ch01-03 stress & strain & properties
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02 Solutions 46060 5/6/10 1:45 PM Page 19<br />
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2–34. If the normal <strong>strain</strong> is defined in reference to the<br />
final length, that is,<br />
P n<br />
œ<br />
¢s¿ -¢s<br />
= lim a b<br />
p:p¿ ¢s¿<br />
instead of in reference to the original length, Eq. 2–2, show<br />
that the difference in these <strong>strain</strong>s is represented as a<br />
œ<br />
second-order term, namely, P n -P n =P n P œ n .<br />
e B =<br />
¢S¿ -¢S<br />
¢S<br />
e B - e A<br />
œ<br />
=<br />
¢S¿ -¢S<br />
¢S<br />
-<br />
¢S¿ -¢S<br />
¢S¿<br />
= ¢S¿2 -¢S¢S¿ -¢S¿¢S +¢S 2<br />
¢S¢S¿<br />
= ¢S¿2 +¢S 2 - 2¢S¿¢S<br />
¢S¢S¿<br />
=<br />
(¢S¿ -¢S)2<br />
¢S¢S¿<br />
= ¢<br />
¢S¿ -¢S ¢S¿ -¢S<br />
≤¢ ≤<br />
¢S ¢S¿<br />
= e A e B<br />
œ<br />
(Q.E.D)<br />
19
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