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SSG 516 Mechanics of ContinuaLecture Two
Kinematics: Study of Displacements & Motionthe various possible types of motion in themselves, leaving out
the causes to which the initiation of motion may be ascribed constitutes the science of Kinematics.ET Whittaker
Helpful Reading: Bower pp 13-27, Fakinlede 146-177
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Kinematics is an organized geometrical description of
displacement and motion. The emphasis here is the fact that displacement and
motion are independent of the material constitution.While we may use the terminology of solid mechanics
the concepts and the ensuing relationships are
independent of the particular materials involved.
We shall also define the concepts of strain and strainrates as deriving from displacements and motion.
Kinematics
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Our cube, for simplicity will have the dimension of unity andwith one edge at the origin.
Let , represent the initial Cartesian coordinates in anundeformed body. We consider the transformation,
= 1 0
1
00
00
,01
01
,10
1
tan,
11
1
1 + t a n
And for small values of, we can easily see that,
00
00
,01
01
,10
1
,11
1
1 +
Simple Shear of a Cube
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Similar to the above, the transformation,
=1 0 1
Transforms the edges as,
00
00
,01
1
,10
10
,11
1 +
1
The square element in the above diagram has therefore
undergone a shearing in the horizontal axes as shown.
Horizontal Shear
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Mathematica Code
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We have seen how a simple tensor can be used totransform a simple shape into a new shape. We are now togeneralise this concept using the figure (Bower) below:
General Deformation
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As a result of the transformation, such that,(, ) = + (, )
Using the linear gradient operator we defined last term,we can take the gradient of the above equation andobtain,
grad (, ) = grad + , (, ) = + grad , [In component form,
=
+
= +
]
Recall that by definition, + , , = grad + , + ( ) So that as we can write that =
Or in full in full component form, =
Transformation equations
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The equation, = transforms an element in the
original configuration to . Note that while we havesuppressed the dependency of the deformation gradient on the location and time, it is general a function of both. Itis therefore more accurate to write, = (, )
This tensor contains all the necessary information aboutthe deformation. Once the deformation gradient isspecified for all locations and time, we can always find thenew configuration based on an initial state and a giventime .
The Deformation Gradient
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The mapping = is necessarily one to onebecause of the physical implications. It necessarily hasan inverse. Therefore, given a new configuration, wecan obtain the old configuration that resulted in it,as =
Furthermore, for two successive deformationgradients, and ,
= , and = we can easily see that,
= = or that =
Inverse & Multiple Transformations
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Transformation of a line element
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Consider the simple parallelepiped shown
here. The base area,A = sin is or = in indexnotation, this cross product is =
the perpendicular height is cos so that the
volume is = A cos =
= =
Volume as a triple product
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Jacobian
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Recall that in component form, = +
so that
the Jacobian determinant of the deformation,
= det = det( +
)
This is a measure of the change in volume as a resultof the deformation. If the three sides of a
parallelepiped deform as, = , = and = , then
= and =
The Jacobian of F
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Clearly,
= = = det =
So that,
= [To understand the above expression, note that in two-D,
det F = and that = det F. It is easy to work
the 2-D version out manually so understand the 3-D]
The Jacobian
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Furthermore, it is easily shown that mass density is
related as
=
This follows from the above derivation. Show this to
be true.
Mass density
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= =1
3!
Differentiating wrt , we obtain,
=
1
3!
+
+
=1
3! + +
=1
3!
+ +
=1
2!
The cofactor of
Lagrangian Strain Tensor
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The tensor Tplays and important role in
continuum mechanics. It is called the right Cauchy-Green Tensor.
The concept of strain is used to evaluate how much agiven displacement differs locally from a rigid bodydisplacement. Consequently, a strain function is apoint function that vanishes in a state consisting ofonly rigid body displacements.
The Right Cauchy-Green Tensor
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We will introduce the two most common strain
functions and demonstrate what they mean in onedimension: Lagrangian and Eulerian Strains.
First the Lagrangian Strain: =
()
Notice that when there is no deformation, we can assumethat the deformation gradient is the identity tensor(Why?). In this case, T = .Clearly = , the zerotensor.
Strain Functions
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First, show that
= 12 = 12
+
+
For example let = 1, = 1:1
2 =
1
2 + +
= 12
+
+
+
+
=1
22
+
+
+
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=
where is the deformation gradient tensor and is its
inverse. If we take the coordinates , = 1, , 3 as the
material coordinates and , = 1, , 3 as spatial
coordinates, then simple chain rule implies,
=
=
Eulerian Strain
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We can therefore write that,
=
If the unit vector along the deformed element is ,
then the length of the element is given by: =
Similarly, in the undeformed system, we have:
= with as the unit vector aligned with thematerial element . In this case, we can write,
= =
=
Eulerian Strain
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The strain,
=
2=
1
2
=
where is the Eulerian strain tensor,
( ). In
component form, following the same arguments it may
similarly be proved that in one dimensional strain,
=
2 =
Eulerian Strain
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Again, beginning with unit vectors ,, = and
= . Recall that with the deformation gradienttransformation,
= = squaring, we have that,
= =
from which we can see that
=
()
=
=
=
Strain of a line element
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()
2=
+ ( )
2
2( )
2=
( )
Which is the elementary definition of strain. We can seefrom here that this definition is only approximatelycorrect when strain is small and will not be a valid strain
function if the difference between and isappreciable. When that happens, we have a situationcalledgeometric nonlinearity in the sense that a linearfunction no longer defines the strain-displacementrelation
Infinitesimal Strain
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Another way to arrive at this same conclusion is to note
that for small strains, the second-order terms in theLagrangian function
=
=
+
+
Is small so that,
[
+
]
Both the Lagrangian as well as the Eulerian strain becomeindistinguishable from small strain.
Infinitesimal Strain