Cementitious materials such as mortar or concrete are brittle and have an inherent weakness in resisting tensile stresses. The addition of discontinuous fibers to such matrices leads to a dramatic improvement in their toughness and remedies their deficiencies. It is generally agreed that the fibers contribute primarily to the post-cracking response of the composite by bridging the cracks and providing resistance to crack opening (Suwaka & Fukuyama 2006).
On the other hand, the multifield theory is a mathematical tool able to describe materials which contain a complex substructure (Mariano & Stazi 2005). This substructure is endowed with its own properties and it interacts with the macrostructure and influences drastically its behavior. Under this mathematical framework, materials such as cement composites can be seen as a continuum with a microstructure. Therefore, the whole continuum damage mechanics theory, incorporating a new microstructure, is still applicable.
A formulation, initially based on the theory of continua with microstructure Capriz (Capriz 1989), has been developed to model the mechanical behavior of the high performance fiber cement composites with arbitrarily oriented fibers. This formulation approaches a continuum with microstructure, in which the microstructure takes into account the fibermatrix interface bond/slip processes, which have been recognized for several authors (Li 2003, Naaman 2007b) as the principal mechanism increasing the ductility of the quasi-brittle cement response. In fact, the interfaces between the fiber and the matrix become a limiting factor in improving mechanical properties such as the tensile strength. Particularly, in short fiber composites is desired to have a strong interface to transfer effectively load from the matrix to the fiber. However, a strong interface will make difficult to relieve fiber stress concentration in front of the approaching crack. According to Naaman (Naaman 2003), in order to develop a better mechanical bond between the fiber and the matrix, the fiber should be modified along its length by roughening its surface or by inducing mechanical deformations. Thus, the premise of the model is to take into account this process considering a micro field that represents the slipping fiber-cement displacement. The conjugate generalized stress to the gradient of this micro-field verifies a balance equation and has a physical meaning.
This contribution includes the computational modeling aspects of the high fiber reinforced cement composites (HFRCC) model. To simulate the composite material, a finite element discretization is used to solve the set of equations given by the multifield approach for this particular case. A two field discretization: the standard macroscopic and the microscopic displacements, is proposed through a mixed finite element methodology. Furthermore, a splitting procedure for uncoupling both fields is proposed, which provides a more convenient numerical treatment of the discrete equation system.
The initiation of failure in HPFRCC at the constitutive level identified as the onset of strain localization depends on the mechanical properties of the all compounds and not only on the matrix ones. As localization criteria is considered the bifurcation analysis in combination with the localized strain injection technique presented by Oliver et al. (Oliver et al. 2010a). It consists of injecting a specific localization mode during the localization stage, via mixed finite element formulations, to the path of elements that are going to capture the cracks, and, in this way, the spurious mesh orientation dependence is removed.
Model validation was performed using a selected set of experiments that proves the viability of this approach. The numerical examples of the proposed formulation illustrated two relevant aspects, namely: 1) the role of the bonding mechanism in the strain hardening behavior after cracking in the HPFRCC and 2) the role that plays the finite element formulation in capturing the displacement localization in the localization stage.
162-TDF, R. T. (2002) Recommendations of RILEM TC 162-TDF: Test and design methods for steel fibre reinforced concrete: bending test. Materials and Structures, 35, 579-582.
Aboudi, J. (1989) Micromechanical Analysis of Composites by the Method of Cells. Applied Mechanics Reviews, 42, 193-221.
Armero, F. & Garikipati, K. (1996) An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. International Journal of Solids and Structures, 33, 2863-2885.
Aveston, J. & Kelly, A. (1973) Theory of Multiple Fracture of Fibrous Composites. Journal of Materials Science, 8, 352-362.
Banthia, N., Bentur, A. & Mufti, A. A. (1998) Fiber reinforced concrete: present and future, Canadian Society for Civil Engineering.
Banthia, N. & Trottier, J.-F. (1994) Concrete Reinforced with Deformed Steel Fibers, Part I: Bond-Slip Mechanisms. ACI Materials Journal, 91, 435-446.
