Exact solution for the mode III stress fields ahead of cracks initiated at sharp notch tips

Salviato, Marco; Zappalorto, Michele; Maragoni, L.

doi:10.1016/j.euromechsol.2018.04.014

Cited by 14 publications

(8 citation statements)

References 38 publications

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“…Under these particular conditions, taking advantage of the properties of conformal mapping, the harmonicity of the displacement function ω ( u , v ) = w { x ( u , v ), y ( u , v ) } is maintained when curvilinear coordinates are used:

\frac{\partial^{2} normalω}{\partial u^{2}} + \frac{\partial^{2} normalω}{\partial v^{2}} = 0 .

Equation can be solved with the method of separation of variables:

ω = f ((), u) g ((), v) .

Substituting Equation into Equation leads to

f^{''} ((), u) g ((), v) + f ((), u) g^{''} ((), v) = 0

or, better:

\frac{f^{' normal'} ((), u)}{normalf ((), u)} = - \frac{g^{''} ((), v)}{g ((), v)} = λ^{2}

…”

Section: Preliminary Remarksmentioning

confidence: 99%

“…22 In addition, suppose that in the (x,y) plane the specific notch profile under analysis can be described mathematically using a certain conformal map z = z(ξ) (with ξ = u + iv and z = x + iy) in combination with the condition u(x, y) = u 0 >0 ( Figure 1B). 22 Under these particular conditions, taking advantage of the properties of conformal mapping, the harmonicity of the displacement function ω (u, v) = w{ x(u, v), y(u, v) } is maintained when curvilinear coordinates are used 22,33 :…”

Section: Preliminary Remarksmentioning

confidence: 99%

“…Equation 33 represents the general solution, valid for any value of the geometrical parameters. However, it is also interesting to determine the two limit solutions valid for short (t C << t n ) and long (t C >> t n ) cracks.…”

Section: Closed-form Solution For the Stress Intensity Factor For Umentioning

confidence: 99%

“…Substituting the previous equation into Equation 33, also considering that b a ¼ a 1 − k 2 À Á , one obtains…”

Section: Closed-form Solution For the Stress Intensity Factor For Umentioning

confidence: 99%

“…One possible solution to overcome this issue and to apply Equation 30 also to finite bodies is to re-write the whole stress field as a function of the stress intensity factor, K III , as commonly done in the context of conventional linear elastic fracture mechanics. To this end, Equation 33 can be substituted within Equation 30, to give…”

Section: Extension Of the Solution To Finite Casesmentioning

confidence: 99%

See 4 more Smart Citations

Analytical study on the mode III stress fields due to blunt notches with cracks

Zappalorto

Salviato

Maragoni

2018

Fatigue Fract Eng Mat Struct

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This work presents an analytical study on the stress fields induced by cracks nucleated at the tip of radiused notches under antiplane shear and torsion loadings. Closed‐form solutions for both shallow and deep notches are derived leveraging a combination of proper curvilinear coordinate systems and the complex potential method for antiplane elasticity. Based on the exact stress field solutions, convenient expressions for the stress intensity factors are derived. The accuracy of all the analytical formulations is verified by comparison with the results from a bulk of numerical analyses.

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