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E22,3131 = (E23.1-E31.2 + E12,3),123 E13,1212 = (E23,1 + E31.2 - E12,3),123 to) Exercises 1. The deformation of a body is defined by displacement components 41 = K(3 x? +X). uy = K(2 x} + X3), uz = K(4X3 + X1) where K is a positive constant. Compute the extension of a line element that passes 17k Ans. through the point (1,1,1) in the direction 3 2. The displacement field for small deformation theory is given by u = (x,- X;)u2 = (X,+ X)?. Uz = - X,X2 Determine the strain tensor, rotation tensor and rotation vector at the point (0.2. - 1) 2 0 -2 0 Ans. (E;) = 0 2 :(R;) = 0 1.R,(-2) -2 1 0 0 -1 0 3. A displacement field is given by x, = X, - CX, + BX3, X2 = CX, +X-AX3. I3 = - BX, + AX, + X3. Show that this displacement represents a finite rigid body rotation only if the constants A, B, C are very small. Determine rotation vector R for infinitesimal rigid body rotation. Hint. r = 0, for small A, B, C, R = 2, 2A +, 2B + , 2C 4. For the deformation u, = A, X; where A, are constants, determine an expression or change of volume per unit original volume. If A, are very small, show that it reduces to a cubical dilatation. 0 0 1 0 jace X A, +1 A21 dV A31 A 32 [Hint: x; = u, + X V A22+1 Change in volume A12 A3 A33 Az + 1 and M at dV-dV. per unit original volume = = J-1A1 + A22 + A 33 = E1 + E22 + E331 dV 5. For the displacement field u = (x,- X;)?. Uz = (X2 + X:)?, uz = - X,X2, Compute the change in right angle between N = s, - z + 48, 4i - 4n + Tz 9 9 the point (0.2.-1). (Hint: Yun=2E,N,M,= 21E,,N,M,+ E2(N, M2 + N2M.) + Ej3(N, Mz+N,M,) + E22NM2 + E23(N,M,+N;M) + E33N,IM3) 318 :) 81 6. The displacement field for a continuum body is given by u = l X? + X} + 2). Theory of Strain 61 K, X 0 0 V 10. Given the strain field (E,) = 0 -KX 0 What should 0 0 - KX be the relation between K, and K, such that there will be no volume change. (Ans. k, = 2K) W. Calculate the strain invariants from strain tensor 5 - 1 -1) (E)) = 1 4 0 - 1 0 4 Determine the Principal strains. Obtain strain invariants from them. Show the equivalence of strain invariants (Ans. @ = 13. = 54, 8, = 72, E, = 6, E2 = 4. Ez = 3] 1 -3 V2 12. The strain tensor at a point is (E.) = 1 -2 V2 - 2 4 Determine (a) extension of a line element in the direction ta) (b) shear between the directions and V2 and (c) principal strains, 2 maximum normal strain, maximum shearing strain and strain invariants. [Ans. En = 6. YMN) = 0, E = 6, E, = 2, E, = - 2. Maximum normal strain = 6. Max shearing strain = 6-(-2) = 4, 8, = 6, 8, = - 4, 0, = -24] 13. Determine principal strains and corresponding direction from strain tensors 1 5 1 (a) (E, ) = 1 3 1 Ana. 6.3. - 2. to Battiat.] 5 2 2 (b) (E, ) = 2 2 - 4 2 (Ans. 6,6. - 3;}.}--}-}}}] 13x7 +4x3), ) = (2x' + 4X,), where e = .0001. What is the deformed position of a point originally at (1, 2, 3)? What are strain components there? .0002 .0007 .0006 (Ans. 1.0007, 2.0019, 3.0014. (E.) = .0007 .0016 0 0 .0006 0004 0 1. The displacement field is given by u = KOX? + 2X3), uz = K[4X, + 2 X} +Xz). W3 = 4Kxwhere K is a very small quantity. What are normal strains at (2, 2, 3) in directions o. test).1,0,0) (1, 0, 0), (0.6,0,0.8)? 33 Ans. IK, 4K, 17.76 K 8. For the displacement field u = XXz, uz = X2 - X}.u3 = X X, determine the unit relative displacement vector at P(1, 2, -1) with respect to Q(4,2,3) du; (Hint: Unit relative displacement du; dx, Ni dL dL ax, du, ax; N Now N2 2-2 N, 3+1 + du dL du N; 2X = = 1/3, N., = 23, N2 = 0, N = 1/3 = 2x,x, x { +0x, X2 = (1.2) = !? -N, = 0 + 0 + (-2X,) - N, duz duz loo dL 10o axi = du dL dun ax 5 3 3 5 9. The strain tensor at a point is given by (E.) = 3 0 4 -1 Determine the - 1 2 extension of a line element in the direction of 2 2 1 333 What is the change of angle 2 2 between two perpendicular line elements in the directions