Barros, J., Cunha, V., Ribeiro, A. & Antunes, J. (2005) Post-cracking behaviour of steel fibre reinforced concrete. Materials and Structures, 38, 47-56.
Bažant, Z. & Oh, B. (1983) Crack band theory for fracture of concrete. Materials and Structures, 16, 155-177.
Bažant, Z. & Planas, J. (1998) Fracture and size effects in concrete and other quasibrittle materials Boca Raton, CRC Press.
Beghini, A., Bazant, Z., Zhou, Y., Gouirand, O. & Caner, F. (2007) Microplane Model M5f for Multiaxial Behavior and Fracture of Fiber-Reinforced Concrete. Journal of Engineering Mechanics, 133, 66-75.
Bencardino, F., Rizzuti, L., Spadea, G. & Swamy, R. N. (2010) Experimental evaluation of fiber reinforced concrete fracture properties. Composites Part B: Engineering, 41, 17-24.
Bensoussan, A., Lions, J.-L. & Papanicolaou, G. (1978) Asymptotic analysis for periodic structures, Elsevier.
Benssousan, A., Lions. J.L & G., P. (1978) Asymptotic Analysis for Periodic Structure, Amsterdam North-Holland.
Benvenuti, E. (2008) A regularized XFEM framework for embedded cohesive interfaces. Computer Methods in Applied Mechanics and Engineering, 197, 4367-4378.
Beyerlein, I. J. & Phoenix, S. L. (1996) Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition. Journal of the Mechanics and Physics of Solids, 44, 1997 - 2039.
Bilby, B. A., Cottrell, A. H. & Swinden, K. H. (1963) The Spread of Plastic Yield from a Notch. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272, 304-314.
Bolander, J., Choi, S. & Duddukuri, S. (2008) Fracture of fiber-reinforced cement composites: effects of fiber dispersion. International Journal of Fracture, 154, 73-86.
Bolander, J. E. & Sukumar, N. (2005) Irregular lattice model for quasistatic crack propagation. Physical Review B, 71, 094106.
Bongué Boma, M. & Brocato, M. (2010) A continuum model of micro-cracks in concrete. Continuum Mechanics and Thermodynamics, 22, 137-161.
Borst, R. d., Remmers, J. J. C., Needleman, A. & Abellan, M.-A. (2004) Discrete vs smeared crack models for concrete fracture: bridging the gap. International Journal for Numerical and Analytical Methods in Geomechanics, 28, 583-607.
Boulfiza, M. (1998) Constitutive Modeling of Fiber Reinforced Cement Composites. Departament of Civil Engineering. Vancouver, The University of British Columbie.
Broch, E. & Franklin, J. A. (1972) The point-load strength test. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 9, 669-676.
Capriz, G. (1989) Continua with microstructure, Berlin, Springer Verlag.
Capriz, G. & Mariano, P. M. (2001) Multifield theories: an introduction. International Journal of Solids and Structures, 38, 939-941.
Car, E., Zalamea, F., Oller, S., Miquel, J. & Oñate, E. (2002) Numerical simulation of fiber reinforced composite materials--two procedures. International Journal of Solids and Structures, 39, 1967-1986.
Carpinteri, A., Cornetti, P., Barpi, F. & Valente, S. (2003) Cohesive crack model description of ductile to brittle size-scale transition: dimensional analysis vs. renormalization group theory. Engineering Fracture Mechanics, 70, 1809-1839.
Colin, J. (2000) Fiber-reinforced cements and concrete, Gorgon and Breach Science publishers.
Coto Roquet, L. A. (2007) Ecuaciones constitutivas para el análisis de secciones de HRFA. Ingeniería de la Construcción, Universitat Politècnica de Catalunya.
Cox, H. L. (1952) The Elasticity and Strength of Paper and Other Fibrous Materials. British Journal of Applied Physics 3, 72-79.
Cunha, V. M. C. F., Barros, J. A. O. & Sena-Cruz, J. M. (2007) Pullout behaviour of hookedend steel fibres in self-compacting concrete. Guimarães.
