混凝土分形断裂行为及损伤本构研究
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摘要
近年来,混凝土材料的开裂行为已引起越来越多研究人员的关注。不少研究人员通过素混凝土梁的三点弯曲试验研究其裂缝的开展,其理论分析基础主要来源于Griffith经典断裂力学,即假定材料的断裂面和裂缝是平滑的,并得到许多重要的结论。但鉴于混凝土的断裂面及裂缝的粗糙性,将经典断裂理论直接用于研究混凝土的断裂行为可能会造成所得结论的不精确。为了充分考虑材料断裂的粗糙性,本文引入分形几何学理论用以描述粗糙裂缝和断裂面,通过理论分析和数值计算,以及与有关试验数据的对比研究了混凝土材料断裂及损伤行为。本文主要完成以下创新性工作:
1、以标准Koch分形曲线构造混凝土的开裂边界,根据分形几何理论中给出的分形区域面积与周长的关系,推导了混凝土材料断裂区域内弹性应变能与表面能的数学表达式,并将Griffith断裂理论推广至考虑非光滑裂纹的分形空间内。在此基础上分析了混凝土材料的临界开裂应力、断裂韧性、裂纹驱动力以及断裂能的性质。
2、考虑到混凝土材料的准脆性性质,利用布朗分形曲面模拟断裂面,以布朗指数表示Carpinteri多重分形尺寸效应法则,并由断裂能推导出断裂韧性的尺寸效应;将Ba?ant的尺寸效应法则和Carpinteri尺寸效应法则应用于应力强度因子的表述式,推导出断裂韧性与试件尺寸及裂纹尺寸的关系;利用Carpinteri多重分形尺寸效应法则中的强度公式验证我国《普通混凝土力学性能试验方法标准》中给出的尺寸效应系数。
3、用Menger海绵模拟水泥石的孔隙结构,将材料的孔隙率表示为分形维数的函数。在此基础上,考虑孔隙率与强度的关系,定量地分析了各构造参数对水泥石强度的影响。根据水泥石与混凝土之间的强度关系,推导出考虑孔隙分形分布情况下的混凝土强度理论公式。
4、视混凝土材料视为水泥浆、水以及骨料三相材料组成的,以三层球体模型表示这三相材料的组成形式。随后分析了模型中各相材料的应力状态。对不考虑分形效应的情形应用Griffith断裂准则给出混凝土断裂韧性随温度变化的表达式。当考虑分形效应时,通过量纲分析定义三维分形裂纹的表面能和应力强度因子,并将修正后的Griffith断裂准则应用于该模型,从而得到混凝土断裂韧性与温度及断裂面分形维数之间的关系式。当已知任一温度下混凝土材料的断裂韧性时,可由该公式求出其它温度下的断裂韧性,所得计算结果与与相关试验结果吻合。
5、利用脆性材料断裂韧性与分形维数之间的关系确定了混凝土初始损伤和临界断裂状态时的分形维数和相应的应力,从而给出了由分形维数表示损伤变量。定义了包含各向异性损伤和不可逆变形的Helmholtz自由能,从而推导出了考虑各向异性损伤的混凝土弹性本构模型,随后给出了相应的损伤演化方程。通过数值计算分析了混凝土试件在单轴受拉/压和双轴受拉/压时的变形曲线,与相关试验数据对比验证了模型的正确性。
6、根据分形区域面积与周长之间的关系,推导了分形空间内损伤变量与欧氏空间内损伤变量之间的关系。随后将分形损伤变量引入到混凝土经典弹塑性损伤本构模型中,以便考虑混凝土裂纹分形分布的特征。利用ABAQUS软件对混凝土梁进行数值计算并与现有试验对比,验证了所提出的考虑分形效应的混凝土弹塑性本构模型的正确性。
Cracking behavior of concrete material has been concerned by more and more researchers in recent years. Many researchers studied the crack development through the 3-point bending tests of the plain concrete beams. Their theorical foundation comes from classic Griffith fracture mechanics, which asumes the fracture surface and crack of concrete are smooth. They got amount of important conclusions. But imprecise result may be obtained by applying the classic fracture theory to studying the fracture behavior of concrete for the roughness of its fracture surface and cracks. In order to take this feature into account, fractal geometry theory is introduced in this work to discribe its fracture surface and crack. Fracture and damage behaviors of concrete are studied by theorical analysis and numerical calculation. The main innovative work completed in this paper is included as follows:
1. By constructing the cracking boundary of concrete with a standard Koch fractal curve, the mathematic expressions of the elastic strain energy and the surface energy in the fracture zone are derived based on the area-perimeter relation given in fractal geometry. And the classic Griffith fracture theory is extended to the fractal space in which the non-smooth crack can be discribed. Behaviors of the critical cracking stress, the fracture toughness, the driving force and the fracture energy are analyzed.
2. Considering the quasi-brittle feature of concrete material, the fracture surface is simulated by a Brownian fractal surface. The multi-fractal size effect law of Carpinteri is expressed as a function of Brownian index. The size effect of the fracture toughness is obtained from the fracture energy. The size effect laws of Ba?ant and Carpinteri are applied in the formula of stress strength factor; and the relation between the fracture toughness and size of the specimen and crack. The statistical size coefficients between concrete specimens given in the Standard for test method of mechanical properties on ordinary concrete are proofed by using the strength size effect law of the size effect law by Carpinteri.
3. By simulating the pore structure of cement paste with Menger sponge, porosity can be exoressed as a function of fractal dimension. Based on this, the relationships between strength of cement paste and the construction parameters are analyzed quantitatively considering the relation between the porosity and strength. The expression of concrete strength considering the pore distribution is obtained based on the strength relations of cement paste and concrete.
4. A three-phase micro fracture spherical model for concrete considering the influence of temperature is established based on a rule that volumes of each component are the same as those of a real material. The components of concrete is simplified as aggregate, water and cement paste, and each part is replaced by a spherical layer. The stress states of the aggregate and cement paste layers are studied. For the smooth cracking case, the relation of the fracture toughness and temperature is given based on the Griffith fracture criterion. For the fractal cracking case, the surface energy and stress strength factor of a 3-dimension crack is defined by the dimension analysis. The updated fracture criterion is employed in this model and the expression between the fracture toughness and temperature is got therefore. The presented model inherits an advantage that only fracture toughness at two different temperatures are needed to be given to predict other temperature fracture toughness for concrete.
5. The fractal dimension and stress of concrete coresponding to the initial damage and critical fracture state are determined by using the relation between the fracture toughness and the fractal dimension of quasi-brittle material. And the damage variable is written in term of fractal dimension. The Helmholtz free energy including isotropic damage, anisotropic damage and irreversible deformation parts is defined and thus a damage constitutive model of concrete considering the fractal feature is derived with a corresponding damage evlution law followed. Numerical computation is conducted for concrete under the uniaxial and the biaxial compression which shows the validation of the presented model is proofed by comparing results with the experimental data. This model provides an approach to link the macro properties of a material with its micro-structure change.
6. The perimeter-area relation is adopted to derive the transformation rule between damage variables in the fractal space and Euclidean space. A plastic damage constitutive model for concrete in the Euclidean space is generalized to fractal case according to the transformation rule of damage variables. The present model considering the fractal effect is used to analyze a notched plain concrete beam under 3-point bending. The numerical results show the efficiency and validation of the present model for structural analysis.
1、以标准Koch分形曲线构造混凝土的开裂边界,根据分形几何理论中给出的分形区域面积与周长的关系,推导了混凝土材料断裂区域内弹性应变能与表面能的数学表达式,并将Griffith断裂理论推广至考虑非光滑裂纹的分形空间内。在此基础上分析了混凝土材料的临界开裂应力、断裂韧性、裂纹驱动力以及断裂能的性质。
2、考虑到混凝土材料的准脆性性质,利用布朗分形曲面模拟断裂面,以布朗指数表示Carpinteri多重分形尺寸效应法则,并由断裂能推导出断裂韧性的尺寸效应;将Ba?ant的尺寸效应法则和Carpinteri尺寸效应法则应用于应力强度因子的表述式,推导出断裂韧性与试件尺寸及裂纹尺寸的关系;利用Carpinteri多重分形尺寸效应法则中的强度公式验证我国《普通混凝土力学性能试验方法标准》中给出的尺寸效应系数。
3、用Menger海绵模拟水泥石的孔隙结构,将材料的孔隙率表示为分形维数的函数。在此基础上,考虑孔隙率与强度的关系,定量地分析了各构造参数对水泥石强度的影响。根据水泥石与混凝土之间的强度关系,推导出考虑孔隙分形分布情况下的混凝土强度理论公式。
4、视混凝土材料视为水泥浆、水以及骨料三相材料组成的,以三层球体模型表示这三相材料的组成形式。随后分析了模型中各相材料的应力状态。对不考虑分形效应的情形应用Griffith断裂准则给出混凝土断裂韧性随温度变化的表达式。当考虑分形效应时,通过量纲分析定义三维分形裂纹的表面能和应力强度因子,并将修正后的Griffith断裂准则应用于该模型,从而得到混凝土断裂韧性与温度及断裂面分形维数之间的关系式。当已知任一温度下混凝土材料的断裂韧性时,可由该公式求出其它温度下的断裂韧性,所得计算结果与与相关试验结果吻合。
5、利用脆性材料断裂韧性与分形维数之间的关系确定了混凝土初始损伤和临界断裂状态时的分形维数和相应的应力,从而给出了由分形维数表示损伤变量。定义了包含各向异性损伤和不可逆变形的Helmholtz自由能,从而推导出了考虑各向异性损伤的混凝土弹性本构模型,随后给出了相应的损伤演化方程。通过数值计算分析了混凝土试件在单轴受拉/压和双轴受拉/压时的变形曲线,与相关试验数据对比验证了模型的正确性。
6、根据分形区域面积与周长之间的关系,推导了分形空间内损伤变量与欧氏空间内损伤变量之间的关系。随后将分形损伤变量引入到混凝土经典弹塑性损伤本构模型中,以便考虑混凝土裂纹分形分布的特征。利用ABAQUS软件对混凝土梁进行数值计算并与现有试验对比,验证了所提出的考虑分形效应的混凝土弹塑性本构模型的正确性。
Cracking behavior of concrete material has been concerned by more and more researchers in recent years. Many researchers studied the crack development through the 3-point bending tests of the plain concrete beams. Their theorical foundation comes from classic Griffith fracture mechanics, which asumes the fracture surface and crack of concrete are smooth. They got amount of important conclusions. But imprecise result may be obtained by applying the classic fracture theory to studying the fracture behavior of concrete for the roughness of its fracture surface and cracks. In order to take this feature into account, fractal geometry theory is introduced in this work to discribe its fracture surface and crack. Fracture and damage behaviors of concrete are studied by theorical analysis and numerical calculation. The main innovative work completed in this paper is included as follows:
1. By constructing the cracking boundary of concrete with a standard Koch fractal curve, the mathematic expressions of the elastic strain energy and the surface energy in the fracture zone are derived based on the area-perimeter relation given in fractal geometry. And the classic Griffith fracture theory is extended to the fractal space in which the non-smooth crack can be discribed. Behaviors of the critical cracking stress, the fracture toughness, the driving force and the fracture energy are analyzed.