of and 333 (Ans. : 58 32 35 E22,3131 = (E23.1-E31.2 + E12,3),123 E13,1212 = (E23,1 + E31.2 - E12,3),123 to) Exercises 1. The deformation of a body is defined by displacement components 41 = K(3 x? +X). uy = K(2 x} + X3), uz = K(4X3 + X1) where K is a positive constant. Compute the extension of a line element that passes 17k Ans. through the point (1,1,1) in the direction 3 2. The displacement field for small deformation theory is given by u = (x,- X;)u2 = (X,+ X)?. Uz = - X,X2 Determine the strain tensor, rotation tensor and rotation vector at the point (0.2. - 1) 2 0 -2 0 Ans. (E;) = 0 2 :(R;) = 0 1.R,(-2) -2 1 0 0 -1 0 3. A displacement field is given by x, = X, - CX, + BX3, X2 = CX, +X-AX3. I3 = - BX, + AX, + X3. Show that this displacement represents a finite rigid body rotation only if the constants A, B, C are very small. Determine rotation vector R for infinitesimal rigid body rotation. Hint. r = 0, for small A, B, C, R = 2, 2A +, 2B + , 2C 4. For the deformation u, = A, X; where A, are constants, determine an expression or change of volume per unit original volume. If A, are very small, show that it reduces to a cubical dilatation. 0 0 1 0 jace X A, +1 A21 dV A31 A 32 [Hint: x; = u, + X V A22+1 Change in volume A12 A3 A33 Az + 1 and M at dV-dV. per unit original volume = = J-1A1 + A22 + A 33 = E1 + E22 + E331 dV 5. For the displacement field u = (x,- X;)?. Uz = (X2 + X:)?, uz = - X,X2, Compute the change in right angle between N = s, - z + 48, 4i - 4n + Tz 9 9 the point (0.2.-1). (Hint: Yun=2E,N,M,= 21E,,N,M,+ E2(N, M2 + N2M.) + Ej3(N, Mz+N,M,) + E22NM2 + E23(N,M,+N;M) + E33N,IM3) 318 :) 81 6. The displacement field for a continuum body is given by u = l X? + X} + 2). Theory of Strain 61 K, X 0 0 V 10. Given the strain field (E,) = 0 -KX 0 What should 0 0 - KX be the relation between K, and K, such that there will be no volume change. (Ans. k, = 2K) W. Calculate the strain invariants from strain tensor 5 - 1 -1) (E)) = 1 4 0 - 1 0 4 Determine the Principal strains. Obtain strain invariants from them. Show the equivalence of strain invariants (Ans. @ = 13. = 54, 8, = 72, E, = 6, E2 = 4. Ez = 3] 1 -3 V2 12. The strain tensor at a point is (E.) = 1 -2 V2 - 2 4 Determine (a) extension of a line element in the direction ta) (b) shear between the directions and V2 and (c) principal strains, 2 maximum normal strain, maximum shearing strain and strain invariants. [Ans. En = 6. YMN) = 0, E = 6, E, = 2, E, = - 2. Maximum normal strain = 6. Max shearing strain = 6-(-2) = 4, 8, = 6, 8, = - 4, 0, = -24] 13. Determine principal strains and corresponding direction from strain tensors 1 5 1 (a) (E, ) = 1 3 1 Ana. 6.3. - 2. to Battiat.] 5 2 2 (b) (E, ) = 2 2 - 4 2 (Ans. 6,6. - 3;}.}--}-}}}] 13x7 +4x3), ) = (2x' + 4X,), where e = .0001. What is the deformed position of a point originally at (1, 2, 3)? What are strain components there? .0002 .0007 .0006 (Ans. 1.0007, 2.0019, 3.0014. (E.) = .0007 .0016 0 0 .0006 0004 0 1. The displacement field is given by u = KOX? + 2X3), uz = K[4X, + 2 X} +Xz). W3 = 4Kxwhere K is a very small quantity. What are normal strains at (2, 2, 3) in directions o. test).1,0,0) (1, 0, 0), (0.6,0,0.8)? 33 Ans. IK, 4K, 17.76 K 8. For the displacement field u = XXz, uz = X2 - X}.u3 = X X, determine the unit relative displacement vector at P(1, 2, -1) with respect to Q(4,2,3) du; (Hint: Unit relative displacement du; dx, Ni dL dL ax, du, ax; N Now N2 2-2 N, 3+1 + du dL du N; 2X = = 1/3, N., = 23, N2 = 0, N = 1/3 = 2x,x, x { +0x, X2 = (1.2) = !? -N, = 0 + 0 + (-2X,) - N, duz duz loo dL 10o axi = du dL dun ax 5 3 3 5 9. The strain tensor at a point is given by (E.) = 3 0 4 -1 Determine the - 1 2 extension of a line element in the direction of 2 2 1 333 What is the change of angle 2 2 between two perpendicular line elements in the directions of and 333 (Ans. : 58 32 35