2. Continuous model#

2.1. Behavioral equations#

The stress-strain relationship takes the following form, in which \({⟨x⟩}_{\text{-}}\) and \({⟨x⟩}_{\text{+}}\) respectively designate the negative and positive parts of a symmetric scalar or tensor:

(2.1)#\[ \ mathrm {\ sigma} (\ mathrm {\ epsilon}, b)\ mathrm {\ epsilon}, b)\)\ left [\ mathrm {\ lambda} {\ mathit {tr} (\ mathrm {\ epsilon}} (\ mathrm {\ epsilon}), b)\)\ text {}} =\ text {} B (b)\ left [\ mathrm {\ lambda} {} {silon} ⟩} _ {\ text {+}}\ right]\ text {}}\ right]\ text {} +\ text {} +\ text {}\ left [\ mathrm {\ epsilon} {\ epsilon} (\ mathrm {\ epsilon}) ⟩} __}} _ {\ text {-}} _ {-}} +2\ mathrm {\ mu} {-}}\ right]\]

As for the stiffness function \(B(b)\), it is equal to:

(2.2)#\[ B (b)\ text {} =\ text {}\ frac {1-b} {\ left (1-b\ right) +\ frac {\ mathrm {\ kappa}} {2} b\ left ({m} _ {0} + ({m} _ {0}} + ({m} _ {1}} - {m} _ {0}) {b} ^ {r-1}\ right)}\]

The evolution of damage is governed by the following Kuhn and Tucker conditions:

(2.3)#\[ \ dot {b} 0\ text {;} Q (\ mathrm {\ epsilon}) -\ mathrm {\ varphi} (b) ≤0\ text {;}\ dot {b}\ left [Q (\ mathrm {\ epsilon}) -\ mathrm {\ varphi} (b)\ right] =0\]

The expression for threshold function \(\mathrm{\varphi }(b)\) is:

\[\]

: label: eq-4

mathrm {varphi} (b)text {=} {=} {left [(1-b) +frac {mathrm {kappa}} {2} bleft ({m} _ {0}} + ({D} _ {0}} + ({D} _ {0}} - {m} _ {1}} - {m} _ {1}} - {m} _ {0}) {b} ^ {r-1}right)right]} ^ {text {0}}} ^ {text {2}}}

As for the function of deformation \(Q(\mathrm{\epsilon })\), it is written as:

\[\]

: label: eq-5

Q (epsilon)text {=} {epsilon} _ {text {*}} ^ {2}}

In this expression, pseudo-norm \({\mathrm{\epsilon }}_{\text{*}}\) is the solution to the following scalar equation:

(2.4)#\[ {f} _ {\ mathrm {\ sigma}}\ left (\ frac {\ mathrm {\ sigma}}\ mathrm {\ epsilon})\ text {Id}\ text {Id}\ text {+} 2\ mathrm {+}} 2\ mathrm {\ mu}\ mathrm {\ epsilon}} {{\ mathrm {\ epsilon}}\ text {\ epsilon}}\ text {\ epsilon}}\ text {\ epsilon}}\ text {\ epsilon}}\ text {Id}\ text {+}\ text {+} 2\ mathrm {\ mu}\ mathrm {\ epsilon}} {*}}}\ right)\ text {=} 0\]

Finally, the initiation threshold is expressed as:

(2.5)#\[\begin{split} {f} _ {\ mathrm {\ sigma}} (\ mathrm {\ sigma}})\ text {\ sigma})\ text {=}\ vert\ frac {\ mathrm {\ sigma}} {{\ mathrm {\ sigma}}} _ {\ sigma}} _ {\ sigma}}} {\ mathrm {\ sigma}}} {\ mathrm {\ sigma}}} {{\ mathrm {\ sigma}}} {{\ mathrm {\ sigma}}} {\ mathrm {\ sigma}}} {{\ mathrm {\ sigma}}} (\ mathrm {\ sigma}}} (\ mathrm {\ sigma}}} (\ mathrm {\ sigma}}})\ mathrm {tr}\ left (\ frac {\ mathrm {\ sigma}}} {{\ mathrm {\ sigma}} _ {0}}\ right)\ text {Id}\ green\ text {+}\\ green\ mathrm {\ sigma}}\ green\ mathrm {\ sigma}}\ green\ mathrm {+}}\ green\ mathrm {\ sigma}}\ green\ mathrm {\ sigma}}\ green\ mathrm {\ exp}}\ left (\ frac {\ mathrm {\ sigma}}} _ {0}}\ right)\ Vert\ text {-} {\ mathrm {\ gamma}} _ {0}\end{split}\]

The stiffness function \(B(b)\), which is involved in the stress-deformation relationship, measures the impact of the level of damage on the residual stiffness (under tension). Its particular shape as well as that of threshold \(\mathrm{\varphi }(b)\) are guided by the consistency with the response of the ENDO_FISS_EXP law in a confined uniaxial tensile test. It plays the same role as the more classical form \((1-b)\) used in many models in literature.

