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Show that two distinct Poisson coefficients ve,and vie are necessary to charac- terize the elastic behavior of this unidirectional layer.Numerical application:a layer of glass/epoxy.V=60%fiber volume fraction. Solution: Let the plate be subjected to two steps of loading as follows: b 1.A uniform stress oe along the e direction:the changes in lengths of the sides can be written as: b 2.A uniform stress o,along the t direction:for a relatively important elongation of the resin,one can only observe a weak shortening of the fibers along e.Using then another notation for the Poisson coefficient,the change in length can be written as: b a Now calculating the accumulated elastic energy under the two loadings above: When o is applied first,and then o,is applied, 1 W=soxaxexAb+-ic,×bxexha,.+axaxex△b, When o,is applied first,and then o is applied, W'=30×bxexha2+50×a×e×△b,+o,×bxe×△@ The final energy is the same: W=W' 2003 by CRC Press LLCShow that two distinct Poisson coefficients nt and nt are necessary to characterize the elastic behavior of this unidirectional layer. Numerical application: a layer of glass/epoxy. Vf = 60% fiber volume fraction. Solution: Let the plate be subjected to two steps of loading as follows: 1. A uniform stress s along the direction: the changes in lengths of the sides can be written as: 2. A uniform stress st along the t direction: for a relatively important elongation of the resin, one can only observe a weak shortening of the fibers along . Using then another notation for the Poisson coefficient, the change in length can be written as: Now calculating the accumulated elastic energy under the two loadings above: When s is applied first, and then st is applied, When st is applied first, and then s is applied, The final energy is the same: Db1 b -------- s E ----- ; Da1 a --------- nt E = = –------s Db2 b -------- nt Et –------st ; Da2 a --------- st Et = = ---- W 1 2 --s a e Db1 1 2 = ¥ ¥ ¥ + --st ¥ b e ¥ ¥ Da2 + s ¥ a e ¥ ¥ Db2 W¢ 1 2 --st b e Da2 1 2 = ¥ ¥ ¥ + --s ¥ a e ¥ ¥ Db1 + st ¥ b e ¥ ¥ Da1 W W= ¢ TX846_Frame_C18a Page 346 Monday, November 18, 2002 12:40 PM © 2003 by CRC Press LLC