Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

time t

lal = 1

time l = 0

AH- 1 = F- 1a

IA;l·I=A-I

Figure 2.7 Deformation of a spatial line element with length d£ into n mate1ial line clement with length A- l d€.

In order to perform the relationship between ).no and a we recall eqs. (2.59)i,

(2.61):.i and definition (2.69h to give Fa0 = ...\a. Hence, from (2.60) we deduce that

Note that the vector quantities a and dx are naturally associated with the current con- jigu111tion of the body.

Using Taylor's expansion, the associated position vector Y E S20 , which is described by the inverse motion x-1 (y' ·t)' may be express-ed by -means of (2.68), (2.69) and the inverse of the deformation _gradient (2.40). By the chain rule we find, by anal-

::·.·:..::··· :x;1 ( eb), wh.ich provide the relations.hips between the -material and spatial quantities, _:_:·_:·:.:::.... ~~e. e = F-TEF- 1 and E = FTeF, respectively.

·However, the push-forward and pull-back operations on con.travariant second-order

tensors (suc11 as (c-1) tt, bu and most of the common stress tensors) are according to

. .. ... . ~: :·.

(2.85)

·· :: ·. ~-~ t~e following chapters we use covariant strain tensors in combination with con..

"travariant stress tensors.

/·_:._._.....: · For completeness we write down the push-forward and pull-back operations on

.._covariant vectors~ i.e..

(2.86)

..

... and of contravariant vectors, i.e.

(2.87)

·:..:·.- _.. Finally we provide the so-called P.iol.a transformation of a spatial vector field

if~ u(x, t), with components u 0 , i..e.

(2.88)

2 Kinematics

where U - U(X, t) denotes a material vector field with the components UA. The transformation from u to U involves the pull-back of .u scaled by the volume ratio J,.

The inverse of eq. (2.88), i.e. u = .1- 1x*{U), involves the push-forward operation on

U here scaled by the inverse of the volume ratio, i.e. .1-1•

For a more complete source on the underlying concept the =mathematically oriented reader is referred to the book by MARSDEN and HUGHES (1994, .and references therein].

EXERCISES

1.For a given material point ex.press conditions on the right Cauchy-Green tensor C which ensure that

(a)no stretch occurs in a specified directfon ao of a fiber,

(b)no change in the angle between a pair of specified directions (a0 1, a0 2 ) takes place, and

(c)no change occurs in an infinitesimal surface element els placed in a plane

perpendicular .to a given direction (set ds = dS)..

2. Let a body undergo a homogeneous deformation defined by

where a, /J, 'Y are constants.

(a)Determine· the components of the material and spatial strain tensors C, E and b? e, respectively.

(b)Take the values /J = -cos() and 'Y = --sinO and show that if a = 1 the

strains are zero. Explain that for a = 1 the map corresponds to a rotation of magnitude fJ about the ){1-axis.

3. If the deformation of a body is defined by

the body is said to be in a state of plane strain. Defonnations and strains occur only .in planes :i:a = const .and do not depend on

x3 are zero,. Determine the matrix representations of tensors F, E, e.

·.·...-·

4.Using relations (2.45h and (2.46:h show that the strain tensors E and =e may be expressed in terms of the displacement gradient tensor according to

......::.:-:.-·----------~------------~------~

..:::_/·:.

:.._:._.:{in the following we decompose a local motion, characterized by the nonsingular (in-

'./ vertible) tensor F(X, t), into a pure stretch and a pure rotation. As above, the argu-

::.....\)iients of the tensors are omitted, for convenience. However, they will be employed

;:.::·;:-.:·:::\Vhen additional information is .needed.

·:.· ......

;/ P~tardecomposition. At each point X E 0 0 and each time t, we have the following i: (~nique polar decomposition of the deformation gradient F:

/_:·/._the right (or material) stretch tensor and the left (or -spatial) stretch tensor, respec-

:·_:.·:.:.:_:_·_:tively. They =measure local stretching (or contraction) along their mutually orthogonal

·.·. :eigenvectors, that .is a change of local shape. The right stretc·h tensor U (w:ith compo- ·:-..'.. ·.~ents U;u1) is defined with respect to the reference configuration while the left stretch ):):.~te_~s~r v (with components 'V.ab) acts on the current configuration. Note that in this

.-//·\·.~ext the symbol U also stands for the material displacement field and v for the spatial

/·'.(._"ye"locityfield, respectively.

