TENSORS/ 3-D STRESS STATE

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TENSORS/ 3-D STRESS STATE
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• Tensors
• Tensors are specified in the following manner
• A zero-rank tensor is specified by a sole
  component, independent of the system of reference
  (e.g., mass, density).
• A first-rank tensor is specified by three (3)
  components, each associated with one reference
  axis (e.g., force).
• A second-rank tensor is specified by nine (9)
  components, each associated simultaneously with
  two reference axes (e.g., stress, strain).
• A fourth-rank tensor is specified by 81
  components, each associated simultaneously with
  four reference axes (e.g., elastic stiffness,
  compliance).

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• The Number of Components (N) required for the
  description of a TENSOR of the nth Rank in a
  k-dimensional space is
• N kn
• EXAMPLES
• (a) For a 2-D space, only four components are
  required to describe a second rank tensor.
• (b) For a 3-D space, the number of components N
  3n
• Scalar quantities ? 30 ? Rank Zero
• Vector quantities ? 31 ? Rank One
• Stress, Strain ? 32 ? Rank Two
• Elastic Moduli ? 34 ? Rank Four

(4-1)
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• The indicial (also called dummy suffix) notation
  will be used.
• The number of indices (subscripts) associated
  with a tensor is equal to its rank. It is noted
  that
• density (?) does not have a subscript
• force has one (F1, F2, etc.)
• stress has two (?12, ?22, etc.)
• The easiest way of representing the components of
  a second- rank tensor is as a matrix
• For the tensor T, we have

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The collection of stresses on an elemental
volume of a body is called stress tensor,
designated as ?ij. In tensor notation, this is
expressed as where i and j are iterated
over x, y, and z, respectively.
(4-2)
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• Here, two identical subscripts (e.g., ?xx)
  indicate a normal stress, while a differing pair
  (e.g., ?xy) indicate a shear stress. It is also
  possible to simplify the notation with normal
  stress designated by a single subscript and shear
  stresses denoted by ?, so
• ?x ? ?xx ?xy ? ?xy

(4-3)
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• In general, a property T that relates two vectors
  p p1, p2, p3 and q q1, q2, q3 in such a
  way that
• where T11, T12, . T33 are constants in a second
  rank tensor.

(4-4)
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• (Eqn. 4-4) can be expressed matricially as
• Equation 4-5 can be expressed in indicial
  notation, where
• The symbol is usually omitted, and the
  Einsteins summation rule used.

(4-5)
(4-6)
(4-7)
dummy Subscript (appears twice)
Free Subscript
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Transformations

• Transformation of vector p p1, p2, p3 from
  reference system x1, x2, x3 to reference x1,
  x2, x3 can be carried out as follows
• where

X3
p
X3
X2
X2
X1
X1
Angle between XiXj
Old
New
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• In vector notation
• p p1i1 p2i2 p3i3
• where i1, i2, and i3 are unit vectors
• p p1 cos(X1X1) p2 cos(X1X1) p3
  cos(X1X1)
• a11 p1 a12 p2 a13 p3
• where aij cos (XiXj) is the direction cosine
  between Xi and Xj.

(4-8)
(4-9)
Old
New
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• The nine angles that the two systems form are as
  follows
• 
  
• This is known as the TRANSFORMATION Matrix

Old System
New System
(4-10)
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• What is the transformation matrix for a simple
  rotation of 30o about the z-direction?

30o
30o
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• For any Transformation from p to p,determine the
  Transformation Matrix and use as follows
• This can be written as
• It is also possible to perform the opposite
  operation, i.e., new to old

(4-11)
(4-12)
(4-13)
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• The Transformation of a second rank Tensor Tkl
  from one reference frame to another is given as
• OR, for stress
• Eqn. 4-14(a) is the transformation law for
  tensors and the letters and subscripts are
  immaterial.
• Transformation from new to old system is given
  as

(4-14a)
(4-14b)
(4-15)
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NOTES on Transformation

• Transformation does not change the physical
  integrity of the tensor, only the components are
  transformed.
• Stress/strain Transformation results in nine
  components.
• Each component of the transformed 2nd rank tensor
  has nine terms.
• lij and Tij are completely different, although
  both have nine components.
• Lij is the relationship between two systems of
  reference.
• Tij is a physical entity related to a specific
  system of reference.

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• Transformation of the stress tensor ?ij from the
  system of axes to the
• We use eqn. 4-14
  
• First sum over j 1, 2, 3
• Then sum over i 1, 2, 3

(4-16)
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• For each value of k and l there will be an
  equation similar to eqn. 4-16.
• To find the equation for the normal stress in the
  x1 direction, let m 1 and n 1.
• Let us determine the shear stress on the x
  plane and the z direction, that is ?13 or
  ?xz for which m 1 and n 3

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• The General definition of the Transformation of
  an nth-rank tensor from one reference system to
  another (i.e., T?T) is given by
• Tmno. lmilnjlok.Tijk..
• Note that aij lmi (the letters are
  immaterial)
• The transformation does not change the physical
  integrity of the tensor, only the components are
  transformed

(4-16)