In the tension - tension - tension regimes, F takes the form

(2–30)

and S is defined as

(2–31)

If the failure criterion is satisfied in directions 1, 2, and 3, cracking occurs in the planes perpendicular to principal stresses σ₁, σ₂, and σ₃.

If the failure criterion is satisfied in directions 1 and 2, cracking occurs in the plane perpendicular to principal stresses σ₁ and σ₂.

If the failure criterion is satisfied only in direction 1, cracking occurs in the plane perpendicular to principal stress σ₁.

Figure 2.3: Failure Surface in Principal Stress Space with Nearly Biaxial Stress

Figure 2.3: Failure Surface in Principal Stress Space with Nearly Biaxial Stress represents the 3D failure surface for states of stress that are biaxial or nearly biaxial. If the most significant nonzero principal stresses are in the σ_xp and σ_yp directions, the three surfaces presented are for σ_zp slightly greater than zero, σ_zp equal to zero, and σ_zp slightly less than zero. Although the three surfaces, shown as projections on the σ_xp - σ_yp plane, are nearly equivalent and the 3D failure surface is continuous, the mode of material failure is a function of the sign of σ_zp. For example, if σ_xp and σ_yp are both negative and σ_zp is slightly positive, cracking would be predicted in a direction perpendicular to the σ_zp direction. However, if σ_zp is zero or slightly negative, the material is assumed to crush.