2.2. Elastic energy and traction — compression transition#

To take into account the restoration of stiffness under compression, the law of elasticity is no longer linear, even if it remains hyperelastic (i.e. it derives from energy) in order to avoid any parasitic production or consumption of energy. It is similar to the formulation adopted for law ENDO_ISOT_BETON [R7.01.04]; it takes into account not only the sign of the deformations itself but also that of the trace of the deformations. In practice, it therefore requires the determination of the specific deformations. Its value is provided among the internal variables (ENERELAS).

Its properties are studied in detail in [Badel et al., 2007]. It should be noted that it is continuously differentiable, which ensures the continuity of the stress-deformation relationship. On the other hand, it is not continuously differentiable twice, which means that stiffness jumps when crossing the tension — compression transition. This may hinder the convergence of the calculation; this is why it is proposed to regularize this transition.

The idea, explained in [Lorentz, 2017] and [Lorentz, 2020], consists in regulating elastic energy. As a consequence of the stress-deformation relationship (), the positive and negative parts are approximated by infinitely differentiable functions:

(2.6)#\[ {⟨x⟩} _ {\ text {-}}}\ simeq\ text {} {} {N} _ {\ mathrm {\ gamma}} (x)\ text {;} {\ text {-}} _ {\ text {+}}}\ simeq\ text {} {\ text {+}}}}\ simeq\ text {} {+}}}\ simeq\ text {} {+}}}\ simeq\ text {} {N} _ {\ mathrm {\ gamma}} (x)\]

Approximation \({N}_{\mathrm{\gamma }}\) depends on a parameter \(\mathrm{\gamma }>0\) that tends to infinity in the absence of regularization. Its expression is:

(2.7)#\[\begin{split} {N} _ {\ mathrm {\ gamma}} (x) =\ {\ begin {array} {cc}\ left (x+\ frac {1} {2\ mathrm {\ gamma}}\ right)\ right)\ mathrm {exp}}\ right)\ mathrm {exp}\ right)\ mathrm {\ exp}\ right)\ mathrm {exp}\ right)\ mathrm {exp}\ left (-\ text {}\ frac {1} {\ mathrm {\ gamma}}}\ right)\ mathrm {exp}\ right)\ mathrm {exp}\ right)\ mathrm {exp}\ left (-\ text {}\ exp}\ left (-\ text {}\ array {x<0\\ 0&\ text {if} x≥0\ end {array}\end{split}\]

2.3. Model parameters#

It is possible to identify the parameters used in the behavioral equations. These are \(\mathrm{\lambda }\), \(\mathrm{\mu }\), \({\mathrm{\sigma }}_{0}\), \({\mathrm{\gamma }}_{0}\),, \({\mathrm{\beta }}_{0}\),,, \(\mathrm{\kappa }\), \({m}_{0}\), \({D}_{1}\), and \(r\). We propose to give their meaning and their expression, to finally list those to be filled in by the user.

Elasticity — The elastic characteristics, which are involved in the expression of deformation energy, are classically reduced to Lamé coefficients \(\lambda\) and \(\mu\) or, equivalently, to the Young’s modulus \(E\) and to the Poisson’s ratio \(\nu\). In particular, confined stiffness \({E}_{c}=\lambda +2\mu\) is deduced therefrom.

Damage initiation threshold — The initiation threshold is defined via the \({f}_{\sigma }\) function, which depends on three parameters: \({\sigma }_{0}\), \({\gamma }_{0}\), and \({\beta }_{0}\). In practice, we fix \({\beta }_{0}\) to 0.1 and we determine \({\sigma }_{0}\) and \({\tau }_{0}\) according to the strengths of concrete in tension \({f}_{t}\) and in compression \({f}_{c}\), according to the methodology presented in [Lorentz, 2017]. These data then make it possible to calculate \(Q\left(\mathrm{\epsilon }\right)\) for any deformation. In particular, it is possible to determine the initiation stress in confined uniaxial tension \({\sigma }_{c}\) by solving the following equation, in which \(n\) designates the (arbitrary) direction of traction:

(2.8)#\[ {f} _ {\ sigma}\ left (\ left (\ frac {\ lambda} {{E}} _ {c}}\ mathit {Id} +\ frac {2\ mu} {{E}} {{E} _ {c}}} n\ mathrm {\ c}}}} n\ mathrm {\ otimes} n\ right) {\ sigma} _ {c}\ right) =0\]

\({w}_{c}\) the deformation energy is then accessed when the initiation threshold is reached in uniaxial traction confined via \({w}_{c}={\sigma }_{c}^{2}/2{E}_{c}\).

