Advanced Solid Mechanics
Assignment - 2: Kinematics
1. A body in the form of a cube, B = {(X, Y, Z)|0 ≤ X ≤ 1cm, 0 ≤ Y ≤
1cm, 0 ≤ Z ≤ 1cm} in the reference configuration, is subjected to the
following deformation field: x = X, y = Y + A ∗ Z, z = Z + A ∗ Y ,
where A is a constant and (X, Y, Z) are the Cartesian coordinates of a
material point before deformation and (x, y, z) are the Cartesian coordinates of the same material point after deformation. For the specified
deformation field:
(a) Determine the displacement vector components in both the material and spatial forms.
(b) Determine the location of the particle originally at Cartesian coordinates (1, 0, 1)
(c) Determine the location of the particle in the reference configuration, if its current Cartesian coordinates are (1, 0, 1)
(d) Determine the displacement of the particle originally at Cartesian
coordinates (1, 0, 1)
(e) Determine the displacement of the particle currently at Cartesian
coordinates (1, 0, 1)
(f) Determine the deformation gradient and Eulerian and Lagrangian
displacement gradients
(g) Calculate the right Cauchy-Green deformation tensor
(h) Calculate the linearized Lagrangian strain and linearized Eulerian
strain. Compare and comment on the value of the strain measures
(i) Calculate the change in the angle between two line segments initially oriented along Ey and Ez directions in the reference configuration
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(j) Calculate the change in the volume of the cube
(k) Calculate the deformed surface area and its orientation for each
of the six faces of a cube
(l) Calculate the change in length of the straight line segments of
length
√ 1 mm initially oriented along (i) Ey (ii) Ez (iii) (Ey +
Ez )/ 2
(m) Is the motion field realizable for any value of A in the cube? Justify. If not, find the values that A can take.
(n) Determine the displaced location of the material particles which
originally comprise
(i) The plane circular surface X = 0, Y 2 + Z 2 = 0.25,
(ii) The plane elliptical surface X = 0, 9Y 2 + 4Z 2 = 1.
(iii) The plane elliptical surface X = 0, 4Y 2 + 9Z 2 = 1.
(o) Sketch the displaced configurations for (i), (ii) and (iii) in the
above problem if A = 0.1.
(p) Sketch the deformed configuration of the cube assuming A = 0.1.
2. Rework the parts (a) to (p) in the above problem if the cube is subjected
to a displacement field of the form, u = (A ∗ Y + 2A ∗ Z)ey + (3A ∗
Y − A ∗ Z)ez , where (X, Y, Z) denotes the coordinates of a typical
material particle in the reference configuration and {ei } the Cartesian
coordinate basis.
3. Which of the following displacement fields of a cube is homogeneous?
(a) u = A ∗ Zey + A ∗ Zez , where A is a constant
(b) u = [(cos(θ) − 1)X + sin(θ)Y ]ex + [− sin(θ)X + (cos(θ) − 1)Y ]ey ,
where θ is some constant
(c) u = 3XY 2 ex + 2XZey + (Z 2 − XY )ez
Here as usual (X, Y, Z) denotes the coordinates of a typical material
particle in the reference configuration and {ei } the Cartesian coordinate
basis. For these deformation fields find parts (a) to (l) in problem 1.
4. A body in the form of an annular cylinder, B = {(R, Θ, Z)|0.5 ≤ R ≤
1cm, 0 ≤ Θ ≤ 2π, 0 ≤qZ ≤ 10cm} is subjected to the following deformation field: r =
ro2 −
1
Λ
+
2
R2
,
Λ
θ = Θ + ΩZ, z = ΛZ, where
(R, Θ, Z) denote the coordinates of a typical material particle in the
reference configuration, (r, θ, z) denote the coordinates of the same material particle in the current configuration, ro , Ω and Λ are constants.
For this deformation field:
(a) Determine the displacement vector components in both the material and spatial forms.
(b) Determine the location of the particle originally at cylindrical polar coordinates (1, 0, 5)
(c) Determine the location of the particle in the reference configuration, if its current cylindrical polar coordinates are (1, 0, 5)
(d) Determine the displacement of the particle originally at cylindrical
polar coordinates (1, 0, 5)
(e) Determine the displacement of the particle currently at cylindrical
polar coordinates (1, 0, 5)
(f) Determine the deformation gradient and Eulerian and Lagrangian
displacement gradients
(g) Calculate the right Cauchy-Green deformation tensor
(h) Calculate the linearized Lagrangian strain and linearized Eulerian
strain. Compare and comment on the value of the strain measures
(i) Sketch the deformed shape of the annular cylinder assuming ro =
1.1, Λ = 1.2 and Ω = 0.1
(j) Calculate the change in the volume of the annular cylinder
(k) Calculate the deformed surface area and its orientation for each of
the two lateral faces and top and bottom surfaces of the cylinder.
Assume ro = 1.1, Λ = 1.2 and Ω = 0.1
(l) Calculate the change in the angle between two line segments oriented along ER and EZ directions in the reference configuration
at (0.5, 0, 5). Assume ro = 1.1, Λ = 1.2 and Ω = 0.1
(m) Calculate the change in length of the straight line segments of 1
mm length located
√ at (1, 0, 5) and oriented along (i) ER (ii) Ez
(iii) (ER + EΘ )/ 2. Assume ro = 1.1, Λ = 1.2 and Ω = 0.1
(n) Is the motion field realizable for any value of the constants - ro , Λ
and Ω - in the cylinder? Justify. If not, find the values that these
constants can take.
