APPROACH Theories have been developed for the static failure of metals based upon the two classes of material failure; ductile metals yield while brittle metals fracture. Thus separate failure theories exist for ductile and brittle metals: Failure Theories for Ductile Materials 1. Maximum Shear Stress (MSS)2. Distortion Energy (DE) 3. Ductile Coulomb-Mohr (DCM) Failure Theories for Brittle Materials 1. Maximum Normal Stress (MNS) 2. Brittle Coulomb-Mohr (BCM) These theories have grown out of hypotheses and experimental data in the following manner. 1. 2. 3. 4. Experimental failure data is first collected through tensile tests. The state of stress is correlated to the experimental data using Mohr’scircle plots. A failure theory is developed from a concept of the responsible failure mechanism. A design envelope is established based upon the theoretical and empirical design equations.
In light of the extensive dependence of failure theories on experimental data, we will first review the acquisition and correlation of tensile test data to failure theory. Subsequently, the criteria andapplication of specific failure theories will be discussed. TENSILE TEST
l0 TEXT FIGURE 3-1: A typical tension-test specimen. Some of the standard dimensions used for d0 are 2.5, 6.25, and 12.5 mm and 0.505 in, but other sections and sizes are in use. Common gauge lengths l0 used are 10, 25, and 50 mm and 1 and 2 in.
Text refers to Mechanical Engineering Design, 7th edition text byJoseph Edward Shigley, Charles R. Mischke and Richard G. Budynas; equations and examples with the prefix T refer to the present tutorial.
The tensile test is a standardized test (ASTM Standard E8 or E8m) and thus allows for the sharing of experimental data amongst researchers, typically in the form of stress-strain curves. Standard dimensions for the test-specimen are provided in Text Figure 3-1while a comparison of characteristic stress-strain curves for ductile and brittle materials are shown in Text Figure 3-2. These engineering stress-strain diagrams graphically demonstrate the difference in the failure behavior of ductile and brittle metals, and the need for separate failure criteria. However, the curves do not represent true values of stress and strain; rather, they are calculatedbased upon the original specimen cross-sectional area, prior to loading. Engineering Stress
P 4P = 2 A0 π d 0 l − l0 l0
(Text Eq. 3-1)
(Text Eq. 3-2)
Su Sf Sy Stress σ = P/A0
u f el pl y Sut Sy u, f y
εu Strain ε (a)
a Strain ε (b)
TEXT FIGURE 3-2: Stress-strain diagram obtained from the standard tensile test (a) Ductilematerial; (b) brittle material. pl marks the proportional limit; el, the elastic limit; y, the offset yield strength as defined by offset strain Oa; u, the maximum or ultimate strength; and f, the fracture strength.
Referring to Text Fig. 3-2 (a), point el, the elastic limit, defines the onset of permanent set while point a represents 0.2 percent permanent set with respect to the originalgauge length (ε = 0.002). A measure of the “true” stress and strain can be obtained by taking simultaneous measurements of the load and cross-sectional area during the tensile test experiment. Text
Shigley, Mischke & Budynas
Machine Design Tutorial 6: Static Failure Theories
Figure 3-4 shows a typical true stress-strain diagram for a ductile material. The curve between points u and fcorresponds to a reduction in stress as the specimen necks down.
σf σu True stress u
εu True strain
TEXT FIGURE 3-4: True stressstrain diagram plotted in Cartesian coordinates.
TEXT FIGURE 3-3: Tension specimen after necking.
CORRELATION OF STATE OF STRESS WITH TEST DATA For design, we need to relate the expected state of stress in a part to the actual state of stress...