In the compression - compression - compression regime, the failure criterion of Willam and Warnke([37]) is implemented. In this case, F takes the form

(4–312)

and S is defined as

(4–313)

Terms used to define S are:

(4–314)

(4–315)

(4–316)

(4–317)

σ_h is defined by Equation 4–309 and the undetermined coefficients a₀, a₁, a₂, b₀, b₁, and b₂ are discussed below.

This failure surface is shown as Figure 4.29: 3-D Failure Surface in Principal Stress Space. The angle of similarity η describes the relative magnitudes of the principal stresses. From Equation 4–314, η = 0° refers to any stress state such that σ₃ = σ₂ > σ₁ (e.g. uniaxial compression, biaxial tension) while ξ = 60° for any stress state where σ₃ >σ₂ = σ₁ (e.g. uniaxial tension, biaxial compression). All other multiaxial stress states have angles of similarity such that 0° η 60°. When η = 0°, S₁ Equation 4–313 equals r₁ while if η = 60°, S₁ equals r₂. Therefore, the function r₁ represents the failure surface of all stress states with η = 0°. The functions r₁, r₂ and the angle η are depicted on Figure 4.29: 3-D Failure Surface in Principal Stress Space.

Figure 4.29:  3-D Failure Surface in Principal Stress Space

It may be seen that the cross-section of the failure plane has cyclic symmetry about each 120° sector of the octahedral plane due to the range 0° < η < 60° of the angle of similitude. The function r₁ is determined by adjusting a₀, a₁, and a₂ such that f_t, f_cb, and f₁ all lie on the failure surface. The proper values for these coefficients are determined through solution of the simultaneous equations:

(4–318)

with

(4–319)

The function r₂ is calculated by adjusting b₀, b₁, and b₂ to satisfy the conditions:

(4–320)

ξ₂ is defined by:

(4–321)

and ξ₀ is the positive root of the equation

(4–322)

where a₀, a₁, and a₂ are evaluated by Equation 4–318.

Since the failure surface must remain convex, the ratio r₁ / r₂ is restricted to the range

(4–323)

although the upper bound is not considered to be restrictive since r₁ / r₂ < 1 for most materials (Willam([36])). Also, the coefficients a₀, a₁, a₂, b₀, b₁, and b₂ must satisfy the conditions (Willam and Warnke([37])):

(4–324)

(4–325)

Therefore, the failure surface is closed and predicts failure under high hydrostatic pressure (ξ > ξ₂). This closure of the failure surface has not been verified experimentally and it has been suggested that a von Mises type cylinder is a more valid failure surface for large compressive σ_h values (Willam([36])). Consequently, it is recommended that values of f₁ and f₂ are selected at a hydrostatic stress level in the vicinity of or above the expected maximum hydrostatic stress encountered in the structure.

Equation 4–322 expresses the condition that the failure surface has an apex at ξ = ξ₀. A profile of r₁ and r₂ as a function of ξ is shown in Figure 4.30: A Profile of the Failure Surface.

Figure 4.30:  A Profile of the Failure Surface

As a Function of ξ_α

The lower curve represents all stress states such that η = 0° while the upper curve represents stress states such that η = 60°. If the failure criterion is satisfied, the material is assumed to crush.

In the regime, F takes the form

(4–326)

and S is defined as

(4–327)

where cos η is defined by Equation 4–314 and

(4–328)

(4–329)

The coefficients a₀, a₁, a₂, b₀, b₁, b₂ are defined by Equation 4–318 and Equation 4–320 while

(4–330)

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

This domain can also crush. See (Willam and Warnke([37])) for details.

In the tension - tension - compression regime, F takes the form

(4–331)

and S is defined as

(4–332)

If the failure criterion for both i = 1, 2 is satisfied, cracking occurs in the planes perpendicular to principal stresses σ₁ and σ₂. If the failure criterion is satisfied only for i = 1, cracking occurs only in the plane perpendicular to principal stress σ₁.

This domain can also crush. See (Willam and Warnke([37])) for details.

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

(4–333)

and S is defined as

(4–334)

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 4.31:  Failure Surface in Principal Stress Space with Nearly Biaxial Stress

Figure 4.31: Failure Surface in Principal Stress Space with Nearly Biaxial Stress represents the 3-D 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 3-D 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.