By Louis M. Milne-Thomson

The time period antiplane used to be brought through L. N. G. FlLON to explain such difficulties as rigidity, push, bending by way of undefined, torsion, and flexure via a transverse load. checked out bodily those difficulties fluctuate from these of airplane elasticity already taken care of * in that convinced shearing stresses now not vanish. This ebook is worried with antiplane elastic platforms in equilibrium or in regular movement in the framework of the linear concept, and is predicated upon lectures given on the Royal Naval university, Greenwich, to officials of the Royal Corps of Naval Constructors, and on technical stories lately released on the arithmetic learn heart, usa military. My goal has been to take on each one challenge, so far as attainable, via direct instead of inverse or guessing tools. right here the advanced variable back assumes a massive position through simplifying equations and by means of introducing order into a lot of the remedy of anisotropic fabric. The paintings starts with an advent to tensors through an intrinsic approach which starts off from a brand new and straightforward definition. this permits elastic houses to be acknowledged with conciseness and actual readability. This direction by no means commits the reader to the specific use of tensor calculus, for the constitution so equipped up merges right into a extra regularly occurring shape. however it is assumed that the tensor tools defined right here will turn out important additionally in different branches of utilized mathematics.

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Thus if we write (4) we have 8~33 lJI=Xz-iyz=2i 00: - {,8(z2+ z2)+2Pzz+4(C1-e)mz}. 1 (15) we can take lJIo=- 8~ 33 {,8(z2+z2)+2,Bzz+4(C1-e)mz}. 3. The displacement We continue to take R e"'fJ= = 0 as the antiplane. 2 (10) e~fJ + K;fJ3333 (A1X + ElY + C1m) R, (1) so that for example o + -kk13 exx = exx (A1X 33 + ElY + C1m) R (2) For brevity write k' 3 Vs= -k-' Va= 33 (3) 1. 3. The displacement 49 Integrating (5)1 with respect to R we get W = ~ (A1X + B 1y + C1m) R2+ e~zR + wo(x,y) , (6) here Wo= Wo (x, y) is an arbitrary function independent of R.

Fig. 9 shows triads at P and Q which are homethetic. The triads at P and R are not homothetic, for the lines 1 and 3 are parallel and in the same sense but the lines 2 and 2', while parallel, are in opposite senses. 3 3 3 Q 2 2' P 1 Fig. 9 Definition: Equivalent triads are triads referred to which, Hooke's moduli of elasticity are the same. Equivalent directions are the directions of corresponding lines in equivalent triads. 82 that the form of the expression for the strainenergy is the same when referred to equivalent triads.

H46 h51 h•• lt53 h.. h5• h' G h61 h62 h6a h6' hS5 h66 xy 2eu z 2ezx 2e. v r hll h12 h13 h14 h,. h l6 h21 h22 h'3 h24 h•• h' 6 (1) 2e zx 2eXY ~6 h2• h'3 h.. h.. h26 ~3 ~. ~5 ~6 (2) h" h42 h43 h.. h' 5 h' 6 h51 h' 2 h5a k 5 • h•• k. 6 h61 h6• k6a k .. k6• k66 zx xy The second formulation is obtained by inverting the square matrix of the first. ;i{J;iy;i6) by numbering the pairs rx{J, yb as follows Pair Number I I 11 22 33 23 2 3 4 I I 12 31 5 I 6 We then have for example and so on. S. Anisotropy 23 We note that the coefficients h rs are of the physical dimensions of stress [M L-l T-2].