Criteria for Failure and Fracture

Abstract

Structural members may have geometric features (including holes, notches, and corners) that result in localized regions of high stress called stress concentrations. The ratio C of the maximum stress resulting from a stress concentration to a defined nominal stress is called the stress concentration factor. Graphs of stress concentration factors are presented for an axially loaded bar, a stepped circular bar subjected to torsion, and a rectangular beam subjected to bending. Failure in brittle materials occurs when a component breaks or fractures, whereas in ductile materials failure is usually defined to occur with the onset of yielding. For a brittle material, the maximum normal stress and Mohr’s failure criteria and for a ductile material the Tresca and von Mises yield criteria are discussed. Fatigue is subdivided into low-cycle, high-cycle, and fatigue crack growth. In low-cycle fatigue, stress levels exceed the yield stress, and the number of cycles to failure is relatively low (<10³). High-cycle fatigue can occur when stress levels are lower than the yield stress, and failure may require 10³ to 10⁶ cycles. An endurance curve is a graph of the stress amplitude or fatigue strength as a function of the number of cycles to failure at zero mean stress. For some materials, there is a stress amplitude, the fatigue limit, below which fatigue life is essentially infinite. Keeping stress levels below the fatigue limit is known as safe life design. When a structural component contains a preexisting crack, fracture mechanics can be used to determine the state of stress and predict when failure will occur. The stress intensity factor K is a measure of the magnitude of the stress in the neighborhood of a crack tip. In many cases stress intensity factors can be expressed in the form $K=\sigma \sqrt{\pi a}\;Q\left(a/W\right),$ where σ is the stress, a is the crack length, and Q(a/W) is a function called the configuration factor. The value of the stress intensity factor at which fast crack growth begins is called the fracture toughness. Under cyclic or repeated loading, cracks can grow at stress intensity factor levels that are lower than the fracture toughness. In this situation, the resulting fatigue crack growth or slow growth is governed by the Paris law da/dN = A(ΔK)ⁿ, and the number of loading cycles required for failure is determined by calculating the number of cycles necessary for the crack length to reach the critical value for fast growth.

Chapter Summary

12.1.1 Stress Concentrations

Structural members may have geometric features (including holes, notches, and corners) that result in localized regions of high stress called stress concentrations. The ratioC of the maximum stress resulting from a stress concentration to a defined nominal stress is called the stress concentration factor. Stress concentration factors are given for an axially loaded bar in Figs. 12.3, 12.4, and 12.5, for a stepped circular bar subjected to torsion in Fig. 12.7, and for a rectangular beam subjected to bending in Figs. 12.10 and 12.11.

12.1.2 Failure

Failure in brittlematerials occurs when a component breaks or fractures, whereas in ductilematerials failure is usually defined to occur with the onset of yielding.

The maximum normal stress criterionstates that fracture of a brittle material occurs when

$$ \operatorname{Max}\left(|{\sigma}_1|, |{\sigma}_2|, |{\sigma}_3|\right)={\sigma}_{\mathrm{U}}, $$

(12.5)

where σ₁, σ₂, σ₃ are the principal stresses and σ_U is the ultimate stress. In plane stress, the values of σ₁ and σ₂ for which fracture will not occur are bounded by the square region shown in Fig. 12.13. It is bounded by the lines

$$ {\sigma}_1=\pm {\sigma}_{\mathrm{U}}, {\sigma}_2=\pm {\sigma}_{\mathrm{U}}. $$

(12.6)

When the ultimate stress of a brittle material subjected to plane stress differs in tension and compression, Mohr’s failure criterion states that fracture occurs when σ₁ and σ₂ lie on the boundaries shown in Fig. 12.15(a) and described by Eqs. (12.8)–(12.11).

The Tresca criterion states that yielding of a ductile material occurs when the absolute maximum shear stress equals the yield stress in shear τ_Y. It can also be expressed in terms of the tensile yield stress σ_Y:

$$ \operatorname{Max}\left(|{\sigma}_1-{\sigma}_2|, |{\sigma}_2-{\sigma}_3|, |{\sigma}_1-{\sigma}_3|\right)={\sigma}_{\mathrm{Y}}. $$

(12.15)

The Tresca factor of safety is

$$ {\mathrm{FS}}_{\mathrm{T}}=\frac{\sigma_{\mathrm{Y}}}{\sigma_{\mathrm{T}}}, $$

(12.19)

where

$$ {\sigma}_{\mathrm{T}}=\operatorname{Max}\left(|{\sigma}_1-{\sigma}_2|,|{\sigma}_2-{\sigma}_3|,|{\sigma}_1-{\sigma}_3|\right). $$

(12.18)

In plane stress, the safe region according to the Tresca criterion is bounded by the hexagon in Fig. (a).

