— David Wagner 2009/09/21 00:12
ME 5463-001 Fracture Mechanics, Homework #4
Homework 3.4 for β = 30° only. (Hint: determine the eigenvalue of λ with respect to the included angle β.)
The plate shown in Figure E3.1 contains a sharp notch with an included angle, β. Assuming that the plate is subjected to in-plane forces that produce only symmetric displacements with respect to the bisector of the wedge angle, (a) use the Williams formulation to determine the order of the singularity at the tip of the notch, and (b) derive expressions for the stresses in polar coordinates, retaining only the first three nonzero terms.
![delim{[}{ matrix{2}{2}{
{cos(lambda-1)alpha} {cos(lambda+1)alpha}
{(lambda-1)sin(lambda-1)alpha} {(lambda+1)sin(lambda+1)alpha}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_1 C_3
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_f2192e73be5e4a07eb433f7337a4bfbd.png "delim{[}{ matrix{2}{2}{
{cos(lambda-1)alpha} {cos(lambda+1)alpha}
{(lambda-1)sin(lambda-1)alpha} {(lambda+1)sin(lambda+1)alpha}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_1 C_3
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}")
=
![delim{[}{ matrix{2}{2}{
{sin(lambda-1)alpha} {sin(lambda+1)alpha}
{(lambda-1)cos(lambda-1)alpha} {(lambda+1)cos(lambda+1)alpha}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_2 C_4
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_7d11770b92554e44be8f98141803054b.png "delim{[}{ matrix{2}{2}{
{sin(lambda-1)alpha} {sin(lambda+1)alpha}
{(lambda-1)cos(lambda-1)alpha} {(lambda+1)cos(lambda+1)alpha}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_2 C_4
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}")
⇒
=
=
⇒ λ= (9 solutions, 4 are >0: 0.6, 1, 1.84, 1.94)
⇒ λ= (7 solutions, 3 are >0: 0.5, 1.2, 1.5)
This indicates the general solution is (at most) order 7 at the tip. (Some terms may vanish as r approaches 0 in the particular solution.)
![delim{[}{ matrix{2}{2}{
{cos(lambda-1){{11 pi}/{12}}} {cos(lambda+1){{11 pi}/{12}}}
{(lambda-1)sin(lambda-1){{11 pi}/{12}}} {(lambda+1)sin(lambda+1){{11 pi}/{12}}}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_1 C_3
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_a800ea1d2d052932fef32ff3c91c5820.png "delim{[}{ matrix{2}{2}{
{cos(lambda-1){{11 pi}/{12}}} {cos(lambda+1){{11 pi}/{12}}}
{(lambda-1)sin(lambda-1){{11 pi}/{12}}} {(lambda+1)sin(lambda+1){{11 pi}/{12}}}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_1 C_3
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}")
![delim{[}{ matrix{2}{2}{
{sin(lambda-1){{11 pi}/{12}}} {sin(lambda+1){{11 pi}/{12}}}
{(lambda-1)cos(lambda-1){{11 pi}/{12}}} {(lambda+1)cos(lambda+1){{11 pi}/{12}}}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_2 C_4
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}](http://wiki.waggy.org/dokuwiki/lib/exe/fetch.php?w=&h=&cache=cache&media=cache_mathplugin%3amath_951.5_b01b302aa37e3e252a90d370710d3231.png "delim{[}{ matrix{2}{2}{
{sin(lambda-1){{11 pi}/{12}}} {sin(lambda+1){{11 pi}/{12}}}
{(lambda-1)cos(lambda-1){{11 pi}/{12}}} {(lambda+1)cos(lambda+1){{11 pi}/{12}}}
} }{]}
delim{lbrace}{ matrix{2}{1}{C_2 C_4
} }{rbrace}
=
delim{lbrace}{ matrix{2}{1}{0 0
} }{rbrace}")
⇒
