175 10.3 The stretch ratio and rotation

and using this:

1

» = »( e) = √ . (10.25)

e · B’1 · e

The tensor B is called the left Cauchy Green deformation tensor. In component

form Eq. (10.25) can be written as

1

» = »( e) = , (10.26)

∼

eT B’1

¸ e

∼ ∼

with

B = F FT . (10.27)

The direction change (rotation) of a material line segment with direction e in the

current con¬guration with respect to the reference con¬guration can formally be

calculated with

F’1 · e

d

’1 ’1

e0 = F · e =F ·e»= √ , (10.28)

e · B’1 · e

d0

and alternatively, using components:

F ’1 e

d

’1

= F ’1 e » =

e =F ∼

e . (10.29)

∼0 ∼ ∼

d0

B’1

eT e

∼ ∼

At the end of the present section we will investigate the in¬‚uence of an extra

displacement as a rigid body of the current con¬guration for the tensors that were

introduced above, see Section 9.7. Properties with respect to the extra rotated (via

the rotation tensor P) and translated (via the translation vector ») virtual state are

denoted by the superscript — . Because

F— = P · F, (10.30)

it can immediately be veri¬ed that

C— = F—T · F— = FT · PT · P · F = FT · P’1 · P · F

= FT · I · F = FT · F = C, (10.31)

and

B— = F— · F—T = P · F · FT · PT = P · B · PT . (10.32)

Based on Eq. (10.31) the right Cauchy Green tensor C is called invariant for extra

displacements of the current state as a rigid body. This invariance is completely

trivial if the Lagrange description » = »( e0 ) for the stretch ratio is taken into

consideration.

Deformation and rotation, deformation rate and spin

176

The left Cauchy Green tensor B is, based on the transformation relation Eq.

(10.32), called objective. In calculating the stretch ratio » with the earlier derived

Eulerian description, the result in the direction e in the current state (with asso-

ciated tensor B) has to be identical to the result in the direction e — = P · e in

the virtual (extra rotated and translated) current state (using tensor B— ). That this

demand is satis¬ed can be veri¬ed easily.

Because the Cauchy stress tensor σ transforms in a similar way as tensor B (see

Section 9.7) the tensor σ is also objective. The deformation tensor F is neither

invariant, nor objective.

10.4 Strain measures and strain tensors and matrices

In the preceding sections the stretch ratio » is considered to be a measure for

the relative length change of a material line segment in the transition from the

reference con¬guration to the current con¬guration. If there is no deformation

» = 1. Often it is more convenient to introduce a variable equal to zero when there

is no deformation: the strain. In the present section several different, generally

accepted strain measures are treated.

In the previous section it was found, using the Lagrangian description:

»2 = »2 ( e0 ) = e0 · FT · F · e0 = e0 · C · e0 . (10.33)

Coupled to this relation, the Green Lagrange strain µGL is de¬ned by:

»2 ’ 1 1

µGL = µGL ( e0 ) = = e0 · FT · F ’ I · e0

2 2

1

= e0 · (C ’ I) · e0 . (10.34)

2

This result invites us to introduce the symmetrical Green Lagrange strain tensor

E according to:

1 1

E= FT · F ’ I = (C ’ I) , (10.35)

2 2

which implies

µGL = µGL ( e0 ) = e0 · E · e0 . (10.36)

When using matrix notation, Eq. (10.35) can be formulated as

1 1

E= FT F ’ I = C’I , (10.37)

2 2

implying

µGL = µGL ( ∼0 ) = eT E e0 .

e (10.38)

∼0 ∼

177 10.4 Strain measures and strain tensors and matrices

From a mathematical perspective these are well manageable relations. The Green

Lagrange strain tensor E (with matrix representation E) is invariant for extra rigid

body motions of the current state. The components of the (symmetrical, 3 — 3)

Green Lagrange strain matrix E can be interpreted as follows:

• The terms on the diagonal are the Green Lagrange strains of material line segments of

the reference con¬guration in the x-, y- and z-directions respectively (the component

on the ¬rst row in the ¬rst column is the Green Lagrange strain of a line segment that

is oriented in the x-direction in the reference con¬guration).

• The off-diagonal terms determine the shear of the material (the component on the

¬rst row of the second column is a measure for the change of the angle enclosed by

material line segments that are oriented in the x- and y-direction in the undeformed

con¬guration).