De Souza Neto, E., Perić, D. & Owen, D. (2008) Computational methods for plasticity. Theory and applications, Chichester, John Willey & Sons Ltd.
De Souza Neto, E. A. & Feijóo, R. A. (2010) Variational Foundations of Large Strain Multiscale Solid Constitutive Models: Kinematical Formulation. Advanced Computational Materials Modeling. Wiley-VCH Verlag GmbH & Co. KGaA.
Dias, I. (2012) Strain injection techniques in numerical modeling of propagating material failure. Departamento de resistencia de materiales y estructuras en la ingeniería. Barcelona, Universidad Politécnica de Cataluña.
Dias, I. F., Oliver, J. & Huespe, A. E. (2011a) Strain injection, mixed formulation and strong discontinuities in fracture modeling of quasi-brittle materials. Congress in numerical methods in engineering. Coimbra.
Dias, I. F., Oliver, J. & Huespe, A. E. (2011b) Strain injection, mixed formulations and strong discontinuities in fracture modeling of quasi-brittle materiales. IN ANTÓNIO., T., ISABEL, N. F., FELIPE, M. L., ANTONIO, R.-F., IRENE, A. & JESUS, B. (Eds.) Congress on numerical methods in engineering. Coimbra
Dupont, D. & Vandewalle, L. (2005) Distribution of steel fibres in rectangular sections. Cement and Concrete Composites, 27, 391-398.
Ericksen, J. L. (1974) Liquid crystals and cosserat surfaces. Mechanics and applied mathematics, 27, 213-219.
Fantilli, A., Mihashi, H. & Vallini, P. (2007) Crack profile in RC, R/FRCC and R/HPFRCC members in tension.
Fantilli, A. P., Mihashi, H. & Vallini, P. (2009) Multiple cracking and strain hardening in fiber-reinforced concrete under uniaxial tension. Cement and Concrete Research, 39, 1217-1229.
Fantilli, A. P. & Vallini, P. (2007) A Cohesive Interface Model for the Pullout of Inclined Steel Fibers in Cementitious Matrixes. Journal of Advanced Concrete Technology, 5, 247-258.
Felippa, C., Park, K. C. & Farhat, C. (2001) Partitioned analysis of coupled mechanical systems. Computer Methods in Applied Mechanics and Engineering, 190, 3247-3270.
Ferrara Liberato, R. G. (2000) Non-local damage analysis of three-point on SFRC notched beams. Fiber Reinforced concrete (FRC) BEFIB' 2000.
Ferreira, L. (2007) Fracture analysis of a high-strength concrete and a high-strength steelfiber-reinforced concrete. Mechanics of Composite Materials, 43, 479-486.
Fischer, J. Y. a. G. (2006) Investigation of the Fiber Bridging Stress-Crack Opening Relationship of Fiber Reinforced Cementitious Composites. IN LI, G. F. A. V. C. (Ed.) International RILEM Workshop on High Performance Fiber Reinforced Cementitious Composites in Structural Applications.
Fish, J., Shek, K., Pandheeradi, M. & Shephard, M. S. (1997) Computational plasticity for composite structures based on mathematical homogenization: Theory and practice. Computer Methods in Applied Mechanics and Engineering, 148, 53-73.
Ghosh, S., Lee, K. & Moorthy, S. (1995) Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method. International Journal of Solids and Structures, 32, 27-62.
Ghosh, S., Lee, K. & Raghavan, P. (2001) A multi-level computational model for multi-scale damage analysis in composite and porous materials. International Journal of Solids and Structures, 38, 2335-2385.
Grammenoudis, P. & Tsakmakis, C. (2009) Micromorphic continuum Part I: Strain and stress tensors and their associated rates. International Journal of Non-Linear Mechanics, 44, 943-956.
Guerrero, P. & Naaman, A. E. (2000) Effect of mortar fineness and adhesive agents on the pull-out response of steel fibers. ACI Materials Journal, 97, 12-20.
Guerrero Z, A. P. (1999) Bond stress-slip mechanisms in high performance fiber reinforced cement composites.
