Crack propagation principle of heavy duty gear based on extended finite element method

In order to simulate the singularity and discontinuity of element displacement stress, it is realized by constructing the extended functions f (x) and H (x) with discontinuity characteristics. The calculated displacement field is:

Where, NJ (x), NK (x) – unit decomposition shape function; SH, SC – the set of H (x) and f (x) extension nodes; α J – crack penetration element; BK is the additional degree of freedom of a node.

Crack propagation is analyzed in ABAQUS, which needs to be simulated by damage model. It is divided into two parts: one is the damage process of element; The second is the final failure process. Therefore, according to the damage evolution law and the damage initial criterion as the failure mechanism, the damage model of the material is given. According to the failure principle of maximum principal stress, when the stress or strain of the material exceeds the initial critical criterion, the degradation begins. The formula is:

Where, σ Max – maximum allowable stress:

The softening rate of material stiffness can be expressed according to the damage evolution law. Under the influence of damage, the components of tangential and normal stress can be expressed as:

Where, D – average damage value, the initial value D is 0, and after the damage model is established, the value range of D is 0-1; TN, TX, Ti – normal stress and shear stress before damage; TN, TX, Ti – normal stress and shear stress after damage;

Energy based damage evolution is the most suitable type of damage evolution law in ABAQUS. Based on the state judgment of crack propagation by its type, the values of gequiv and gequivc in the type are compared, where gequiv is the equivalent strain energy release rate and gequivc is the critical equivalent strain energy release rate. For gequivc, the power criterion is adopted, and the formula is:

Where, G Ⅰ, G Ⅱ and G Ⅲ – crack energy release rate; G Ⅰ C, G Ⅱ C, G Ⅲ C – limit value of crack energy release rate; α m、 α n、 α O is the exponent of the power function.