  • Damage Energy — Cracking Energy \({G}_{F}\) measures the energy consumed per unit of cracked area. It should be compared to the average distance \(L\) that separates two cracks to build the volume energy \(k={G}_{F}/L\) consumed by the damage model; we choose to normalize the latter via \(\kappa ={G}_{F}/\left(L{w}_{c}\right)\). The two parameters \({G}_{F}\) and \(L\) therefore only intervene in the damage model through their ratio even if both have their own physical meaning.

  • Form factor — A final parameter, \(p⩾1\), affects the shape of the response during the damage phase (response is all the more curved in a \(\sigma -\epsilon\) diagram as \(p\) is large). From the value of \(p\), the parameters \({m}_{0}\), \({D}_{1}\) and \(r\) are deduced according to the following expressions:

(2.9)#\[ {m} _ {0}\ text {=}\ frac {3\ mathrm {\ pi}} {2} {\ left (p+2\ right)} ^ {-\ text {}\ text {}\ frac {3} {2}}\]
(2.10)#\[ {D} _ {1}\ text {=}\ frac {3\ mathrm {\ pi}} {4}\ sqrt {p+1}\]
(2.11)#\[ r\ text {=}\ frac {2 ({D} _ {1} -1) - {m} _ {0}} {2- {m} _ {0}}\]

And from there, also knowing the normalized volume energy at break \(\kappa\), it is possible to express for any damage value the stiffness function \(B(b)\) and the threshold \(\varphi \left(b\right)\). The data from \(p\) also limits the possible values for \(\mathrm{\kappa }\) for reasons of stability of the material point, which, indirectly, sets an upper bound \({L}_{c}\) to the inter-crack distance \(L\) via the following inequality:

(2.12)#\[ L\ text {}\ text {} {L} _ {L} _ {c}\ c}\ text {=}\ frac {{G} _ {c}}\ frac {3\ mathrm {\ pi}} {3\ mathrm {\ pi}}} {4}} {4} {(p+2)} {(p+2)}} ^ {-\ text {}}\ frac {3} {2}}\]

  • Regularization of the tensile — compression transition — The last parameter, \(\mathrm{\gamma }\), conditions the regularization of the positive and negative part functions involved in the stress-deformation relationship, as explained in § 2.2. As an alternative to giving a value for \(\mathrm{\gamma }\), it is preferable to set the proportion of stiffness (without regularization) that is found for a deformation (in compression) of the order of \({f}_{c}/E\), a proportion that is noted as \({\mathrm{\rho }}_{\mathrm{\gamma }}\).

In the end, using the DEFI_MATERIAU command, we are asked to fill in the following nine parameters: \(E\), \(\mathrm{\nu }\), \({\mathrm{\sigma }}_{\mathrm{c}}\), \({\mathrm{\sigma }}_{0}\), \({\gamma }_{0}\), \({\mathrm{\beta }}_{0}\), \(\mathrm{\kappa }\),,, \(p\) and \(\mathrm{\gamma }\). Values by default are proposed: \({\mathrm{\beta }}_{0}\text{=}0.1\) and \(\mathrm{\gamma }\text{=}0\) (no regularization step). Alternatively, the DEFI_MATER_GC command makes it possible to fill in these parameters in a more common way. In fact, the following ten parameters are provided: \(E\), \(\mathrm{\nu }\),, \({f}_{t}\),,, \({f}_{c}\),, \({\gamma }_{0}\), \({\mathrm{\beta }}_{0}\), \({G}_{F}\), \(L\), \(p\) and \({\mathrm{\rho }}_{\mathrm{\gamma }}\). Again, two default values are proposed: \({\mathrm{\beta }}_{0}\text{=}0.1\) and \({\mathrm{\rho }}_{\mathrm{\gamma }}\text{=}0.95\) (a regularization considered reasonable).