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Practice Set#3

This document contains 5 practice problems related to advanced mechanics of solids: 1. Find the deformation gradient tensor for a radially symmetric deformation of a spherical shell. 2. Calculate the deformation gradient and displacement of a square under a given displacement field, and validate the results by sketching the deformed shape. 3. Calculate the deformation gradient, Lagrangian strain, and infinitesimal strain tensors for a beam bent into a circular arc, and compare the strain tensors at two points. 4. Find the increase in length of a diagonal of a unit cube under a given displacement field, using small strain and geometric approaches. 5. Determine the displacement components by integrating the given strain field for

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112 views3 pages

Practice Set#3

This document contains 5 practice problems related to advanced mechanics of solids: 1. Find the deformation gradient tensor for a radially symmetric deformation of a spherical shell. 2. Calculate the deformation gradient and displacement of a square under a given displacement field, and validate the results by sketching the deformed shape. 3. Calculate the deformation gradient, Lagrangian strain, and infinitesimal strain tensors for a beam bent into a circular arc, and compare the strain tensors at two points. 4. Find the increase in length of a diagonal of a unit cube under a given displacement field, using small strain and geometric approaches. 5. Determine the displacement components by integrating the given strain field for

Uploaded by

shreya sri
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Practice set-3

ME321A: Advanced Mechanics of Solids

Note:

1. As discussed in class, the answer must be written clearly with the same convention for
tensor and vector.

Pr 1: A spherical shell is made from an incompressible material. In its undeformed state,


the inner and outer radii of the shell are A and B respectivelly. After deformation, the
new values are a and b, respectively. The deformation in the shell are can be described
by the equation

xi
yi = r ,
R
√
where r = (R3 + a3 − A3 )1/3 and R = xk xk . Find the components of the deformation
gradient tensor.
Note that the deformation is radially symmetric and as such one is required to compute
the determinant along any radial line.

Pr 2: Given below are the components of a displacement field u = (u1 , u2 , u3 ),

u1 = kx2 , u2 = u3 = 0,

where x = (x1 , x2 , x3 ) is position vector.

(a) Sketch the deformed shape of the unit square OABC. see Figure 1
−→
(b) Find ∇u, calculate vector a = (0, 1, 0) = OC after deformation and validate the answer
by comparing it to sketch.

Pr 3: An initially straight beam is bent into a circular arc with radius R. Material fibres
that are perpendicular to the axis of the undeformed beam are assumed to remain
perpendicular to the axis after deformation, and the beam’s thickness and the length
of its axis are assumed to be unchanged. Under these conditions the deformation can
be described as

y1 = (R − x2 )sin(x1 /R), y2 = R − (R − x2 )cos(x1 /R)

1
Figure 1: Figure for Pr 2

where x is the position vector of a material particle in the undeformed beam and y is
the the position vector of the same particle after deformation.

(a) Calculate the deformation gradient field in the beam.

(b) Calculate the Lagrangian strain tensor.

(c) Calculate the infinitesimal strain field in the beam.

(d) Compare the value of Lagrangian strain tensor and infinitesimal strain tensor at the
points (x1 = 0, x2 = h) and (x1 = L, x2 = 0).

Pr 4: A unit cube with edges parallel to coordinate axes, is given a displacement field

u1 = kx1 , u2 = u3 = 0, k = 10−4

Figure 2: Figure for Pr 4

Find increase in length of diagonal AB (Figure 2)

2
(a) By using small strain tensor
(b) By geometry.

Pr 5: A three-dimensional elasticity problem of a uniformed bar stretched under its own


weight gives the following strain field:
 
Az 0 0
 
 0 Az 0 
 
0 0 Bz

where A and B are constants. Integrate the strain-displacement relations to determine


the displacement components.

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