Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

spatial velocity gradient I and the material time derivative -of the right stretch tensor, iJ, which is given in terms of material coordinates. By means of I = F.F- 1, the polar decomposition F = RU, the pro.duct rule of differentiation .and the property _RTR = I, we conclude that

(2.157)

where R characterizes the time rate of change of the proper orthogonal rotation tensor R. From RRT = I we deduce directly upon differentiation that RRT + (RRT)'r = 0. Thus,

(2.158.)

which clearly s.hows that RRT is a skew tensor.

In order to express the rate of defonnation tensor d and the spin tensor w in terms of Uwe adopt the additive decomposition of the spatial velocity gradient (2.1.45). Hence, using (2.l57h and property (2.158_), the definitions (2J46)1 and (2.147)1 g.ive

These relations show that d is not a pure rate of strain and w is not a pure rate of rotation.

......, ..................................... :.····:········.· ·······················································."····:·········."····:····.···'·."························:········. ······:·.··:·:··:········

EXAMPLE 2,12 Show that for a rigid-body .rotation (.motion) the .rate of deformation tensor d vanishes and the identity w = ·R.RT holds.

which means lhat for the case of a rigid-body rotation the spin tensor w coincides with the skew tensor RRT. •

EXAMPLE 2.13 Suppose a rigid-body is rotating about an axis. The rotation is characterized by the angular velocity vector w = w(t_), as-depicted in Figure 2.8.

An arbitrary point x of the body with current position x E n is moving around a circle with the spatial velocity v(x, t) relative to a fixed point 0. Show that the ·velocity

9

= w = w(t)

v=wxx

n·.-._·.·._·.._·:_.

·Figure 2.8 Velocity v of a particle relative to a fixed po.int 0.

where w is the time-dependent antisymmetric spin tensor.

Solution. The rotation of the rigid-body may be described by the linear transformation x = R(t)X (compare with eq. (2.95)), with the proper orthogonal rotation tensor R and the referential position X of the point. The material time dedvative of x yields, by means of the product rule, eq. (2.28)i and X = RT(t)x7

For a rigid-body .rotation we know from the last example that the skew tensor RRT coincides with the spin tensor w (see eq. (2.160)), which gives the desired expression (2J61)1• Relation v{x, t) = w{t) x x .is well-known from rigid-body dynamics.

In conclusion, relation (2.161) shows that the spin tensor w is associated with the angular velocity of a .rotating rigid-body -characterized by the vector w. In fact, w is simply the axial vector of the skew tensor w. Eq. (2.161) represents a nice physical interpretation of relations (l.118) and (2.152). •

·Material time derivatives -of some strain -tensors. Our present starting point involves material strain tensors and the.ir derivatives with respect to time t. In particular, we compute the material time derivative of the Green-Lagrange strain tensor, E, the

·_.. right Cauchy-Green tensor, -C, and introduce the rotated rate of deformation tensor DR.

From the definition of the Green-Lagrange strain tensor (2.67), and with the pro.duct

.:":._rule, eq. (2.139),1 and the rate of deformation -tensor d introduced in (2.146)i, we find

-...-_.that

(.2..1.63)

The .material time derivative -of tensor E is also known as the material str.ain rate tensor E. As can be seen from (2.163)..i, it is simply the pull-pack of the covariant rate of deformation tensor d, which we may write as E = x;- 1( d~) = FTdF (see rule

(2.84h).

Now we may show that the directional derivative of the Green-Lagrange strain tensor E in the direction of v equals lhe material strain rate tensor E, as given in (2.163). Using eq.. (2.138) and the common properties of the product rule, we find using defini-

:tion (2.67) that

(2.:164)

and with (2.163) 1,

The resulting relation (2.16.5) may also be found immediately from (2 ..20).

which is known as the rotated rate of deformation tensor D1t. Using the polar decomposition F = RU and the sym.metry UT = U we find from (2."166)2 that C = 2U(RTdR)U = 2UDnU. FinaUy, we have an alternative expression for Dn_., i.e.

where c-1/ 2 = u-1 is in accordance with (2.93J1.

In the following we consider spatial strain tensors and compute a relationship between the material time derivative of the le.ft Cauchy-Green tensor, b, and the spatial tensors I and b. Recall the definition (2.78) of b and use the product rule :in order to

where the definition (2.139),1 of the spatial velocity gradient I is to be used.

Hence, with the derived relations (2.144h and (2.143.h and the definition of the EulerAlmansi st.rain tensor (2.81), we obtain

where the definition (2.146)1 of the rate of deformation tensor d is to be used. II

l\tlaterial time derivatives of spatial line, surface and volume elements.. We

consider "first spatial and material line elements dx E Hand dX E Q0 , as introduced in (2.37), and compute the material time derivative of dx. ·we know from (2.38) that line elements map via the deformation gradient according to dx = F(X_, t)dX. By means of .the product rule .and relation (2..l 39),1 we find that

As can be s-een, I = .gradv is a spatial tensor field transforming dx in.to dx.

/ fonsequently, with relation (2.174), F = IF and properties (l.95), (l.94), (l.279) we

.-:...-:... find expressions for the material time derivative of the scalar field J, namely

}= JF-T : .F = JF-T : IF

= JF-T.FT : I = JI : gradv

.·_._..··using the additive split 1 = gTadv = d +wand knowing that the trace of a skew tensor

....·.·.is zero, we deduce from (2.l 75)n an important alternative expression for .i, namely

··_::_-:·:< ~e also may write DJ(X, t)/.Dt = DvJ(X1 t).