.86 2 Kinema.tics

which is based on the square-root theorem; see, for example, GURTIN [198 la, p. 13]. Consider the relation detC = detb = J 2 we deduce with (2.93) and rule (1..101) the important property

The unique R is a proper orthogonal tensor, with detR = 1_, called the rotation tensor. It measures the local rotation that is a change of local orientation. A rigidbody -rotation about a fixed origin is then characterized {f and only {f U = v = I, so that F = R. Hence, each material line element dX is rotated into a unique spatial 1.ine

element dx (and vice versa) according to (2.38), i.e.

dx = R{X, -t)dX or cb;a = RaA cLY_.\ . (2.95)

On the other hand if R = I, the .deformation is called pure stretch, for which ·.F = U =

·v. Using (2.95), the pointwise polar decomposition (2.91.) may be written as

.Relation (2.96) describes a pure stretch of dX by the material tensor U, (UAu), followed by a pure rotation performed by the rotation tensor R, (RaA)· The rotation tensor transforms the material vector UdX into the spatial vector dx. However, relation (2.97) describes a pure rotation of dX performed by the same rotation tensor R, (RaA), which transforms .the material vector dX into the spatial vector RdX. Th.is rotation is followed by .a pure stretch of RdX with the spatial tensor v, (vab), which gives the spatial vector dx. In both cases the rotation tensor R maps between the reference and the current configuration and, therefore, R is a two-point tensor like F.

Relation F = RU is also known as the unique right polar decomposition, while

F = vR is referred to as the unique left polar decomposition.

Show that R is proper orthogonal, i.e. RTR = I and dPtR = 1.

Show further that the multiplicative decomposition F = RU is unique.

Solution. Using R = Fu- 1 and eqs. (2.93)1. and (2.9?h we find that

In addition, computing d<;~tR = det (Fu-1 ) we deduce with rule ( 1.10 O and relations (2.51) and (2.94) that

.. .. . In order to show uniqueness of the polar decomposition we suppose that there exist

.(i. ·pos.itive definite, second-order tensors R and UT so that

.·...·Then, with ·C = .FTF and eqs. (2.100)2 and (2.101) it follows that

(2.102)

·::...· ·\vhich means using (2.93) 1 that U = U, since C has a unique square root, and hence

.:.: R = R. The proof is similar for F = vR. •

.. We discuss further a physical .interpretation of the right stretch tensor U. Look in ::... · itie direction of n0 at point X E n0 , with laoI = 1. According to (2.60) we find using

.( ¥. = RU that the stretch vector A00 may be expressed as

;>·Then, the square of the stretch ratio ,.\ = (A00 • ~ao)112 may be computed with (2.103 ),

.-._:·.identities ( 1.8 .I), ( 1. .84) and relation RTR = I. Thus, we have

A2 = RU.ao · RUa0 = n0 • U'I'(.RT R)Ua0

. Note that the length of A.110 acting .at X E Ho along the direction of the unit vector ao depends only on a state of pure stretch, i.e. U; therefore we call U the stretch tensor.

Consequently, the right Cauchy-Green tensor C (recall eq. (2.93)i) and the Gree.n- Lag.range strain tensor "E = (C - I) /2 do not include information about the rotation,

that is experienced by a particle during motion. The deformation gradient F ·contains more information th.an strains do in general (rotation and strain-like information).

Finaily we find the relation between v and U, and between b and C. By me.ans of

the polar decomposition (2.91) we have

88 2 Kinematics

Consequently, with v2 = RU2 RT and eq. (2.93) we find an important relation between the left and right Cauchy-Green tensors, namely

(2.107)

A general formula for strain measures was introduced by SETH [1964] and HILL

The generalized strain measures in the Lagrangian and Eulerian descriptions

are defined as

where n is a rea] number (not necessarily an integer).

For the special cases n = 0 and n = 1 we obtain strain tensors associated with the names .Henck)' and Blot, respectively, while for n = 2 and n = -2 we obtain the

Green-Lagrange strain tensor (U2 - 1)/2 (or also (v2 - 1)/2) and the Euler-Almansi

.strain tensor (I - u- (or also (I - v-2 )/2), which we have introduced in (2.67) and (2.81 ), respectively. For more details see, for example, MAN and Guo [1993].

The Hencky strain tensor in the material and spatial form, Le. lnU and Inv, is of particular interest in .nonlinear constitutive theories. Hecause of the logarithmic func- tions, these tensors are decomposed additively into so-called volumetric and isochoric parts (for a more detailed exposition of the decomposition .procedure of strain measures see, for example, p.

EXAMPLE 2.7 Assume a certain two-dimens.ional ·motion which is given in the form of the deformation gradient F(X, t) = Bx(X, 't)/DX. All tensor quantities are defined with respect to the orthonormal basis {ea}, c\! = l_, 2.

Show that fo.r the case of two dimensions the polar decomposition F = RU may be

for the right stretch tensor, and by the rotation tensor R which follows by means of the inverse of expression (2.109), i.e. R = F0- 1• In eq. (2.109), / 1(C) = C11 + C22 and

J(C) = (C11 C22 )"112 characterize the first scalar invariant of the right Cauchy-Green tensor C and the volume ratio, respectively.