Constitutive Equations based on Strain

The author is proposing the following constitutive equations;

sx = A'(ex) ex + B ey,
sy = A'(ey) ey + B ex,

where s is stress, e is strain, B is a elastic constant and A'(e) is a modulus of elasticity which decreases with extensile strain. These constitutive equations were derived from an observation that the trace of the axial strain, circumferential strain and axial stress is almost on a plane in the three dimensional space defined by the strains and stress, and an assumption that extensile strain is mainly due to crack opening which decreases elastic modulus.

The following characteristics are observed in the results of simulation by the above constitutive equations:

  • Strain-softening behavior including Class II;
  • Larger compressive strength than tensile strength; and
  • Increase in strength with confining pressure.
    Stress-strain curves for actual rocks can also be simulated by adjusting four constants. The critical extensile strain theory stands for the results of simulation.

    Example of stress-strai curves. Classes I and II appears depending on a constant B.