Document Type : Article

**Authors**

Civil Engineering Faculty, K. N. Toosi University of Technology, No. 1346, ValiAsr Ave., Mirdamad Cross, Tehran, Iran.

**Abstract**

Mathematical simulation of the nonlinear tri-dimensional mechanical behavior of quasi-brittle materials like concrete is one of the biggest challenges in the engineering science. It is vital to have the knowledge of the response of concrete specimens subjected to low and high strain rate deformation for the analysis of concrete structures under the static and dynamic loading cases. The behavior of this material is generally known to be strain rate sensitive. Among phenomena of different orientation, the multi plane models, like multi-laminate models using a constitutive equation in a vectorial form rather than tensorial form by means of capturing interactions, can meet this goal adequately. This paper suggests a robust rate dependent damage based model in the multi-planes framework accomplished with minimum parameters for calibration and appropriate for engineering purposes. This damage formulation has been built on the basis of two types of essential damage, axial damage and shear damage, that basically can happen on each sampling plane and based on this concept two new axial and shear damage functions are proposed. Model verification has been studied under different compressive and tensile loading rates, comparing the results of the proposed model with the experimental data and Mohr-Coulomb failure criterion envelope line.

**Keywords**

**Main Subjects**

References:

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26. Pande, G.N. and Sharma, K.G. "Multilaminate model of clays - a numerical evaluation of the influence of rotation of principal stress axes", International Journal of Numerical and Analytical Methods in Geomechanics, 7(4), pp. 397-418 (1983).

27. Cusatis, G. and Zhou, X. "High-order microplane theory for quasi-brittle materials with multiple characteristic lengths", Journal of Engineering Mechanics, 140(7), pp. 1-10 (2013).

28. Bazant, Z.P., Adley, M., Carol, I., Jirasek, M., Akers, S., Rohani, B., Cargile, J. and Caner, F. "Largestrain generalization of microplane model for concrete and application", Journal of Engineering Mechanics, 126(9), pp. 971-980 (2000a).

29. Brocca, M. and Bazant, Z.P. "Microplane constitutive model and metal plasticity", Applied Mechanics Reviews, 53(10), pp. 265-281 (2000).

30. Caner, F.C. and Bazant, Z.P. "Microplane model M7 for plain concrete. I: Formulation", Journal of Engineering Mechanics, 139(12), pp. 1714-1723 (2013a).

31. Sadrnejad, S.A. "A general multi-plane model for postliquefaction of sand", Iranian Journal of Science and Technology, Transaction B, Engineering, 31(2), pp. 123-141 (2007).

32. Caner, F.C., Bazant, Z.P., and Cervenka, J. "Vertex effect in strain-softening concrete at rotating principal axes", Journal of Engineering Mechanics, 128(1), pp. 24-33 (2002).

33. Galavi, V., and Schweiger, H.F. "Nonlocal multilaminate model for strain softening analysis", International Journal of Geomechanics, 10(1), pp. 30-44 (2010).

34. Bazant, Z.P. and Oh, B.H. "Microplane model for progressive fracture of concrete and rock", Journal of Engineering Mechanics, 111(4), pp. 559-582 (1985).

35. Sadrnejad, S.A. and Hoseinzadeh, M.R. "An anisotropic multi-plane elastic-damage model with axial and shear damage and its application to concrete behavior", Scientia Iranica, 24(5), pp. 2202-2212 (2017).

36. Bazant, Z.P. and Prat, P.C. "Microplane model for brittle plastic material: I. Theory", Journal of Engineering Mechanics, 114(10), pp. 1672-1688 (1988a).

37. Kazemi, M.T. and Vossough Shahvari, F. "Mixed mode fracture of concrete: An experimental investigation", Scientia Iranica, 11(4), pp. 378-385 (2004).

38. Ozbolt, J., Sharma, A., Irhan, B., and Sola, E. "Tensile behavior of concrete under high loading rates", International Journal of Impact Engineering, 69, pp. 55-68 (2014).