{:::_<·....·_·_·.... A motion with J = 1 we called isochoric (keeping the volume constant, dv =

:;/·:··... we say that the body is subjected to so-called internal ·constraints. In particular, the :_.-·._.· conditions formulated in (2.177) characterize the most important .internal constraint ·.... known as the internal kinematic constraint, or more precisely the incompressibility

. ·:···constraint.

104 2 Kinematics

Now we perform the material ti-me derivative of a··vector element ds of infinites-

imally sma11 area -defined in the current configuration. Nanson's formula (2.54), i.e. ds = JF-TdS, the product rule and eqs. (2.l75)n and (2.144h, we obtain

with the second-order unit tensor I (note that the spatial velocity gradient I has a si-milar symbol as I),. In .index notation eq. (2. l78h reads as

Expression (2.178.h (and (2.179)) represent the relation between the rate of change of the infinitesimal spatial vector area in terms of ds, the divergence of the spatial velocity

field the transpose spatial ·velocity gradient I.

Finally, the material ti.me derivative of the spatial volume element dv = J d 1/ gives

of the product rule

EXERCisgs

1.Show that in a moving continuum the spatial velocity field v and the spatial vorticity fie"ld .2w are related to .their material time derivatives by the identity

.Hint: Take the curl of relation (2.1.51) by considering eqs. {2.153) and (2.154),

and then us-e property ( 1.292) with ( 1.275).

2.Consider a deformed fiber characterized by the vector a, with fal = 1. Show that the material time derivative of the logarithmic stretch ratio ,.\ at a particle along that direction a is given by

Project the tensor d onto the orthonormal basis vectors e.u and give a physical interpretation of the rate of deformation tensor d. In particular, discuss the diagonal components daa and the off-diagonal components dab (.a # b), a, lJ = 1, 2, 3, of the matrix [d] .

Compute the dot product of vector a and the

and use the fact that a· a = 0 (since a· a = 1), and a · wa = a · -(w x a) = -0

(since u · (u x v) = O_, whic.h should be compared with -eq. ( l.31 )).

3.By combining eq. (2..141) with eq. {2.182) show that the material time derivative of a unit vector a may be expressed as

We now assume that ~a' a = 1, 2, 3; denote the three eigenvalues of d and fia its three associated normalized eigenvectors. For the particular case in which a is an eigenvector of the rate of deformation tensor d, show by means of (2.183)

Obviously, the spin tensor w is a measure for the rate of change of the eigenvectors of d, which gives a physical interpretation of the spin tensor w.

4.Recall Exercise 2, p. 93, with the motion x = X + c(t) (e2 • X)e1, the -orthogonal unit vectors -ei, e2, e3 = e1 x e2 and the parameter c(t) = tanO(t) > 0.

(a)Show that the spat!al velocity field v(x, t) .may be given by v = i~(t)(e2·x)e·.z, where c(t) = tanO(t) is called the shear rate (for typical ranges of shear

rates of-some specific materials see, for example, BARNES et al. [1989]).

(b)Based on result (a) compute the rate of deformation tensor d(x, t), the spin tensor w(x, t) and the angular velocity vect:or w{x, t), i.e. the axial vector of the spin tensor, in terms of ei, e2 and e3 .

(c)Show that the rate of -de.formation tensor .may be expressed in its spectral

form

:1

d = L ltaDa 0 D.a , a=l

and hence compute the three eigenvalues of d~ .i.e. O:a, .and the three normalized eigenvectors of dt i.e. the set {Da}.

5. Consider the representation of the .deformation gradient (compare with (2.1.21)2):

o=l

where the principal referential directions Na (orthonormal eigenvectors) are as- sumed not to change in time.

(a)Compute the rate of deformation tensor d(x, t), the spin tensor w(x, t) and the .angular velocity vector w(x, t)~

(b)Bas-e.d on representation (2.185), find uu-1 and show that the antisymmetric tensors w and RRT coincide.

wd + dw = (divv)w .

2.8 Lie Time Derivatives

Consider a spatial field f f (x, "/;) characterizing some physical scalarT vector or tensor quantity in space and time (for the relevant notation recall Section 2.3). .I.n the following we compute the change of f relative to a vector field v which is commonly known as the Lie time derivative off, denoted by 11v(f).

The Lie time derivative of a spatial field f is obtained using the following concept:

(i)compute the pull-back ope.ration of f to the reference configuration; as a result we obtain the associated material fie:ld :F(X, t;) ~ x; 1(/ (x, t) );

(ii)take the .material time derivative of :F, i.e. :F, and

(iii)carry out the push:fonvard operation of the result to the current configuration. This technique is simply summarized as·

(2.186)

Since the material time derivative can be obtained from the directional derivative according to relation (2.20) we may apply the concept of directional derivative to eq. (2.186). Consequently, we express the Lie time derivative of .f = f (x, t) as the directional derivative (se-e also WRIGGERS [1988]). Hence, eq. (2.186) reads equivalently as

(.2.187)

In summary: the Lie time derivative of the spatial field .f is the push-forward of the directional derivative of the associated material field :F = x.:;- 1(.f) in the direction of the vector v, identified as the velocity vector.

By recalli.ng definition (I. .266), we may specify the directional derivative of :F at

108 2 Kinematics

2. Consider a contravariant spatial vector field u with its material time derivative

.u = au/at+ (gradu)v (see eq. (2.30)).

Recall relations (2.87) and (2. l43h and show that the Lie time derivative of u .is,

according to the rule (2..186),