1. Prove the formulae As= λi (∂*s/∂*i)
2. A body undergoes the homogeneous deformation
x1 = N/2,C1+11/2X2, x2 = -X1+1X2+14V2X3 , x3 = X1 - 4X2 + 1-V2X3
Find: (a) the direction after the deformation of a line element with direction ratios 1 : 1 : 1 in the reference configuration; (b) the stretch of this line element.
3. Find the components of the tensors F, C, B, f' and 11 for the deformation
xl= a,(Xi+ aX2), X2 = a2X2, X3 = a3X3
where al, a2, a3 and a are constants. Find the conditions on these constants for the deformation to be possible in an incom¬pressible material. A body which in the reference configuration is a unit cube with its edges parallel to the coordinate axes under¬goes this deformation. Determine the lengths of its edges, and the angles between the edges, after the deformation. Sketch the deformed body.
4. A circular cylinder in its reference configuration has radius A and its axis lies along the X3-axis. It undergoes the deformation
x1 = AIX1 cos (0X3) + X2 sin (t/IX3)}
x2 = μ{-X1sin(ΨX3)+X2cos(ΨX3)}
X3 = λX3
Find the conditions on the constants A, p, and for this deformation to be possible in an incompressible material. A line drawn on the surface of the cylinder has unit. length and is parallel to the axis of the cylinder in the reference configuration. Find its length after the deformation. Find also the initial length of a line on the surface which has unit length and is parallel to the axis after the deformation.
5. Show that the condition for a material line element to he unchanged in direction during a deformation is (Fir-λδiR)AR= 0. Deduce that the only lines which do not rotate in the simple shear deformation (6.44) are lines which are perpendicular to the X2-axis. For the deformation
X =µ(X1+ X2 tan y), x2 = µ zi v X3 -= X3 / 1)
show that there are three directions which remain constant. Find these directions and the corresponding stretches.
6. Prove that in the homogeneous deformation (6.46), particles which after the deformation lie on the surface of a sphere of radius b originally lay on the surface of an ellipsoid. Prove that this ellipsoid is a sphere of radius a if a2AijAik = b2δik.
7. A rod of circular cross-section with its axis coincident with the x3-axis is given a small twist so that its displacement is given by
u1= -Ψx2x3, u2=Ψx1x3 , u3=0
where Ψ is constant. Find the components of infinitesimal- strain and infinitesimal rotation. Show that one of the principal compo¬nents of infinitesimal strain is always zero and find the other two principal components. Find also the principal axes of the infinites¬imal strain tensor.
8. For the deformation ul= AX1 + BX1(.X12+X22)-1 , u2 = AX2 + BX2(.X12+X22)-1 , u3 = CX3
where A, B and C are constants, find the components of the tensors F, E and Ω. Also find the principal values and principal axes of E.
9. prove that the rate of change of the angle θ between two material line elements whose direction in the current configuration are determined by unit vectors a and b is given by
θsinθ = (aiaj+bibj)Dij cos θ -2aibjDij
Deduce that -2Dij (i ≠j) is the rate of change of the angle be-tween two material line elements which instantaneously lie along the xi- and xi-axes.
10. An incompressible body is reinforced by embedding in it two families of straight inextensible fibres whose directions in the re¬ference configuration are given by Al= cos β, A2 = ±sin β, A3 = 0, where β is constant. The body undergoes the homogeneous de¬formation
x1 = µ-1/2aX1, x2 = µ-1/2a-1X2, X3 =µX3
where a and β. are constants. Show that the condition λ= 1 for inextensibility in the fibre direction requires that a2 cos2 a3+a-2sin-2β=µ.
Deduce that: (a) the extent to which the body can contract in the x3 direction is limited by the inequality g 2(3;
(b) when this maximum contraction is achieved, the two families of fibres are orthogonal in the deformed configuration.