39. Zheng, D. and Li, Q. "An explanation for rate effect of concrete strength based on fracture toughness including free water viscosity", Engineering Fracture Mechanics, 71(16-17), pp. 2319-2327 (2004).

40. Zhang, XX., Ruiz, G., Yu, R.C. and Tarifa, M. "Fracture behavior of high-strength concrete at a wide range of loading rates", International Journal of Impact Engineering, 36(10-11), pp. 1204-1209 (2009).

41. Marzec, I. and Tejchman, J. "Coupled elsto-plastic model with non-local softening enhanced by viscosity to describe dynamic concrete behavior", In Proceeding of the 8th International Conference on Fracture Mechanics of Concrete and Concrete Structures, Barcelona (2013).

42. Kwak, H.G. and Gang, H. "An improved criterion to minimize FE mesh-dependency in concrete structures under high strain rate conditions", International Journal of Impact Engineering, 86, pp. 84-95 (2015).

43. Eibl, J. and Schmidt-Hurtienne, B. "Strain ratesensitive constitutive law for concrete", Journal of Engineering Mechanics, 125(12), pp. 1411-1420 (1999).

44. Haussler-Combe, U. and Kitzig, M. "Modeling of concrete behavior under high strain rates with inertially retarded damage", International Journal of Impact Engineering, 36(9), pp. 1106-1115 (2009).

45. Malvar, L.J. and Ross, C.A. "Review of strain rate effects for concrete in tension", American Concrete Institute Materials Journal, 95(6), pp. 735-739 (1998).

46. Haussler-Combe, U. and Kuhn, T. "Modeling of strain rate effects for concrete with viscoelasticity and retarded damage", International Journal of Impact Engineering, 50, pp. 17-28 (2012).

47. Bischoff, P.H. and Perry, S.H. "Compressive behavior of concrete at high strain rates", Materials and Structures, 24(6), pp. 425-450 (1991).

48. CEB (Comite Euro-International du Beton), CEB-FIP Model Code 2010 - Design Code, 1st draft, Thomas Telford, Lausanne, Switzerland (2010).

49. Gernay, T., Millard, A., and Franssen, J. "A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation", International Journal of Solids and Structures, 50(22-23), pp. 3659-3673 (2013).

50. Grassl, P., Xenos, D., Nystrom, U., Rempling, R., and Gylltoft, K. "CDPM2: A damage-plasticity approach to modelling the failure of concrete", International Journal of Solids and Structures, 50(24), pp. 3805- 3816 (2013).

51. Soltanzadeh, F., Edalat-Behbahani, A., Barros, J.A.O., and Mazaheripour, H. "Effect of fiber dosage and prestress level on shear behavior of hybrid GFRPsteel reinforced concrete I-shape beams without stirrups", Composites, Part B: Engineering, 102, pp. 57- 77 (2016).

52. Edalat-Behbahani, A., Barros, J.A.O., and Ventura- Gouveia, A. "Three dimensional plastic-damage multidirectional fixed smeared crack approach for modelling concrete structures", International Journal of Solids and Structures, 115(116), pp. 104-125, (2017).

53. Grassl, P. and Jirasek, M. "Damage-plastic model for concrete failure", International Journal of Solids and Structures, 43, pp. 7166-7196 (2006).

54. Carrazedo, R., Mirmiran, A., and Hanai, J.B. "Plasticity based stress-strain model for concrete confinement", Engineering Structures, 48, pp. 645-657 (2013).

55. Edalat-Behbahani, A., Barros, J.A.O., and Ventura- Gouveia, A. "Application of plastic-damage multidirectional fixed smeared crack model in analysis of RC structures", Engineering Structures, 125, pp. 374-391 (2016).

56. Edalat Behbahani, A., Barros, J.A.O., and Ventura- Gouveia, A. "Plastic-damage smeared crack model to simulate the behavior of structures made by cement based materials", International Journal of Solids and Structures, 73(74), pp. 20-40 (2015).

