Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

U11 lf22 - Ur2 ·

Knowing from .the Cayley..Hamilton equation (1.174) that any (second-order) tensor satisfies its own characteristic -equation, ·we may write instead of (2.110), U2 -

= 0. By use of U2 = C and rearranging we ·may write the expli-cit i.e.

(2.l I 1)

which coincides with the relation obtained in, for -example, HOGER and CARLSON

[1984] and TING [1985].

On taking the trace of the last equation (2.111) and knowing that trC = 11 ( C) and ~rl = 2 in two dimensions we are able to ex_press the first invariant of U in .terms of the (irst invariant of C.. Thus,

fo addition, using the relation det·C = det{UU) = (detU)2 , where the property (l.101) is applied, and recalling (2..65h we conclude that

Eigenvalues .and eigenvectors -of strain tensors" We introduce the mutually or-

~hogonal and normalized set of eigenvectors {Na} and their corresponding eigenvalues Aa, a = 1, 2, .3,, of the material tensor U as

UNa ·=Aa.Na with :jNal = 1 , (L = 1, 2, 3 . (2.114)

Furthermore, after combining (2.93) 1 with (2.114) we obtain the eigenvalue prob- lem for C, i.e.

Clearly U and C have the same orthonormal -eigenvectors, i.e. the set {Na}, called

~he princi-pal referential directions (or principal referential axes). However, the corresponding positive and real eigenvalues d.iffer. The eigenvalues of the symmetric tensor U are Aa, a = 1, 2, 3, called the principal s.tretches, while for the symmetric tensor C we find the squares of the principal stretches denoted by,,\~.

2.6 Ro.tation, Stretch Tensors

.find by means of (2.118)-1 that

.a=1

Note that both tensors involve principal directions in the reference and current configuration, which emphasizes the two-point character of these tensors .. Since tensors If and R are, in _general, non-symmetri~, the representations (2.121) and (2.. 122) may not be viewed as spectral decompositions in the sense introduced on p. 25. Hence, it is clear that the principal stretches ,\1' a = 1, 2, 3~ .in eq.. (2.121 h may not be interpreted ~s the eigenvalues of the deformation gradient F.

EXAMPLE 2.·s Suppose that the spectral decomposition of the right Cauchy-Green tensor C, Le. eq. (2. "119), is given. Since the principal stretches are functions of C we ·muywrite /\, = ,.\a (C). For later use we introduce the following relations

(compare also with SIMO and TAYLOR [19.9la] or SALEEB et al. ..[.1992]). Prove this set of equations for these disti net cases.

n==I

Recall that the three vectors Nn., with INuJ = 1, form an orthonormal bas·is (Nn ·Nb = c~ab), so that ·r'fu · dNa = 0 (the change of a vector with constant length is always

a:rthogonal to the vector itselt). lf we preand postmultiply eq. c2~126) with N" we find that

(provided that the plane deformation is isochoric), known as pure shear (or strip- biaxial extension). The associated principal stretches are ,.\ 1, ,.\2 = 1 and ,.\a = while R = I. The area remains constant in the plane spanned by the orthogonal eigenvectors n1 and n3 • Experimental results on a thin sheet of rubber under pure shear were "first published by RIVLIN and SAUNDERS . [.1951], see also the book by TRELOAR fl 975, Chapter 5].

The volume of a ·continuum body under biaxial or pure shear deformations according to kinematic relations (2.1.30) and (2.131) .remains constant, as we will see later in the text..

EXERCISES

1.Use (2..106) to show that vn = RUnRT., where n .is an integer.

2.The motion presented by eq. (.2.3) may be given in symbolic notation as x = x(X, t) = X + c(t ){e2 • X}e1 , e1 and e2 denoting -orthogonal unit vectors fixed in space, with the components (1, 0, 0) and (0, 1, 0), .respectively, and the parame- ter c(t) = tanlJ(t) > 0. The motion described causes a so-called simple shear

deformation (also known as a uniform shear deformation), where the planes

;i;2 = const. .are the shear planes and the direction along ~i·1 is the shear direction. The .an_gle B{t_) is a measure of the amount of shear.

(a)Compute the matrix representations of tensors F, ·C, b, c-1, b-1..

