1

TENSORS/ 3-D STRESS STATE

2

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:

333231

232221

131211

TTT

TTT

TTT

T

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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)ij

zzyzxz

zyyyxy

zxyxxx

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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.

3332321313

3232221212

3132121111

qTqTqTp

qTqTqTp

qTqTqTp

(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 Einstein’s summation rule used.

3

2

1

333231

232221

131211

3

2

1

q

q

q

TTT

TTT

TTT

p

p

p

(4-5)

3

1

3

1i jjiji qTp (4-6)

)3,2,1,( jiqTp jiji(4-7)

Free Subscript “dummy” Subscript (appears twice)

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Transformations

• Transformation of vector p [p1, p2, p3 ] from reference system x1, x2, x3 to reference x’

1, x’2, x’

3 can be carried out as follows

where

=

1112

2223

33

31

X1

X’1

X2

X3

X’2

X’3

ij Angle between X’iXj

New Old

p

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• In vector notation:

p = p1i1 + p2i2 + p3i3

where i1, i2, and i3 are unit vectors

p’ = p1 cos(X’1X1) + p2 cos(X’

1X1) + p3 cos(X’1X1)

= a11 p1 + a12 p2 + a13 p3

where aij = cos (X’iXj) is the direction cosine

between X’i and Xj.

NewOld

(4-8)

(4-9)

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• The nine angles that the two systems form are as follows:

=

This is known as the TRANSFORMATION Matrix

333231'3

232221'2

131211'1

321

aaaX

aaaX

aaaX

XXX

Old System

New

System ijL (4-10)

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• What is the transformation matrix for a simple rotation of 30o about the z-direction?

30o

30o

1X

2X

3X

'1X

'2X

'3X

ijL

000

030cos30sin

030sin30cos

333231

232221

131211

lll

lll

lll

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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

pLp '

jiji plp '

'jjii plp

(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:

kljlikij TaaT '(4-14a)

'klljkiij TaaT (4-15)

ijnjmimn ll '(4-14b)

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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

321 ,, xxx '3

'2

'1 ,, xxx

ijnjmimn ll '

332211'

inmiinmiinmimn llllll

333332233113

233222222112

133112211111'

nmnmnm

nmnmnm

nmnmnmmn

llllll

llllll

llllll

(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 x’1

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 x’z’ for which m = 1 and n = 3

331313321213311113

231312221212211112

131311121211111111'

llllll

llllll

llllllmn

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• The General definition of the Transformation of an nth-rank tensor from one reference system to another (i.e., TT’) is given by:

T’mno……. = 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)