57. Van Mier, J.G.M. "Multi-axial strain-softening of concrete", Materials and Structures, 19(3), pp. 179-200 (1986).

58. Karsan, I.D. and Jirsa, J.O. "Behavior of concrete under compressive loading", Journal of Structural Division, ASCE, 95(12), pp. 2535-2563 (1969).

59. Van Mier, J.G.M. "Strain-softening of concrete under multi-axial loading conditions", Ph.D. Thesis, Die Technische Hogeschool Eindhoven, Eindhoven, Netherlands (1984).

60. TH Tan, Andrew "Effects of tri-axial stress on concrete", 30th Conference on Our World in Concrete & Structures, pp. 23-24, August, Singapore (2005).

61. Petersson, P.E. "Crack growth and development of fracture zones in plain concrete and similar materials", Report Number TVBM 1006, Lund Institute of Technology, Lund, Sweden (1981).

62. Grote, D.L., Park, S.W., and Zhou, M. "Dynamic behavior of concrete at high strain rates and pressures: I. Experimental characterization", International Journal of Impact Engineering, 25(9), pp. 869-886 (2001).

63. Song, Z., and Lu, Y. "Mesoscopic analysis of concrete under excessively high strain rate compression and implications on interpretation of test data", International Journal of Impact Engineering, 46, pp. 41-55 (2012).

64. Ragueneau, F., Gatuingt, F., and Bailly, P. "Inelastic behavior modelling of concrete in low and high strain rate dynamics", Computers and Structures, 81(12), pp. 1287-1299 (2003).

65. Omidi, O. and Lotfi, V. "Numerical analysis of cyclically loaded concrete under large tensile strains by the plastic-damage model", Scientia Iranica, 17(3), pp. 194-208 (2010).

2. Cusatis, G., Pelessone, D., and Mencarelli, A. "Lattice discrete particle model (LDPM) for failure behavior of concrete", Cement and Concrete Composites, 33, pp. 881-890 (2011).

3. Bousikhane, F., Rezakhani, R., Smith J., and Cusatis, G. "Calibration and validation of concrete model for the simulation of the quasi-static and dynamic response of concrete structures", 9th International Conference on Fracture Mechanics of Concrete and Concrete Structures (FraMCoS-9), Brekeley, USA (2016).

4. Rezakhani, R. and Cusatis, G. "Asymptotic expansion homogenization of discrete fine-scale models with rotational degrees of freedom for the simulation of quasi-brittle materials", Journal of the Mechanics and Physics of Solids, 88, pp. 320-345 (2016).

5. Rezakhani, R., Zhou, X., and Cusatis, G. "Adaptive multiscale homogenization of the lattice discrete particle model for the analysis of damage and fracture in concrete", Available on: http://arxiv.org/abs/1702.00695.

6. Shahbazi, S. and Rasoolan, I. "Meso-scale finite element modeling of non-homogeneous three-phase concrete", Case Studies in Construction Materials, 6, pp. 29-42 (2017).

7. Le Nard, H. and Bailly, P. "Dynamic behavior of concrete: the structural effects on compressive strength increase", Mechanics of Cohesive-Frictional Materials, 5(6), pp. 491-510 (2000).

8. Tu, Z. and Lu, Y. "Modifications of RHT material model for improved numerical simulation of dynamic response of concrete", International Journal of Impact Engineering, 37(10), pp. 1072-1082 (2010).

9. Abrams, D.A. "Effect of rate of application of load on the compressive strength of concrete-Part 2", ASTM Journal, 17, pp. 364-377 (1917).

10. Reinhardt, H.W. and Weerheijm, J. "Tensile fracture of concrete at high loading rates taking account of inertia and crack velocity effects", International Journal of Fracture, 51(1), pp. 31-42 (1991).

