In the compression - compression - compression regime, the failure criterion of Willam and Warnke is implemented. In this case, F takes the form:

(2–9)

and S is defined as:

(2–10)

Terms used to define S are:

(2–11)

(2–12)

(2–13)

(2–14)

σ_h is defined by Equation 2–6 and the undetermined coefficients a₀, a₁, a₂, b₀, b₁, and b₂ are discussed below.

This failure surface is shown as Figure 2.1: 3D Failure Surface in Principal Stress Space. The angle of similarity η describes the relative magnitudes of the principal stresses. From Equation 2–11, 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 2–10 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 2.1: 3D Failure Surface in Principal Stress Space.

Figure 2.1: 3D 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:

(2–15)

with

(2–16)

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

(2–17)

ξ₂ is defined by:

(2–18)

and ξ₀ is the positive root of the equation

(2–19)

where a₀, a₁, and a₂ are evaluated by Equation 2–15.

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

(2–20)

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

(2–21)

(2–22)

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). 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 2–19 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 2.2: A Profile of the Failure Surface.

Figure 2.2: 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.