2. Considering the quasi-brittle feature of concrete material, the fracture surface is simulated by a Brownian fractal surface. The multi-fractal size effect law of Carpinteri is expressed as a function of Brownian index. The size effect of the fracture toughness is obtained from the fracture energy. The size effect laws of Ba?ant and Carpinteri are applied in the formula of stress strength factor; and the relation between the fracture toughness and size of the specimen and crack. The statistical size coefficients between concrete specimens given in the Standard for test method of mechanical properties on ordinary concrete are proofed by using the strength size effect law of the size effect law by Carpinteri.
3. By simulating the pore structure of cement paste with Menger sponge, porosity can be exoressed as a function of fractal dimension. Based on this, the relationships between strength of cement paste and the construction parameters are analyzed quantitatively considering the relation between the porosity and strength. The expression of concrete strength considering the pore distribution is obtained based on the strength relations of cement paste and concrete.
4. A three-phase micro fracture spherical model for concrete considering the influence of temperature is established based on a rule that volumes of each component are the same as those of a real material. The components of concrete is simplified as aggregate, water and cement paste, and each part is replaced by a spherical layer. The stress states of the aggregate and cement paste layers are studied. For the smooth cracking case, the relation of the fracture toughness and temperature is given based on the Griffith fracture criterion. For the fractal cracking case, the surface energy and stress strength factor of a 3-dimension crack is defined by the dimension analysis. The updated fracture criterion is employed in this model and the expression between the fracture toughness and temperature is got therefore. The presented model inherits an advantage that only fracture toughness at two different temperatures are needed to be given to predict other temperature fracture toughness for concrete.
5. The fractal dimension and stress of concrete coresponding to the initial damage and critical fracture state are determined by using the relation between the fracture toughness and the fractal dimension of quasi-brittle material. And the damage variable is written in term of fractal dimension. The Helmholtz free energy including isotropic damage, anisotropic damage and irreversible deformation parts is defined and thus a damage constitutive model of concrete considering the fractal feature is derived with a corresponding damage evlution law followed. Numerical computation is conducted for concrete under the uniaxial and the biaxial compression which shows the validation of the presented model is proofed by comparing results with the experimental data. This model provides an approach to link the macro properties of a material with its micro-structure change.
6. The perimeter-area relation is adopted to derive the transformation rule between damage variables in the fractal space and Euclidean space. A plastic damage constitutive model for concrete in the Euclidean space is generalized to fractal case according to the transformation rule of damage variables. The present model considering the fractal effect is used to analyze a notched plain concrete beam under 3-point bending. The numerical results show the efficiency and validation of the present model for structural analysis.
引文
[1] GB 50010-2002,混凝土结构设计规范[S].北京:中国建筑工业出版社, 2002.
[2]谢和平,薛秀谦.分形应用中的数学基础与方法[M].北京:科学出版社, 1997.
[3] Weierstrass K.über Continuirliche Functionen eines reellen Arguments, die für keinen Werth des Letzteren einen bestimmten Differentialquotien besitzen[J]. Royal Prussian Academy of Science. 1872.
[4] Mandelbrot B.B. Fractals: form, chance, and dimension[M]. San Francisco: W. H. Freeman and Company, 1977.
[5] Mandelbrot B.B. The Fractal Geometry of Nature[M]. CA: W.H. Freeman and Company, 1983.
[6] Minkowski H. Ber die Ann herung an eine reelle Gr sse durch rationale Zahlen[J]. Mathematische Annalen. 1901, 54: 91-124.
[7] Hausdorff F. Dimension und ?usseres Mass[J]. Mathematische Annalen. 1919, 79: 157-179.
[8] Besicovitch A. On the sum of digits of real numbers represented in the dyadic system[J]. Mathematische Annalen. 1934, 110: 321-330.
[9] Bouligand G. Ensembles impropres et nombre dimensionnel[J]. Bulletin des Sciences Mathématiques. 1928, 52: 320-334.
[10] Pontrjagin L., Schnirelmann L. Sur une propriétémétrique de la dimension[J]. Annals of Mathematics. 1932, 33(2): 152-162.
[11] Kolmogorov A.N., Tikhomirov V.M.ε-Entropy andε-capacity[J]. Uspekhi Mat. Nauk. 1959, 14: 3-86.
[12] Lévy P. Les courbes planes ou gauches et les surfaces composées de parties semblales au tout[J]. Journal of L'Ecole Polytechnique. 1938, 111(78): 227-247, 249-291.
[13] Kahane J.P., Salem R. Ensembles parfaits et series trigonometriques[M]. Paris: Herman, 1963.
[14] Mandelbrot B.B., Passoja D.E., Paullay A.J. Fractal character of fracture surfaces of metals[J]. Nature. 1984, 308: 721-722.
[15] Falconer K. Fractal geometry: Mathematical Foundations and Applications[M]. 2 ed.England: John Wiley & Sons Ltd, 2003.
[16] Bouchaud J.P., Bouchaud E., Lapasset G., et al. Models of fractal crack[J]. Physical Review Letter. 1993, 71(14): 2240-2243.
[17] Mecholsky J.J. Fracture of Ceramics by Single-Pulse 10.6-μm Irradiation[J]. Journal of Applied Physics. 1979, 50(8): 5488-5491.
[18] Mecholsky Jr J.J. Estimating theoretical strength of brittle materials using fractal geometry[J]. Materials Letters. 2006, 60: 2485-2488.
[19] Bouchaud E., Lapasset G., Planès J. Fractal dimension of fractured surfaces: a universal value?[J]. Europhysical Letter. 1990, 13: 73-79.
[20] Mandelbrot B.B. Fractals and an art for the sake of art[J]. Leonardo; Suppl. 1989: 21-24.
[21] Mandelbrot B.B. Multifractal measures, especially for the geophysicist[J]. Pure and applied geophysics. 1989, 131: 5-42.
[22] Bouchaud E., Lapasset G., Planès J. Statistics of branched fracture surfaces[J]. Physical Review B. 1993, 48(5): 2917-2928.
[23] Mosolov A.A. Cracks with fractal surfaces[J]. Doklady Akademii Nauk SSSR. 1991, 319(4): 840-844.
[24] Bouchaud E., Bouchaud J.P. Fracture surfaces: Apparent roughness, relevant length scales, and fracture toughness[J]. Physics Review B. 1994, 50(23): 17752-17755.
[25] Shi D., Jiang J., Lung C. Correlation between the scale-dependent fractal dimension of fracture surfaces and the fracture toughness[J]. Physical Review B. 1996, 54(24): 17355-17358.
[26] Milman V.Y., Stelmashenko N.A., Blumenfeld R. Fracture surfaces: a critical review of fractal studies and novel morphological analysis of scanning tunneling microscopy measurements[J]. Progress in Materials Science. 1994, 38: 425-474.
[27] Nagahama H. A fractal criterion for ductile and brittle fracture[J]. Journal of Applied Physics. 1994, 75(6): 3220-3222.
[28] Mu Z.Q., Lung C.W. Studies on the fractal dimension and fracture toughness of steel[J]. Journal of Physics D: Applied Physics. 1988, 21(5): 848-850.
[29] Hillerborg A., Modeer M., Petersson P.E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[J]. Cement and ConcreteResearch. 1976, 6: 773-782.
[30] Wittmann F.H., Mihashi H., Nomura N. Size effect on fracture energy of concrete[J]. Engineering Fracture Mechanics. 1990, 35: 107-115.
[31] Wu K.R., Yan A., Liu J.Y., et al. Reconstruction and analysis of 3-D profile of fracture surface of concrete[J]. Cement and Concrete Research. 2000, 30: 981-987.
[32] Griffith A.A. The phenomena of rupture and flow in solids[J]. Philosophical Transactions of the Royal Society. 1920, 221: 163-198.
[33] Weibull W. A statistical representation of fatigue failures in solids[M]. Sweden: Transactions of the Royal Institute of Technology, 1949.
[34] Weibull W. A statistical distribution function of wide applicability[J]. Journal of Applied Mechanics, ASME. 1951, 18: 293-297.