The von Mises criterionstates that yielding of a ductilematerial occurs when

$$ \frac{1}{2}\left[{\left({\sigma}_1-{\sigma}_2\right)}^2+{\left({\sigma}_2-{\sigma}_3\right)}^2+{\left({\sigma}_1-{\sigma}_3\right)}^2\right]={\sigma}_{\mathrm{Y}}^2. $$

(12.20)

The von Mises factor of safety is

$$ {\mathrm{FS}}_{\mathrm{M}}=\frac{\sigma_{\mathrm{Y}}}{\sigma_{\mathrm{M}}}, $$

(12.24)

where σ_M is the von Mises equivalent stress

$$ {\sigma}_{\mathrm{M}}=\frac{1}{\sqrt{2}}\sqrt{{\left({\sigma}_1-{\sigma}_2\right)}^2+{\left({\sigma}_2-{\sigma}_3\right)}^2+{\left({\sigma}_1-{\sigma}_3\right)}^2}. $$

(12.22)

In plane stress, the safe region according to the von Mises criterion is bounded by the ellipse in Fig. (a).

Fatigue is subdivided into low-cycle, high-cycle, and fatigue crack growth. In low-cycle fatigue, stress levels exceed the yield stress, and the number of cycles to failure is relatively low (<10³). High-cycle fatigue can occur when stress levels are lower than the yield stress, and failure may require 10³ to 10⁶ cycles.

Denoting the maximum and minimum values of the stress in constant frequency and amplitude loading by σ_max and σ_min, the stress amplitude σ_a and mean stress σ_m are

$$ {\sigma}_{\mathrm{a}}=\frac{\sigma_{\mathrm{m}\mathrm{ax}}-{\sigma}_{\mathrm{m}\mathrm{in}}}{2}, {\sigma}_{\mathrm{m}}=\frac{\sigma_{\mathrm{m}\mathrm{ax}}+{\sigma}_{\mathrm{m}\mathrm{in}}}{2}. $$

(12.25)

An S-Ncurve, or endurance curve, is a graph of the stress amplitude or fatigue strength as a function of the number of cycles N to failure at zero mean stress. For some materials, there is a stress amplitude, the fatigue limitσ_fat, below which fatigue life is essentially infinite. Keeping stress levels below the fatigue limit is known as safe life design. An empirical expression used to fit S-N curves is

$$ {\sigma}_{\mathrm{a}}={\sigma}_{\mathrm{fat}}+\frac{b}{N^c}. $$

(12.26)

Solving this equation for N yields

$$ N={\left(\frac{b}{\sigma_{\mathrm{a}}-{\sigma}_{\mathrm{fat}}}\right)}^{1/c}. $$

(12.27)

Empirical equations that can be used to determine the effect of a constant level of mean stress on the S-N curve include the Goodman relation (12.28), the Gerber parabola (12.29), and the Soderberg line (12.30).

Suppose that a component is subjected to a sequence of blocks of loading. Let n_i be the number of cycles applied in the ith block, and let N_i be the number of cycles required to fail the undamaged component at that stress amplitude and mean stress. Miner’s lawstates that the accumulated damage after k blocks of loading is

$$ D=\sum \limits_{i=1}^k\;{D}_i=\sum \limits_{i=1}^k\;\frac{n_i}{N_i}, $$

(12.31)

and that failure occurs when D = 1.

12.1.3 Fracture

The stress intensity factor K is a measure of the magnitude of the stress in the neighborhood of a crack tip (see Eq. 12.35). Stress intensity factors for a centrally cracked plate subjected to tension and a plate subjected to a central load on a crack face are given by Eqs. (12.37) and (12.38), respectively. In many cases stress intensity factors can be expressed in the form

$$ K=\sigma \sqrt{\pi a}Q\ \left(\frac{a}{W}\right), $$

(12.39)

where σ represents the stress, a is the crack length, and Q(a/W) is a function called the configuration factor.

The value of the stress intensity factor at which fast crack growth begins is called the fracture toughnessK_c. From Eq. (12.39), the criterion for fast crack growth is

$$ \sigma \sqrt{\pi a}Q\ \left(\frac{a}{W}\right)={K}_{\mathrm{c}}. $$

(12.40)

Let ΔK be the change in the stress intensity factor over one cycle in a cracked component subjected to cyclic loading. The derivative of the crack length a with respect to the number N of loading cycles is given by the Paris law

$$ \frac{da}{dN}=A{\left(\Delta K\right)}^n. $$

(12.41)

The constants A and n are material properties reflecting resistance to fatigue crack growth. Rearranging Eq. (12.41) and integrating give a relation between the number of cycles N and the crack length a:

$$ N=\frac{1}{A}{\int}_{a_0}^a\;\frac{da}{{\left(\Delta K\right)}^n}. $$

(12.42)

The number of cycles at which failure occurs is obtained by setting a equal to the critical crack length for fast crack growth. The critical crack length is determined from Eq. (12.40) with σ = σ_max, the maximum magnitude of the constant amplitude loading.