Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

It is trivial but worthy of mention that any material field F(X, t) of some physical scalar, vector or tens.or quantity, which is characterized as a function of the referential is unaffected by a ri_gid-body motion superimposed on n. Hence,

J=-+(X, t+) = :F(X, t).

Euclidean transformation of various ki:nematical quantities. The follow·ing dis-

cuss.ion is concerned with the behavior of various .kinematical quantities during a super.imposed rig.id-body motion.

for the deformation gradient (for convenience, we will not indicate subsequently the dependence for the above functions on space and time). Note .that the second-order tensor F is objective ·even though (5.33) does not coincide with the fundamentaJ (objectivity) requirement (5.28)i. However, recall that the deformation gradient F is a two-point tensor field, in which one index describes material coordinates ..X.4 which are intrinsically independent of the observer. That is why the deformation gradient

.transforms like a vector according to (5.28)2 and why .F is regarded as objective. ·Moreover, let .J = detF and ,J+ = detF+. Since the tensor Q is proper orthogonal

Hence., the scalar field ,J remains unaltered by a superimposed rigid-body motion. AJso the sign of the volume ratio J is preserved, since detQ = +1.

Next, we recall the unique polar decomposition of the deformatio·n gradient at x E

!l and x+ E n+, i.e.

Apply.ing (5.33h to (5.35), we arrive at the representations

(5.3~)

Since the tensor QR is orthogonal it follows from (5.36) 1 that the transformation roles for the rotation tensor R and the right strekh tensor U are

.190 5 Some Aspects of Obje-ctivity

By analogy with the deformation gradient, R is an objective two-point tensor field. The right stretch tensor is defined with respect to the reference configuration. Hence,

U remains unaltered by a superimposed rigid-body motion and U is therefore also objective.

From eq. (5.36h, we obtain, using result (5.37) 1 and the orthogonality condition

RTR = I, the transformation rule

(5.38)

Clearly, the left stretch tensor vis objective.

Next, we discuss the spatial velocity gradient according to (2.139)4 , i.e. I·= FF-1 •

In an analogous manner, the spatial velocity gradient generated by the motion x+ reads

(5.39)

By der.iving-eq. (5.33h with respect to time, using the product rule, i.e. F+ = QF+QF,".·. and the inverse relation of (5.33):1, i.e. -(F+_)- 1 = F- 1QT, the spatial velocity gradient follows from (5.39h as

(5.40)

with the skew tensor n = QQT. Sinc·e ·n is present, the spatial velocity .gradient I fails to satisfy the obj-ectivity requirement (I+ :f QIQ1 \ Hence, the kinematical

quantity l is not a suitable candidate for formulating constitutive equations, which must be objective.

Euclidean transformation of stress tensors. Let dynamical processes be given by the pairs (u+, x+) and (u, x), where x+ and x are related through (5.10). ·we now want to show how the Cauchy stress tensors .u+ and .u are re.lated.

We recall the Cauchy trac~ion vector t = un with the unit vector n at point x directed along the outward norma·1 to the boundary surface .an of an arbitrary region n at time t. A superi.mposed rigid-body motion trans~~~nns region Q lo a new region n+ which is bounded by the associated boundary surface an+ at a later time ·t.+ = t + (\;.

The Cauchy traction vector transforms tot+ = u+n+ with the unit vector n+ at point x+ normal to an+. By taking note that the vectors t and n transform according to the objectivity requirement (528)2, we obtain that Qt = u+Qn. A comparison with t = crn gives the fundamental transformation rule

for the stress tensor. This means that the Cauchy stress tensor is objective..

In order to describe the first Piola-Kirchhoff stress tensor which is generated by the motion x+ = x+ (X, we may write -the .Piola transformation (3.8) as p+ (F+yr =

,J+ u+. Knowing Lhat the scalar J is objective according to eq. (5.34), and using (5.33h and (5.4·1 ), we find, with the help of (3.8), that

p+(QF)T = JQuQT '

Since the two-point stress tensor field P transforms .like .a vector fie.Id according to the objectivity requirement (5.28:h, P is objective. The second Piola-Kirchhoff stress tensor Sis parameterized by material coordinates only. Therefore, the material tensor field does not depend on any superimposed rigid-body motion~ and hence S = s+.

Note that all the stress tensors u, P and S discussed are suitable candidates for the description of material response, which fundamentally .is required to be independent of the observer.

