^{d}\) is the damagable elasticity tensor that depends on the damage variables \({d}_{1}\), \({d}_{2}\) and also on the signs of some components of \({\kappa }^{e}=\kappa -{\kappa }^{p}\) (of the trace and of the eigenvalues, in particular). We recall that in [éq. 2.4.1] and [éq. 2.4.2] the membrane-elastic flexure coupling is neglected (see [§2.2]). In addition, because of the presence of the eigenvalues of the elastic curvatures in the expression for free energy (see [éq. 2.3.4]), the generalized stresses are calculated using the equations [éq. 2.4.1], [éq. 2.4.2] in the natural coordinate system. Details of the transformation between the guides are available in [R7.01.32]. It is also pointed out that according to the isotropic elasticity hypothesis, the membrane-flexure coupling is due solely to the elasto-plastic process through \({\epsilon }^{p}\) and \({\kappa }^{p}\) (see §2.5).

Note:

It can be seen that the specific coordinate system of moments is the same as that of elastic curvatures. Likewise, the specific frame of reference of membrane forces is the same as that of.elastic deformations. In the absence of damage, :math:`{xi }_{f}(x,{d}_{mathrm{1,}}{d}_{2})mathrm{=}1`: we indeed find isotropic elastic plate behavior. *

Damaging forces, for \(j\mathrm{=}\mathrm{1,2}\):

(2.24)#\[ {Y} _ {j}\ mathrm {=}\ mathrm {-}\ mathrm {-}\ frac {\ mathrm {\ partial} {\ phi} _ {\ mathit {epd}}}} {\ mathrm {\ partial} {\ partial} {d} {d}}\ mathrm {-}}} {\ mathrm {-}}} {\ mathrm {\ partial}}} {\ mathrm {\ partial} {d} _ {j})} ^ {2}} (\ frac {{\ lambda}} _ {f}} {2}\ mathit {tr} {({\ kappa} ^ {e})} ^ {2} H ({(\ mathrm {-} 1) {(\ mathrm {-} 1)} _ {1)}} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {f}\ mathrm {\ sum} _ {i} {i} {({\ tilde {\ kappa}}} _ {i} ^ {e})} ^ {2} H ({(\ mathrm {-} 1)}} ^ {i} {-} 1)} ^ {1)}} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {1)} ^ {\]

We note that the \({Y}_{j}\) defined by [éq. 2.4.3] are positive (it is a restoration of surface energy, whose SI unit is J/m²) if \(\gamma \mathrm{\in }\mathrm{[}\mathrm{0,1}\mathrm{]}\).

Efforts and irreversible moments of plasticity:

(2.24)#\[ {N} ^ {p}\ mathrm {=}\ frac {\ mathrm {\ partial} {\ Phi} _ {\ mathit {epd}} ^ {S}} {\ mathrm {\ partial} {\ partial} {\ partial} {\ epsilon} {\ epsilon} {\ epsilon}} ^ {p}}}\ mathrm {:} (\ epsilon\ mathrm}} {-} {\ epsilon} ^ {p})\ mathrm {=} N\]

Kinematic work hardening return tensors:

(2.24)#\[ {X} ^ {m}\ mathrm {=}\ frac {\ mathrm {\ partial} {\ partial} {\ Phi} _ {\ mathit {epd}} ^ {S}} {\ mathrm {\ partial} {\ partial} {\ partial} {\} {\ partial} {\ alpha} {\ alpha} {\ alpha}\ frac {\ alpha}}\ frac {\ mathrm {=}} {\ mathrm {\ partial}} {\ partial} {\ partial} {\ alpha} {\ alpha} _ {m}\]

2.5. Criteria — threshold surfaces#

2.5.1. Plasticity criterion#

Johansen’s plasticity criterion for kinematic work hardening is split for the plasticity of the upper part (index 1) and the lower part (index 2) of the plate. This criterion couples membrane plasticity with flexural plasticity. If \(x\) and \(y\) designate the directions of the orthogonal reinforcement of the concrete plate, for \(j\mathrm{=}\mathrm{1,2}\), the criterion is written as:

(2.24)#\[\begin{split} \ begin {array} {ccc} {f} _ {j} _ {j} ^ {p} (N\ mathrm {-} {x} ^ {m}, M\ mathrm {-} {X} ^ {f}) &\ mathrm {=}} &\ mathrm {=}} &\ mathrm {=}}\ mathrm {=}} &\ mathrm {=}} &\ mathrm {=}} &\ mathrm {-}} &\ mathrm {-}} &\ mathrm {=}} &\ mathrm {-}} &\ mathrm {-}}\ mathrm {=}} &\ mathrm {-}} &\ mathrm {-}} {X} _ {\ mathit {xx}} ^ {f}\ mathrm {-} {X} {M} {M} _ {\ mathit {jx}}} ^ {p} ({N} _ {\ mathit {xx}}}\ mathrm {-} {X}} {X}} _ {X} _ {X} _ {X} _ {X} _ {X} _ {\ mathit {xx}} _ {X} _ {\ mathit {xx}} _ {X} _ {\ mathit {xx}} _ {\ mathit {xx}}} ^ {m}))\\ &\ mathrm {\ times}} & ({M} _ {X}} _ {X} _ {X} _ {X} _ {X} _ {X} _ {\ mathit {xx}}}\ mathrm {-} {X} _ {\ mathit {yy}} _ {\ mathit {yy}}} ^ {x}} _ {\ mathit {jy}} ^ {p} ({N} _ {\ mathit {yy}} _ {\ mathit {yy}}}} _ {\ mathit {yy}}} ^ {m})) ({N} _ {p} _ {p}) _ {N} _ {p} _ {p} ({N} _ _ {\ mathit {xy}}\ mathrm {-} {X} _ {\ mathit {xy}} ^ {f})} ^ {2}\ mathrm {\ le} 0\ end {array}\end{split}\]

The \({f}_{j}^{p}\) define a convex domain (cf. [bib4]) of reversibility, parameterized by 4 functions: \({M}_{\mathit{jx}}^{p}({N}_{\mathit{xx}})\) and \({M}_{\mathit{jy}}^{p}({N}_{\mathit{yy}})\). These functions are constructed using the limit analysis of reinforced concrete beam sections representative of the plate section under study, taken in the direction of the reinforcing bars, cf. [bib1, bib2]. It is observed that only the differences between the stress state and the work-hardening return tensors are involved in the expression of the criterion. This is characteristic of models with kinematic work hardening.

The dissipation potential associated with this criterion is given by:

(2.24)#\[\begin{split} \ begin {array} {c} {\ Psi} _ {\ mathit {tot}} {\ mathit {tot}}\ mathrm {=}\ dot {\ epsilon} +M\ mathrm {::}\ dot {\ kappa}\ dot {\ kappa}}\ dot {\ kappa}}\ mathrm {\ kappa}}\ dot {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {\ kappa}}\ mathrm {} _ {p} + {\ Psi} _ {d}\\ {\ Psi} _ {p}\ mathrm {=} N\ mathrm {:} {\ dot {\ epsilon}}} ^ {p}} +M\ mathrm {:} {:} {\ dot {\ kappa}} _ {m}}\ mathrm {:} {C} _ {m}\ mathrm {:} {\ alpha} _ {m}\ mathrm {-} {\ dot {\ alpha}} _ {f}\ mathrm {:} {:} {C} {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C} _ {C}} _ {1} {\ dot {d}}} _ {1} + {Y} _ {2} {\ dot {d}} _ {2}\ end {array}\end{split}\]

2.5.2. Damage criterion#

The brittle damage criterion without work hardening is defined by a scalar. This criterion is duplicated to differentiate between positive and negative inflections. It is written:

