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Using Equation (3.19a) a similar expression involving the Almansi strain
tensor can be derived as,


1 ds2 ’ dS 2
= n · en (3.24)
ds2
2


where n is a unit vector in the direction of dx.
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EXAMPLE 3.4: Physical interpretation of strain tensors
Refering to Example 3.2 the magnitude of the elemental vector dx2 is
ds2 = 4.51/2 . Using (3.23) the scalar value of Green™s strain associated
with the elemental material vector dX 2 is,
1 ds2 ’ dS 2 7
µG = =
dS 2
2 4
Again using (3.23) and Example 3.3 the same strain can be determined
from Green™s strain tensor E as,
103 7
0
µG = N T EN = [0, 1] =
437 1 4
Using (3.24) the scalar value of the Almansi strain associated with the
elemental spatial vector dx2 is,
1 ds2 ’ dS 2 7
µA = =
ds2
2 18
Alternatively, again using (3.24) and Example 3.3 the same strain is
determined from the Almansi strain tensor e as,
1

1110 7
9 2
µA = nT en = √ , √ =
1
2 2 18 9 ’4 18

2




Remark 3.5.2. In terms of the language of pull back and push forward,
the material and spatial strain measures can be related through the operator
φ— . Precisely how this operator works in this case can be discovered by
recognizing, because of their de¬nitions, the equality,


dx1 · e dx2 = dX 1 · E dX 2 (3.25)


for any corresponding pairs of elemental vectors. Recalling Equations (3.12“
13) enables the push forward and pull back operations to be written as,


Push forward
e = φ— [E] = F ’T EF ’1 (3.26a)
Pull back
E = φ’1 [e] = F T eF (3.26b)

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3.6 POLAR DECOMPOSITION



3.6 POLAR DECOMPOSITION

The deformation gradient tensor F discussed in the previous sections trans-
forms a material vector dX into the corresponding spatial vector dx. The
crucial role of F is further disclosed in terms of its decomposition into stretch
and rotation components. The use of the physical terminology stretch and
rotation will become clearer later. For the moment, from a purely mathe-
matical point of view, the tensor F is expressed as the product of a rotation
tensor R times a stretch tensor U to de¬ne the polar decomposition as,
F = RU (3.27)
For the purpose of evaluating these tensors, recall the de¬nition of the right
Cauchy“Green tensor C as,
C = F T F = U T RT R U (3.28)
Given that R is an orthogonal rotation tensor as de¬ned in Equation (2.27),
that is, RT R = I, and choosing U to be a symmetric tensor gives a unique
de¬nition of the material stretch tensor U in terms of C as,
U2 = UU = C (3.29)
In order to actually obtain U from this equation, it is ¬rst necessary to
evaluate the principal directions of C, denoted here by the eigenvector triad
{N 1 , N 2 , N 3 } and their corresponding eigenvalues »2 , »2 , and »2 , which
1 2 3
enable C to be expressed as,
3
»2 N ± — N ±
C= (3.30)
±
±=1

where, because of the symmetry of C, the triad {N 1 , N 2 , N 3 } are orthog-
onal unit vectors. Combining Equations (3.29) and (3.30), the material
stretch tensor U can be easily obtained as,
3
U= »± N ± — N ± (3.31)
±=1

Once the stretch tensor U is known, the rotation tensor R can be easily
evaluated from Equation (3.27) as R = F U ’1 .
In terms of this polar decomposition, typical material and spatial ele-
mental vectors are related as,
dx = F dX = R(U dX) (3.32)
In the above equation, the material vector dX is ¬rst stretched to give
U dX and then rotated to the spatial con¬guration by R. Note that U is a
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material tensor whereas R transforms material vectors into spatial vectors
and is therefore, like F , a two-point tensor.