[35] Kaplan M.F. Strains and stresses of concrete at initiation of cracking and near failure[J]. Journal of the American Concrete Institute Proceedings. 1963, 60(7): 853-880.
[36] Bazant Z.P., Kaplan M.F. Concrete at High Temperatures: Material Properties and Mathematical Models[M]. Essex: Longman House, 1996.
[37] Kesler C.E., Naus D.J., Lott J.L. Fracture mechanics-its applicability to concrete[J]. International Conference on the Mechanical Behaviour of Materials. 1972, 4: 113-124.
[38] Walsh P.F. Crack initiation in plain concrete[J]. Magazine of Concrete Research. 1976, 28(94): 37-41.
[39] Ba?ant Z. Size effect in blunt fracture: Concrete, rock, metal[J]. Journal of Engineering Mechanics, ASCE. 1984, 110: 518-535.
[40] Ba?ant Z.P., Pfeiffer P.A. Determination of fracture energy from size effect and brittleness number[J]. ACI Material Journal. 1987, 84(6): 463-480.
[41] Ba?ant Z.P., Kazemi M.T. Size effect in fracture of ceramics and its use to determine fracture energy and effective process zone length[J]. Journal of the American Ceramic Society. 1990, 73(7): 1841-1853.
[42] Ba?ant Z.P. Is size effect caused by fractal nature of crack surfaces[R]. Preliminary Report, 1994.
[43] Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials[J]. International Journal of Solids and Structures. 1994, 31:291-302.
[44] Carpinteri A., Chiaia B. Multifractal nature of concrete fracture surfaces and size effects on nominal fractal fracture energy[J]. Materials and Structures. 1995, 28: 435-443.
[45] Carpinteri A., Chiaia B. Size effects on concrete fracture energy: dimensional transition from order to disorder[J]. Materials and Structures. 1996, 29: 259-266.
[46]谢和平,鞠杨.分数维空间中的损伤力学研究初探[J].力学学报. 1999, 31(3): 300-310.
[47] Carpinteri A., Chiaia B. Crack-resistance behavior as a consequence of self-similar fracture topologies[J]. International Journal of Fracture. 1996, 76(4): 327-340.
[48] Carpinteri A., Chiaia B., Nemati K.M. Complex fracture energy dissipation in concrete under different loading conditions[J]. Mechanics of Material. 1997, 26: 93-108.
[49] Carpinteri A., Chiaia B., Invernizzi S. Three dimensional fractal analysis of concrete fracture at the mesolevel[J]. Theoretical and Applied Fracture Mechanics. 1999, 31: 163-172.
[50] Carpinteri A., Chiaia B., Cornetti P. On the mechanics of quasi-brittle materials with a fractal microstructure[J]. Engineering Fracture Mechanics. 2003, 70: 2321-2349.
[51] Carpinteri A., Chiaia B., Cornetti P. A fractal theory for the mechanics of elastic materials[J]. Materials Science and Engineering A. 2004, 365: 235-240.
[52] Carpinteri A., Cornetti P., Puzzi S. Scaling laws and multiscale approach in the mechanics of heterogeneous and disordered materials[J]. Applied Mechanics Reviews. 2006, 59: 283-305.
[53] Carpinteri A., Puzzi S. A fractal approach to indentation size effect[J]. Engineering Fracture Mechanics. 2006, 73: 2110-2122.
[54] Carpinteri A., Paggi M. A fractal interpretation of size-scale effects on strength, friction and fracture energy of faults[J]. Chaos Solitons and Fractals. 2009, 39(2): 540-546.
[55] Saouma V.E., Barton C.C., Gamaleldin N.A. Fractal characterization of fracture surfaces in concrete[J]. Engineering Fracture Mechanics. 1990, 35: 47-53.
[56] Issa M.A., Hammad A.M. Assessment and evaluation of fractal dimension of concrete fracture surface digitized images[J]. Cement and Concrete Research. 1994, 24(2): 325-334.
[57] Baran G.R., Roques-Carmes C., Wehbi D., et al. Fractal characteristics of fracture surfaces[J]. Journal of the American Ceramic Society. 1992, 75(10): 2687-2691.
[58] Lung C.W., Jiang J., Tian E.K., et al. Relation between fractal dimension and roughness index for fractal surfaces[J]. Physical Review E. 1999, 60(5): 5121-5125.
[59] Borodich F.M. Some fractal models of fracture[J]. Journal of the Mechanics and Physics of Solids. 1997, 45(2): 239-259.
[60] Borodich F.M. Fractals and fractal scaling in fracture mechanics[J]. International Journal of Fracture. 1995, 95: 239-259.
[61] Borodich F.M. Fractals and fractal scaling in fracture mechanics[J]. International Journal of Fracture. 1999, 95: 239-259.
[62] Kachanov L.M. Introduction to Continuum Damage Mechanics[M]. New York: Kluwer Academic Pub, 1986.
[63] Lee J., Fenves G.L. Plastic-damage model for cyclic loading of concrete structures[J]. Journal of Engineering Mechanics, ASCE. 1998, 124: 892-900.
[64] Chaboche J.L. Continuum Damage Mechanics: Part 1 - General Concepts[J]. Journal of Applied Mechanics. 1988, 55: 59-72.
[65] Dyson B.F., Mclean D. Creep of nimonic 80A in torsion and tension[J]. Metal Science. 1977, 11: 37-45.
[66] Nemat-Nasset S., Iwakuma T., Accorsi M. Cavity growth and grain boundary sliding in polycrystalline solids[J]. Mechanics of Materials. 1986, 5: 317-329.
[67] Grassl P., Lundgren K., Gylltoft K. Concrete in compression: a plasticity theory with a novel hardening law[J]. International Journal of Solids and Structures. 2002, 39: 5205-5223.
[68] Grassl P., Jirásek M. Damage-plastic model for concrete failure[J]. International Journal of Solids and Structures. 2006, 43: 7166-7196.
[69] Cicekli U., Voyiadjis G.Z., Al-Rub R.K.A. A plasticity and anisotropic damage model for plain concrete[J]. International Journal of Plasticity. 2007, 23: 1874-1900.
[70]范天佑.断裂理论基础[M].北京:科学出版社, 2003.
[71] Wu Z., Yang S., Hu X., et al. An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams[J]. Engineering FractureMechanics. 2006, 73: 2166-2191.
[72] Brandt A.M., Prokopski G. On the fractal dimension of fracture surfaces of concrete elements[J]. Journal of Materials Science. 1993, 28(17): 4762-4766.
[73] Issa M.A., Issa M.A., Islam M.S., et al. Fractal dimension-a measure of fracture roughness and toughness of concrete[J]. Engineering Fracture Mechanics. 2003, 70: 125-137.
[74] Kornev V.M., Kurguzov V.D. Multiparametric sufficient criterion of quasi-brittle fracture for complicated stress state[J]. Engineering Fracture Mechanics. 2008, 75: 1099-1113.
[75] Ewalds H.L., Wanhill R.J.H. Fracture Mechanics[M]. London: Edward Arnold, 1984.
[76] Borodich F.M. Fracture energy in a fractal crack propagating in concrete or rock[J]. Doklady Rossiyskoy Akademii Nauk. 1992, 325: 1138-1141.
[77] Borodich F.M. Fracture energy of brittle and quasi-brittle fractal cracks[Z]. North-Holland: Elsevier, 199461-68.
[78] Irwin G.R. Analysis of stresses and strains near the end of a crack traversing a plate[J]. Journal of Applied Mechanics. 1957, 24: 361-364.
[79] Yavari A. Generalization of Barenblatt’s cohesive fracture theory for fractal cracks[J]. Fractals. 2002, 10(2): 189-198.
[80] Yavari A., Sarkani S., Moyer Jr E.T. The mechanics of self-similar and self-affine fractal cracks[J]. International Journal of Fracture. 2002, 114: 1-27.
[81] Wnuk M.P., Yavari A. On estimating stress intensity factors and modulus of cohesion for fractal cracks[J]. Engineering Fracture Mechancis. 2003, 70: 1659-1674.
[82] Gong B., Lai Z.H. Fractal characteristics of J-R resistance curves of Ti-6A1-4V alloys[J]. Engineering Fracture Mechanics. 1993, 44: 991-995.
[83] Ba?ant Z.P. Analysis of work-of-fracture method for measuring fracture energy of concrete[J]. Journal of Engineering Mechanics, ASCE. 1996, 122(2): 138-144.
[84] Ba?ant Z.P. Size effect on structural strength: a review[J]. Archive of Applied Mechanics. 1999, 69: 703-725.
[85] Ba?ant Z.P. Size effect[J]. International Journal of Solids and Structures. 2000, 37: 69-80.
[86] Ba?ant Z.P. Concrete fracture models: testing and practice[J]. Engineering Fracture Mechanics. 2002, 69(2): 165-205.
[87] Chiaia B., Van Mier J.G.M., Vervuurt A. Crack growth mechanisms in four different concretes: microscopic observation and fractal analysis[J]. Cement and Concrete Research. 1998, 28: 103-114.
[88] Saouma V.E., Barton C.C. Fractals, fractures and size effects in concrete[J]. Journal of Engineering Mechanics, ASCE. 1994, 120: 835-854.
[89] Trunk B., Wittmann F.H. Influence of size on fracture energy of concrete[J]. Materials and Structures. 2001, 31: 260-265.
[90]谢和平.分形-岩石力学导论[M].北京:科学出版社, 1996.
[91] Goldshteǐn R.V., Mosolov A.A. Cracks with a fractal surface[J]. Soviet Physics Doklady. 1991, 38(8): 603-605.