Hariri, K. (2001) Fracture Mechanics Behaviour of Concrete at Early Age. IN ELFGREN, L. (Ed.) IMPROVED PRODUCTION OF ADVANCED CONCRETE STRUCTURES. Technical University of Braunschweig.
Herakovich, C. (1998) Mechanics of Fibrous Composites, New York John Wiley and Sons Inc
Hernández, J. (2008) Numerical modeling of crack formation in powder compaction based manufacturing processes. Universitat Politècnica de Catalunya Escola Tècnica Superior D’enginyers de Camins, Canals I Ports.
Hill, R. (1964) Theory of Mechanical Properties of Fibre-strengthened Materials: I. Elastic Behaviour. Journal of the Mechanics and Physics of Solids, 12, 199-212.
Hillerborg, A., Modéer, M. & Petersson, P. E. (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 6, 773-781.
Hofstetter, G. & Meschke, G. (2011) Numerical Modeling of Concrete Cracking, Springer.
Holzapfel, G. (2000) Nonlinear solids mechanics, A continuum approach for engineering, Chichester, Wiley.
Hu, X., Day, R. & Dux, P. (2003) Biaxial failure model for fiber reinforced concrete. Journal of Materials in Civil Engineering, 15, 609-615.
Jiang, H., Valdez, J. A., Zhu, Y. T., Beyerlein, I. J. & Lowe, T. C. (2000) The strength and toughness of cement reinforced with bone-shaped steel wires. Composites Science and Technology, 60, 1753-1761.
Kabele, P. (2007a) Multiscale framework for modeling of fracture in high performance fiber reinforced cementitious composites. Engineering Fracture Mechanics, 74, 194-209.
Kabele, P. (2007b) Multiscale framework for modeling of fracture in high performance fiber reinforced cementitious composites. Engineering Fracture Mechanics, 74, 194-209.
Kim, D., Naaman, A. E. & El-Tawil, S. (2008) Comparative flexural behavior of four fiber reinforced cementitious composites. Cement and Concrete Composites, 30, 917-928.
Kim, D. J., Naaman, A. E. & S, E.-T. (2009) High Performance Fiber Reinforced Cement Composites with Innovative Slip Hardending Twisted Steel Fibers. International Journal of Concrete Structures and Materials, 3, 119-126.
Kim, P., Li, V. & Kamada, T. (2002) Fracture Toughness of Microfiber Reinforced Cement Composites. Journal of Materials in Civil Engineering, 14, 384-391.
Kulla, J. (1998) Constitutive modeling of fiber reinforced brittle materials. Helsinki university of technology.
Lange-Kornbak, D. & Karihaloo, B. L. (1997) Tension softening of fibre-reinforced cementitious composites. Cement and Concrete Composites, 19, 315-328.
Laranjeira, F., Aguado, A. & Molins, C. (2010) Predicting the pullout response of inclined straight steel fibers. Materials and Structures, 43, 875-895.
Lee, Y., Kang, S.-T. & Kim, J.-K. (2010) Pullout behavior of inclined steel fiber in an ultrahigh strength cementitious matrix. Construction and Building Materials, 24, 2030-2041.
Leung, C. K. Y. & Geng, Y. P. (1998) Micromechanical modeling of softening behavior in steel fiber reinforced cementitious composites. International Journal of Solids and Structures, 35, 4205-4222.
Leung, C. K. Y. & Li, V. C. (1990) Applications of a two-way debonding theory to short fiber composites. Composites 21, 305-317.
Li, F. & Li, Z. (2000) Continuum damage mechanics based modeling of fiber reinforced concrete in tension. International Journal of Solids and Structures, 38, 777-793.
Li, F. & Li, Z. (2001) Continuum damage mechanics based modeling of fiber reinforced concrete in tension. International Journal of Solids and Structures, 38, 777-793.
Li, Q. M. (2001) Strain energy density failure criterion. International Journal of Solids and Structures, 38, 6997-7013.
Li, V. C. (1992) A simplified micromechanical model of compressive strength of fiberreinforced cementitious composites. Cement and Concrete Composites, 14, 131 -141.
