Study on the Macro-Meso Constitutive Law of Concrete Material

In mesoscopic, concrete can be taken as three-phase composites consisting of mortar matrix, aggregate and the bond between them. So, the overall behavior of concrete depends on its meso structures such as aggregate shape, interface status, mortar matrix property and so on. In this paper, to investigate the intrinsic mechanism of the macro mechanical characteristic, the nonlinear property of matrix and the bond status of interface are considered in focus. Firstly, a continuous slippery power- exponent function that can reflect soft sect is constructed to describe the nonlinear constitutive law of mortar matrix. Secondly, the variational principle proposed by P. Ponte Castaneda is adopted to study the nonlinear behavior of concrete by selecting linear reference composite material where the distribution and the shape of aggregate are same as the concrete. And the overall property of linear reference composite can be estimated by classical estimation method such as Mori-Tanaka method. This variational structure method involves an optimum problem ultimately. In order to make the optimization successfully, the compressible property of material is assumed to only stem from the micro holes contained in it, and the single variable optimization scheme is adopted by three stages. Finally, by optimizing the shear modulus of matrix of the linear reference composite, the macro-meso constitutive law of concrete is established as follows. 1 $$ \bar \sigma _e = 2\tilde G_m \bar \varepsilon _e \exp \left( {a - \frac{a} {{\varepsilon _s }}\sqrt {\frac{{3(1 + \sum\limits_i {c_i ^\prime (1 - s_2^i )/s_2^i } )}} {{(3 + 2c)(1 - \sum\limits_i {c_i ^\prime } )^2 }}\bar \varepsilon _e^2 + \frac{4} {{3c}}\bar \varepsilon _m^2 } } \right) $$ Where $$ \bar \sigma _e $$ and $$ \bar \varepsilon _e $$ are the macro equivalent stress and strain, respectively, and $$ \bar \varepsilon _m $$ the macro average strain, $$ \tilde G_m $$ the effective shear modulus of concrete, c the cavity ratio, c i ′ the volume occupancy of the ith phase inclusion, s 2 i the shear component of the modified Eshelby tensor of the ith phase inclusion, a and εs the experiment parameters. By analyzing the constitutive relation above, we find that the micro holes contained in concrete are a deep reason why concrete material is of shear dilation property.