By Louis M. Milne-Thomson
The time period antiplane was once brought via L. N. G. FlLON to explain such difficulties as pressure, push, bending by means of undefined, torsion, and flexure by way of a transverse load. checked out bodily those difficulties fluctuate from these of airplane elasticity already taken care of * in that sure shearing stresses not vanish. This e-book is anxious with antiplane elastic structures in equilibrium or in regular movement in the framework of the linear concept, and relies upon lectures given on the Royal Naval collage, Greenwich, to officials of the Royal Corps of Naval Constructors, and on technical reviews lately released on the arithmetic study heart, usa military. My goal has been to take on each one challenge, so far as attainable, through direct instead of inverse or guessing tools. the following the complicated variable back assumes a big function by means of simplifying equations and by means of introducing order into a lot of the therapy of anisotropic fabric. The paintings starts with an advent to tensors by way of an intrinsic approach which starts off from a brand new and easy definition. this permits elastic houses to be acknowledged with conciseness and actual readability. This path on no account commits the reader to the specific use of tensor calculus, for the constitution so outfitted up merges right into a extra frequent shape. nonetheless it is assumed that the tensor equipment defined the following will end up valuable additionally in different branches of utilized mathematics.
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4) This means that the antiplane stress situation can exist only in body-fields in which (4) is satisfied. We shall complete this result by an assumption which will apply to all our subsequent work. Assumption. The body-field is independent of distance from the antiplane. 1. jaR is independent of R and therefore that = Rf(x, y) + g (x, y) (5) zz where the functions f and g are independent of R. All the foregoing applies independently of elastic response. 7 (5) D= K(4)' • S. Therefore the deformation tensor D is at most a linear function of R, and if S is independent of R, so is D.
1 (11), (12) would depend on R which is impossible since 8, CP, P are independent of R. Thus we must have Vo= V(x, y) + RV1 (x, y) = V + RV1 so that b - i b = ~ _ i ~ + R (0 i 0 1 2 ox VI _ oyVI) . oy ox Since b1 - ib 2 is independent of R we must have ° 1 oV ox = ' I oV oy = ° and therefore VI must be a constant, say c. Therefore Vo= V(x, y) + Rc and therefore (2) ozZ Ba=c- oR . (3) · b 'b oV,. 1 (13) by taking 8 0 = 2V, CPo = 0, so that in this case (5) 8 =4 (6) o· X ozoz + 2 V, The foregoing results do not depend on the assumption of elastic response of the material.
Then u + iv = - iyR + iiXR = iz-rR . Therefore iRis an angle of rotation and thus i is to be interpreted as an angle of twist per unit length measured along a line perpendicular to the antiplane. Here (IX, (J, y) is an arbitrary translation and (Wl> w 2, wa) an arbitrary small rotation so that the last three terms on the right-hand side of the expressions for u, v, ware the components of an arbitrary rigid body movement. We note that, since ezz = aw/aR, + Bly + Clm)R ezz = (AlX + ~ 'V SAlX2+ ~ ('V4Al+'Vs Bl)XY+ ~ B l 'V 4y 2 + A 2x (22) + B 2y + C2m.