11. Mechteherine, V., Millon, O., Butler, M., and Thoma, K. "Mechanical behavior of strain hardening cementbased composite under impact loading", Cement and Concrete Composites, 33(1), pp. 1-11 (2011).

12. Chen, X., Bu, J., and Xu, L. "Effect of strain rate on post-peak cyclic behavior of concrete in direct tension", Construction and Building Materials, 124(16), pp. 746-754 (2016).

13. Chen, X., Huang, Y., Chen, C., Xu, L., and Fan, X. "Experimental study and analytical modeling on hysteresis behavior of plain concrete in uniaxial cyclic tension", International Journal of Fatigue, 96, pp. 261- 269 (2017).

14. Freund, L.B. "Crack propagation in an elastic solid subjected to general loading-I. Constant rate of extension", Journal of the Mechanics and Physics of Solids, 20(3), pp. 129-140 (1972a).

15. Freund, L.B. "Crack propagation in an elastic solid subjected to general loading-II. Non-uniform rate of extension", Journal of the Mechanics and Physics of Solids, 20(3), pp. 141-152 (1972b).

16. Ozbolt, J., Rah, K.K., and Mestrovic, D. "Influence of loading rate on concrete cone failure", International Journal of Fracture, 139(2), pp. 239-52 (2006).

17. Ozbolt, J., Sharma, A., and Reinhardt, H.W. "Dynamic fracture of concrete-compact tension specimen", International Journal of Fracture, 48(10), pp. 1534- 1543 (2011).

18. Xiao, J., Li, Z., Xie, Q., and Shen, L. "Effect of strain rate on compressive behaviour of high-strength concrete after exposure to elevated temperatures", Fire Safety Journal, 83, pp. 25-37 (2016).

19. Yu, X., Chen, L., Fang, Q., Ruan, Z., Hong, J., and Xiang, J. "A concrete constitutive model considering coupled effects of high temperature and high strain rate", International Journal of Impact Engineering, 101, pp. 66-77 (2017).

20. CEB (Comite Euro-International du Beton), CEB-FIP Model Code 1990-Design Code, 6th edition, Thomas Telford, Lausanne, Switzerland (1993).

21. Hao, Y., and Hao, H. "Influence of the concrete DIF model on the numerical predictions of RC wall responses to blast loadings", Engineering Structures, 73, pp. 24-38 (2014).

22. Huang, L.C., Li, J., Tue, N.V., Nemecek, J., and Puschel, T. "Numerical aspects of micro-plane constitutive models for concrete", Applied Mathematical Modeling, 41, pp. 530-548 (2017).

23. Sanchez, F., Prat, P.C., Galavi, V., and Schweiger, H.F. "Multilaminate and Microplane Models: Same Principles and Different Solutions for Constitutive Behaviour of Geomaterials", 12th International Conference on International Association for Computer Methods and Advances in Geomechanics (IACMAG),

Goa, India (2008).

24. Labibzadeh, M. and Sadrnejad, S.A. "Mesoscopic damage based model for plane concrete under static and dynamic loadings", American Journal of Applied Sciences, 3(9), pp. 2011-2019 (2006).

25. Bazant, Z.P. and Oh, B.H. "Microplane model for fracture analysis of concrete structures", Report No. ADP001715, Technological Institute, Northwestern University, Evanston. pp. 49-55 (1983).

26. Pande, G.N. and Sharma, K.G. "Multilaminate model of clays - a numerical evaluation of the influence of rotation of principal stress axes", International Journal of Numerical and Analytical Methods in Geomechanics, 7(4), pp. 397-418 (1983).

27. Cusatis, G. and Zhou, X. "High-order microplane theory for quasi-brittle materials with multiple characteristic lengths", Journal of Engineering Mechanics, 140(7), pp. 1-10 (2013).

28. Bazant, Z.P., Adley, M., Carol, I., Jirasek, M., Akers, S., Rohani, B., Cargile, J. and Caner, F. "Largestrain generalization of microplane model for concrete and application", Journal of Engineering Mechanics, 126(9), pp. 971-980 (2000a).

