A CUSP CATASTROPHE MODEL OF UNSTABLE FAILURE PROCESS OF QUASI-BRITTLE MATERIALS
准脆性材料破裂过程失稳的尖点突变模型

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.