For the deformation tensor F and the displacement vector u, both applying to

the current con¬guration and related to the reference con¬guration, the following

relation was derived in Section 9.6:

T

F = I + ∇ 0u . (10.39)

Substitution into Eq. (10.35) yields

1 T T

E= ∇ 0u + ∇ 0u + ∇ 0u · ∇ 0u . (10.40)

2

It can be observed that the ¬rst two terms on the right-hand side of this equation

are linear in the displacements, while the third term is non-linear (quadratic).

The linear strain µlin is de¬ned, according to:

µlin = µlin ( e0 ) = » ’ 1 = e0 · FT · F · e0 ’ 1. (10.41)

This expression is not easily manageable for mathematical elaborations.

At small deformations and small rotations, for which: F ≈ I (and therefore the

T

components of the tensor ∇ 0 u are much smaller than 1), it can be written:

µlin = µlin ( e0 )

= 1 + e0 · FT · F ’ I · e0 ’ 1

1

≈ e0 · FT · F ’ I · e0

2

1

≈ e0 · FT + F ’ 2I · e0 . (10.42)

2

The last found approximation for the linear strain is denoted by the symbol µ. This

strain de¬nition is used on a broad scale. Therefore, the assumption F ≈ I leads

to the following, mathematically well manageable relation:

Deformation and rotation, deformation rate and spin

178

1

µ = µ( e0 ) = e0 · µ · e0 with µ = FT + F ’ 2I , (10.43)

2

where the symmetrical tensor µ is called the linear strain tensor. In displace-

ments this tensor can also be expressed as

1 T

µ= ∇ 0u + ∇ 0u . (10.44)

2

The strain tensor µ is linear in the displacements and can be considered (with

respect to the displacements) as a linearized form of the Green Lagrange strain

tensor E. In component form this results in the well-known and often-used

formulation:

⎡ ¤

‚uy ‚uz

‚ux 1 ‚ux 1 ‚ux

2 ‚y0 + ‚x0 2 ‚z0 + ‚x0

‚x0

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ 1 ‚uy ‚ux ⎥

‚uy 1 ‚uy ‚uz

µ = ⎢ 2 ‚x0 + ‚y0 2 ‚z0 + ‚y0 ⎥. (10.45)

‚y0

⎢ ⎥

⎢ ⎥

⎣ ¦

‚uy

1 ‚uz 1 ‚uz ‚uz

‚ux

2 ‚x0 + ‚z0 2 ‚y0 + ‚z0 ‚z0

The components of the (symmetrical, 3 — 3) linear strain matrix µ can be

interpreted as follows:

• The terms on the diagonal are the linear strains of material line segments of the ref-

erence con¬guration in the x-, y- and z-directions respectively (the component on the

¬rst row in the ¬rst column is the linear strain of a line segment that is oriented in the

x-direction in the reference con¬guration).

• The off-diagonal terms determine the shear of the material (the component on the

¬rst row of the second column is a measure for the change of the angle enclosed by

material line segments that are oriented in the x- and y-direction in the undeformed

con¬guration. See Fig. 10.3.

y

‚ux

dy

‚y

‚uy

dx

dy ux ‚x

uy

x

dx

Figure 10.3

Interpretation of linear strain components.

179 10.4 Strain measures and strain tensors and matrices

In the previous section the Eulerian description for the stretch ratio was derived:

»’2 = »’2 ( e) = e · F’T · F’1 · e = e · B’1 · e . (10.46)

Coupled to this relation, the Almansi Euler strain µAE is de¬ned by

1 ’ »’2 1

= e · I ’ F’T · F’1 · e

µAE = µAE ( e) =

2 2

1

= e · I ’ B’1 · e. (10.47)

2

This relation gives rise to introduce the symmetrical Almansi Euler strain tensor

A according to

1 1

I ’ F’T · F’1 = I ’ B’1 ,

A= (10.48)

2 2

which implies

µAE = µAE ( e) = e · A · e . (10.49)

When using matrix notation this may be formulated as

1 1

I ’ F ’T F ’1 = I ’ B’1 ,

A= (10.50)

2 2

implying

µAE = µAE ( ∼) = eT A ∼.

e e (10.51)

∼

Again from a mathematical perspective this represents well manageable relations.

The Almansi Euler strain is used only sporadically, in contrast to the related

symmetrical and objective strain tensor µF (Finger) de¬ned by

1 1

µF = F · FT ’ I = ( B ’ I) = B · A = A · B. (10.52)

2 2