(2.24)#\[ {f} _ {j} ^ {d} ({Y} _ {j})\ mathrm {=} {Y} _ {j} (\ kappa, {d} _ {j})\ mathrm {-} {j})\ mathrm {-} {k} {k} _ {k} _ {j}\ mathrm {\ the} 0\]

This criterion represents a convex range (cf. [bib2]) of reversibility parameterized by the thresholds \({k}_{1}\) and \({k}_{2}\) which define the appearance of the first flexural cracks in the reinforced concrete plate. Their SI unit is \(J\mathrm{/}{m}^{2}\). They correspond to a limitation of the elastic energy surface density. This criterion is associated with the positive dissipation potential:

(2.24)#\[ {\ Psi} _ {d} ({d} _ {j}}, {\ dot {d}}} _ {j})\ mathrm {=} {k} _ {j} {\ dot {d}} {\ dot {d}}} _ {j}} _ {j}\ mathrm {\ ge} 0\]

This damage criterion is basic, but it is its combination with the effect of the damage on elastic stiffness, cf. function \({\xi }_{f}(x,{d}_{\mathrm{1,}}{d}_{2})\) [éq. 2.3.5], that affects the response of the model.

2.6. Plastic flow laws#

The law of plastic flow is written (according to the rule of normality in criterion [éq. 2.5.1]):

(2.24)#\[ {\ dot {\ epsilon}} ^ {p}\ mathrm {=} {\ dot {\ epsilon}} _ {1} ^ {p} + {\ dot {\ epsilon}}} _ {2} ^ {p}\ mathrm {=}\ mathrm {=} {\ lambda} _ {1} ^ {p}\ frac {\ mathrm {\ partial} {f} _ {f} _ {1} ^ {p}} {\ mathrm {\ partial} N} N} + {\ lambda} _ {2} ^ {p}\ frac {\ mathrm {\ partial} {f} {f} _ {2} ^ {p}} _ {2} ^ {p}} {\ mathrm {\ partial} N}\]

(2.24)#\[ {\ dot {\ kappa}} ^ {p}\ mathrm {=} {=} {\ dot {\ kappa}} _ {1} ^ {p} + {\ dot {\ kappa}} _ {2} ^ {p}\ mathrm {p}\ mathrm {=} {=} {=} {\ lambda} _ {\ lambda} _ {\ lambda} _ {1} ^ {p}\ frac {\ mathrm {\ partial} {f} ^ {f} {p} {f} ^ {p} {f} ^ {p} {p}} {\ mathrm {\ partial} M}} + {\ lambda} _ {2} ^ {p}\ frac {\ mathrm {\ partial} {f} {f} _ {2} ^ {p}} {2} ^ {p}}} {\ mathrm {\ partial} M}\]

(2.24)#\[ {\ dot {\ alpha}} ^ {m}\ mathrm {=} {=} {\ dot {\ alpha}} _ {1} ^ {m} + {\ dot {\ alpha}} _ {2} ^ {m}\ mathrm {=}\ mathrm {=} {=} {=} {\ lambda} _ {=} {=} {\ lambda} _ {1} ^ {p} _ {1} ^ {p} _ {1} ^ {p}} {\ mathrm {\ partial} {X} {X} ^ {m}}} + {\ lambda}} _ {2} ^ {p}\ frac {\ mathrm {\ partial} {f} _ {2} ^ {p}}} {\ mathrm {{p}}} {\ mathrm {-}}\ mathrm {-} (\ mathrm {-}) {\ dot {\ epsilon}} _ {1} ^ {p}\ mathrm {-} {\ dot {\ epsilon}} _ {2} ^ {p})\ mathrm {=} {=} {\ dot {\ epsilon}} ^ {p})\ mathrm {=} {=} {\ p}\]