Li, V. C. (2003) On High Performance Fiber Reinforced Cementitious Composite. JCI Symposium on DFRC, 13-23.
Li, V. C. & Stang, H. (1997) Interface property characterization and strengthening mechanisms in fiber reinforced cement based composites. Advanced Cement Based Materials, 6, 1-20.
Li, V. C. & Wang, S. (2006) Microstructure variability and macroscopic composite properties of high performance fiber reinforced cementitious composites. Probabilistic Engineering Mechanics, 21, 201-206.
Lin, Z. & Li, V. C. (1997) Crack bridging in fiber reinforced cementitious composites with slip-hardening interfaces. Journal of the Mechanics and Physics of Solids, 45, 763-787.
Linero, S. D. L. (2006) Un modelo del fallo material en el hormigón armado, mediante la metodología de discontinuidades fuertes de continuo y la teoría de mezclas. Universitat Politècnica De Catalunya Escola Tècnica Superior D’enginyers De Camins, Canals I Ports.
Mariano, P. (2007) Roubícek, T., Nonlinear partial differential equations with applications. Meccanica, 42, 615-616.
Mariano, P. (2008a) Cracks in Complex Bodies: Covariance of Tip Balances. Journal of Nonlinear Science, 18, 99-141.
Mariano, P. & Stazi, F. (2005) Computational aspects of the mechanics of complex mmaterials. Archives of Computational Methods in Engineering, 12, 391-478.
Mariano, P. M. (2000) Configurational forces in continua with microstructure. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 51, 752-791.
Mariano, P. M. (2008b) Mechanics of complex bodies: Commentary on the unified modeling of material substructures. Theoretical and Applied Mechanics, 35, 235-254.
Martínez, X. (2008) Micromechanical simulation of composite materials using the serial/parallel mixing theory. Departament de resistència de materials i estructures a l'enginyeria. Barcelona, Universistat Politècnica de Catalunya.
Maugin, G. (1992) The thermomechanics of plasticity and fracture, Cambridge University Press.
Mindlin, R. D. (1964) Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51-78.
Möes, N., Dolbow, J. & Belytschko, T. (1999) A Finite Element Method for Crack Growth Remesching. International Journal for Numerical Methods in Engineering, 46, 131-150.
Mori, T. & Tanaka, K. (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21, 571-574.
Mosconi, M. (2005) Multifield hyperelasticity: variational theorems for complex bodies. Mechanics Research Communications, 32, 525 - 535.
Muliana, A. H. (2008) Multi-scale framework for the thermo-viscoelastic analyses of polymer composites. Mechanics Research Communications, 35, 89-95.
Muliana, A. H. & Haj-Ali, R. (2008) A multi-scale framework for layered composites with thermo-rheologically complex behaviors. International Journal of Solids and Structures, 45, 2937 - 2963.
Naaman, A. & Reinhardt, H. (2006) Proposed classification of HPFRC composites based on their tensile response. Materials and Structures, 39, 547-555.
Naaman A. E & Alwan J.A (1993) Comment on "Characterization of Interfacial Properties in Fiber-Reinforced Cementitious Composites". Journal of american ceramic society, 76, 1645-1646.
Naaman A. E, R. H. W. (1995) High performance fiber reinforced cement composites 2 (HPFRCC 2): proceedings of the Second International RILEM Workshop 'High Performance Fiber Reinforced Cement Composites'. IN A E NAAMAN, R. H. W. (Ed.).
Naaman A. E, S. S. P. (1976) Pull-out Mechanisms in Steel Fiber Reinforced Concrete. Journal of the structural division, 1537-1548.
Naaman, A. E. (2003) Engineered Steel Fibers with Optimal Properties for Reinforcement of Cement Composites. Journal of Advanced Concrete Technology, 1, 241-252.
Naaman, A. E. (2007a) High Performance Fiber Reinforced Cement Composites. IN CAIJUN & MO, Y. L. (Eds.) High Performance Construction Materials – Science and Applications. World Scientific Publishing Co. Pte. Ltd.