EXERCISES

1.Recal1 the kinematic relations (2.63), (2.67) and (2.77), .(2.81 ). Using the trans- formation ru1e F+ = QFt show that the material strain tensors C and E are

unaffected by any possible rigid-body motion., i.e.

and that the spatial strain tensors band e transform according to the rules

Note that .all these kinematical quantities are objective, since C and .E are defined with respect to the .reference configuration and the second-order tensor fields b and e conform with the requirement of objectivity given in eq. (5.28) 1•

where d is objective. Note that w, which is expressed through the skew tensor n, is affected by rigid-body motions, and hence w .is not objective.

known as the Jaumann~Zaremba rate, which is often used in plasticity theory. Obvi- ously, in regard to eqs. {5.49) and (5.52), the co-rotational rates of u and A are indeed objective.

If A is a symmetric tensor we can easily show an interesting property connecting the Jaumann-Zaremba rate and the material time derivative of A. Using (5.53) and the

property of double contraction according to (1.95), i.e. A : wA =A: Aw, we obtain

Finally we define the convected rates ..of u and A. These are the ot?jective .fields

where the accent (•) indicates convected rates. The rate A is also called the CotterRivlin rate..

Objective stress rates. We now focus attention on some of the infinitely ·many pos- sible ·objective stress rates that may be defined. The choke of suitable, .i.e. objective, stress rates is essential in the formulation of .constitutive rate equations, which must be objective.

The ·Oldroyd stress rate of a spatial stress field .is defined to be the Lie tim.e deriva- tive of that field. We shall indicate Oldroyd stress rates by the abbreviation Oldr. By recalling the concept of Lie time derivatives from Section 2.8, in particular, rule (2.186) 1, the Lie time derivative of the contravariant Cauchy stress tensor u is given by

where the transformations (2.85) and the product rule of differentiation are to be used. Hence, using the identities (2. l43h and (2 .. 144h, we conclude that the Oldroyd stress rate of the Cauchy stress u is

where i:r .denotes the m~~etial time derivative -of the Cauchy stress tensor.

We now show h~)\i/the Oldroyd stress rate, Oldr(u), generated by the m·otion x, is related to its cquriterpart Oldr( u )+, generated by x+. Considering Oldr(.u) + =

However, setting d = 0 :in the Lie time derivative (5.56), which equivalently means that I ·== w, we deduce from (5~57) that

with the skew tensor w = -wT.

The Green-Naghdi stress rate and the Jaumann-Zaremba stress rnte clearly coin.. cide for w = RRT. This is the case for a rigid-body rotation (recall Example 2.12~

eq. (2.160)). IJ

The Truesdell stress rate of the Cauchy stress, denoted by Trues(u), is defined

Hence, using the Piola transfonnation (3.65h and the product rule of differentiation we find from (5.64) that

By comparing eqs. (.5.57) and (~..~.59)°with (5.66) we may easily deduce the relation-

.,..·

EXA1"1PLE 5.4 Suppose that the transformation .(5.68) .is given. Express u as J- 1r and derive the relation (5.67) by simply applying the product rule of differentiation to the Oldroyd stress rate.

Solution. With u = J-1r and the fact that the directional derivative (in our case the Lie time derivative) satisfies the common rules of differentiation, for example, .the

Accord.in_g to considerations of Section 2.8, the Lie time derivative of a scalar fie!~ is equal to the ·material time -derivative of that scalar fieldt and hence £ v{ ,J- l) = J- l. Therefore, with the chain rule and eqs. (2. l 76h and (5 ..68), relation (5.6.9)2 leads to the desired result,

.,,..::,.,..,,,,..,.,,..,..,,,,,.,..,.,.,..,..,.~ ...,,,.,..,,.:,•,.:,·,,...,.,,.,,.,,,,.,.,,,:•':'•'..''"•"'•"''''·'"'"''"'"'":"..,..,,,.,,,.''"'"''···,. ....,,.... ...,,.. . .,..,,,,. .,.,...,..., ...,,.,,,...,,...,,,..,..,,,..,,.,.,.,...,,_,,.,...,,,. ..,.....,....-..-...,,.....' .•. .,.,.,.::'."''···:•.·•'."•,'(•·:'·"."·"'''·"........,...,.:.,,.:,,,.,.,...,,,..,.,..,,,,........... ,,,....,..

EXERCISES

.u

I.. Consider the Cotter-Rivlin rate A defined by (5.55)2 •

~

(a)Show that A .is the Lie time derivative of a cnvariant tensor field A..

(b)With (5.53) and I = d + w show that the connection between the CotterRivlin and Jaumann-Zaremba rates is

6 ~,)

A =A+dA+Ad !