[92] Balankin A.S. Physics of fracture and mechanics of self-affine cracks[J]. Engineering Fracture Mechanics. 1997, 57(2): 135-203.
[93] Antoun T.H. Constitutive/failure model for the static and dynamic behaviors of concrete incorporating effects of damage and anisotropy[D]. Dayton, Ohio: The University of Dayton, 1991.
[94] Yavari A., Hockett K.G., Sarkani S. The fourth mode of fracture in fractal fracture mechanics[J]. International Journal of Fracture. 2000, 101(4): 365-384.
[95] Yavari A., Sarkani S., Moyer E.T. On fractal cracks in micropolar elastic solids[J]. Journal of Applied Mechanics. 2002, 69(1): 45-54.
[96] Ba?ant Z.P., Yavari A. The cause of size effect on structural strength: Is it fractal or energetic-statistical?[J]. Engineering Fracture Mechanics. 2005, 72(8): 1-31.
[97] Ba?ant Z.P., Yavari A. Response to A. Carpinteri, B. Chiaia, P. Cornetti and S. Puzzi's Comments on“Is The Cause of Size Effect on Structural Strength Fractal or Energetic-Statistical?”[J]. Engineering Fracture Mechanics. 2007, 74: 2897-2910.
[98] Higgins D.D., Bailey J.E. Fracture measurements on cement paste[J]. Journal of Materials Science. 1976, 11: 1995-2003.
[99] Cotterell B., Mai Y.W. Crack growth resistance curve and size effect in the fracture of cement paste[J]. Journal of Materials Science. 1987, 22: 2734-2738.
[100] Duan K., Hu X.Z., Wittmann F.H. Size effect on fracture resistance and fracture energy of concrete[J]. Materials and Structures. 2003, 38: 74-80.
[101] Tada H., Paris P.C., Irwin G.R. The Stress Analysis of Cracks Handbook[M]. 3 ed. New York: American Society of Mechanical Engineers, 2000.
[102] Saouma V.E., Fava G. On fractals and size effects[J]. International Journal of Fracture. 2006, 137: 231-249.
[103] GB/T 50081-2002,普通混凝土力学性能试验方法标准[S].北京:中国建筑工业出版社, 2002.
[104]唐明.混凝土孔隙分形特征的研究[J].混凝土. 2000, 8: 3-5.
[105] Pfeifer P., Avnir D. Chemistry nonintegral dimensions between two and three[J]. Journal of Chemical Physics. 1983, 79(7): 3369-3558.
[106] Katz A.J., Thompson A.H. Fractal Sandstone Pores: Implications for Conductivity and Pore Formation[J]. Physical Review Letters. 1985, 54(12): 1325-1328.
[107] Krohn C.E. Sandstone fractal and Euclidean pore volume distribution[J]. Geophysical Research. 1988, 93(B4): 3286-3296.
[108] Krohn C.E. Fractal measurements of sandstone, shales and carbonate[J]. Geophysical Research. 1988, 93(B4): 3297-3305.
[109] Rieu M., Sposito G. Fractal fragmentation, soil porosity, and soil water properties: I. Theory[J]. Soil Science Society of America Journal. 1991, 55: 1231-1238.
[110] Mandelbrot B.B. The fractal geometry of nature[M]. San Francisco, CA: W.H. Freeman and Company, 1983.
[111]谢和平.岩土介质的分形孔隙和分形粒子[J].力学进展. 1993, 23(2): 145-164.
[112]周宏伟,谢和平.多孔介质孔隙度与比表面积的分形描述[J].西安矿业学院学报. 1997, 17(2): 97-102.
[113]郑瑛,周英彪,郑楚光.多孔CaO孔隙结构的分形描述[J].华中科技大学学报. 2001, 29(3): 82-84.
[114] Guarracino L. A fractal constitutive model for unsaturated flow in fracture hard rocks[J]. Journal of Hydrology. 2006, 324: 154-162.
[115] Czernin W. Cement Chemistry and Physics for Civil Engineers[M]. Berlin: Bauverlag GMBH, 1980.
[116] Ficker T. Fractal strength of cement gels and universal dimension of fracture surfaces[J]. Theoretical and Applied Fractuer Mechanics. 2008, 50: 167-171.
[117] Hildgen P., Nekka F., Hildgen F., et al. Macroporosity measurement by fractal analysis[J]. Physica A. 1997, 234: 593-603.
[118] Robler M., Odler I. Investigations on the relationship between porosity, structure and strength of hydrated Portland cement pastes: I. Effect of porosity[J]. Cement and Concrete Research. 1985, 15(2): 320-330.
[119] Fagerlund G. Strength and porosity of concrete[Z]. Academia, Prague: 1973: 4, D51-D73.
[120] Kearsley E.P., Wainwright P.J. The effect of porosity on the strength of foamed concrete[J]. Cement and Concrete Research. 2002, 32(2): 233-239.
[121] JGJ 55-2000,普通混凝土配合比设计规程[S].北京:中国建筑工业出版社, 2001.
[122] Prokopski G. Fracture toughness of concretes at high temperature[J]. Journal of Materials Science. 1995, 30: 1609-1612.
[123] Baker G. The effect of exposure to elevated temperatures on the fracture energy of plain concrete[J]. Materials and Structures. 1996, 29: 383-388.
[124] Abdel-Fattah H., Hamoush S.A. Variation of the fracture toughness of concrete with temperature[J]. Construction and Building Materials. 1997, 11(2): 105-108.
[125] Wang X., Wu B., Wang Q. Online SEM investigation of microcrack characteristics of concretes at various temperatures[J]. Cement and Concrete Research. 2005, 35: 1385-1390.
[126] Nielsen C.V., Bi?ani? N. Residual fracture energy of high-performance and normal concrete subject to high temperatures[J]. Materials and Structures. 2003, 36: 515-521.
[127] Menou A., Mounajed G., Boussa H. Residual fracture energy of cement paste, mortar and concrete subject to high temperature[J]. Theoretical and Applied Fracture Mechanics. 2006, 45: 64-71.
[128] Bear T.J., Barr B. Fracture toughness tests for concrete[J]. International Journal of Fracture. 1977, 13: 92-96.
[129] Top?u I.B. Fracture toughness of a solidified composite residual material[J]. Cement and Concrete Research. 1996, 26(4): 521-527.
[130] Mechtcherine V. Fracture mechanical behavior of concrete and the condition of its fracture surface[J]. Cement and Concrete Research. 2009, 39(7): 620-628.
[131] Yan A., Wu K., Zhang D., et al. Influence of concrete composition on the characterization of fracture surface[J]. Cement and Concrete Composites. 2003, 25(1): 153-157.
[132] Carpinteri A., Brighenti R., Davoli A., et al. A micromechanical model for the prediction of the temperature fracture behaviour dependence in metallic alloys[J]. Engineering Fracture Mechanics. 2008, 75: 3646-3662.
[133] Timoshenko S.P., Goodier J.N. Theory of elasticity[M]. 3 ed. USA: Mc Graw-Hill Book Co, 1970.
[134] Sneddon I.N., Lowengrub M. Crack problems in the Classical theory of elasticity[M]. New York: John Wiles & Sons. Inc, 1969.
[135] Xie H.P., Wang J.A. Direct fractal measurement of fracture surfaces[J]. International Journal of Solids and Structures. 1999, 36: 3073-3084.
[136] Zhou H.W., Xie H.P. Direct estimation of the fractal dimensions of a fracture surface of rock[J]. Surface Review and Letters. 2003, 10(5): 751-762.
[137] Xie H.P., Wang J.A., Stein E. Direct fractal measurement and multifractal properties of fracture surfaces[J]. Physics Letters A. 1998, 242: 41-50.
[138] Goldshteǐn R.V., Mosolov A.A. Fractal cracks[J]. Journal of Applied Mathematics and Mechanics. 1992, 51(4): 563-571.
[139]谢和平.岩石、混凝土损伤力学[M].徐州:中国矿业大学出版社, 1990.
[140] Kashtanov A.V., Petrov Y.V. Fractal models in fracture mechanics[J]. International Journal of Fracture. 2004, 128: 271-276.
[141] Mehta P.K., Monteiro P.J.M. Concrete: Microstructure, Properties and Materials[M]. 3 ed. New York: MeGraw Hill, 2005.
[142] John P., Shah S.P. Fracture mechanics analysis of high-strength concrete[J]. International Journal of Materials in Civil Engineering. 1989, 1(4): 185-198.
[143] Hsu T.T.C. Unified theory of reinforced concrete[M]. Florida: Boca Raton, 1993.
[144] Belarbi A., Hsu T.T.C. Constitutive laws of concrete in tension and reinforced bars stiffened by concrete[J]. Structural Journal, American concrete Institute. 1994, 91(4):465-474.
[145] Pang X.B., Hsu T.T.C. Behavior of reinforced concrete membrane elements in shear[J]. Structural Journal, American concrete Institute. 1995, 92(6): 665-679.
[146] Zhang L.X., Hsu T.T.C. Behavior and analysis of 100MPa concrete membrane elements[J]. Journal of Structure engineering. 1998, 124(1): 24-34.
[147] Zhu R.H., Hsu T.T.C., Lee J.Y. Rational shear modulus for smeared-crack analysis of reinforced concrete[J]. Structural Journal, American concrete Institute. 2001, 98(4): 443-450.
[148] Mattei N.J., Mehrabadi M.M., Zhu H.N. A micromechanical constitutive model for the behavior of concrete[J]. Mechanics of Materials. 2007, 39: 357-379.
[149] Karihaloo B.L., Santhikumar S. Application of a visco-elastic tension-softening constitutive model to cracked and ageing concrete[J]. Construction and Building Materials. 1999, 13: 15-21.