Naaman, A. E. (2007b) Tensile strain-hardening FRC composites: Historical evolution since the 1960. IN GROSSE, C. U. (Ed.) Advances in Construction Materials 2007. Springer Berlin Heidelberg.
Naaman, A. E. & Najm, H. (1991) Bond-slip Mechanisms of Steel Fibers in Concrete. ACI Materials Journal, 88, 135-145.
Naaman, A. E., Namur, G. J., Alwan, J. & Najm, H. (1991) Fiber Pull-Out and Bond Slip. Part I: Analytical Study. ASCE Journal of Structural Engineering, 117, 2769-2790.
Nammur, G. G. J., and Naaman, A.E. (1989) A Bond Stress Model for Fiber Reinforced Concrete Based on Bond Stress Slip Relationship. ACI Materials Journal, 86, 45-57.
Nguyen, B. N. & Khaleel, M. A. (2004) A mechanistic approach to damage in short-fiber composites based on micromechanical and continuum damage mechanics descriptions. Composites Science and Technology, 64, 607-617.
Oliver, J. (1989) A consistent characteristic length for smeared cracking models. International Journal for Numerical Methods in Engineering, 28, 461-474.
Oliver, J., Cervera, M., Oller, S. & Lubliner, J. (1990) Isotropic damage models and smeared crack analysis of concrete. IN NENAD BICANIC, H. M. (Ed.) Computer aided analysis and design of concrete structures. Zell am See, Austria.
Oliver, J., Dias, I. F. & Huespe, A. E. (2010a) Strong discontinuities, mixed finite element formulations and localized strain injection, in fracture modeling of quasi-brittle materials. Computational Modelling of Concrete Structures: EURO-C 2010. CRC Press.
Oliver, J. & Huespe, A. E. (2004a) Continuum approach to material failure in strong discontinuity settings. Computer Methods in Applied Mechanics and Engineering, 193, 3195-3220.
Oliver, J. & Huespe, A. E. (2004b) Theoretical and computational issues in modelling material failure in strong discontinuity scenarios. Computer Methods in Applied Mechanics and Engineering, 193, 2987-3014.
Oliver, J., Huespe, A. E., Blanco, S. & Linero, D. L. (2006a) Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach. Computer Methods in Applied Mechanics and Engineering, 195, 7093-7114.
Oliver, J., Huespe, A. E. & Cante, J. C. (2008) An implicit/explicit integration scheme to increase computability of non-linear material and contact/friction problems. Computer Methods in Applied Mechanics and Engineering, 197, 1865-1889.
Oliver, J., Huespe, A. E., Cante, J. C. & Díaz, G. (2010b) On the numerical resolution of the discontinuous material bifurcation problem. International Journal for Numerical Methods in Engineering, 83, 786-804.
Oliver, J., Huespe, A. E., Pulido, M. D. G. & Chaves, E. (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Engineering Fracture Mechanics, 69, 113-136.
Oliver, J., Huespe, A. E. & Sánchez, P. J. (2006b) A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM. Computer Methods in Applied Mechanics and Engineering, 195, 4732-4752.
Oller, S. (2003) Simulación Númerica del Comportamiento Mecánico de los Materiales Compuestos, Barcelona, Centro Internacional de Metodos Numéricos en Ingeniría (CIMNE).
Oller, S., Oñate, E., Miquel, J. & Botello, S. (1996) A plastic damage constitutive model for composite materials. International Journal of Solids and Structures, 33, 2501-2518.
Ostoja-Starzewski, M. (2002) Microstructural randomness versus representative volome element in thermomechanics. Journal of applied mechanics, 69, 25-35.
P.J. Sánchez, A. E. H., J. Oliver G. Diaz & Sonzogni, V. E. A macroscopic damage-plastic constitutive model for concrete failure simulation.
Paley, M. & Aboudi, J. (1992) Micromechanical analysis of composites by the generalized cells model. Mechanics of Materials, 14, 127-139. Patrick, G. W., Marsden, J. E. & Shadwick, W. F. (Eds.) (1993) Integration Algorithms And Classical Mechanics, Providence, R.I., American Mathematical Society.
