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ch4-LEFM3

# ch4-LEFM3 - Principle of Superposition For a linear elastic...

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Principle of Superposition For a linear elastic system, the individual components of stress, strain, displacement are additive. For example, two normal stress in x direction caused by two different external load can be added to get total stress P M P M P M P M a M M P P a x total I I a b total I a P M y A I On simillar lines stress intensity factors can also added K K (a) K (b) = Y a Y a K K (a) 0 = Y a 0 σ = = + σ π + σ π = + σ π +

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Principle of Superposition For example consider a plate having semicircular crack subjected to an internal pressure I 1/4 2 2 2 2 2 2 1.12 a K g( ) 3 a a , g( ) sin cos 8 8 c c σ π = φ ψ π π ψ = + φ = φ+ φ a 2c I I I For Semi circular crack , g( ) 1 2 K (total) K (b) K (c) 2 1.12 p a 0 2.24 = p a π ψ = φ = = - = π - π π π
Crack Tip Plasticity LEFM assumes a sharp crack tip, inducing infinite stress at the crack tip. But in real materials, the crack tip radius is finite and hence the crack tip stresses are also finite. In addition, inelastic deformations due to plasticity in metals, crazing in polymers leads to further relaxation of stresses. For metals with yielding, LEFM solutions are not accurate. A small region around the crack tip yields leading to a small plastic zone around it. For moderately yielding metals, LEFM solutions can be used with simple correction. For extensively yielding metals, alternative fracture parameters like CTOD, J-integral are to be used taking into account material non-linearity.

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Small Scale Plasticity Irwin’s Approach Normal stress σ yy based on elastic analysis is given by On the crack plane θ = 0 As a first approximation yielding occurs when σ yy = σ ys r y = first order estimate of plastic zone size. This is approximate I yy K 3 cos 1 sin sin 2 2 2 2 r θ θ θ � � � � � � σ = + � � � � � � π � � � � � � I yy K 2 r σ = π 2 2 I I ys y y ys ys y K 1 K a , r , or r 2 2 2 r σ σ = = = π σ σ π
Irwin’s Approach When yielding occurs, stresses redistribute in order to satisfy equilibrium conditions. The cross hatched regions represents forces active in the elastic analysis that cannot be carried in elasto-plastic analysis, because of the reason that the stresses cannot exceed σ ys .To redistribute this excessive force, the plastic zone size must increase. This is possible if the material immediately ahead of plastic zone carries more stress. Irwin proposed that plasticity makes the crack behave as if it were larger than actual physical size. Let the effective crack size be a eff , such that a eff = a+ δ, where δ is the correction. or

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To permit redistribution of stresses , the areas A and B must be the same 2 I ys 2 I eff y ys I eff K plastic constraint factor 1 K effective crack size = a a 2r = a+ Effective stress in tensity factor K Y a = σ = + π σ = σ π
A long rectangular plate has a width of 100 mm, thickness of 5 mm and an axial load of 50 kN. If the plate is made of titanium Ti-6AL-4V,(K IC =115 MPa-m 1/2 , σ ys =p10 MPa)

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