(2.24)#\[ {\ dot {\ alpha}} ^ {f}\ mathrm {=} {=} {\ dot {\ alpha}} _ {1} ^ {f} + {\ dot {\ alpha}} _ {2} ^ {f}\ mathrm {=}\ mathrm {=} {=} {=} {\ lambda} _ {=} {=} {\ lambda} _ {1} ^ {p} _ {1} ^ {p} _ {1} ^ {p}} {\ mathrm {\ partial} {X} {X} ^ {f}}} + {\ lambda}} _ {2} ^ {p}\ frac {\ mathrm {\ partial} {f} _ {2} ^ {p}}} {\ mathrm {{p}}} {\ mathrm {-}}\ mathrm {-} (\ mathrm {-}) {\ dot {\ kappa}} _ {1} ^ {p}\ mathrm {-} {\ dot {\ kappa}} _ {2} ^ {p})\ mathrm {=} {=} {\ dot {\ kappa}} ^ {p})\ mathrm {=} {=} {\ dot {\ kappa}}\]

where the \({\lambda }_{j}^{p}\) are the plastic multipliers, positive or zero, for positive inflections and negative inflections. They are shared by membrane flow and by flexure flow. It is deduced from [éq. 2.6.3] and [éq. 2.6.4], in the usual way in linear kinematic work hardening, that the internal variables of membrane and flexural work hardening are equal respectively to the deformations and to the plastic curvatures. This results in the following relationships on the return tensors in membrane and flexure:

(2.24)#\[ {X} ^ {m}\ mathrm {=}\ mathrm {=}\ mathrm {-} {C} _ {m}\ mathrm {:} {\ epsilon} ^ {p}\]

(2.24)#\[ {X} ^ {f}\ mathrm {=}\ mathrm {=}\ mathrm {-} {C} _ {f}\ mathrm {:} {\ kappa}} ^ {p}\]

It should be noted that this choice of a Prager \({C}_{m}\) tensor identical both in tension and in compression is questionable. In fact, in plastic compression, concrete and steel are involved, while in traction, only steel contributes (the concrete being broken).

The plasticity criteria [éq. 2.4.1] can be reached at the same time (for particular bi-flexure regimes), so flow can take place from the two criteria reached at the same time. The consistency condition gives two additional relationships:

(2.24)#\[ \ begin {array} {ccccc} {\ lambda} _ {j} ^ {p} _ {j}} {\ dot {f}} _ {j} ^ {p} (N, M)\ mathrm {=} 0&\ text {if}} & {\ text {si}} & {\ text {si}} 0&\ text {f}} 0&\ text {f}} 0&\ text {f}} 0&\ text {f}} 0&\ text {f} 0&\ text {f}} 0&\ text {if}} 0&\ text {si} & {\ text {si}} & {\ text {if}} & {j} ^ {p} _ {j} ^ {p}} _ {j} ^ {p} (N, M)\ mathrm {=} 0\ end {array}\]

2.7. Law of evolution of damage variables#

The law of evolution of flexural damage is written, for positive flexures and negative flexions (according to the rule of normality to criterion [éq. 2.5.3]):

(2.24)#\[ {\ dot {d}} _ {j}\ mathrm {=} {\ lambda} _ {j} ^ {d}\ frac {\ mathrm {\ partial} {f} _ {j} ^ {d}}} {\ mathrm {\ partial} {Y}} {\ partial} _ {j}}\]

The \({\lambda }_{j}^{d}\) are the damage multipliers, positive or zero. We recall that \({Y}_{j}\), defined by [éq. 2.4.3], are positive by construction (energy return).

The consistency condition gives two additional relationships:

(2.24)#\[ \ begin {array} {ccccc} {\ lambda} _ {\ lambda} _ {j}} ^ {d}} _ {j} ^ {d} ({Y} _ {j})\ mathrm {=} 0&\ text {=} 0&\ text {if}} 0&\ text {f}} _ {j}} _ {j} ^ {d} ({Y} _ {j})\ mathrm {=} 0\ mathrm {\ iff} {\ dot {Y}} _ {j}\ mathrm {=}} 0\ end {array}\]

The two damage variables can change simultaneously.