EXAMPLE 3.5: Polar decomposition (i)
This example illustrates the decomposition of the deformation gradient
tensor
F = RU using the deformation shown below as,
x1 = 1 (4X1 + (9 ’ 3X1 ’ 5X2 ’ X1 X2 )t)
4
1
x2 = 4 (4X2 + (16 + 8X1 )t)
For X = (0, 0) and time t = 1 the deformation gradient F and right
Cauchy“Green tensor C are,
1 1 ’5 1 65 27
F= ; C=
48 4 16 27 41
from which the stretches »1 and »2 and principal material vectors N 1
and N 2 are found as,
0.8385
»1 = 2.2714; »2 = 1.2107; N1 = ;
0.5449
’0.5449
N2 =
0.8385
Hence using (3.31) and R = F U ’1 , the stretch and rotation tensors
can be found as,
1.9564 0.4846 0.3590 ’0.9333
U= ; R=
0.4846 1.5257 0.9333 0.3590


It is also possible to decompose F in terms of the same rotation tensor R
followed by a stretch in the spatial con¬guration as,

F =VR (3.33)

which can now be interpreted as ¬rst rotating the material vector dX to the
spatial con¬guration, where it is then stretched to give dx as,

dx = F dX = V (RdX) (3.34)

where the spatial stretch tensor V can be obtained in terms of U by com-
bining Equations (3.27) and (3.33) to give,

V = RU RT (3.35)
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3.6 POLAR DECOMPOSITION



Additionally, recalling Equation (3.17) for the left Cauchy“Green or Fin-
ger tensor b gives,
b = F F T = (V R)(RT V ) = V 2 (3.36)
Consequently, if the principal directions of b are given by the orthogonal
¯¯ ¯
spatial vectors {n1 , n2 , n3 } with associated eigenvalues »2 , »2 , and »2 , then
1 2 3
the spatial stretch tensor can be expressed as,
3
¯
V= »± n± — n± (3.37)
±=1

Substituting Equation (3.31) for U into Expression (3.35) for V gives V in
terms of the vector triad in the undeformed con¬guration as,
3
V= »± (RN ± ) — (RN ± ) (3.38)
±=1

Comparing this expression with Equation (3.37) and noting that (RN ± )
remain unit vectors, it must follow that,
¯
»± = »± ; n± = RN ± ; ± = 1, 2, 3 (3.39a,b)
This equation implies that the two-point tensor R rotates the material vec-
tor triad {N 1 , N 2 , N 3 } into the spatial triad {n1 , n2 , n3 } as shown in Fig-
ure 3.4. Furthermore, the unique eigenvalues »2 , »2 , and »2 are the squares
1 2 3
of the stretches in the principal directions in the sense that taking a mate-
rial vector dX 1 of length dS1 in the direction of N 1 , its corresponding push
forward spatial vector dx1 of length ds1 is given as,
dx1 = F dX 1 = RU (dS1 N 1 ) (3.40)
Given that U N 1 = »1 N 1 and recalling Equation (3.39) gives the spatial
vector dx1 as,
dx1 = (»1 dS1 )n1 = ds1 n1 (3.41)
Hence, the stretch »1 gives the ratio between current and initial lengths as,
»1 = ds1 /dS1 (3.42)
It is instructive to express the deformation gradient tensor in terms
of the principal stretches and principal directions. To this end, substitute
Equation (3.31) for U into Equation (3.27) for F and use (3.39) to give,
3
F= »± n± — N ± (3.43)
±=1
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n3

R
‚3 dX N3
N3
UdX dx
N2
dX
‚3 d X N3
p
d X N3 ‚2 d X N2 n2
‚2 d X N2
d X N2
‚1d X N1 ‚1d X N1
P
d X N1
N1
n1


time = t
time = 0

FIGURE 3.4 Material and spatial vector triads.




This expression clearly reveals the two-point nature of the deformation gra-
dient tensor in that it involves both the eigenvectors in the initial and ¬nal
con¬gurations.
It will be seen later that it is often convenient, and indeed more natural,
to describe the material behavior in terms of principal directions. Con-
sequently, it is pertinent to develop the relationships inherent in Equa-
tion (3.43) a little further. For this purpose, consider the mapping of the
unit vector N ± given by the tensor F , which on substituting the polar
decomposition F = RU gives,




F N ± = RU N ±
= »± RN ±
= »± n± (3.44a)
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3.6 POLAR DECOMPOSITION



Alternative expressions relating N ± and n± can be similarly obtained as,

1

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