(b)Show that F, C and b may be expressed as

(c)Show that a simple shear defommtion is iso.choric.

(d)Compute the principal stretches /\1 , a = 1, 2, 3, and the three principal referential directions N,1' i.e.. the normalized eigenvectors of C.

(e)Compute the angle between the principal referential directions N(1 and el, e2~ ea = e1 x -e2 in terms of c(t).

(f)Compute the three principal spatial directions Da, i.-e. the normalized eigenvectors of b.

Hint: The principal directions nu are simply obtained from N(1 by inter- changing .-e1 and e2.

(g)By using results {d) and (t) determine the spectral decompositions of the Cauchy-Green tensors C = U2 , b = v2 , i.e. (2.11.9), (2.120), and the

94 2 Kinematics

representations of the deformation gradient F and the rotation tensor R, as derived in eqs. (2.121 h and (2.122)3, respectively.

(h)From (g) we have basically obtained the polar decomposition ·F = RU via the spectral decomposition of C. For the particu tar deformation of simple shear, com.pare (and check) your results for the right stretch tensor U and

the rotation tensor R w.ith those obtained from the closed-form expression

(2.109) and R = Fu- 1•

3.Rec.all the representation (2.l2lh, the polardecomposifion F =RU and (2.l 18)t and (2.119). Establish the following expressions relating .the two sets of principal directions {N.a} and {Dn}, a= 1, 2, 3, i.e..

(2.132)

4.A body undergoes uniform extension in all three directions .according to (2.129). Find the matrix representations of tensors F, R, U, E" c.

5.Consider the deformation gradient Fin the form of a 3 x 3 matrix with detF > 0. Write a computer program in -order to determine the right stretch tensor U and the rotation tensor R.

Start the procedure by computing C = FTF and the eigenvalues and eigenvectors

of C. Then use the spectral decomposition (2.119) for the tensor U, and by means of R = Fu- 1 finally find R.

(a = 1, 2, 3; no summation) where the scalar D" must be nonzero, with Da = 2.-\;! - / 1,,\~ + Ia>...;;2 • The first and third invariants of C, and also of b, are J.1 and

13_, respectively. The closed form expressions for Na ®Na and fia@n,1 circumvent the explicit computation of the ei_genvectors.

Hint: Relations (2.133) and (2.] 34) foUow from the Rivlin-Ericksen representation theorem (compare with p. 201 of this text), see BOWEN and WANG [1.976b],

MORMAN -{1986] for an analytical treatment and inter alia SIMO and TAYLOR

[ 199 .la] and MIEHE [ 1994] for a numerical treatment.

EXAM.PLE 2.9 Obtain the relationship which expresses the rate .of change of a unit vector a (characterizing the direction of a fiber in the deformed configuration, as introduced in (2.69h, with reference to Figure 2.7) .in terms of the spatial velocity gradient I in the form

(compare also with relation (1.237)). Consequently, from e.q.. (2. l43h we obtain :1 =

98 2 Kinematics

The antisymmetric spin tensor w may also be represented by its axial vector w,

where 2w is referred to as the vorticity vector (or s·p·in), which is of fundamental importance in fluid mechanics. The spatial vector field w = (cuilv) /2 is -called the angular velocity vector. Fo.r an frrotatio.nal motion, Le. curlv = o, the vorticity vector

2-w and (therefore) the sp.i n tensor w vanish .at each point.

Note that by ·means of eq. (2.152) and the vorticity vector, as given in (2..15.3), the

EXAMPLE 2.11 Ass~me a certain motion wherein the spatial velocity field v{xl t) is the gradient of a given.spatial scalar field <I>(x, t), known us the potential of v (v = grad<I>).

Compute the spatial acc-deration field a(x., t) and show that a may also be derived

from that potential <I>.

Solution. Knowing that every potential motion is an irrotational motion, i.e. curlv = o (or equivalently w = o, s-ee eq. (2.154))~ we find that the third term in (2.151) van- ishes. Consequently we have

and with the property that .the spatial time derivative commutes with the spatial gradi .. ent_,

Relation (2.156) shows that if v is the gradient of a :potential ~I>, then a may be derived from <I>. II