Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Section 10.3
10.3 Rheological Models
In this section, a number of onedimensional linear viscoelastic models are discussed.
10.3.1
Mechanical (rheological) models
The word viscoelastic is derived from the words "viscous" + "elastic"; a viscoelastic
materi
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Answers to Selected Problems: Chapter 8
8.1
1. U
1
1
3
3x1 x1
2. W mgh , Wgrav mgh U
3.
1
x gt , x gt 2 h
2
Wcon mg ( x h) , K
Wcon K
1 2
mx
2
1
mg 2 t 2 , U mg ( x h) , 0 U K .
2
4.
2
2
2
K 1 mx 2 1 mx0 , W 1 kx 2 1 kx0 , W K , U 1 kx 2 1 kx0 ,
2
2
2
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Problem 1. Is the following structure (Fig.3) statically determinate or indeterminate? Why? What
additional relation can be introduced to make it determinate?
rigid
P
P
rods of dierent material and dierent crosssection
Figure 1:
b) Which of the following
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Problem Set 1 (Kinematics)
1. Consider a small tetrahedron OABC with OA = OB = OC and OA, OB and OC mutually orthogonal. For
a homogeneous deformation, the small strain tensor in a cartesian basis, with the basis vectors along OA,
OB and OC, is given by
0
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Problem 1.
Figure 1:
A curved beam, as shown in Fig. 1, is xed at one end and subjected to a radial force (per unit
thickness of the beam) of magnitude P at the free end ( = 0). The inner and outer radii are a and
b respectively. The Airy stress function
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Q. 1. An axisymmetric composite cylinder is composed of a solid inner shaft, of radius a shear
modulus 1 , and an outer sleeve of outer radius b and shear modulus 2 . The shaft and sleeve
are ideally bonded at their interface and the composite cylinder is
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 1: ME 321 (Total 10 Marks) ( S 0 L U T' W")
Name: Roll Number:
Let u, v, and W be arbitrary vectors.
Problem 1 (2): Show that det(u (8 v) = 0.
AJ(961)= [wager (Q®!)é;_ @Cg‘ilgzl (41%” M'm'h'm)
= [Cy£09 (2949 (2.9.921
61:.) (y. at) 91.5.1) C E 9 a1
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 1 Makeup: ME 321 (Total 10 Marks) C S OLUT (0N)
Name: Roll Number:
Problem 1 (2): Show that if A  B 2 0 is true for every tensor B then tensor A = 0.
Mam 2
E Can/n bu. My. (3. (dims.
*3
WG Cam m {giLMH ML W CLIVE“. 13.: gg' ®3=J m
“J z"! ng
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 4: ME 321 (Total 10 Marks) «C  ﬁat 7*{3 . mi. :5)
Name: a Roll Number: ' "H
T :. K "
Problem 1 (5):  I4 In.
‘H‘ “’1.
‘r 69
f3 = :
th ‘13
I“. ._ 1.4.x A.
d ‘i’z
7" = "f
Find expression for the shear stresses in a thin walled tube of the sec
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
/ Quiz 5: ME 321 (Total 10 Marks)
Name: Roll Number:
Problem 1 (3): Prove that 15% : a’  (13, i = 1,.,3, where ('1; = a x d.;, d; = 9 x di, and Q = midi. [Hintz
b><c><g=(bg)ci[b'c)gl '
Ln’ = to a
1.33114? “rent, = illXi,1—J}xgli
W L._—.—._.a
ext
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Answers to Selected Problems: Chapter 3
3.1
1.
2.
3.
4.
(a) F 0.8 N , (b) M 0.0213 Nm , (c) M 0.0053Nm
F 0, M 0.0053Nm
RA 2000N/m, RB 1600N/m
2kPa
3.2
1.
1
xc ,
3
yc
1
3
3.3
2.
