Theory of Elasticity
CEng 6501
Instructor
Dr. L. N. Ojha
M. Sc. (Structural Engineering)
Structural Engineering Chair
SoCE, EiT-M
2017-18
Contents
Kinematics
1 General Deformations
2 Small Deformation Theory-Strain
3 Strain- Displacement Relationships
4 Strain Transformation
5 Principal Strains
6 Spherical and Deviatoric Strains
7 Strain Compatibility
Types of Deformations
kinematics of material deformation
The response of a material to applied forces
depends on the type and nature of the bond and
the structural arrangement of atoms, molecules or
ions.
Basic Types:
1. Elastic deformation (instantaneously recoverable)
2. Plastic deformation (non-recoverable)
3. Viscous deformation (time dependent )
KINEMATICS OF MATERIAL DEFORMATION Deformational Displacements
Deformations can be quantified by knowing the displacements of
material points in the body.
Body-Displacement
(Translation + Rotation)
Change in
Position
Points of Body-Deformation
( Translation + Rotation)
An elastic solid is said to be deformed or strained when the
relative displacements between points in the body are changed.
This is in contrast to rigid-body motion where the distance
between points remains the same.
General Deformational Displacements
General deformation between
two neighbouring points.
2D- Deformation Example
General Deformational Displacements
TAYLOR SERIES EXPANSION
Note that the higher-
order terms of the
expansion have been
dropped since the
components of r are
small.
where u, v, w are the Cartesian components of the displacement vector
General Deformational Displacements
SYMMETRIC-6 Independent Components
ANTISYMMETRIC, (diagonal is zero,
3 independent components)
STRAIN TENSOR
ROTATION TENSOR
General Deformational Displacements
imply that for small deformation theory,
the change in the relative position vector between neighbouring points
can be expressed in terms of a sum of strain and rotation components
Because we are considering a general displacement field, these
results include both strain deformation and rigid-body motion.
Example
Small deformation theory
General Displacements = Deformation (Large or Small) +
Rigid Body Motion
Small Deformation Theory
The usual types of motion include rigid-body rotation and extensional and shearing
deformations as illustrated.
Rigid-body motion does not
Rigid Body
contribute to the strain field, and Rotation
thus also does not affect the stresses.
We therefore focus our study
primarily on the extensional and
shearing deformation
Small deformation theory
The undeformed and deformed configurations can
be significantly different, for finite or large
deformation.
A distinction between these two configurations
must be maintained leading to Lagrangian and
Eulerian descriptions.
Linear elasticity, which uses only small deformation
theory, the distinction between undeformed and
deformed configurations can be dropped.
Engineering Strains- Small Deformations
Small Deformations and Small Strains
Large Deformation and Large Strain
Strain
Strain = “% Deformation of an infinitesimal element”
The strain component that measures elongation or
construction – Normal Strain
The strain component that measures rotation of any
two Perpendicular lines is – Shearing Strain
Stretch = Dimensionless ratio of Different Measure of Strain
deformed length to original length
Strain
Geometric Construction of Small Deformation Theory
Geometric Construction of Small Deformation Theory
Geometric Construction of Small Deformation Theory
Geometric Construction of Small Deformation Theory
Strain Displacement Relationships
Strain Displacement Relationships
COMPACT TENSOR NOTATION
The strain is a symmetric second-order tensor
and is commonly written in matrix
format:
DIRECT VECTOR/MATRIX NOTATION
Strain Displacement Relationships
Strain Displacement Relationships
BOTTOM EDGE
LEFT EDGE
These two expressions are
of course the same;
Tensor and Engineering Strain
Tensor and Engineering Strain
Example
Determine the components of strain for the following displacement field
u Axz , v B ( x 2 y 2 ) , w Cxy , where A, B, C are constants
_________________________________________
u
ex Az
x
v
ey 2 By
y
w
ez 0
z
1 u v 1
exy 0 2 Bx Bx
2 y x 2
1 v w
e yz
1
0 Cx 1 Cx
2 z y 2 2
1 w u
ezx
1
Cy Ax
2 x z 2
Example
Determine the strain and rotation tensors for the following
displacement fields
Example
A two-dimensional displacement field is given by
where k is a constant.