[150] Li Q.B., Zhang L.X., Ansari F. Damage constitutive for high strength concrete in triaxial cyclic compression[J]. International Journal of Solids and Structures. 2002, 39: 4013-4025.
[151] Nguyen G.D., Korsunsky A.M. Development of an approach to constitutive modelling of concrete: Isotropic damage coupled with plasticity[J]. International Journal of Solids and Structures. 2008, 45: 5483-5501.
[152] Voyiadjis G.Z., Taqieddin Z.N., Kattan P.I. Anisotropic damage–plasticity model for concrete[J]. International Journal of Plasticity. 2008, 24: 1946-1965.
[153] Badel P., Godard V., Leblond J. Application of some anisotropic damage model to the prediction of the failure of some complex industrial concrete structure[J]. International Journal of Solids and Structures. 2007, 44: 5848-5874.
[154] Epstein M., ?niatycki J. Fractal mechanics[J]. Physica D. 2006, 220: 54-68.
[155] Cherepanov G.P., Balankin A.S., Ivanova V.S. Fractal fracture mechanics-a review[J]. Engineering Fracture Mechanics. 1995, 51(6): 997-1033.
[156] Wang S.G. The dependence of the fractal dimension of fractured surface on material in three dimensions[J]. Physica B. 2004, 348: 183-189.
[157] West J.K., Mecholsky Jr J.J., Hench L.L. The application of fractal and quantumgeometry to brittle fracture[J]. Journal of Non-Crystalline Solids. 1999, 260: 99-108.
[158] Hadjileontiadis L.J., Douka E. Crack detection in plates using fractal dimension[J]. Engineering Structures. 2007, 29: 1612-1625.
[159] Wang Y.T., Diamond S. A fractal study of the fracture surfaces of cement pastes and mortars using a stereoscopic SEM method[J]. Cement and Concrete Research. 2001, 31: 1385-1392.
[160] Zhao Y.H. Crack pattern evolution and a fractal damage constitutive model for rock[J]. International Journal of Rock Mechanics and Mining Science. 1998, 35(3): 349-366.
[161] Nechnech W., Meftah F., Reynouard J.M. An elasto-plastic damage model for plain concrete subject to high temperatures[J]. Engineering Structures. 2002, 24: 597-611.
[162] Wu J.Y., Li J., Faria R. An energy release rate-based plastic-damage model for concrete[J]. International Journal of Solids and Structures. 2006, 43: 583-612.
[163] Guarracino L. A fractal constitutive model for unsaturated flow in fractured hard rocks[J]. Journal of Hydrology. 2006, 324: 154-162.
[164]钱济成,周建方.混凝土的两种损伤模型及其应用[J].河海大学学报. 1989, 3: 40-47.
[165] Wriggers P., Moftah S.O. Mesoscale models for concrete: Homogenisation and damage behavior[J]. Finite Elements in Analysis and Design. 2006, 42: 623-636.
[166] Kupfer H., Hilsdorf H.K., Rusch H. Behavior of concrete under biaxial stress[J]. Proc. ACI 66. 1969: 656-666.
[167] Hsieh S.S., Ting E.C., Chen W.F. A plasticity-fracture model for concrete[J]. International Journal of Solids and Structures. 1982, 18(3): 181-197.
[168] Han D.J., Chen W.F. A nonuniform hardening plasticity model for concrete materials[J]. Journal of Mechanics of Materials. 1985, 4: 283-302.
[169] Kang H.D. Triaxial constitutive model for plain and reinforced concrete behavior[D]. Colorado: University of Colorado, Boulder, 1997.
[170] Mazars J. Application de la mécanique de l’endommagement au comportement non linéaire etàla rupture du béton de structure[D]. Fracnce: UniversitéParis VI, 1984.
[171] Halm D., Dragon A. A model of anisotropic damage by mesocrack growth; unilateral effect[J]. International Journal of Damage Mechanics. 1996, 5(4): 384-402.
[172] Comi C., Perego U. Fracture energy based bi-dissipative damage model for concrete[J]. International Journal of Solids and Structures. 2001, 38: 6427-6454.
[173] Kr?tzig W., Polling R. An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[J]. Computers and Structures. 2004, 82: 1201-1215.
[174] Jason L., Huerta A., Pijaudier-Cabot G., et al. An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[J]. Computer Methods in Applied Mechanics and Engineering. 2006, 195(52): 7077-7092.
[175] Carol I., Rizzi E., Willam K.J. On the formulation of anisotropic elastics degradation. II. Generalized pseudo-Rankine model for tensile damage[J]. International Journal of Solids and Structures. 2001, 38: 519-546.
[176] Xie H.P., Gao F. The mechanics of cracks and a statistical strength theory for rocks[J]. International Journal of Rock Mechanics and Mining Sciences. 2000, 37: 477-488.
[177] Ortiz M. A constitutive theory for inelastic behavior of concrete[J]. Mechanics of Materials. 1985, 4: 67-93.
[178] Faria R., Oliver J., Cervera M. A strain-based plastic viscous-damage model for massive concrete structures[J]. International Journal of Solids and Structures. 1998, 35(14): 1533-1558.
[179] Faria R., Oliver J., Cervera M. On isotropic scalar damage models for the numerical analysis of concrete structures[Z]. Barcelona, Spain: CIMNE, 2000422-426.
[180] Lubliner J., Oliver S., O?ate E. A plastic-damage model for concrete[J]. International Journal of Solids and Structures. 1989, 25(3): 299-326.
[181] Simo J.C., Hughes T.J.R. Computational Inelasticity[M]. New York: Springer-Verlag, 1998.
[182]吴建营.基于损伤能释放率的混凝土弹塑性损伤本构模型及其在结构非线性分析中的应用[D].上海:同济大学, 2004.
[183]叶志明.各向异性材料与混凝土材料断裂力学引论[M].北京:中国铁道出版社, 2000.
[184] Gopalaratnam V.S., Shah S.P. Softening respose of plain concrete in direct tension[J].ACI Journal. 1985, 82(3): 310-323.
[185] Karsan A.I., Jirsa J.O. Behavior of concrete under compressive loadings[J]. Journal of the Structural Division, ASCE. 1969, 95: 2535-2563.
[186] Petersson P.E. Crack growth and development of fracture zones in plain concrete and similar materials[Z]. Sweden: University of Lund: 1981.
[187] Rots J.G., Kusters G.M.A., Blaauwendraad J. The need for fracture mechanics options in finite element models for concrete structures[Z]. United Kingdom. Swansea: Pineridge Press, 198419-32.
[188] Rots J., Nauta P., Kursters G., et al. Smeared crack approach and fracture localization in concrete[Z]. The Netherlands: Delft University of Technology: 19851-48.
[189] De Borst R. Non-linear analysis of factional materials[D]. The Netherlands: Delft University of Technology, 1986.
[190] Meyer R., Ahrens H., Duddeck H. Material model for concrete in cracked and uncracked states[J]. Journal of Engineering Mechanics Division, ASCE. 1994, 120(9): 1877-1895.
[2]谢和平,薛秀谦.分形应用中的数学基础与方法[M].北京:科学出版社, 1997.
[3] Weierstrass K.über Continuirliche Functionen eines reellen Arguments, die für keinen Werth des Letzteren einen bestimmten Differentialquotien besitzen[J]. Royal Prussian Academy of Science. 1872.
[4] Mandelbrot B.B. Fractals: form, chance, and dimension[M]. San Francisco: W. H. Freeman and Company, 1977.
[5] Mandelbrot B.B. The Fractal Geometry of Nature[M]. CA: W.H. Freeman and Company, 1983.
[6] Minkowski H. Ber die Ann herung an eine reelle Gr sse durch rationale Zahlen[J]. Mathematische Annalen. 1901, 54: 91-124.
[7] Hausdorff F. Dimension und ?usseres Mass[J]. Mathematische Annalen. 1919, 79: 157-179.
[8] Besicovitch A. On the sum of digits of real numbers represented in the dyadic system[J]. Mathematische Annalen. 1934, 110: 321-330.
[9] Bouligand G. Ensembles impropres et nombre dimensionnel[J]. Bulletin des Sciences Mathématiques. 1928, 52: 320-334.
[10] Pontrjagin L., Schnirelmann L. Sur une propriétémétrique de la dimension[J]. Annals of Mathematics. 1932, 33(2): 152-162.
[11] Kolmogorov A.N., Tikhomirov V.M.ε-Entropy andε-capacity[J]. Uspekhi Mat. Nauk. 1959, 14: 3-86.
[12] Lévy P. Les courbes planes ou gauches et les surfaces composées de parties semblales au tout[J]. Journal of L'Ecole Polytechnique. 1938, 111(78): 227-247, 249-291.
[13] Kahane J.P., Salem R. Ensembles parfaits et series trigonometriques[M]. Paris: Herman, 1963.
[14] Mandelbrot B.B., Passoja D.E., Paullay A.J. Fractal character of fracture surfaces of metals[J]. Nature. 1984, 308: 721-722.
[15] Falconer K. Fractal geometry: Mathematical Foundations and Applications[M]. 2 ed.England: John Wiley & Sons Ltd, 2003.
[16] Bouchaud J.P., Bouchaud E., Lapasset G., et al. Models of fractal crack[J]. Physical Review Letter. 1993, 71(14): 2240-2243.
[17] Mecholsky J.J. Fracture of Ceramics by Single-Pulse 10.6-μm Irradiation[J]. Journal of Applied Physics. 1979, 50(8): 5488-5491.
[18] Mecholsky Jr J.J. Estimating theoretical strength of brittle materials using fractal geometry[J]. Materials Letters. 2006, 60: 2485-2488.