N S / l, s 0
3.
at A:
4.
yz is negative
5.
yz (positive)
bottom left: xz (
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Section 10.4
10.4 The Hereditary Integral
In the previous section, it was shown that the constitutive relation for a linear viscoelastic
material can be expressed in the form of a linear differential equation, Eqn. 10.3.19. Here
it is shown that the stres
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Section 10.5
10.5 Linear Viscoelasticity and the Laplace Transform
The Laplace transform is very useful in constructing and analysing linear viscoelastic
models.
10.5.1
The Laplace Transform
The formula for the Laplace transform of the derivative of a fun
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Section 10.7
10.7 Temperaturedependent Viscoelastic Materials
Many materials, for example polymeric materials, have a response which is strongly
temperaturedependent. Temperature effects can be incorporated into the theory
discussed thus far in a simple
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Section 10.6
10.6 Oscillatory Stress, Dynamic Loading and
Vibrations
Creep and relaxation experiments do not provide complete information concerning the
mechanical behaviour of viscoelastic materials. These experiments usually provide test
data in the tim
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Assignment Guidelines
(1) No late submission of assignment problems will be accepted unless previously
arranged with the lecturer.
(2) Each assignment question will be marked as follows:
0 for totally incorrect solution
19 for partially incorrect/correct
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Assignment Guidelines
(1) No late submission of assignment problems will be accepted unless previously arranged
with the lecturer.
(2) Each assignment question will be marked as follows:
0 for totally incorrect solution
19 for partially incorrect/correct
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Assignment Guidelines
(1) No late submission of assignment problems will be accepted unless previously
arranged with the lecturer.
(2) Each assignment question will be marked as follows:
0 for totally incorrect solution
19 for partially incorrect/correct
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Midterm Exam: ME 321, Total Marks 75
Problem1 (15 Marks)
Consider a uniform crosssectional cantilever beam (thin and long) with exural rigidity EI. There
are two loading congurations (1 & 2) as shown in Figure 1. Consider two points A and B at x = L1
a
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
The Deborah Number
The following lines are from an afterdinner talk presented at the Fourth
International Congress on Rheology,
which took place last August in Providence, R. I. Marcus Reiner, research
professor at the Israel Institute of
Technology, is c
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 14
THERMOELASTICITY
Most materials tend to expand iftheir temperature rises and, to a first approximation,
the expansion is proportional to the temperature change. If the expansion is unre
strained, all dimensions will expand equally 7 i.e. there
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
CHAPTER 4
STRESS AND STRAIN
Objectives
STRESS
To know state of stress at a point
To solve for plane stress condition
applications
STRAIN
To know state of stress at a point
To solve for plane stress condition
applications
STRESS
Lets considered that th
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 5
Suitability of a structure or machine may
depend on the deformations in the structure as
well as the stresses induced under loading.
Sometimes structures must be designed to
accommodate or produce certain deformations
Certain structural elements
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter: 1
Fundamental Principles of
Mechanics
Mechanics of Solids
Vikas Cahudhari
1
Mechanics
Study of Force & Motion
Mechanics of Solids
Vikas Cahudhari
2
Mechanics
Study of Force & Motion
Gross/Overall Motion
(Dynamics)
KINETICS
Mechanics of Solids
KIN
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 2
Introduction to Mechanics
of deformable bodies
Analysis of deformable bodies
1. Identification of a system
2. Simplification of this system
3. To develop model which can be analyzed
Steps to analysis System
1. Study of forces and equilibrium req
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Stresses in Bending
DR PRAVIN SINGRU
INTRODUCTION
IN ORDER TO MAINTAIN EQUILIBRIUM, A SHEAR FORCE
,V & BENDING MOMENT, M HAVE TO ACT ON THE
CROSS SECTION.
GEOMETRY OF DEFORMATION OF A
SYMMETRICAL BEAM SUBJECTED TO PURE
BENDING
ASSUMPTIONS
STRAIGHT UNIFO