Journal of Engineering Mechanics ASCE, 124 (1998), 842-851.

## ANALYSIS OF ROTATING CRACK MODEL

Milan Jirásek and Thomas Zimmermann
Swiss Federal Institute of Technology
LSC -DGC, EPFL, 1015 Lausanne, Switzerland

### Abstract

The paper extends the standard rotating crack model to a formulation with multiple orthogonal cracks. The corresponding stress evaluation algorithm is described, and a new derivation of the tangent stiffness matrix is presented. The derivation is extended to the case of equal principal strains, in which the classical formula for the tangent shear modulus fails. A condition excluding snapback of the stress-strain diagram for an arbitrary loading path is derived. Attention then shifts to stress locking, meaning here spurious stress transfer across widely opening cracks. The problem is illustrated by numerical examples. The mechanism of stress transfer is thoroughly analyzed, and the source of locking is detected. A remedy and extension of the model to a nonlocal formulation will be described in a separate paper.

### Summary and Conclusions

The rotating crack model has been extended to a formulation with multiple orthogonal cracks. If the stress-cracking strain curve is not too steep, the response of the model is unique in the sense that, for any strain increment, only one combination of loading/unloading assumptions for the cracking directions leads to an admissible solution.

In certain situations, the rotating crack model (even with a single crack) fails to give a consistently defined response. This happens if two or three principal strains are equal while the strain histories in the corresponding principal directions are different. If the strains are equal only at one isolated time instant, the tangent shear stiffness has a singularity but the problem can still be handled numerically. However, if the strains remain equal for a certain period of time, the generated stress cannot be consistently derived from the basic assumptions of the model.

Misalignment of the macroscopic crack (cracking band) direction with the sides of the finite elements leads to spurious stress transfer across a widely opening crack, which might produce a completely unrealistic structural response. The source of this type of stress locking is the normal stress in the lateral principal direction. Due to misalignment of the principal axes with the overall crack direction, the lateral stress contributes to internal forces orthogonal to the macroscopic crack. Spurious cohesive stress transfer across the cracking band induces secondary cracking, and it might even affect the final failure pattern.

Spurious stress transfer can be alleviated by combining the rotating crack model with a scalar damage model.

Please send me an email if you wish to receive the complete paper.

EPFL / 24 August 1998 / Milan.Jirasek@epfl.ch