[19] Bouchaud E., Lapasset G., Planès J. Fractal dimension of fractured surfaces: a universal value?[J]. Europhysical Letter. 1990, 13: 73-79.
[20] Mandelbrot B.B. Fractals and an art for the sake of art[J]. Leonardo; Suppl. 1989: 21-24.
[21] Mandelbrot B.B. Multifractal measures, especially for the geophysicist[J]. Pure and applied geophysics. 1989, 131: 5-42.
[22] Bouchaud E., Lapasset G., Planès J. Statistics of branched fracture surfaces[J]. Physical Review B. 1993, 48(5): 2917-2928.
[23] Mosolov A.A. Cracks with fractal surfaces[J]. Doklady Akademii Nauk SSSR. 1991, 319(4): 840-844.
[24] Bouchaud E., Bouchaud J.P. Fracture surfaces: Apparent roughness, relevant length scales, and fracture toughness[J]. Physics Review B. 1994, 50(23): 17752-17755.
[25] Shi D., Jiang J., Lung C. Correlation between the scale-dependent fractal dimension of fracture surfaces and the fracture toughness[J]. Physical Review B. 1996, 54(24): 17355-17358.
[26] Milman V.Y., Stelmashenko N.A., Blumenfeld R. Fracture surfaces: a critical review of fractal studies and novel morphological analysis of scanning tunneling microscopy measurements[J]. Progress in Materials Science. 1994, 38: 425-474.
[27] Nagahama H. A fractal criterion for ductile and brittle fracture[J]. Journal of Applied Physics. 1994, 75(6): 3220-3222.
[28] Mu Z.Q., Lung C.W. Studies on the fractal dimension and fracture toughness of steel[J]. Journal of Physics D: Applied Physics. 1988, 21(5): 848-850.
[29] Hillerborg A., Modeer M., Petersson P.E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[J]. Cement and ConcreteResearch. 1976, 6: 773-782.
[30] Wittmann F.H., Mihashi H., Nomura N. Size effect on fracture energy of concrete[J]. Engineering Fracture Mechanics. 1990, 35: 107-115.
[31] Wu K.R., Yan A., Liu J.Y., et al. Reconstruction and analysis of 3-D profile of fracture surface of concrete[J]. Cement and Concrete Research. 2000, 30: 981-987.
[32] Griffith A.A. The phenomena of rupture and flow in solids[J]. Philosophical Transactions of the Royal Society. 1920, 221: 163-198.
[33] Weibull W. A statistical representation of fatigue failures in solids[M]. Sweden: Transactions of the Royal Institute of Technology, 1949.
[34] Weibull W. A statistical distribution function of wide applicability[J]. Journal of Applied Mechanics, ASME. 1951, 18: 293-297.
[35] Kaplan M.F. Strains and stresses of concrete at initiation of cracking and near failure[J]. Journal of the American Concrete Institute Proceedings. 1963, 60(7): 853-880.
[36] Bazant Z.P., Kaplan M.F. Concrete at High Temperatures: Material Properties and Mathematical Models[M]. Essex: Longman House, 1996.
[37] Kesler C.E., Naus D.J., Lott J.L. Fracture mechanics-its applicability to concrete[J]. International Conference on the Mechanical Behaviour of Materials. 1972, 4: 113-124.
[38] Walsh P.F. Crack initiation in plain concrete[J]. Magazine of Concrete Research. 1976, 28(94): 37-41.
[39] Ba?ant Z. Size effect in blunt fracture: Concrete, rock, metal[J]. Journal of Engineering Mechanics, ASCE. 1984, 110: 518-535.
[40] Ba?ant Z.P., Pfeiffer P.A. Determination of fracture energy from size effect and brittleness number[J]. ACI Material Journal. 1987, 84(6): 463-480.
[41] Ba?ant Z.P., Kazemi M.T. Size effect in fracture of ceramics and its use to determine fracture energy and effective process zone length[J]. Journal of the American Ceramic Society. 1990, 73(7): 1841-1853.
[42] Ba?ant Z.P. Is size effect caused by fractal nature of crack surfaces[R]. Preliminary Report, 1994.
[43] Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials[J]. International Journal of Solids and Structures. 1994, 31:291-302.
[44] Carpinteri A., Chiaia B. Multifractal nature of concrete fracture surfaces and size effects on nominal fractal fracture energy[J]. Materials and Structures. 1995, 28: 435-443.
[45] Carpinteri A., Chiaia B. Size effects on concrete fracture energy: dimensional transition from order to disorder[J]. Materials and Structures. 1996, 29: 259-266.
[46]谢和平,鞠杨.分数维空间中的损伤力学研究初探[J].力学学报. 1999, 31(3): 300-310.
[47] Carpinteri A., Chiaia B. Crack-resistance behavior as a consequence of self-similar fracture topologies[J]. International Journal of Fracture. 1996, 76(4): 327-340.
[48] Carpinteri A., Chiaia B., Nemati K.M. Complex fracture energy dissipation in concrete under different loading conditions[J]. Mechanics of Material. 1997, 26: 93-108.
[49] Carpinteri A., Chiaia B., Invernizzi S. Three dimensional fractal analysis of concrete fracture at the mesolevel[J]. Theoretical and Applied Fracture Mechanics. 1999, 31: 163-172.
[50] Carpinteri A., Chiaia B., Cornetti P. On the mechanics of quasi-brittle materials with a fractal microstructure[J]. Engineering Fracture Mechanics. 2003, 70: 2321-2349.
[51] Carpinteri A., Chiaia B., Cornetti P. A fractal theory for the mechanics of elastic materials[J]. Materials Science and Engineering A. 2004, 365: 235-240.
[52] Carpinteri A., Cornetti P., Puzzi S. Scaling laws and multiscale approach in the mechanics of heterogeneous and disordered materials[J]. Applied Mechanics Reviews. 2006, 59: 283-305.
[53] Carpinteri A., Puzzi S. A fractal approach to indentation size effect[J]. Engineering Fracture Mechanics. 2006, 73: 2110-2122.
[54] Carpinteri A., Paggi M. A fractal interpretation of size-scale effects on strength, friction and fracture energy of faults[J]. Chaos Solitons and Fractals. 2009, 39(2): 540-546.
[55] Saouma V.E., Barton C.C., Gamaleldin N.A. Fractal characterization of fracture surfaces in concrete[J]. Engineering Fracture Mechanics. 1990, 35: 47-53.
[56] Issa M.A., Hammad A.M. Assessment and evaluation of fractal dimension of concrete fracture surface digitized images[J]. Cement and Concrete Research. 1994, 24(2): 325-334.
[57] Baran G.R., Roques-Carmes C., Wehbi D., et al. Fractal characteristics of fracture surfaces[J]. Journal of the American Ceramic Society. 1992, 75(10): 2687-2691.
[58] Lung C.W., Jiang J., Tian E.K., et al. Relation between fractal dimension and roughness index for fractal surfaces[J]. Physical Review E. 1999, 60(5): 5121-5125.
[59] Borodich F.M. Some fractal models of fracture[J]. Journal of the Mechanics and Physics of Solids. 1997, 45(2): 239-259.
[60] Borodich F.M. Fractals and fractal scaling in fracture mechanics[J]. International Journal of Fracture. 1995, 95: 239-259.
[61] Borodich F.M. Fractals and fractal scaling in fracture mechanics[J]. International Journal of Fracture. 1999, 95: 239-259.
[62] Kachanov L.M. Introduction to Continuum Damage Mechanics[M]. New York: Kluwer Academic Pub, 1986.
[63] Lee J., Fenves G.L. Plastic-damage model for cyclic loading of concrete structures[J]. Journal of Engineering Mechanics, ASCE. 1998, 124: 892-900.
[64] Chaboche J.L. Continuum Damage Mechanics: Part 1 - General Concepts[J]. Journal of Applied Mechanics. 1988, 55: 59-72.
[65] Dyson B.F., Mclean D. Creep of nimonic 80A in torsion and tension[J]. Metal Science. 1977, 11: 37-45.
[66] Nemat-Nasset S., Iwakuma T., Accorsi M. Cavity growth and grain boundary sliding in polycrystalline solids[J]. Mechanics of Materials. 1986, 5: 317-329.
[67] Grassl P., Lundgren K., Gylltoft K. Concrete in compression: a plasticity theory with a novel hardening law[J]. International Journal of Solids and Structures. 2002, 39: 5205-5223.
[68] Grassl P., Jirásek M. Damage-plastic model for concrete failure[J]. International Journal of Solids and Structures. 2006, 43: 7166-7196.
[69] Cicekli U., Voyiadjis G.Z., Al-Rub R.K.A. A plasticity and anisotropic damage model for plain concrete[J]. International Journal of Plasticity. 2007, 23: 1874-1900.
[70]范天佑.断裂理论基础[M].北京:科学出版社, 2003.
[71] Wu Z., Yang S., Hu X., et al. An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams[J]. Engineering FractureMechanics. 2006, 73: 2166-2191.
[72] Brandt A.M., Prokopski G. On the fractal dimension of fracture surfaces of concrete elements[J]. Journal of Materials Science. 1993, 28(17): 4762-4766.
[73] Issa M.A., Issa M.A., Islam M.S., et al. Fractal dimension-a measure of fracture roughness and toughness of concrete[J]. Engineering Fracture Mechanics. 2003, 70: 125-137.
[74] Kornev V.M., Kurguzov V.D. Multiparametric sufficient criterion of quasi-brittle fracture for complicated stress state[J]. Engineering Fracture Mechanics. 2008, 75: 1099-1113.
[75] Ewalds H.L., Wanhill R.J.H. Fracture Mechanics[M]. London: Edward Arnold, 1984.
