(a) Extension to the torsion of multi-material bars Consider now the case in which the cross...

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(a) Extension to the torsion of multi-material bars Consider now the case in which the cross section is composed by N different materials, with shear modulus G; (i=1,...,N), connected by smooth interfaces. With reference to figure la, let us denote by 2; the domain that hosts the ith material, by to the intersection of the boundary of 2; with the external boundary o, and by I; the interface between materials i and j (i #j). We make a distinction between 'open' bi-material interfaces, i.e. open curves whose extremal points lay on Io, and 'closed' interfaces, if they are closed curves inside the cross section. To emphasize this distinction, if Iij is a closed interface it will be formally denoted by Iij. Introduce the stress function as in (2.1). Using the same arguments of the classical approach, for the ith material, one obtains AF;(x, y)=-2G; in 2; and F(x, y) = 0 on Fo (2.5) On the generic interface I' define the normal and tangential unit vectors n and t such that (n, t, k) is right-handed. Bi-material matching conditions [14] require the continuity of the normal traction Tnz = F(x,y)/at and the continuity of the strain Yytz = Ttz/G in tangential direction. In terms of stress function, one finds and F;(x, y) aF;(x,y) F(x, y) = F(x, y) on Fij at at (2.6) 1 aF(x,y) 1 F;(x,y) Gi an G; an on Tij. If the jth material domain 2; is enclosed by closed curve fij, as for materials 2 and 4 in figure 1a, recalling (2.5), one obtains 1 f aF(x,y) 1 F; (x,y) Gi fr dl= dl n Gj n 21 = AF;(x, y) dA = -2A;, (2.7) where A; is the area of 2;. Once the stress function has been determined, the torsional stiffness is again obtained through (2.4). A prismatic shaft is made of two materials. The cross section of the two-material prismatic shaft has an outer equilateral triangle made of Material-1 (green area in Figure) and an inner square made of Material-2 (yellow area in Figure). Material-1 has shear modulus G and Material-2 has shear modulus G2. Both materials have the same Poisson's ratio, V = V = V . - side 1 x = -2a side 2 side 3 x = a A torque M is applied and the shaft deforms in torsion with an angle of twist per unit length a. The equations describing the 3 sides of the triangle are: x - 3y+ 2a = 0 (side 1) x + 3y+2a = 0 (side 2) x-a=0 (side 3) The equations that bound the 4 sides of the square are: x-0.5a=0 x +0.5a=0 y-0.5a=0 y + 0.5a= 0 For this two-material prismatic shaft, the company needs to find: (a) The shear stress distribution xz (x,y) and yz(x, y). (b) The torsional rigidity D. (c) The warping function x(x, y). x (a) Extension to the torsion of multi-material bars Consider now the case in which the cross section is composed by N different materials, with shear modulus G; (i=1,...,N), connected by smooth interfaces. With reference to figure la, let us denote by 2; the domain that hosts the ith material, by to the intersection of the boundary of 2; with the external boundary o, and by I; the interface between materials i and j (i #j). We make a distinction between 'open' bi-material interfaces, i.e. open curves whose extremal points lay on Io, and 'closed' interfaces, if they are closed curves inside the cross section. To emphasize this distinction, if Iij is a closed interface it will be formally denoted by Iij. Introduce the stress function as in (2.1). Using the same arguments of the classical approach, for the ith material, one obtains AF;(x, y)=-2G; in 2; and F(x, y) = 0 on Fo (2.5) On the generic interface I' define the normal and tangential unit vectors n and t such that (n, t, k) is right-handed. Bi-material matching conditions [14] require the continuity of the normal traction Tnz = F(x,y)/at and the continuity of the strain Yytz = Ttz/G in tangential direction. In terms of stress function, one finds and F;(x, y) aF;(x,y) F(x, y) = F(x, y) on Fij at at (2.6) 1 aF(x,y) 1 F;(x,y) Gi an G; an on Tij. If the jth material domain 2; is enclosed by closed curve fij, as for materials 2 and 4 in figure 1a, recalling (2.5), one obtains 1 f aF(x,y) 1 F; (x,y) Gi fr dl= dl n Gj n 21 = AF;(x, y) dA = -2A;, (2.7) where A; is the area of 2;. Once the stress function has been determined, the torsional stiffness is again obtained through (2.4). A prismatic shaft is made of two materials. The cross section of the two-material prismatic shaft has an outer equilateral triangle made of Material-1 (green area in Figure) and an inner square made of Material-2 (yellow area in Figure). Material-1 has shear modulus G and Material-2 has shear modulus G2. Both materials have the same Poisson's ratio, V = V = V . - side 1 x = -2a side 2 side 3 x = a A torque M is applied and the shaft deforms in torsion with an angle of twist per unit length a. The equations describing the 3 sides of the triangle are: x - 3y+ 2a = 0 (side 1) x + 3y+2a = 0 (side 2) x-a=0 (side 3) The equations that bound the 4 sides of the square are: x-0.5a=0 x +0.5a=0 y-0.5a=0 y + 0.5a= 0 For this two-material prismatic shaft, the company needs to find: (a) The shear stress distribution xz (x,y) and yz(x, y). (b) The torsional rigidity D. (c) The warping function x(x, y). x