22.1 Elasto-Plastic Damage Model

The Maekawa Concrete model is derived from two-dimensional and three-dimensional cyclic loading data. The derivation uses four material parameters (K , F , H , and D ) for concrete with normal aggregate and strength ranging from 15 MPa to 50 MPa.

Once the elastic strain vector has been determined, then the so-called fracture parameter K is calculated as a function of the invariants of the elastic strain tensor and a number of elastic parameters. Due to damage it is assumed that the shear modulus will be reduced by a factor K , i.e., the initial shear modulus G is multiplied with K . Factor K runs from 1 to 0, where 0 stands for complete deterioration and 1 for no damage. To determine the actual damage shear modulus the actual stresses and the elastic-moduli matrix of damaged concrete are formulated as a function of the damage parameter K , the invariants of the elastic strain tensor, and a number of material parameters. These relations are defined by Meakawa et. al [66,67,68] as:

$$\left(\vphantom{ - \frac{ F }{ 3.25 } \left( 1 - \exp \left( - \frac{ F }{ 0.8 } \right) \right) }\right. {\frac{{ F }}{{ 3.25 }}} \left(\vphantom{ 1 - \exp \left( - \frac{ F }{ 0.8 } \right) }\right. \left(\vphantom{ - \frac{ F }{ 0.8 } }\right. {\frac{{ F }}{{ 0.8 }}} \left.\vphantom{ - \frac{ F }{ 0.8 } }\right) \left.\vphantom{ 1 - \exp \left( - \frac{ F }{ 0.8 } \right) }\right) \left.\vphantom{ - \frac{ F }{ 3.25 } \left( 1 - \exp \left( - \frac{ F }{ 0.8 } \right) \right) }\right)$$ K (22.1)

$$\left(\vphantom{ I_{\mathrm{1e}}, J_{\mathrm{2e}}, J_{\mathrm{3e}} }\right. \left.\vphantom{ I_{\mathrm{1e}}, J_{\mathrm{2e}}, J_{\mathrm{3e}} }\right)$$ F

$${\frac{{ \sqrt{2} \, J_{\mathrm{2e}} }}{{ 0.23 \, \varepsilon _{0} - \sqrt{3} \, I_{\mathrm{1e}} }}} {\frac{{ 1 }}{{ 5 }}} \left(\vphantom{ \frac{ 3 \sqrt{3} }{ 2 } \left( \frac{ J_{\mathrm{3e}} }{ J_{\mathrm{2e}} } \right)^{\!3} + 6 }\right. {\frac{{ 3 \sqrt{3} }}{{ 2 }}} \left(\vphantom{ \frac{ J_{\mathrm{3e}} }{ J_{\mathrm{2e}} } }\right. {\frac{{ J_{\mathrm{3e}} }}{{ J_{\mathrm{2e}} }}} \left.\vphantom{ \frac{ J_{\mathrm{3e}} }{ J_{\mathrm{2e}} } }\right)^{{\!3}}_{} \left.\vphantom{ \frac{ 3 \sqrt{3} }{ 2 } \left( \frac{ J_{\mathrm{3e}} }{ J_{\mathrm{2e}} } \right)^{\!3} + 6 }\right)$$ (22.2)

$${\frac{{ 9 }}{{ 10 }}} \varepsilon_{{0}}^{} \left(\vphantom{ \frac{ J_{\mathrm{2e}} }{ \varepsilon _{0} } }\right. {\frac{{ J_{\mathrm{2e}} }}{{ \varepsilon _{0} }}} \left.\vphantom{ \frac{ J_{\mathrm{2e}} }{ \varepsilon _{0} } }\right)^{{\!3}}_{}$$ H (22.3)

D = D(I_1e, K)

$$\left(\vphantom{ \frac{ -1 + 2 \nu }{ \sqrt{3} \, ( 1 + \nu ) } \... ...repsilon _{0} }{ 0.28 \, \varepsilon _{0} } \left( 1 - 4 K^{2} \right) }\right. {\frac{{ -1 + 2 \nu }}{{ \sqrt{3} \, ( 1 + \nu ) }}} {\frac{{ \sqrt{2} \, I_{\mathrm{1e}} + 0.38 \, \varepsilon _{0} }}{{ 0.28 \, \varepsilon _{0} }}} \left(\vphantom{ 1 - 4 K^{2} }\right. \left.\vphantom{ 1 - 4 K^{2} }\right) \left.\vphantom{ \frac{ -1 + 2 \nu }{ \sqrt{3} \, ( 1 + \nu ) } \... ...repsilon _{0} }{ 0.28 \, \varepsilon _{0} } \left( 1 - 4 K^{2} \right) }\right)$$ (22.4)

Scalars I_1e , J_2e , and J_3e respectively are the first, second, and third elastic strain invariants:

$${\frac{{1}}{{3}}} \varepsilon_{{\mathrm{e}ii}}^{}$$ (22.5)

$$\sqrt{{\frac{1}{2} \, e_{\mathrm{e}ij} \, e_{\mathrm{e}ij}}}$$ (22.6)

$$\sqrt[3]{{\frac{1}{3} \, e_{\mathrm{e}ij} \, e_{\mathrm{e}jk} \, e_{\mathrm{e}ki}}}$$ (22.7)

with

$$\varepsilon_{{\mathrm{e}ij}}^{} \delta_{{ij}}^{}$$ (22.8)

being the elastic deviatoric tensor and $\varepsilon_{{\mathrm{e}ij}}^{}$ the elastic strain tensor. Equations (22.1) to (22.4) include the material constant $\varepsilon_{{0}}^{}$ which was adopted as a function of the compressive strength f_c , Young's modulus E , and Poisson's ratio $\nu$ :

$$\varepsilon_{{0}}^{} \nu {\frac{{ f_{\mathrm{c}} }}{{ E }}}$$ (22.9)

so that these material functions would be applicable to concrete of normal aggregate a1Gnd strength.

The fracture function K [Eq.(22.1)] represents the degradation of the shear elastic strain energy of concrete including defects. The parameter F [Eq.(22.2)] is the indicator (equivalent elastic strain) to express the macroscopic intensity of internal stress which advances the damage under an arbitrary level of confinement (F = 0 and $\dot{{F}}$ > 0 ). The function H [Eq.(22.3)] indicates the plastic hardening of the internal plastic element in the damaged concrete, with b being the user-defined correction factor for plastic evolution, which has a default value of 1.0. The derivative D [Eq.(22.4)] indicates the plastic dilatancy induced by the shear plastic dislocation along the internal defects.