Issue 29

M.L. De Bellis et alii, Frattura ed Integrità Strutturale, 29 (2014) 37-48; DOI: 10.3221/IGF-ESIS.29.05 46 before), as the aspect ratio changes, are reported for UC1 and UC2, respectively. It can be remarked that Procedure B confirms to be the most suitable to reproduce the actual behavior of the heterogeneous medium. Moreover, the slenderer the wall ( B/H 1.3  ) the closer the values obtained with the first order technique are to the response of the micromechanichal models. The micropolar effects are, thus, more pronounced for the squat wall ( B/H 2  ). Also in this case, Procedure B gives results which best match those obtained with the micromechanical model. Simple shear test of a composite strip As a second example, a displacement driven shear test on a 2D structure, made from the same composite medium considered in the previous example, is presented. A strip with B/H 0.1  is considered assuming plane strain condition. In Fig. 4 the schematic of the geometry and boundary conditions is shown for two possible strips, corresponding to assemblages of UC1 and UC2 in Fig. 2-a and e 2-b , respectively. The strips are fixed at both bottom and top edges and a horizontal displacement d=H/100 is prescribed at the top. Heter Cauchy Cos A Cos B Cos C B/H 1.3  147.82 0.990 1.228 1.007 1.016 B/H 1.6  153.81 0.986 1.349 1.013 1.036 B/H 2  162.43 0.931 1.560 1.012 1.024 Table 3 : Structural stiffness for different ratios of B/H in the case of UC1: Heter = micromechanical model; Cauchy= homogenized first order model; Cos A = homogenized Cosserat with Procedure A ; Cos B = homogenized Cosserat with Procedure B ; Cos C = homogenized Cosserat with Procedure C . Heter Cauchy Cos A Cos B Cos C B/H 1.3  142.22 1.029 0.876 0.992 0.987 B/H 1.6  143.74 1.033 0.808 0.993 0.982 B/H 2  143.85 1.051 0.708 0.995 0.982 Table 4 Structural stiffness for different ratios of B/H in the case of UC2: Heter = micromechanical model; Cauchy= homogenized first order model; Cos A = homogenized Cosserat with Procedure A ; Cos B = homogenized Cosserat with Procedure B ; Cos C = homogenized Cosserat with Procedure C . In Fig. 5-a the displacement horizontal component, evaluated along the vertical symmetry axis of each strip along the vertical axis of the strip is shown. The responses of the micromechanical models are represented in solid line (heter1) and in dash-dot line (heter2) for the strips reported in Fig. 4-a and 4-b, respectively. The homogenized constitutive coefficients adopted in the micropolar models are evaluated using Procedure B . The squares represent the response of the homogenized micropolar model adopting the UC1 in Fig. 2-a (CosB_UC1) and the circles represent the response with UC2 in Fig. 2-b (CosB_UC2). Finally, the dotted line refers to the response obtained by the standard first order computational homogenization (Cauchy), able to reproduce a linear variation of the horizontal displacement component along the height of the strip. No relevant differences between the results obtained with the two micromechanical models arise and both the homogenized micropolar models can satisfactorily follow the expected displacement distribution. In Fig. 5-b the rotation evaluated along the vertical symmetry axis of each strip versus the vertical axis is reported. Owing to the symmetry of this displacement component with respect to a horizontal axis located at the mid height, only one half of the strip height is depicted. Due to the different natures of the compared models, the rigid rotation  , i.e. the skew- symmetric part of the displacement gradient, is reported for micromechanical and Cauchy models, while the Cosserat rotation  is shown for the micropolar homogenized model. Line styles and symbols have the same meaning as in Fig. 5-