The MAZARS models ===================== Original Mazars Model -------------------------- The MAZARS model was developed as part of damage mechanics. This model is detailed in the thesis of MAZARS] The constraint is given by the following relationship: .. math:: : label: EQ-None \ sigma = (1-D)\ text {E} {\ epsilon} ^ {\ mathrm {e}} with: • :math:`E` the Hooke matrix, • :math:`D` the damage variable • :math:`{\varepsilon }^{e}` elastic deformation :math:`{\varepsilon }^{e}=\varepsilon -{\varepsilon }^{\text{th}}-{\varepsilon }^{\text{rd}}-{\varepsilon }^{\text{re}}` *•* :math:`{\varepsilon }^{\text{th}}=\alpha (T-{T}_{\text{ref}}){I}_{d}` thermal expansion *•* :math:`{\varepsilon }^{\text{re}}\mathrm{=}\mathrm{-}\beta \xi {\mathrm{I}}_{\mathrm{d}}` endogenous withdrawal (linked to hydration) *•* :math:`{\varepsilon }_{\text{rd}}=-\kappa ({C}_{\text{ref}}-C){I}_{d}` desiccation shrinkage (related to drying) :math:`D` is the damage variable. It is between :math:`0`, healthy material, and :math:`1`, broken material. The damage is driven by equivalent deformation :math:`{\epsilon }_{\text{eq}}`, which makes it possible to translate a triaxial state into an equivalence to a uniaxial state. As extensions are essential in the phenomenon of concrete cracking, the equivalent deformation introduced is defined on the basis of the positive eigenvalues of the deformation tensor, i.e.: .. csv-table:: ":math:`{\epsilon }_{\text{eq}}=\sqrt{{\langle \epsilon \rangle }_{+}\mathrm{:}{\langle \epsilon \rangle }_{+}}` where in the main coordinate system of the deformation tensor: :math:`{\varepsilon }_{\text{eq}}=\sqrt{{\langle {\varepsilon }_{1}\rangle }_{+}^{2}+{\langle {\varepsilon }_{2}\rangle }_{+}^{2}+{\langle {\varepsilon }_{3}\rangle }_{+}^{2}}` ", "**(eq** 2.1-2 **)**" knowing that the positive part :math:`{\mathrm{\langle }\mathrm{\rangle }}_{+}` is defined so that if :math:`{\varepsilon }_{i}` is the main deformation in the direction :math:`i`: .. math:: : label: EQ-None \ {\ begin {array} {c} {\ langle {\ epsilon} _ {i}\ epsilon} _ {+} = {\ epsilon} _ {i}\ text {\ epsilon} _ {\ epsilon} _ {i}\ epsilon} _ {i}\ epsilon} _ {\ epsilon} _ {+} =0\ text {\ epsilon} _ {\ epsilon} _ {+} =0\ text {\ epsilon} _ {\ epsilon} _ {+} =0\ text {\ epsilon} _ {\ epsilon} _ {+} =0\ text {\ epsilon} _ {\ epsilon} _ {+} =0\ text {\ epsilon} _ {\ epsilon} _ {silon} _ {i} <0\ end {array} **Note:** *In the case of thermo-mechanical loading, only elastic deformation* :math:`{\varepsilon }^{e}\mathrm{=}\varepsilon \mathrm{-}{\varepsilon }^{\text{th}}` *contributes to the evolution of the damage, whence:* :math:`{\varepsilon }_{\text{eq}}\mathrm{=}\sqrt{{\mathrm{\langle }{\varepsilon }^{e}\mathrm{\rangle }}_{+}\mathrm{:}{\mathrm{\langle }{\varepsilon }^{e}\mathrm{\rangle }}_{+}}` *.* :math:`{\epsilon }_{\text{eq}}` is an indicator of the state of tension in the material that is causing the damage. This quantity defines the load area :math:`f` as: .. math:: : label: EQ-None f\ mathrm {=} {\ varepsilon} _ {\ text {eq}}\ mathrm {-} K (D)\ mathrm {=} 0 *(4)* with :math:`K(D)={\epsilon }_{\text{d0}}` if :math:`D=0`. :math:`{\epsilon }_{\text{d0}}` the deformation damage threshold. When the equivalent deformation reaches this value, the damage is activated. :math:`D` is defined as a combination of two damage