A unified solution approach for a large variety of antiplane shear and torsion notch problems: Theory and examples

Salviato, Marco; Zappalorto, Michele

doi:10.1016/j.ijsolstr.2016.10.022

Cited by 9 publications

(12 citation statements)

References 29 publications

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“…Consider the homogeneous, isotropic, and linear elastic notched solid shown in Figure A, subjected to a uniform remote shear stress τ ∞ . This loading condition gives rise only to an out‐of‐plane displacement field w (in the z‐direction), which was shown to be a harmonic function …”

Section: Preliminary Remarksmentioning

confidence: 69%

“…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 + i v and z = x + i y ) in combination with the condition u ( x , y ) = u 0 >0 (Figure B) …”

Section: Preliminary Remarksmentioning

confidence: 99%

“…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%

“…Accordingly, Equation simplifies to give

K_{III} ≅ τ_{\infty} \sqrt{πa} \frac{1 + normalk}{\sqrt{normalk}} \approx τ_{\infty} ((), 1 + \sqrt{\frac{t_{n}}{ρ_{n}}}) \sqrt{normalπ t_{c}} .

The latter substitution being valid as Equation , when t C << t n , gives a = t c /k, also accounting for Equation . It is worth noting that

τ_{\infty} ((), 1 + \sqrt{\frac{t_{n}}{ρ_{n}}})

is the elastic peak stress at the tip of the notch in the absence of any crack …”

Section: Stress Fields For a Crack Initiated At The Tip Of A Rounded mentioning

confidence: 99%

“…To this end, the conformal mapping technique is used in combination with a recent approach proposed by the authors based on the complex potential approach for antiplane elasticity. 22 Two relevant cases are addressed and solved separately:…”

Section: Introductionmentioning

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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Section: Preliminary Remarksmentioning

confidence: 69%

Section: Preliminary Remarksmentioning

confidence: 99%