[76] Borodich F.M. Fracture energy in a fractal crack propagating in concrete or rock[J]. Doklady Rossiyskoy Akademii Nauk. 1992, 325: 1138-1141.
[77] Borodich F.M. Fracture energy of brittle and quasi-brittle fractal cracks[Z]. North-Holland: Elsevier, 199461-68.
[78] Irwin G.R. Analysis of stresses and strains near the end of a crack traversing a plate[J]. Journal of Applied Mechanics. 1957, 24: 361-364.
[79] Yavari A. Generalization of Barenblatt’s cohesive fracture theory for fractal cracks[J]. Fractals. 2002, 10(2): 189-198.
[80] Yavari A., Sarkani S., Moyer Jr E.T. The mechanics of self-similar and self-affine fractal cracks[J]. International Journal of Fracture. 2002, 114: 1-27.
[81] Wnuk M.P., Yavari A. On estimating stress intensity factors and modulus of cohesion for fractal cracks[J]. Engineering Fracture Mechancis. 2003, 70: 1659-1674.
[82] Gong B., Lai Z.H. Fractal characteristics of J-R resistance curves of Ti-6A1-4V alloys[J]. Engineering Fracture Mechanics. 1993, 44: 991-995.
[83] Ba?ant Z.P. Analysis of work-of-fracture method for measuring fracture energy of concrete[J]. Journal of Engineering Mechanics, ASCE. 1996, 122(2): 138-144.
[84] Ba?ant Z.P. Size effect on structural strength: a review[J]. Archive of Applied Mechanics. 1999, 69: 703-725.
[85] Ba?ant Z.P. Size effect[J]. International Journal of Solids and Structures. 2000, 37: 69-80.
[86] Ba?ant Z.P. Concrete fracture models: testing and practice[J]. Engineering Fracture Mechanics. 2002, 69(2): 165-205.
[87] Chiaia B., Van Mier J.G.M., Vervuurt A. Crack growth mechanisms in four different concretes: microscopic observation and fractal analysis[J]. Cement and Concrete Research. 1998, 28: 103-114.
[88] Saouma V.E., Barton C.C. Fractals, fractures and size effects in concrete[J]. Journal of Engineering Mechanics, ASCE. 1994, 120: 835-854.
[89] Trunk B., Wittmann F.H. Influence of size on fracture energy of concrete[J]. Materials and Structures. 2001, 31: 260-265.
[90]谢和平.分形-岩石力学导论[M].北京:科学出版社, 1996.
[91] Goldshteǐn R.V., Mosolov A.A. Cracks with a fractal surface[J]. Soviet Physics Doklady. 1991, 38(8): 603-605.
[92] Balankin A.S. Physics of fracture and mechanics of self-affine cracks[J]. Engineering Fracture Mechanics. 1997, 57(2): 135-203.
[93] Antoun T.H. Constitutive/failure model for the static and dynamic behaviors of concrete incorporating effects of damage and anisotropy[D]. Dayton, Ohio: The University of Dayton, 1991.
[94] Yavari A., Hockett K.G., Sarkani S. The fourth mode of fracture in fractal fracture mechanics[J]. International Journal of Fracture. 2000, 101(4): 365-384.
[95] Yavari A., Sarkani S., Moyer E.T. On fractal cracks in micropolar elastic solids[J]. Journal of Applied Mechanics. 2002, 69(1): 45-54.
[96] Ba?ant Z.P., Yavari A. The cause of size effect on structural strength: Is it fractal or energetic-statistical?[J]. Engineering Fracture Mechanics. 2005, 72(8): 1-31.
[97] Ba?ant Z.P., Yavari A. Response to A. Carpinteri, B. Chiaia, P. Cornetti and S. Puzzi's Comments on“Is The Cause of Size Effect on Structural Strength Fractal or Energetic-Statistical?”[J]. Engineering Fracture Mechanics. 2007, 74: 2897-2910.
[98] Higgins D.D., Bailey J.E. Fracture measurements on cement paste[J]. Journal of Materials Science. 1976, 11: 1995-2003.
[99] Cotterell B., Mai Y.W. Crack growth resistance curve and size effect in the fracture of cement paste[J]. Journal of Materials Science. 1987, 22: 2734-2738.
[100] Duan K., Hu X.Z., Wittmann F.H. Size effect on fracture resistance and fracture energy of concrete[J]. Materials and Structures. 2003, 38: 74-80.
[101] Tada H., Paris P.C., Irwin G.R. The Stress Analysis of Cracks Handbook[M]. 3 ed. New York: American Society of Mechanical Engineers, 2000.
[102] Saouma V.E., Fava G. On fractals and size effects[J]. International Journal of Fracture. 2006, 137: 231-249.
[103] GB/T 50081-2002,普通混凝土力学性能试验方法标准[S].北京:中国建筑工业出版社, 2002.
[104]唐明.混凝土孔隙分形特征的研究[J].混凝土. 2000, 8: 3-5.
[105] Pfeifer P., Avnir D. Chemistry nonintegral dimensions between two and three[J]. Journal of Chemical Physics. 1983, 79(7): 3369-3558.
[106] Katz A.J., Thompson A.H. Fractal Sandstone Pores: Implications for Conductivity and Pore Formation[J]. Physical Review Letters. 1985, 54(12): 1325-1328.
[107] Krohn C.E. Sandstone fractal and Euclidean pore volume distribution[J]. Geophysical Research. 1988, 93(B4): 3286-3296.
[108] Krohn C.E. Fractal measurements of sandstone, shales and carbonate[J]. Geophysical Research. 1988, 93(B4): 3297-3305.
[109] Rieu M., Sposito G. Fractal fragmentation, soil porosity, and soil water properties: I. Theory[J]. Soil Science Society of America Journal. 1991, 55: 1231-1238.
[110] Mandelbrot B.B. The fractal geometry of nature[M]. San Francisco, CA: W.H. Freeman and Company, 1983.
[111]谢和平.岩土介质的分形孔隙和分形粒子[J].力学进展. 1993, 23(2): 145-164.
[112]周宏伟,谢和平.多孔介质孔隙度与比表面积的分形描述[J].西安矿业学院学报. 1997, 17(2): 97-102.
[113]郑瑛,周英彪,郑楚光.多孔CaO孔隙结构的分形描述[J].华中科技大学学报. 2001, 29(3): 82-84.
[114] Guarracino L. A fractal constitutive model for unsaturated flow in fracture hard rocks[J]. Journal of Hydrology. 2006, 324: 154-162.
[115] Czernin W. Cement Chemistry and Physics for Civil Engineers[M]. Berlin: Bauverlag GMBH, 1980.
[116] Ficker T. Fractal strength of cement gels and universal dimension of fracture surfaces[J]. Theoretical and Applied Fractuer Mechanics. 2008, 50: 167-171.
[117] Hildgen P., Nekka F., Hildgen F., et al. Macroporosity measurement by fractal analysis[J]. Physica A. 1997, 234: 593-603.
[118] Robler M., Odler I. Investigations on the relationship between porosity, structure and strength of hydrated Portland cement pastes: I. Effect of porosity[J]. Cement and Concrete Research. 1985, 15(2): 320-330.
[119] Fagerlund G. Strength and porosity of concrete[Z]. Academia, Prague: 1973: 4, D51-D73.
[120] Kearsley E.P., Wainwright P.J. The effect of porosity on the strength of foamed concrete[J]. Cement and Concrete Research. 2002, 32(2): 233-239.
[121] JGJ 55-2000,普通混凝土配合比设计规程[S].北京:中国建筑工业出版社, 2001.
[122] Prokopski G. Fracture toughness of concretes at high temperature[J]. Journal of Materials Science. 1995, 30: 1609-1612.
[123] Baker G. The effect of exposure to elevated temperatures on the fracture energy of plain concrete[J]. Materials and Structures. 1996, 29: 383-388.
[124] Abdel-Fattah H., Hamoush S.A. Variation of the fracture toughness of concrete with temperature[J]. Construction and Building Materials. 1997, 11(2): 105-108.
[125] Wang X., Wu B., Wang Q. Online SEM investigation of microcrack characteristics of concretes at various temperatures[J]. Cement and Concrete Research. 2005, 35: 1385-1390.
[126] Nielsen C.V., Bi?ani? N. Residual fracture energy of high-performance and normal concrete subject to high temperatures[J]. Materials and Structures. 2003, 36: 515-521.
[127] Menou A., Mounajed G., Boussa H. Residual fracture energy of cement paste, mortar and concrete subject to high temperature[J]. Theoretical and Applied Fracture Mechanics. 2006, 45: 64-71.
[128] Bear T.J., Barr B. Fracture toughness tests for concrete[J]. International Journal of Fracture. 1977, 13: 92-96.
[129] Top?u I.B. Fracture toughness of a solidified composite residual material[J]. Cement and Concrete Research. 1996, 26(4): 521-527.
[130] Mechtcherine V. Fracture mechanical behavior of concrete and the condition of its fracture surface[J]. Cement and Concrete Research. 2009, 39(7): 620-628.
[131] Yan A., Wu K., Zhang D., et al. Influence of concrete composition on the characterization of fracture surface[J]. Cement and Concrete Composites. 2003, 25(1): 153-157.
[132] Carpinteri A., Brighenti R., Davoli A., et al. A micromechanical model for the prediction of the temperature fracture behaviour dependence in metallic alloys[J]. Engineering Fracture Mechanics. 2008, 75: 3646-3662.
[133] Timoshenko S.P., Goodier J.N. Theory of elasticity[M]. 3 ed. USA: Mc Graw-Hill Book Co, 1970.