~

2.Recall the convected .rates ~and A, as defined in eq. (5_55). Using eqs. (5.40h, (5 .. 15) and (5.46)~ (5.28), show thatthey are objective according to

5..4 Invariance of Elastic Material Response

In this section we introduce the principle of material fr.ame-indifferenc~ which states basically that material properties do not d~pend on the change of observer.. .In particular, we show how it restricts the response of elastic materials and derive objective constitutive equations which are defined to be invariant for all changes of observer. This principle is crucial when constitutive theories such as the theory of elasticity or plasticity are considered.

In the following we consider only the isothermal case for which the absolute temperature e remains constant during the process.

A material is called Cauc.hy-elastic or elastic if the stress field at time t depends only on the state of defonnation (and the state of temperature) at this time 't and not on the deformation history (and temperature history). Hence, the stress field of a Cauchy-elastic material is indepe.ndent of the deformation path (independent .of the time). However, note that the actual work done by the stress field on a Cauchy-elastic material does, in general, depend on the deformation path.

A constitutive equation (or equation of state) represents the intrinsic physical properties of a continuum body. It determines generally the state of stress at any point of that body to any arbitrary motion at time t. A constitutive equation is either regarded as mathematically generalized (ax.iomatic) or is based upon experimental data (empirical).

The constitutive equation of an is.othermal elastic body relates the Cauchy stress

In equation (5.7.l), .u was allowed to depend up.on the referential position X E n0 in addition .to F. Hence, the stress response varie~.:·'from one particle to the other. However, for subsequent introductory treatments, ..~r..·is convenient to restrict our atten.tion on continuum bodies, in which both the CaucJ1.y stress tensor u and the reference mass density p0 are independent of the position-~;''such bodies are called homogeneous.

Hence, .instead of (5.71), we write _thf(constitutive equation in the form

(5.72)

which determines the stresses u from the given deformation _gradient F. From the mechanical point of view g characterizes the material properties of a (isothermal) Cauchyelastic material.., while from the mathematical point of view g is a tensor-valued function of.one tensor variable F. The concept of tensor functions, as we will use it here, is explained in Section 1.7. A constitutive equation of the type of (5. 72) .is often referred to as a stress relation.

Nole that for homogeneous deformations the corresponding stresses are constant (since F has the same value at every point of the body) and, interestingly enough, Cauchy's equation of equilibrium (4.54) trivially .reduces to divu = o. For this case the body force bis zero, which means that a homogeneous deformation of a continuum body occurs without body force.

tions, constitutive equations must be objective (frame-indifferent) with respect to the. Euclidean transformation (5.2). ln other words, if a constitutive equation is satisfied for a dynamical process (u, x) then it must also be satisfied .for any associated (equiv~ alent) dynamical process (u+., x+) which is generated by the transformations {5.41) and (5. l 0). This is a fundamental ax.iom of mechanics which is known as the pri.ndple of mater"ial f.rame-.indifference or the :prindplc of material objectivity or simply as obJectivity (see TRUESDELL and NOLL [1992, Sections .l.9, 19AJ). If this principle -is violated, the constitutive equations are affected by rigid--body motions and meaningless results are obtained.

To begin with, the material frame-indifference of the stres·s relation (5.72) impos·es

certain .restrictions on the response function g. We consider a mo.~ion.x+ which differs from x by a rigid-body motion superimposed on the current co~figura:tion (compare

with Figure 5.2). The rigid-body motion maps the region n to a new r,¢gion n+ and the stress relation (5.72) to .u+ = g(F+). We demand that both .regions/namely n and n+, are associated with the same function D because it is for the same elastic material. Hence, using (5.33)a on the one hand and (5..41 ), {5..72) on the other hand, we arrive at

on g for every nonsingular F and orthogonal Q. In other words, constitutive equation

.(5.72) is independent of the observer if the.response function g satisfies the invariance relation (5.74).

Employing the right polar decomposition F = RU on the right-hand side nf (5.74), we may write Qg(F)QT = g(QRU), where R -is the orthogonal rotation tensor and U the right stretch tensor. Since the latter relation holds .for all proper orthogonal tensors Q, it also holds for the special choice Q =RT. Hence, using the orthogonality

for the function .9 and for every F and R. Therefore, the associated stress relation reads

which shows that the properties of an elastic material are independent of the rotational part of F =RU, characterized by R. Note that the reduced constitutive equation (5.76) is compatible with the principle of material frame...indifterence which can be shown