Derivation of the Weak Form
26 We will now apply the Galerkin method to the equation of elasticity and show that we will retrieve the principle of virtual work which was derived from a variational principle. Hence, in this particular case the two methods are indeed (as anticipated) equivalent.
27 Starting with the equilibrium equation (Eq. ??)
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28 Expanding the equations of equilibrium we have d<J XX I da xy , dgXz I h _ Q
8x dy dz x
^Ttr + ^W + ^+by = 0 'Ml d(jZx I 9czy I daZz h _ n dx dy ^ dz ^ vz ~ u
29 We multiply the first equation in Eq. 11.39 by the arbitrary function wx and we integrate over the volume5 Q:
/daxx f d axy f daxz f wx dfl + / wx ' dfl+ / wx ~dfl+ / wxbxdfl = 0 (11.40)
30 We now apply Green-Gauss theorem (a form of the divergence theorem6)
tVx&xx'FlxU'*- I ,», ~r I WxO'xiiTlytti- I (JxyU>*'
XU XXI(XU,± I ^^ UXXU:ZL I U'XU Xyl(yU:±- J ^^ ^ UX yt dwx
+ f wx(jxziizdr — f ^-^-axzdü+ f wxbxdQ = 0 (11.41-a)
31 Recalling that the tractions are given by Eq. ??
tx — ftxxnx + ft xy ny + ftxz nz ty — ftyxnx + ftyy ny + ftyz nz (11.42)
tz — ftzxnx + ftzyny + ft zz nz then the preceding equation reduces to
32 Similarly, we obtain for the other two equations of equilibrium
WytydT - J ( —+ -Q^-Vyy + -^ftyzj dQ + J wybydQ = 0 (11.44-a)
/f i dw dw dw \ f wztzdT - J \-^-azx + -^aZy + -^azzjdQ + J wzbzd,Q = 0 (11.44-b)
33 Summing those three equations
/ (Wxtx + Wyty + Wz tz )dT + / (Wx^X + Wy by + Wz bz) dQ
WTt wTb dwx dwy dwz (11.45)
'<7xx ftyy 0 ftzz
5 Note similarity
-Ty——11 xx—I-r-j-11 yy—^ zz i fth the derivation of the principle of virtual work.
From E
34 We next focus on the third term in the preceding equation. For w = [ wx wy wz \T, then
35 Hence, substituting Eq. 11.47 into Eq. 11.45 we obtain j (Lw)TadQ. = j wTtdr + / wTbdQ J n J r J n
36 But since the L operator is symmetric, from Eq. 11.2, the above equation can also be written j wT(Lta)dtt = j wTtdr + I wTbdQ J n J r J n
which is the weak form of the differential equation of equilibrium subjected to the traction (natural) boundary conditions.
37 The weight vector w is completely arbitrary. In the principle of virtual displacement, it would correspond to the virtual displacement, compare this last equation with Eq. 5.11.
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