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Z
C
B
O
A
Y
D
X
Figure 1: Figure for problem 5
(o) Find the conditions when this deformation would be homogeneous.
5. Consider the following homogeneous deformation: x = a1 X + k1 Y , y
= a2 Y + k2 Z, z = a3 Z, where ai and ki are constants and (X, Y, Z)
are the Cartesian coordinates of a material particle in the reference
configuration and (x, y, z) are the Cartesian coordinates of the same
material particle in the current configuration. For this deformation
field and the tetrahedron OABC shown in figure ?? such that OA =
OB = OC and AD = DB, compute
a. The change in length of the line segment AB
b. The change in angle between line segment AB and CD
c. The deformed surface area of the faces ABC, OAB and OBC
6. A body in the form of a unit cube, B = {(X, Y, Z)|0 ≤ X ≤ 1, 0 ≤
Y ≤ 1, 0 ≤ Z ≤ 1} in the reference configuration, is subjected to the
following deformation field: x = X, y = Y + A ∗ X, z = Z, where A is a
constant and (X, Y, Z) are the Cartesian coordinates of a material point
before deformation and (x, y, z) are the Cartesian coordinates of the
same material point after deformation. For the specified deformation
field:
(a) Compute the right Cauchy-Green deformation tensor
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(b) Compute the linearized Lagrangian strain tensor
(c) Compute the eigenvalues and eigenvectors for right Cauchy-Green
deformation tensor
(d) Compute the eigenvalues and eigenvectors for linearized strain tensor
(e) Find the direction of the line segments in the reference configuration along which the maximum extension or shortening occurs
and its value
(f) Find the directions of the line segments in the reference configuration along which no extension or shortening takes place in the
cube
(g) Find the maximum change in angle between line segments and the
orientation of the line segments for which this occurs.
(h) Find the directions orthogonal to planes in which no change of
area occurs
(i) Find the change in volume of the cube
(j) Decompose deformation gradient, F as F = RU, where R is the
proper orthogonal tensor and U is a symmetric positive definite
tensor.
7. A body in the form of a unit cube, B = {(X, Y, Z)|0 ≤ X ≤ 1, 0 ≤
Y ≤ 1, 0 ≤ Z ≤ 1} in the reference configuration, is subjected to the
following linearized strain field:


AY 3 + BX 2 CXY (X + Y ) 0
0 ,
= CXY (X + Y ) AX 3 + DY
(1)
0
0
0
where A, B, C, D are constants, find conditions, if any, on the constants if this strain field is to be obtained from a smooth displacement
field of the cube. For this value of the constants, find the smooth displacement field that gives raise to the above linearized strain field in a
cube. Assume the coordinates of the point (0, 0, 0) after deformation is
(10, 0, 0) and that of the point (0, 0, 1) after deformation is (10, 0, 1) to
determine the unknown constants in the displacement field. For what
magnitude of these constants is the use of linearized strain justified.
5
Y
X
Figure 2: Figure for problem 9: Schematic of a strain rosette
8. A cube of side 10 cm, is subjected to a uniform plane state of strain
whose Cartesian components are:


1 −2 0
(2)
= −2 3 0 ∗ 10−4 .
0
0 0
For this constant strain field, find
(a) Eigenvalues and eigenvectors of this strain tensor
(b) Maximum change in length and the direction of the material fiber
along which this occurs
(c) Maximum change in angle and the orientation of the material
fibers along which this occurs
(d) Whether the deformation is isochoric
√
(e) Change in length of a material fiber oriented along (ex + ey )/ 2
and of length 1 mm
√
(f) Change in angle√between line elements oriented along (ex +ey )/ 2
and (ex − ey )/ 2
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9. A rosette strain gauge is an electromechanical device that can measure
relative surface elongations in three directions. Bonding such a device
to the surface of a structure allows determination of elongation along
the direction in which the gauge is located. Figure ?? shows the orientation of gauges in one such rosette along with the coordinate system
used to study the problem. For a particular loading, these gauges measured the strain along their directions as a = 0.001, b = 0.002 and c
= 0.004. Assuming the state of strain at the point of measurement is
plane, find the Cartesian components of the strain: xx , yy and xy for
the orientation of the basis also shown in the figure ??.
10. It is commonly stated that the rigid body rotation undergone by a body
for a given deformation when the components of the displacement gradient are small is the skew-symmetric part of the displacement gradient,
i.e. ω = [grad(u) − grad(u)t ]/2. Show that this is not the case by
computing R, the orthogonal tensor in the polar decomposition of the
deformation gradient, F and ω for the following displacement field of
a cube with sides 10 cm: u = (AX + BY )ex + CY ey + DZez , where
A, B, C, D are constants, (X, Y, Z) are the Cartesian coordinates of
a typical material particle in the reference configuration and ei the
Cartesian coordinate basis. Recollect that R represents the true rigid
body rotation component of the deformation.
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