Section 5–1 illustrated some ways that loss of function is manifested. Events such as distortion, permanent set, cracking, and rupturing are among the ways that a machine element fails. Testing machines appeared in the 1700s, and specimens were pulled, bent, and twisted in simple loading processes. If the failure mechanism is simple, then simple tests can give clues. Just what is simple? The tension test is uniaxial (that’s simple) and elongations are largest in the axial direction, so strains can be measured and stresses inferred up to “failure.” Just what is important: a critical stress, a critical strain, a critical energy? In the next several sections, we shall show failure theories that have helped answer some of these questions. Unfortunately, there is no universal theory of failure for the general case of material properties and stress state. Instead, over the years several hypotheses have been formulated and tested, leading to today’s accepted practices. Being accepted, we will characterize these “practices” as theories as most designers do. Structural metal behavior is typically classified as being ductile or brittle, although under special situations, a material normally considered ductile can fail in a brittle manner (see Sec. 5–12). Ductile materials are normally classified such that εf ≥ 0.05 and have an identifiable yield strength that is often the same in compression as in tension (Syt = Syc = Sy ). Brittle materials, εf < 0.05, do not exhibit an identifiable yield strength, and are typically classified by ultimate tensile and compressive strengths, Sut and Suc, respectively (where Suc is given as a positive quantity). The generally accepted theories are:
Ductile materials (yield criteria)
• Maximum shear stress (MSS), Sec. 5–4
• Distortion energy (DE), Sec. 5–5
• Ductile Coulomb-Mohr (DCM), Sec. 5–6
Brittle materials (fracture criteria)
• Maximum normal stress (MNS), Sec. 5–8
• Brittle Coulomb-Mohr (BCM), Sec. 5–9
• Modified Mohr (MM), Sec. 5–9
It would be inviting if we had one universally accepted theory for each material type, but for one reason or another, they are all used. Later, we will provide rationales for selecting a particular theory. First, we will describe the bases of these theories and apply them to some examples.
Distortion-Energy Theory for Ductile Materials
The distortion-energy theory predicts that yielding occurs when the distortion strain energy per unit volume reaches or exceeds the distortion strain energy per unit volume for yield in simple tension or compression of the same material.
The distortion-energy (DE) theory originated from the observation that ductile materials stressed hydrostatically (equal principal stresses) exhibited yield strengths reatly in excess of the values given by the simple tension test. Therefore it was postulated that yielding was not a simple tensile or compressive phenomenon at all, but, rather, that it was related somehow to the angular distortion of the stressed element. To develop the theory, note, in Fig. 5–8a, the unit volume subjected to any three dimensional stress state designated by the stresses σ1, σ2, and σ3. The stress state shown in Fig. 5–8b is one of hydrostatic normal stresses due to the stresses σav acting in each of the same principal directions as in Fig. 5–8a. The formula for σav is simply σav = σ1 + σ2 + σ33
(a)Thus the element in Fig. 5–8b undergoes pure volume change, that is, no angular distortion. If we regard σav as a component of σ1, σ2, and σ3, then this component can be subtracted from them, resulting in the stress state shown in Fig. 5–8c. This element is subjected to pure angular distortion, that is, no volume change.
This post was written by: Author Name
Your description comes here!
Get Free Email Updates to your Inbox!
Post a Comment