Determine and plot the deformed
shape of a differential rectangular
element originally located with its
left bottom corner at the origin as
shown.
Finally, calculate the rotation
component .
Example
Example
A two-dimensional problem of a rectangular bar stretched
by uniform end loadings results in the following constant
strain field:
Where are constants.
Assuming the field depends only on x and y, integrate the
strain-displacement relations to determine the
displacement components and identify any rigid-body
motion terms.
Example
Example
Strain Transformations
Strain Transformations
Principal Strains
Spherical and Deviatoric Strains
The spherical strain represents only volumetric deformation and is
an isotropic tensor being the same in all coordinate systems.
The deviatoric strain tensor then accounts for changes in shape of
material elements. It can be shown that the principal directions of
the deviatoric strain are the same as those of the strain tensor.
Spherical and Deviatoric Strains
Strain Displacement Relationships-Strain Compatibility
six equations for the six strain components in terms of three displacements.
If we specify continuous, single-valued displacements u,v,w, then through
differentiation the resulting strain field will be equally well behaved.
However, the converse is not necessarily true; that is, given the six strain
components, integration of the strain-displacement relations does not
necessarily produce continuous, single-valued displacements. This should not
be totally surprising since we are trying to solve six equations for only three
unknown displacement components.
In order to ensure continuous, single-valued displacements, the strains must
satisfy additional relations called integrability or compatibility equations
Strain Displacement Relationships-Strain Compatibility
Strain Displacement Relationships-Strain Compatibility
Eliminating the displacements , from the strain-displacement
relations, working in index notation, differentiating twice with
respect to,
Through simple interchange of
subscripts, we can generate the
following additional relations:
Strain Displacement Relationships-Strain Compatibility
Working under the assumption of continuous displacements, we
can interchange the order of differentiation on u, and the
displacements can be eliminated from the preceding set to get
These are called the Saint-Venant compatibility equations.
Although the system would lead to 81 individual equations,
most are either simple identities or repetitions, and
only 6 are meaningful.
Strain Displacement Relationships-Strain Compatibility
Strain Displacement Relationships-Strain Compatibility
These six relations may be determined by letting k = l, and
in scalar notation, they become
Strain Displacement Relationships-Strain Compatibility
Example
Check to see if the following strain field
ex Ay , e y Ax , exy Bxy( x y )
3 3
satisfies the two - dimensional compatibility equation
__________________________________________
ex e y exy
2 2 2
2 6 Ay 6 Ax 2 B(2 x 2 y )
y 2
x 2
xy
2
6 A 4B A B
3
2
only satisfies equation with A B
3
Example
The square block is in plane strain and is subjected to the
following strains
y
x 2 xy 2
1
y 3xy
2
2
2
xy x 2 y 3 3 4
x
Compute the displacement field (i.e., displacement
components u(x,y) and v(x,y)) within the block
Example
x
u
2 xy u ( x, y) x 2 y C1 ( y) u v
x xy x2 y3
v v( x, y) xy C2 ( x)
3
y x
y 3 xy 2
y
x 2 y C1 ( y ) xy3 C2 ( x)
x2 y3
y x
C1 ( y ) C2 ( x)
x
2
y 3
x2 y3
y x
C ( y ) C2 ( x)
1 0
y x
C1 ( y) C2 ( x)
C
y x
C1 ( y ) Cy D1
C2 ( x) Cx D2
u ( x, y ) x y Cy D1
2
v( x, y ) xy3 Cx D2
Example
y u ( x, y ) x 2 y Cy D1
v( x, y ) xy3 Cx D2
2
u (0,0) 0 C 0
2 1
v (0,0) 0 D1 0
D2 0
v ( 2,0) 0
2
4
3
x
u ( x, y ) x y 2
v( x, y ) xy 3
Homework
Homework No. 2
Martin Saad Book PROBLEM No.
2.1(c)
2.4
2.11
2.12
2.13
Questions