29. Brocca, M. and Bazant, Z.P. "Microplane constitutive model and metal plasticity", Applied Mechanics Reviews, 53(10), pp. 265-281 (2000).

30. Caner, F.C. and Bazant, Z.P. "Microplane model M7 for plain concrete. I: Formulation", Journal of Engineering Mechanics, 139(12), pp. 1714-1723 (2013a).

31. Sadrnejad, S.A. "A general multi-plane model for postliquefaction of sand", Iranian Journal of Science and Technology, Transaction B, Engineering, 31(2), pp. 123-141 (2007).

32. Caner, F.C., Bazant, Z.P., and Cervenka, J. "Vertex effect in strain-softening concrete at rotating principal axes", Journal of Engineering Mechanics, 128(1), pp. 24-33 (2002).

33. Galavi, V., and Schweiger, H.F. "Nonlocal multilaminate model for strain softening analysis", International Journal of Geomechanics, 10(1), pp. 30-44 (2010).

34. Bazant, Z.P. and Oh, B.H. "Microplane model for progressive fracture of concrete and rock", Journal of Engineering Mechanics, 111(4), pp. 559-582 (1985).

35. Sadrnejad, S.A. and Hoseinzadeh, M.R. "An anisotropic multi-plane elastic-damage model with axial and shear damage and its application to concrete behavior", Scientia Iranica, 24(5), pp. 2202-2212 (2017).

36. Bazant, Z.P. and Prat, P.C. "Microplane model for brittle plastic material: I. Theory", Journal of Engineering Mechanics, 114(10), pp. 1672-1688 (1988a).

37. Kazemi, M.T. and Vossough Shahvari, F. "Mixed mode fracture of concrete: An experimental investigation", Scientia Iranica, 11(4), pp. 378-385 (2004).

38. Ozbolt, J., Sharma, A., Irhan, B., and Sola, E. "Tensile behavior of concrete under high loading rates", International Journal of Impact Engineering, 69, pp. 55-68 (2014).

39. Zheng, D. and Li, Q. "An explanation for rate effect of concrete strength based on fracture toughness including free water viscosity", Engineering Fracture Mechanics, 71(16-17), pp. 2319-2327 (2004).

40. Zhang, XX., Ruiz, G., Yu, R.C. and Tarifa, M. "Fracture behavior of high-strength concrete at a wide range of loading rates", International Journal of Impact Engineering, 36(10-11), pp. 1204-1209 (2009).

41. Marzec, I. and Tejchman, J. "Coupled elsto-plastic model with non-local softening enhanced by viscosity to describe dynamic concrete behavior", In Proceeding of the 8th International Conference on Fracture Mechanics of Concrete and Concrete Structures, Barcelona (2013).

42. Kwak, H.G. and Gang, H. "An improved criterion to minimize FE mesh-dependency in concrete structures under high strain rate conditions", International Journal of Impact Engineering, 86, pp. 84-95 (2015).

43. Eibl, J. and Schmidt-Hurtienne, B. "Strain ratesensitive constitutive law for concrete", Journal of Engineering Mechanics, 125(12), pp. 1411-1420 (1999).

44. Haussler-Combe, U. and Kitzig, M. "Modeling of concrete behavior under high strain rates with inertially retarded damage", International Journal of Impact Engineering, 36(9), pp. 1106-1115 (2009).

45. Malvar, L.J. and Ross, C.A. "Review of strain rate effects for concrete in tension", American Concrete Institute Materials Journal, 95(6), pp. 735-739 (1998).

46. Haussler-Combe, U. and Kuhn, T. "Modeling of strain rate effects for concrete with viscoelasticity and retarded damage", International Journal of Impact Engineering, 50, pp. 17-28 (2012).

47. Bischoff, P.H. and Perry, S.H. "Compressive behavior of concrete at high strain rates", Materials and Structures, 24(6), pp. 425-450 (1991).

