Issue 58

K. Benyahi et alii, Frattura ed Integrità Strutturale, 58 (2021) 319-343; DOI: 10.3221/IGF-ESIS.58.24

n j : The unit normal, u i : The i th displacement, S j : The outer limit of the RVE, E 11 , E 22 and E 33 : The moduli of elasticity of extensions along directions 1, 2 and 3, respectively,

ν ij (i, j = 1, 2, 3) : Poisson's ratios, G 12 , G 13 , and G 23 : Shear moduli, U i P : The nodal variable at the node P, of degree of freedom i, E : Hooke's matrix, D : The damage variable, ε e : Elastic strain, E b0 : Modulus of longitudinal elasticity of concrete,

ϖ : Percentage by volume of fibers, θ 0 : Fiber orientation coefficient, l r : Reference length, β : Model constant, h : Composite cross section height, l f : Fiber length, E f : Elastic modulus of the fiber, n : Fiber-concrete equivalence coefficient, τ u : Ultimate fiber-concrete bond stress, ϕ : Diameter of a fiber, f ft : Composite tensile strength, ε ft : Composite cracking strain, ε rf : Fiber breaking strain, ε rt : Breaking strain of the composite in tension.

R EFERENCES

[1] Suquet, P. (1987). Elements of homogenization theory for inelastic solid mechanics. Sanchez-Palencia, E., Zaoui, A. (Eds.), Homogenization Techniques for Composite Media. Springer-Verlag, Berlin, pp. 194 – 275. DOI: 10.1007/3-540-17616-0 [2] Mori, T., Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica, 21(5), pp. 571-574. DOI: 10.1016/0001-6160(73)90064-3. [3] Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13, pp. 213-222. DOI: 10.1016/0022-5096(65)90010-4. [4] Eshelby, J.D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 241(1226), pp. 376-396. DOI: 10.1098/rspa.1957.0133. [5] Michel, JC., Moulinec, H., Suquet, P. Effective properties of composite materials with periodic microstructure: a computational approach. Computer methods in applied mechanics and engineering 172 (1-4), pp. 109-143. DOI: 10.1016/s0045-7825(98)00227-8. [6] Sun, C T., Vaidya, R S. (1996). Prediction of composite properties from a representative volume element. Composites Science and Technology, 56(2), pp. 171-179. DOI: 10.1016/0266-3538(95)00141-7. [7] Xia, Z., Zhang, Y., Ellyin F. (2003). A unified periodical boundary conditions for representative volume elements of composites and applications. International Journal of Solids and Structures 40, pp. 1907 – 1921. DOI: 10.1016/s0020-7683(03)00024-6. [8] Yvonnet, J., Auffray, N., and Monchiet, V. (2020). Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior. International Journal of Solids and Structures, 191, pp. 434-448. DOI: 10.1016/j.ijsolstr.2020.01.006.

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