By Louis M. Milne-Thomson
The time period antiplane was once brought by means of L. N. G. FlLON to explain such difficulties as rigidity, push, bending via undefined, torsion, and flexure through a transverse load. checked out bodily those difficulties fluctuate from these of aircraft elasticity already taken care of * in that definite shearing stresses now not vanish. This e-book is anxious with antiplane elastic platforms in equilibrium or in regular movement in the framework of the linear idea, and is predicated upon lectures given on the Royal Naval collage, Greenwich, to officials of the Royal Corps of Naval Constructors, and on technical studies lately released on the arithmetic learn heart, usa military. My objective has been to take on each one challenge, so far as attainable, by means of direct instead of inverse or guessing equipment. the following the complicated variable back assumes a major function by means of simplifying equations and via introducing order into a lot of the therapy of anisotropic fabric. The paintings starts with an advent to tensors by means of an intrinsic process which begins from a brand new and straightforward definition. this allows elastic homes to be acknowledged with conciseness and actual readability. This direction under no circumstances commits the reader to the specific use of tensor calculus, for the constitution so outfitted up merges right into a extra common shape. but it is thought that the tensor equipment defined right here will turn out worthy additionally in different branches of utilized mathematics.
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92 (3) exhibits the scheme of moduli when there are three perpendicular planes of symmetry so that this scheme must apply to isotropic material. 94 (13) must also apply and therefore for * For details of the elastic symmetry of crystals see [LOVE]. 98. Isotropic material complete symmetry the scheme for isotropy must be o o o o o o o o o 2 (hl1-h12) 0 0 1 1 2 (h l1 - h12 ) (1) 0 1 2 (h l1 -h12 ) with two independent moduli. The strain-energy-function is therefore u= ~ hll(eil+e~2+e~3)+h12(e22e33+e33el1+el1ed + (hl1 - h12 ) (e~3+ e~l + er2) .
4. 5. 5. 3 (29) (30). 3 (1), (14) ezz=e~z+ LR, e~z=; (3) 1 (4) [V5AIX2+ (v4Al+V5Bl)XY+V4Bly2]+A2X+B2y+C2m. 32 (6), (7) oUl 1""" 1 ,..... 1 ,..... 5 ZX + vleZZ OVI I ,..... I ,..... I ,..... I""" ,..... 0 ay= 21 XX + 26 XY+ 22YY+ uyz+l25 zx + v2ezz (6) oVl OUI I ,..... I ,..... I ,..... I ,..... I""" 0 a;-+ay= 61 XX + 66 XY+ 62YY+ 64YZ+ 65 zx + v6eZZ OWl I4l XX ,..... Y+ I42YY+ ,..... IuYZ+ ,..... I""" Ty= U zx + V4 e0ZZ- Q'1 OWl ax = I XX ,..... 5l ,..... I ,..... 52. The equations satisfied by the stress functions 55 and (In o~ - 151 0:) Xx + (/46 oOx -/sa 0:) xy + (/42 o~ -/S2 0:) fy + (/44 o~ -/54 OOy) yz + (/4S OOx -Iss 0: )Zx 0 + ( 'V4 ax - 0 ) 0 ay ezz - 'V s 0 Ql' ~ (9) 0 Ql + ---ay = O.
It is also found to be characteristic of such problems that isotropic material which occupies planes perpendicular to the generators before deformation in general becomes warped after (6) Fig. 0. Introductory notions deformation, so as no longer to occupy planes perpendicular to the generators, or even to occupy planes at all. When the material is no longer isotropic it is found that the assumption (6) is too restrictive but can be replaced by yy is independent of R . (7) We shall therefore generalize FILON'S definition of antiplane systems in the manner described in the following section.
Antiplane Elastic Systems by Louis M. Milne-Thomson