准脆性材料破裂过程失稳的尖点突变模型

岩石等准脆性材料在非刚性加载条件下的非稳定问题研究，有助于预报材料或结构的失稳破坏、推断灾害等突变现象的剧烈程度。首先推广了基于Weibull分布而建立的特殊的破裂过程失稳的尖点突变模型，得到材料破裂过程的失稳条件、失稳开始时系统的总变形、失稳前后的变形突跳和系统能量释放的一般表达式，从而阐明一般情况下，失稳及其剧烈程度完全由材料性质及加载体的刚度决定，而且失稳程度与干扰程度即总变形无关。结合基于Weibull和对数正态分布的两种本构曲线进行讨论，从中推得系统的起始刚度之比越小，越易发生失稳，系统的总变形越明显，而失稳突跳和能量释放越大;材料均匀程度的影响是双重的，当材料比较均匀时，均匀度减小，整个系统的总变形随之减小，失稳加剧；当材料过于不均匀，继续减小均质度则情况相反，以致不发生失稳。还利用建立的力学模型，给出上述两个材料本构方程的参数之间的关系。

A CUSP CATASTROPHE MODEL OF UNSTABLE FAILURE PROCESS OF QUASI-BRITTLE MATERIALS

The study of the instability of quasi-brittle materials under the action of a non-stiff loading system is helpful to forecast the unstable failure of materials or structures and to extrapolate the intensity of catastrophic failures.Firstly,the cusp catastrophe model established by Tang and Xu for rock unstable failure process is generalized in this paper by adopting a general form of material constitutive equation.The condition for the outbreak of instability,the total deformation and the energy release of the material and the loading system,and the deformation jump of the material in the catastrophe process are obtained.It is found that the catastrophe in the unstable failure of quasi-brittle materials and its intensity are determined completely by the material properties and the stiffness of the loading system and the catastrophe intensity is independent of the total deformation.Secondly,two types of material constitutive relationships based on the Weibull distribution and the lognormal distribution respectively are introduced into the generalized cusp model,from which two factors affecting the instability are founded further.One is the initial ratio of the stiffness of the loading system to the material stiffness.The instability takes place easier and its features become prominent as the ratio decreases.The other is the material homogeneity.As the homogeneity decreases from that of the totally homogeneous material,the total system deformation decreases and the instability is aggravated.But for a real loading system,which has a non-zero stiffness,as the homogeneity drops below a value,the total system deformation increases slightly and the instability weakens.Lastly,the method to find the relationship between the parameters in different types of constitutive equations of a material by means of the results from the generalized model is proposed as well.The relationships between the homogeneity indices of the two constitutive equations discussed in this paper are obtained by this method.These relationships are in excellent agreement with each other.