48. CEB (Comite Euro-International du Beton), CEB-FIP Model Code 2010 - Design Code, 1st draft, Thomas Telford, Lausanne, Switzerland (2010).

49. Gernay, T., Millard, A., and Franssen, J. "A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation", International Journal of Solids and Structures, 50(22-23), pp. 3659-3673 (2013).

50. Grassl, P., Xenos, D., Nystrom, U., Rempling, R., and Gylltoft, K. "CDPM2: A damage-plasticity approach to modelling the failure of concrete", International Journal of Solids and Structures, 50(24), pp. 3805- 3816 (2013).

51. Soltanzadeh, F., Edalat-Behbahani, A., Barros, J.A.O., and Mazaheripour, H. "Effect of fiber dosage and prestress level on shear behavior of hybrid GFRPsteel reinforced concrete I-shape beams without stirrups", Composites, Part B: Engineering, 102, pp. 57- 77 (2016).

52. Edalat-Behbahani, A., Barros, J.A.O., and Ventura- Gouveia, A. "Three dimensional plastic-damage multidirectional fixed smeared crack approach for modelling concrete structures", International Journal of Solids and Structures, 115(116), pp. 104-125, (2017).

53. Grassl, P. and Jirasek, M. "Damage-plastic model for concrete failure", International Journal of Solids and Structures, 43, pp. 7166-7196 (2006).

54. Carrazedo, R., Mirmiran, A., and Hanai, J.B. "Plasticity based stress-strain model for concrete confinement", Engineering Structures, 48, pp. 645-657 (2013).

55. Edalat-Behbahani, A., Barros, J.A.O., and Ventura- Gouveia, A. "Application of plastic-damage multidirectional fixed smeared crack model in analysis of RC structures", Engineering Structures, 125, pp. 374-391 (2016).

56. Edalat Behbahani, A., Barros, J.A.O., and Ventura- Gouveia, A. "Plastic-damage smeared crack model to simulate the behavior of structures made by cement based materials", International Journal of Solids and Structures, 73(74), pp. 20-40 (2015).

57. Van Mier, J.G.M. "Multi-axial strain-softening of concrete", Materials and Structures, 19(3), pp. 179-200 (1986).

58. Karsan, I.D. and Jirsa, J.O. "Behavior of concrete under compressive loading", Journal of Structural Division, ASCE, 95(12), pp. 2535-2563 (1969).

59. Van Mier, J.G.M. "Strain-softening of concrete under multi-axial loading conditions", Ph.D. Thesis, Die Technische Hogeschool Eindhoven, Eindhoven, Netherlands (1984).

60. TH Tan, Andrew "Effects of tri-axial stress on concrete", 30th Conference on Our World in Concrete & Structures, pp. 23-24, August, Singapore (2005).

61. Petersson, P.E. "Crack growth and development of fracture zones in plain concrete and similar materials", Report Number TVBM 1006, Lund Institute of Technology, Lund, Sweden (1981).

62. Grote, D.L., Park, S.W., and Zhou, M. "Dynamic behavior of concrete at high strain rates and pressures: I. Experimental characterization", International Journal of Impact Engineering, 25(9), pp. 869-886 (2001).

63. Song, Z., and Lu, Y. "Mesoscopic analysis of concrete under excessively high strain rate compression and implications on interpretation of test data", International Journal of Impact Engineering, 46, pp. 41-55 (2012).

64. Ragueneau, F., Gatuingt, F., and Bailly, P. "Inelastic behavior modelling of concrete in low and high strain rate dynamics", Computers and Structures, 81(12), pp. 1287-1299 (2003).

65. Omidi, O. and Lotfi, V. "Numerical analysis of cyclically loaded concrete under large tensile strains by the plastic-damage model", Scientia Iranica, 17(3), pp. 194-208 (2010).

Transactions on Civil Engineering (A)

May and June 2019Pages 1194-1205