On the Deformation of Twisted Strips. 477 



of the fibre at distance x from the axis will be 



\°?¥, 



and the principal curvatures will everywhere be 



1 /Pi=- I lp2 = 4>- 

 We now suppose that the strip is permanently twisted so 

 that the helicoidal form may be taken as the unstrained state. 

 We proceed to calculate the potential energy per unit length 

 when the strip undergoes a small axial elongation e per unit 

 length and when its twist is increased by a small amount t. 

 In consequence of the lateral contraction due to longitudinal 

 elongation, the distance of any point from the axis will be 

 altered ; let the point x thus become displaced to a distance 

 x + u from the axis. To our order of approximation, the 

 strains in the middle surface will be 



clu 

 ^=dx 3 " = 0; 



<r 2 = * + i^(<f> + T) 2 -<£ 2 } =e + x 2 $T; 



while, for the changes of curvature, 



©=-*©- 



4 



+ 3 



8 



the directions of principal curvature being unaltered. Thus 

 the usual expressions* give for W, the total potential energy 

 due to stretching and bending, per unit length of the strip, 



nh 3 t t 2 dx ; 



J-a 



where E denotes Young's modulus, n the simple rigidity, and 

 jx Poisson's ratio. 



Since there is no force acting perpendicular to the axis of 

 the strip, we have by variation of u, 



* Basset tinds that the potential energy also contains terms depending 

 on the stretching of the middle surface and involving the cube of the 

 thickness. These can, however, in the present case, be safely neglected 

 in comparison with the corresponding terms involving the first power of 

 the thickness. It is different with the terms depending on bending, as 

 bending may be large in comparison with the stretching. 