Peng, X. & Meyer, C. (2000) A continuum damage mechanics model for concrete reinforced with randomly distributed short fibers. Computers \& Structures, 78, 505-515.
Pietruszczak, S. & Mróz, Z. (1981) Finite element analysis of deformation of strain-softening materials. International Journal for Numerical Methods in Engineering, 17, 327- 334.
Planas, J., Elices, M., Guinea, G. V., Gómez, F. J., Cendón, D. A. & Arbilla, I. (2003) Generalizations and specializations of cohesive crack models. Engineering Fracture Mechanics, 70, 1759-1776.
Pros, A., Diez, P. & Molins, C. (2011) Modeling steel fiber reinforced concrete: numerical immersed boundary approach and a phenomenological mesomodel for concretefiber interaction. International Journal for Numerical Methods in Engineering, n/an/a.
Radtke, F. K. F., Simone, A. & Sluys, L. J. (2010) A computational model for failure analysis of fibre reinforced concrete with discrete treatment of fibres. Engineering Fracture Mechanics, 77, 597-620.
Rajput, R. K. (2008) A Textbook of Manufacturing Technology (Manufacturing Processes), New Delhi, Laxmi Publications.
Rashid, Y. R. (1968) Ultimate strength analysis of prestressed concrete pressure vessels. Nuclear Engineering and Design, 7, 334-344.
Rastellini, F., Oller, S., Salomón, O. & Oñate, E. (2008) Composite materials non-linear modelling for long fibre-reinforced laminates: Continuum basis, computational aspects and validations. Computers \& Structures, 86, 879 - 896.
Rice, J. R. & Rudnicki, J. W. (1980) A note on some features of the theory of localization of deformation. International Journal of Solids and Structures, 16, 597-605.
Robins, P., Austin, S. & Jones, P. (2002) Pull-out behaviour of hooked steel fibres. Materials and Structures, 35, 434-442.
Rodriguez, F. P., D. (1984) Hormigón con la Incorporación de Fibras. Revista de Obras Públicas, 779-796.
Sanchez-Palencia, E. (1987) General introduction to asymptotic methods. IN SANCHEZPALENCIA, E. & ZAOUI, A. (Eds.) Homogenization Techniques for Composite Media. Springer Berlin / Heidelberg.
Sánchez-Palencia, E. (1986) Homogenization in Mechanics A Survey of Solved and Open Problems. Rend. Sem. Mat. Univers. Politecn. Torino, 44, 1-45.
Sanz-Herrera, J. A., García-Aznar, J. M. & Doblaré, M. (2008) Micro-macro numerical modelling of bone regeneration in tissue engineering. Computer Methods in Applied Mechanics and Engineering, 197, 3092-3107.
Shannag, M., Hnasen, W. & Tjiptobroto, P. (1999) Interface debonding in fiber reinfoorced cement-matrix composites. Journal of composites materials, 33, 158-176.
Si-Larbi, A., Ferrier, E. & Hamelin, P. (2006) Flexural behaviour of MRBC beams (multireinforcing bars concrete beams), promoting the use of FRHPC. Composite Structures, 74, 163-174.
Simo, J. C. & Hughes, T. J. R. (1998) Computational inelasticity, New York Springer-Verlag.
Simo, J. C. & Ju, J. W. (1987) Strain- and stress-based continuum damage models--I. Formulation. International Journal of Solids and Structures, 23, 821-840.
Simo, J. C. & Oliver, J. (1994) A new approach to the analysis and simulation of strong discontinuities. IN BAZANT, Z., BITTNAR, B., JIRÁSEK, M. & MAZARS, J. (Eds.) Fracture and Damage in Quasibrittle Materials, Experiments, Modeling and Computer Analysis. E&FN Spon.
Simo, J. C., Oliver, J. & Armero, F. (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Computational Mechanics, 12, 277-296.
Sirijaroonchai, K. (2009) A Macro-scale Plasticity Model for High Performance Fiber Reinforced Cement Composites. Civil \& Environmental Engineering, University of Michigan.
