混凝土各向异性损伤模型的数学和热力学表述

MATHEMATICAL AND THERMODYNAMICAL REPRESENTATIONS OF ANISOTROPIC DAMAGE MODELS FOR CONCRETE

迄今为止,无论是细观损伤模型、宏观(连续)损伤模型还是微平面模型等方法,均难以合理地描述混凝土等准脆性材料的各向异性损伤。为了(部分)解决这一问题并考察这些模型之间的内在联系,首先发展了连续损伤力学模型的数学表述方法即改进的张度表示理论,引入宏观体积模量和剪切模量定向分布函数,并将相应的球量损伤和偏量损伤定向分布函数展开为宏观损伤变量的傅里叶级数形式,建立了弹性损伤材料刚度张量与宏观损伤变量之间的一般关系式。利用上述方法,采用损伤标量描述球量空间各向同性损伤,分别采用损伤标量和二阶损伤张量描述偏量空间的非线性行为,建立了相应的各向同性损伤和正交各向异性损伤模型而无需引入应变等效或能量等效等唯象学假定。其次,基于内变量理论和不可逆热力学基本原理发展了与上述数学表述方法等效的热力学表述方法,给出了材料Helmholtz自由能的一般表达式,并利用最大损伤耗散原理建立了宏观损伤变量的演化法则。作为示例,给出了上述各向同性损伤和正交各向异性损伤模型的热力学表述方法。最后,为了探讨材料微裂缝演化对宏观力学行为的影响,选取反映材料微观结构的组构张量作为宏观损伤变量,发展了混凝土二阶和四阶微平面模型的数学表述和热力学表述,建立了宏观和微平面层次各物理量如损伤变量、损伤能释放率、损伤耗散以及损伤演化法则等之间的内在联系。所建立的损伤力学模型数学和热力学表述方法具有较好的通用性,可以应用于今后混凝土多尺度损伤模型研究。

Despite the substantial research efforts in the mesoscopic damage mechanics, continuum damage mechanics and microplane theory, etc., the modeling of damage induced anisotropy remains a challenging issue. Firstly, to （partially） solve the exhibited problems and to investigate the interrelations among these models, a mathematical representation of continuum damage model was established based on the improved stiffness representation theorem. The orientation distribution functions （ODFs） for the macroscopic bulk and shear moduli were introduced, and the corresponding macroscopic volumetric and deviatoric damage ODFs were expanded into the forms of Fourier serials, from which the coefficients in the irreducible form of stiffness tensor can be uniquely determined. Based on the above approach, the isotropic and anisotropic damage models, in which a damage scalar was used to describe the isotropic volumetric performance whereas an additional damage scalar or a second-order damage tensor was postulated to represent the isotropic or orthotropic deviatoric behavior, respectively, were derived without the introduction of the phenomenological assumptions （e.g. the strain equivalence, the energy equivalence, etc.）. Secondly, the equivalent thermodynamical representation of continuum damage model was developed within the framework of irreversible thermodynamics furnished with the theory of internal variables. The general expression of the Helmholtz free energy potential was proposed and the consistent evolution laws of the involved damage variables were derived from the postulate of maximum damage dissipation. As illustrative examples, the above isotropic and orthotropic damage models were re-derived from the thermodynamieal representation. Finally, to investigate the influences of the microstructural changes on the macroscopic nonlinear material behavior, second-order and fourth-order fabric tensors were proposed to develop the mathematical and thermodynamical representations for the orthotropic and anisotropic microplane models. The interrelations of physical quantities on the macroscopic and microplane levels, such as the damage variables and the conjugated damage energy release rates, the damage dissipations and the evolution laws of the damage variables, were systematically obtained. The proposed approach can be used to develop the multiscale damage model for concrete in the near future.