Things that Did Not Work

⇒ P=Delta/C=Delta/{ka}= {B Delta}/{kA}.

The area under the load-displacement curve is, int{0}{Delta}{P(Delta)}d Delta =int{0}{Delta}{{B}/{kA}}Delta d Delta = {B}/{2kA}}Delta^2.

The crack area A=aB.

If the contour integral defining J is taken in pieces in the usual way,

  • J_Pi = J_pi_1 +J_pi_2 +J_pi_3 +J_pi_4 +J_pi_5.

If the crack is sufficiently deep,

  • J_pi_1 = J_pi_5 = 0.

Since a « W,

  • J_pi_3 = int{0}{L}.

If the load is uniformly distributed over the the top and bottom faces, Also, because of symmetry, the top and bottom faces have the same J, so

  • J_pi_2 = J_pi_4 =

C = Delta/P = ka ⇒

  • {partial C}/{partial a} = k;
  • Delta = Pka⇒ {partial Delta}/{partial a} = Pk.

J = J_P = int{0}{P}{({1/B}{partial Delta}/{partial a})_P} dP ={1/B}int{0}{P}Pk dP ={1/{2B}}k P^2

G = J_el = {{P^2}/{2B}}{{partial C}/{partial a}} = {k P^2}/{2 B}.

J = J_el + J_pl⇒

J_pl = J-J_el ={1/{2B}}k P^2 - {k P^2}/{2 B} =0.

=({1/2}-{1/{2B}})k P^2

  • Assume this is a long plate, Length L » W?
  • Assume plane stress?
  • Assume Ramberg-Osgood stress-strain relationship.

—-

U^circ = int{0}{P}{Delta}dP = int{0}{P}{Pka}dP = {1/2}kaP^2

* P= Delta/{ka}.

({partial G}/{partial a})_P=0.

  • b = W-a.
  • sigma = P/{B W}.
  • E = 1/C = 1/{ka}.
  • C = ka = Delta/P ⇒ Delta = Pka.

U(a) = int{0}{infty}{P Delta} d Delta

  • E = sigma/Delta_el => Delta_el = sigma/E = P/{BWE}

Delta = Pka = Delta_el + Delta_pl = P/{BWE} + Delta_pl

J = J_el + J_pl =int{0}{P}{({partial Delta}/{partial b})_P} dP =int{0}{P}{({partial(Pka)}/{partial b})_P} dP =int{0}{P}{Pk({partial (W-b)}/{partial b})_P} dP =int{0}{P}{Pk({partial W}/{partial b} - {partial b}/{partial b})_P} dP =int{0}{P}{-Pk} dP =-k int{0}{P}{P} dP =-k P^2

J_pl = J - J_el = -k P^2 - {K_I}^2/E

This should end up J_pl = 1/{2b} [ 2 int{0}{P} d Delta_pl - P Delta_pl].

=int{0}{P}{({partial(PkW-Pkb)}/{partial b})_P} dP

=int{0}{P}{({partial(PkW)}/{partial b})_P} dP -int{0}{P}{({partial(Pkb)}/{partial b})_P} dP

={1/2}int{0}{P}[({partial Delta_el}/{partial b})_P + ({partial Delta_pl}/{partial b})_P] dP ={K_I}^2/E + {1/2}int{0}{P}{({partial Delta_pl}/{partial b})_P} dP

FIXME

=ka{K_I}^2 + {1/2}int{0}{P}{({partial Delta_pl}/{partial b})_P} dP


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