modes defined by :math:`{D}_{\text{t}}` and :math:`{D}_{\text{c}}`, varying between 0 and 1 depending on the associated damage state, and corresponding to tensile and compressive damage respectively. The relationship between these variables is as follows: .. math:: : label: EQ-None D\ mathrm {=} {\ alpha} _ {t} _ {t} ^ {\ beta} {D} _ {t} + {\ alpha} _ {c} ^ {\ beta} {\ beta} {D} {D} _ {c} *(5)* :math:`\beta` is a coefficient that was introduced later to improve shear behavior. Usually its value is fixed at 1.06. The coefficients :math:`{\alpha }_{\text{t}}` and :math:`{\alpha }_{\text{c}}` establish a link between the damage and the state of tension or compression. When traction is activated :math:`{\alpha }_{\text{t}}=1` while :math:`{\alpha }_{\text{t}}=0` and vice versa in compression. A particularity of this model is its explicit writing, which implies that all the quantities are calculated directly without using a linearization algorithm such as that of Newton-Raphson. Thus, the laws of evolution of damages :math:`{D}_{\text{t}}` and :math:`{D}_{\text{c}}` are expressed only on the basis of the equivalent deformation :math:`{\epsilon }_{\text{eq}}` .. math:: : label: EQ-None {D} _ {t}\ mathrm {=}} 1\ mathrm {-}\ frac {(1\ mathrm {-} {A} _ {t}) {\ varepsilon} _ {\ mathit {d0}}} {\ mathit {d0}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}\ text {exp} (\ mathrm {-} {B}} _ {B} _ {t} ({\ varepsilon} _ {\ text {eq}}\ mathrm {-} {\ varepsilon} {\ varepsilon}} _ {\ varepsilon} _ {\ mathit {d0}}}) .. math:: : label: EQ-None {D} _ {c}\ mathrm {=}} 1\ mathrm {-}\ frac {(1\ mathrm {-} {A} _ {c}) {\ varepsilon} _ {\ mathit {d0}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0}}}} {\ mathit {d0\ text {exp} (\ mathrm {-} {B}} _ {B} _ {c} ({\ varepsilon} _ {\ text {eq}}\ mathrm {-} {\ varepsilon} {\ varepsilon}} _ {\ varepsilon} _ {\ mathit {d0}}}) with :math:`{A}_{t}`, :math:`{A}_{c}`, :math:`{B}_{t}`, and :math:`{B}_{c}`, material parameters to be identified. These parameters make it possible to modulate the shape of the post-peak curve. They are obtained using tensile tests and a compression test. Mazars Revisited Model ------------------------- Although commonly used, the Mazars Origin model has shortcomings in modeling the behavior of concrete during shear and bi-compression loading. A comparison between the load surfaces of the two models is given in. .. image:: images/1000000000000395000001DC17F7AAC17446E432.jpg :width: 5.3126in :height: 2.7571in .. _RefImage_1000000000000395000001DC17F7AAC17446E432.jpg: **a) Original Mazars Model** **b) Revisited Mazars Model** Figure 2.2-1: Comparison of damage initiation and failure surfaces of Mazars models in plane :math:`{\sigma }_{3}=0` and C30 concrete Thus, a new formulation is proposed through 2 major modifications: 1. improvement of bi-compression behavior, 2. simplification and improvement of shear behavior. The original Mazars model [from the 1980s] greatly underestimates the strength of concrete under bi-compression. The first modification made by the Revisited model therefore improves the bi-compression behavior. This goal is achieved by correcting the equivalent deformation when at least one main stress is negative, using a variable :math:`\gamma`: .. math:: : label: EQ-None {\ epsilon} _ {\ mathit {eq}}} ^ {\ mathit {corrected}} =\ gamma {\ epsilon} _ {\ text {eq}} =\ gamma\ sqrt {{\ langle\ epsilon\ rangle}} ^ {\ langle\ epsilon\ rangle}} ^ {\ langle\ epsilon\ rangle}}} ^ {\ langle\ epsilon\ rangle}} ^ {\ langle\ epsilon\ rangle}} ^ {\ langle\ epsilon\ rangle}} ^ {\ langle\ epsilon\ rangle}} ^ {\ langle\ epsilon\ rangle}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} with: .. math:: : label: EQ-None \ {\ begin {array} {c}\ gamma =-\ frac {\ sqrt {\ sqrt {\ sqrt {\ sum _ {i} {\ sigma}}} _ {i}\ rangle} _ {-}\ rangle} _ {-}} _ {-} ^ {2}}} {-} ^ {2}}}}} {\ sum _ {2}}}} {\ sum _ {i} {2}}}} {\ sum _ {i} {2}}}} {\ sum _ {i} {2}}}} {\ sum _ {i} {i} {2}}}} {\ sum _ {i} {2}}}} {\ sum _ {i} {i} {2}}}} {\ sum _ {i} {2}}}} _ {-}}\ text {if at least one effective constraint is negative}\\\ gamma =1\ text {otherwise}\ end {array} The effective stress in the sense of the mechanics of damage is defined by: .. math:: : label: EQ-None \ underline {\ underline {\ tilde {\ sigma}}}}\ mathrm {=}\ frac {\ underline {\ underline {\ sigma}}} {1\ mathrm {\ sigma}}} {1\ mathrm {-} D} The definition of < > - is similar to: .. math:: : label: EQ-None \ {\ begin {array} {c} {\ langle {\ stackrel {\ stackrel {}} {\ sigma}} _ {i}\ rangle} _ {-} = {\ stackrel {} {\ sigma}}} _ {\ sigma}} _ {\ sigma}} = {i}\ le 0\\ {\ sigma}} _ {\ langle {\ stackrel}} _ {\ langle {\ stackrel {}}} {\ sigma}} _ {i}\ rangle}} _ {-} = 0\ text {\ sigma} {\ sigma}} _ {i} >0\ end {array} >0\ end {array} where :math:`{\tilde{\sigma }}_{i}` is a primary effective constraint. The improvement of shear behavior is achieved by the introduction of a new internal variable: :math:`Y`. It corresponds to the maximum reached during the loading of the equivalent deformation. Its initial value :math:`{Y}_{0}` is :math:`{\epsilon }_{\mathit{d0}}`. :math:`Y` is defined by the following equation: .. math:: : label: EQ-None Y=\ text {max}\ left ({\ epsilon}} _ {\ mathit {d0}}}\ text {, max}\ left ({\ epsilon} _ {\ mathit {eq}}}} ^ {\ mathit {eq}}} ^ {\ mathit {eq}}} ^ {\ mathit {eq}}} ^ {\ mathit {eq}}} ^ {\ mathit {corrected}}\ right)\ right) The charging function is: .. math:: : label: EQ-None f= {\ epsilon} _ {\ mathit {eq}}} ^ {\ mathit {corrected}} -Y The evolution of the damage is given by: .. math:: : label: EQ-None D\ mathrm {=} 1\ mathrm {-}\ frac {(1\ mathrm {-} A) {Y} _ {0}} {Y}\ mathrm {-} A\ text {exp} (\ text {exp}} (\ mathrm {-}}) In this expression, it is the variables :math:`A` and :math:`B` that make it possible to reproduce the almost fragile behavior of concrete under tension and the work-hardened behavior under compression. To best represent the experimental results, the following laws of evolution were chosen for :math:`A` and :math:`B`: .. math:: : label: EQ-None A\ mathrm {=} {A} _ {t} ({\ mathrm {2r}}} ^ {2} (1\ mathrm {-}\ mathrm {2k})\ mathrm {-} r (1\ mathrm {-} r (1\ mathrm {-}\ mathrm {-}}\ mathrm {2r}} r (1\ mathrm {-} r (1\ mathrm {-} r (1\ mathrm {-}} r (1\ mathrm {-} r (1\ mathrm {-}) r (1\ mathrm {-} r (1\ mathrm {-} r (1\ mathrm {-} r (1\ mathrm {-} r (1\ mathrm}\ mathrm {-}\ mathrm {3r} +1) and .. math:: : label: EQ-None B\ mathrm {=} {r} ^ {2} {2} {B} {B} _ {t} + (1\ mathrm {-} {r} ^ {2}) {B} _ {2}) {B} _ {c} where the expression for :math:`r` is: .. math:: : label: EQ-None r=\ frac {\ sum _ {i} {\ langle {\ stackrel {\ stackrel {}} {\ sigma}} _ {i}\ rangle} _ {+}} {\ sum _ {i}\ mid}\ mid}\ mid}\ mid}\ mid}\ mid}\ mid} In these equations, a new variable :math:`r` appears, which tells us about the stress state. When :math:`r` is equal to 1 (corresponding to the traction sector), the variables :math:`A` and :math:`B` are equivalent to the parameters :math:`{A}_{t}` and :math:`{B}_{t}`. So is the same as. Conversely, if :math:`r` is zero (corresponding to the compression sector), then :math:`A={A}_{c}`, :math:`B\mathrm{=}{B}_{c}` and is the same as. In plan :math:`{\sigma }_{3}=0`, the evolution according to the sign of the main constraints of the variables :math:`A`, :math:`B`, :math:`r` and :math:`\gamma` depends on the sign of the main constraints. .. image:: images/100000000000024E000001AD016D3A52FE7FC172.jpg :width: 3.3571in :height: 2.0346in .. _RefImage_100000000000024E000001AD016D3A52FE7FC172.jpg: Figure 2.2-2: Evolution of the variables :math:`A`, :math:`B`,, :math:`r` and :math:`\gamma` in the :math:`{\sigma }_{3}\mathrm{=}0` plane In the equation a new parameter appears: :math:`k`. It introduces an asymptote to curve :math:`\sigma -\epsilon` in shear and is defined by: .. math:: : label: EQ-None k\ mathrm {=}\ frac {{A} _ {\ text {shear}}} {{A} _ {t}} where :math:`{A}_{\text{cisaillement}}` defines residual stress in pure shear. It is similar to :math:`{A}_{t}` for this load case. The recommended value for :math:`k` is :math:`0.7`. The value of :math:`k` less than 1 is very useful in modeling the effects of friction between concrete and reinforcements in reinforced concrete structures because it induces residual shear stress. For the value :math:`k=1` we find the behavior of the Original model (). .. image:: images/10000000000000A00000004109FC1D76969C8820.png :width: 1.25in :height: 0.5075in .. _RefImage_10000000000000A00000004109FC1D76969C8820.png: .. image:: images/10000000000002C00000023D79F55CFEF736E280.png :width: 3.2425in :height: 2.4953in .. _RefImage_10000000000002C00000023D79F55CFEF736E280.png: Asymptote induced by k 1 Figure 2.2-3: Stress-strain curve during a pure shear test on a Gauss point The Origin model underestimates the strength of concrete under pure shear. This new formulation makes it possible to increase this pure shear strength from :math:`2.5\mathit{MPa}` to :math:`3.5\mathit{Mpa}` for C30 concrete. This value depends on those of the material parameters entered (:math:`{A}_{t}`, :math:`{A}_{c}`, :math:`{B}_{t}`, and :math:`{B}_{c}`). The local response of the Mazars Revisited model under successive tensile and compression loading is given by the. .. image:: images/10000000000003C800000271D7263F38B3A7117B.png :width: 5.0354in :height: 3.5102in .. _RefImage_10000000000003C800000271D7263F38B3A7117B.png: Figure 2.2-4: Stress-strain response of the Mazars model for **1D stress.** It allows you to visualize a certain number of characteristics of the Mazars model, namely: • the damage affects the stiffness of the concrete, • there are no irreversible deformations, • the tensile and compression responses are well asymmetric, *Note*: The Mazars Original and Revisited models do not take into account the unilateral nature of concrete, namely the closure of cracks during the transition from a state of tension to a state of compression.