[134] Sneddon I.N., Lowengrub M. Crack problems in the Classical theory of elasticity[M]. New York: John Wiles & Sons. Inc, 1969.
[135] Xie H.P., Wang J.A. Direct fractal measurement of fracture surfaces[J]. International Journal of Solids and Structures. 1999, 36: 3073-3084.
[136] Zhou H.W., Xie H.P. Direct estimation of the fractal dimensions of a fracture surface of rock[J]. Surface Review and Letters. 2003, 10(5): 751-762.
[137] Xie H.P., Wang J.A., Stein E. Direct fractal measurement and multifractal properties of fracture surfaces[J]. Physics Letters A. 1998, 242: 41-50.
[138] Goldshteǐn R.V., Mosolov A.A. Fractal cracks[J]. Journal of Applied Mathematics and Mechanics. 1992, 51(4): 563-571.
[139]谢和平.岩石、混凝土损伤力学[M].徐州:中国矿业大学出版社, 1990.
[140] Kashtanov A.V., Petrov Y.V. Fractal models in fracture mechanics[J]. International Journal of Fracture. 2004, 128: 271-276.
[141] Mehta P.K., Monteiro P.J.M. Concrete: Microstructure, Properties and Materials[M]. 3 ed. New York: MeGraw Hill, 2005.
[142] John P., Shah S.P. Fracture mechanics analysis of high-strength concrete[J]. International Journal of Materials in Civil Engineering. 1989, 1(4): 185-198.
[143] Hsu T.T.C. Unified theory of reinforced concrete[M]. Florida: Boca Raton, 1993.
[144] Belarbi A., Hsu T.T.C. Constitutive laws of concrete in tension and reinforced bars stiffened by concrete[J]. Structural Journal, American concrete Institute. 1994, 91(4):465-474.
[145] Pang X.B., Hsu T.T.C. Behavior of reinforced concrete membrane elements in shear[J]. Structural Journal, American concrete Institute. 1995, 92(6): 665-679.
[146] Zhang L.X., Hsu T.T.C. Behavior and analysis of 100MPa concrete membrane elements[J]. Journal of Structure engineering. 1998, 124(1): 24-34.
[147] Zhu R.H., Hsu T.T.C., Lee J.Y. Rational shear modulus for smeared-crack analysis of reinforced concrete[J]. Structural Journal, American concrete Institute. 2001, 98(4): 443-450.
[148] Mattei N.J., Mehrabadi M.M., Zhu H.N. A micromechanical constitutive model for the behavior of concrete[J]. Mechanics of Materials. 2007, 39: 357-379.
[149] Karihaloo B.L., Santhikumar S. Application of a visco-elastic tension-softening constitutive model to cracked and ageing concrete[J]. Construction and Building Materials. 1999, 13: 15-21.
[150] Li Q.B., Zhang L.X., Ansari F. Damage constitutive for high strength concrete in triaxial cyclic compression[J]. International Journal of Solids and Structures. 2002, 39: 4013-4025.
[151] Nguyen G.D., Korsunsky A.M. Development of an approach to constitutive modelling of concrete: Isotropic damage coupled with plasticity[J]. International Journal of Solids and Structures. 2008, 45: 5483-5501.
[152] Voyiadjis G.Z., Taqieddin Z.N., Kattan P.I. Anisotropic damage–plasticity model for concrete[J]. International Journal of Plasticity. 2008, 24: 1946-1965.
[153] Badel P., Godard V., Leblond J. Application of some anisotropic damage model to the prediction of the failure of some complex industrial concrete structure[J]. International Journal of Solids and Structures. 2007, 44: 5848-5874.
[154] Epstein M., ?niatycki J. Fractal mechanics[J]. Physica D. 2006, 220: 54-68.
[155] Cherepanov G.P., Balankin A.S., Ivanova V.S. Fractal fracture mechanics-a review[J]. Engineering Fracture Mechanics. 1995, 51(6): 997-1033.
[156] Wang S.G. The dependence of the fractal dimension of fractured surface on material in three dimensions[J]. Physica B. 2004, 348: 183-189.
[157] West J.K., Mecholsky Jr J.J., Hench L.L. The application of fractal and quantumgeometry to brittle fracture[J]. Journal of Non-Crystalline Solids. 1999, 260: 99-108.
[158] Hadjileontiadis L.J., Douka E. Crack detection in plates using fractal dimension[J]. Engineering Structures. 2007, 29: 1612-1625.
[159] Wang Y.T., Diamond S. A fractal study of the fracture surfaces of cement pastes and mortars using a stereoscopic SEM method[J]. Cement and Concrete Research. 2001, 31: 1385-1392.
[160] Zhao Y.H. Crack pattern evolution and a fractal damage constitutive model for rock[J]. International Journal of Rock Mechanics and Mining Science. 1998, 35(3): 349-366.
[161] Nechnech W., Meftah F., Reynouard J.M. An elasto-plastic damage model for plain concrete subject to high temperatures[J]. Engineering Structures. 2002, 24: 597-611.
[162] Wu J.Y., Li J., Faria R. An energy release rate-based plastic-damage model for concrete[J]. International Journal of Solids and Structures. 2006, 43: 583-612.
[163] Guarracino L. A fractal constitutive model for unsaturated flow in fractured hard rocks[J]. Journal of Hydrology. 2006, 324: 154-162.
[164]钱济成,周建方.混凝土的两种损伤模型及其应用[J].河海大学学报. 1989, 3: 40-47.
[165] Wriggers P., Moftah S.O. Mesoscale models for concrete: Homogenisation and damage behavior[J]. Finite Elements in Analysis and Design. 2006, 42: 623-636.
[166] Kupfer H., Hilsdorf H.K., Rusch H. Behavior of concrete under biaxial stress[J]. Proc. ACI 66. 1969: 656-666.
[167] Hsieh S.S., Ting E.C., Chen W.F. A plasticity-fracture model for concrete[J]. International Journal of Solids and Structures. 1982, 18(3): 181-197.
[168] Han D.J., Chen W.F. A nonuniform hardening plasticity model for concrete materials[J]. Journal of Mechanics of Materials. 1985, 4: 283-302.
[169] Kang H.D. Triaxial constitutive model for plain and reinforced concrete behavior[D]. Colorado: University of Colorado, Boulder, 1997.
[170] Mazars J. Application de la mécanique de l’endommagement au comportement non linéaire etàla rupture du béton de structure[D]. Fracnce: UniversitéParis VI, 1984.
[171] Halm D., Dragon A. A model of anisotropic damage by mesocrack growth; unilateral effect[J]. International Journal of Damage Mechanics. 1996, 5(4): 384-402.
[172] Comi C., Perego U. Fracture energy based bi-dissipative damage model for concrete[J]. International Journal of Solids and Structures. 2001, 38: 6427-6454.
[173] Kr?tzig W., Polling R. An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[J]. Computers and Structures. 2004, 82: 1201-1215.
[174] Jason L., Huerta A., Pijaudier-Cabot G., et al. An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[J]. Computer Methods in Applied Mechanics and Engineering. 2006, 195(52): 7077-7092.
[175] Carol I., Rizzi E., Willam K.J. On the formulation of anisotropic elastics degradation. II. Generalized pseudo-Rankine model for tensile damage[J]. International Journal of Solids and Structures. 2001, 38: 519-546.
[176] Xie H.P., Gao F. The mechanics of cracks and a statistical strength theory for rocks[J]. International Journal of Rock Mechanics and Mining Sciences. 2000, 37: 477-488.
[177] Ortiz M. A constitutive theory for inelastic behavior of concrete[J]. Mechanics of Materials. 1985, 4: 67-93.
[178] Faria R., Oliver J., Cervera M. A strain-based plastic viscous-damage model for massive concrete structures[J]. International Journal of Solids and Structures. 1998, 35(14): 1533-1558.
[179] Faria R., Oliver J., Cervera M. On isotropic scalar damage models for the numerical analysis of concrete structures[Z]. Barcelona, Spain: CIMNE, 2000422-426.
[180] Lubliner J., Oliver S., O?ate E. A plastic-damage model for concrete[J]. International Journal of Solids and Structures. 1989, 25(3): 299-326.
[181] Simo J.C., Hughes T.J.R. Computational Inelasticity[M]. New York: Springer-Verlag, 1998.
[182]吴建营.基于损伤能释放率的混凝土弹塑性损伤本构模型及其在结构非线性分析中的应用[D].上海:同济大学, 2004.
[183]叶志明.各向异性材料与混凝土材料断裂力学引论[M].北京:中国铁道出版社, 2000.
[184] Gopalaratnam V.S., Shah S.P. Softening respose of plain concrete in direct tension[J].ACI Journal. 1985, 82(3): 310-323.
[185] Karsan A.I., Jirsa J.O. Behavior of concrete under compressive loadings[J]. Journal of the Structural Division, ASCE. 1969, 95: 2535-2563.
[186] Petersson P.E. Crack growth and development of fracture zones in plain concrete and similar materials[Z]. Sweden: University of Lund: 1981.
[187] Rots J.G., Kusters G.M.A., Blaauwendraad J. The need for fracture mechanics options in finite element models for concrete structures[Z]. United Kingdom. Swansea: Pineridge Press, 198419-32.
[188] Rots J., Nauta P., Kursters G., et al. Smeared crack approach and fracture localization in concrete[Z]. The Netherlands: Delft University of Technology: 19851-48.
[189] De Borst R. Non-linear analysis of factional materials[D]. The Netherlands: Delft University of Technology, 1986.
[190] Meyer R., Ahrens H., Duddeck H. Material model for concrete in cracked and uncracked states[J]. Journal of Engineering Mechanics Division, ASCE. 1994, 120(9): 1877-1895.