Sirijaroonchai, K., El-Tawil, S. & Parra-Montesinos, G. (2010) Behavior of high performance fiber reinforced cement composites under multi-axial compressive loading. Cement and Concrete Composites, 32, 62-72.
Sujivorakul, C., Waas, A. M. & Naaman, A. E. (1999) Pullout Response of a Smooth Fiber with an End Anchorage Journal of engineering mechanics, 126, 986-993.
Suquet, P. (1987) Elements of Homogenization for Inelastic solids Mechanics. IN SANCHEZPALENCIA, E. & ZAOUI, A. (Eds.) Homogenization Techniques for Composite Media. Springer Berlin / Heidelberg.
Suwaka, H. & Fukuyama, H. (2006) Nonlinear finite element analysis on shear failure of structural elements using high performance fiber cement composite. Journal of Advanced Concrete Technology, 4, 45-57.
Suwannakarn, S. W. (2009) Post-cracking characteristics of high performance fiber reinforced cementitious composites. University of Michigan.
Svendsen, B. (2001) Formulation of balance relations and configurational fields for continua with microstructure and moving point defects via invariance. International Journal of Solids and Structures, 38, 1183-1200.
Tripp, D. E. H., J. H. Gyekenyesi, J. P. (1989) A Review of Failure Models for Unidirectional Ceramic Matrix Composites Under Monotonic Loads. 34th International Gas Turbine and Aeroengine Congress and Exposition. Toronto, NASA.
Trovalusci, P. & Masiani, R. (2005) A multifield model for blocky materials based on multiscale description. International Journal of Solids and Structures, 42, 5778-5794.
Trusdell, C. & Toupin, R. (1960a) The clasical field theories, Berlin, Springer-Verlag.
Trusdell, C. & Toupin, R. (1960b) The clasical field theories., Berlin, springer-verlag.
Trusdell, C. & Toupin, R. (1960c) Principiples of classical mechanics and field theory, Berlin, Springer-Verlag.
Tucker Iii, C. L. & Liang, E. (1999) Stiffness predictions for unidirectional short-fiber composites: Review and evaluation. Composites Science and Technology, 59, 655-671.
Wang, Y., Backer, S. & Li, V. (1989) A statistical tensile model of fiber reinforced cementitious composites. Journal of composites 20, 265-274.
Welschinger, F. & Miehe, C. (2008) Variational Formulations and FE Active-Set Strategies for Rate-Independent Nonlocal Material Response. PAMM, 8, 10475-10476.
Willis, J. R. (1967) A comparison of the fracture criteria of griffith and barenblatt. Journal of the Mechanics and Physics of Solids, 15, 151-162.
Wittel, F. K., Kun, F. & Herrmann, H. J. Damage Simulation of High Performance Fiber Reinforced Concrete.
Wu, H.-C. & Li, V. C. (1994) Trade-off between strength and ductility of random discontinuous fiber reinforced cementitious composites. Cement and Concrete Composites, 16, 23-29.
Wu, M. S. (2011) Strategies and challenges for the mechanical modeling of biological and bio-inspired materials. Materials Science and Engineering: C, 31, 1209-1220.
Xu, B., Ju, J. W. & Shi, H. (2010) Progressive Micromechanical Modeling for Pullout Energy of Hooked-end Steel Fiber in Cement-based Composites. International Journal of Damage Mechanics.
Zhang, J. & Li, V. C. (2004) Simulation of crack propagation in fiber-reinforced concrete by fracture mechanics. Cement and Concrete Research, 34, 333-339.
Zhang, J., Li, V. C., Nowak, A. S. & Wang, S. (2002) Introducing Ductile Strip for Durability Enhancement of Concrete Slabs. Journal of Materials in Civil Engineering, 14, 253-261.
Zhao, P. & Ji, S. (1997) Refinements of shear-lag model and its applications. Tectonophysics, 279, 37-53.
Zhou, X. F. & Wagner, H. D. (1999) Stress concentrations caused by fiber failure in twodimensional composites. Composites Science and Technology, 59, 1063 - 1071.