Maximum Shear Stress Theory: Maximum Distortion Energy Theory

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4. Total Strain Energy theory or HAIGH’S THEORY
5. Maximum Distortion Energy theory or VONMISES AND HENCKY’S THEORY

1. Maximum Principal Stress theory (M.P.S.T)
According to M.P.S.T
Condition for failure is,
Maximum principal stress (1) failure stresses (Syt or Sut)
And Factor of safety (F.O.S) = 1
If σ1 is +ve then Syt or Sut σ1 is –ve then Syc or Suc
Condition for safe design,
Factor of safety (F.O.S) > 1
Maximum principal stress (1) ≤ Permissible stress (per)
Where permissible stress = Failure stress Factor of safety = Syt/N or Sut/N σ1≤ Syt/N or Sut/N------------------------Eq. (1)

2. Maximum Shear Stress theory (M.S.S.T)
Condition for failure,
Maximum shear stress induced at a critical Yield point under tri-axial combined stress>strength
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Maximum Distortion Energy Theory (M.D.E.T)
Condition for failure,
Maximum Distortion Energy/volume(M.D.E/vol.)>Distortion energy/volume at yield pointunder tension test (D.E/vol.) Y.P.]T.T
Condition for safe design,
Maximum Distortion Energy/volume ≤ Distortion energy/volume at yield pointUnder tension test ------------Eq.(11)
T.S.E/vol. = Volumetric S.E/vol. + D.E/vol.
D.E/vol. = T.S.E/vol. - Volumetric S.E /vol. ------------------Eq.(12)
Under hydrostatic stress condition, D.E/vol. = 0
Under pure shear stress condition, Volumetric S.E/vol. = 0
From equation (8)
T.S.E/vol. = 1/2E [σ12 + σ22 + σ32 - 2μ (σ1 σ2 + σ2 σ3 +σ3 σ1)]
Volumetric S.E/vol. = 1/2 (Average stress) (Volumetric strain)
= 1/2 ((σ1 + σ2 + σ3)/3) [((1-2μ)/E ) (σ1 + σ2 + σ3) ]
Vol. S.E/vol. = (1-2μ)/6E (σ1 + σ2 + σ3)2 ------------------Eq.(13)
From equation (12) and (13)
D.E/vol. = (1+μ)/6E [(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2] ---------------- Eq.(14)
To get [(D.E/vol.) Y.P.]T.T.
Substitute σ1 = σ = Syt/N , σ2 = σ3 = 0 in equation (14)
[(D.E/vol.) Y.P.]T.T.= (1+μ)/3E (Syt/N)^2--------------Eq.(15)
Substituting equation (14) and (15) in the condition for safe design , the following equation is obtained
[(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2] ≤ 2
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An energetic contour path integral (called J) was independent of the path around a crack.
The J-integral is equal to the strain energy release rate for a crack in a body subjected to monotonic loading. This is generally true, under quasi-static conditions, only for linear elastic materials. For materials that experience small-scale yielding at the crack tip, J can be used to compute the energy release rate under special circumstances such as monotonic loading in mode III (anti-plane shear). The strain energy release rate can also be computed from J for pure hardening plastic materials that undergo small-scale yielding at the crack tip.
The quantity J is not path-independent for monotonic mode I and mode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice showed that J is path-independent in plastic materials when there is no non-proportional loading. Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence. Such non-proportional loading is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials.
1.6 Bevel Gear

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