Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

entropy (7. L59) constitute the one-dimensional linear counterparts of eqs.. (7.150) and (7.1.5.1), respectively.. Hence, the purely phenomenological thermoviscoelastic mode.I presented in Example 7.6 can be viewed as a nonlinear multi-dimensio- nal generalization of the linear rheological model, as illustrated in Figure 7.10.

2. Recall the proposed one-dimensional thermoviscoelastic model from the previous exercise. The dashpots in the rheological model (Figure 7. l 0) .characterize the dissipation mechanism. According to a Newtonian viscous fluid we may

"Discuss the thermostatic limit and, in particular, specify in which parts of the device the stresses and the entropy remain.

Note that the evo:Jution equations (7.160) and the .internal dissipation .(7. l 6l) constitute the one-dimensional line~__counterparts of eqs. (7. I53) and (7 .147h.

372 8 Variational Principles

efficient finite element formulations. The powerful concept of li.nearization is reviewed brie·fly and the .principle of virtual work in both the material and spatial descriptions is linearized explicitly. We present som·e of the basic ideas of two and three-field variational principles particularly designed to capture kinematic constraints such as incompressibility.

The reader who wishes for additional information on the rich area of variational principles should consult the books by TRUESDELL and TOUPIN [1960], VAIN.BERG [1964], DUVAUT and LIONS [1972], ODEN and REDDY WASHIZU [1-982].

8.1Virtual Displacements, Variations

Consider a continuum body B with a typical particle P E B at a given instant of time t. As usual, points X E no and x E !1 characterize the positions X and x -of that particle in the reference configuration Q0 at ti.me t = 0 and the current configuration n .at a subsequent ti.met > 0. In the .following we indicate the displacement vector field of P as u, pointing from the reference configuration of the continuum body into the current configuration, i.e. from X to

'~\

\

'\\

'\\

F-igure "8•.1 Virtua".J configuration in Lhe neighborhood of u, given by u = u +-nv.

Next_, consider some arbitrary and entirely new vector field w at point x which yields a virtual, slightly ·modified deformed configuration in the neighborhood of u. The virtual configuration is characterized by the modified displacement vector field u according to

where £ is a scalar parameter.

The displacement vector field is regarded as a continuous and differentiable function of space and time. It may be written in the spatial or mate.rial form, i.e. u(x, t), U(X, t), as introduced in Section 2.2. ln order to keep the notation as simple as possible, we agree not to use this distinction any longer, we write subsequently u(x, t) = u(X, t). It will be clear from the text if the displace:ment field actually depends on spatial or material coordinates. In addition, within this chapter, the space and time 3:~guments will often be omitted, for convenience.

Virtual displacement field.. Following Lagrange we know that the difference .between two neighboring disp.lace.ment .fields, i.e. u .and u, .is called the (first) variation of the displacement field u, denoted by Ju. We write

Jn mechanics 8u is also known as the virtual displacement field. The variation of u is assumed to be an arbitrary, il!fmitesimal (since E ~ 0) and a virtual c.hange, i.e. an imaginary (not a 'real') change. Note that du also characterizes an il~finitesimal change o_f. u. However, du refers to an actual change. The variation of the time-dependent ~~splacement vector field u .is always performed at.fixed instant of time.

·.The .virtual displacement field Ju is totally independent of the actual displacement field u and .may be expressed in terms of spatial coordinates or material coordinates.

For" simplicity, we agree, by analogy with the relevant notation introduced above, to write c5u(x) for the virtual displace·me.ncfield ..Ju.

·.."> We discuss brie.fly the two fundamental commutative properties of the <~-process. For-the gradient of c5u we find by means of relation (8.2)1 that

grad(c)u) = gradu - gradu ., (SA)

while, on the other hand, the variation of gradu =Du/ax yields by analogy with (8.2) 1

~:~ich shows that the variation of the derivative of a function (•)is equal to the der.iva- ~;~~:~ 'ofthe variation of that function (•). In an analogous manner we may derive the

374 8 Varfational Principles

second characteristic commutative property of the 8-process, namely, that the order of variation and definite integral is ·interchangeable.

By analogy with the transformation (2.48) we may relate the gradient with respect to the current position of a particle to the material gradient, defined on region 0 0 • For subsequent use we note the relation

For more details on the variation see COURANT and HILBERT [I 968a, 1968b]. A clearly arranged summary of the calculus of variations may also be found in the book

F(u) be a smooth (possibly time-dependent) vector function. The single argument of :Fis the displacement vector variable u given in the material description. We agree that the value of :F, which characterizes some physical quantity, is either .a scalar, vector or tensor. (Note the abuse of notation in regard to Section 2.3 where :F = :F(X, t)

characterizes a smooth material field).

In order to obtain the first variation of the vector function :F we must eva:Juate simply the directional derivative (or Gateaux derivative) of F(u) at any ·fixed u in the direction of <Su, which we denote as D 0u:F(u) (recall the concept .introduced in Section 1.8). We may cons.ider the definition

the virtual displacement field 6u. It is the ordinary differentiation of F( u + £~U) with respect to the scalar parameter e.

Note that the ·variational operator .8(•) and the Gateaux operator D( •) are Linear.

The usual properties of differentiation are valid, i.e.. the chain .rufo, product rule and so forth.

EXAMPLE 8.1 Show that the first variation c>.F of the deformation gradient F may be expressed as

In addition, verify that the first variation c5F-1 of the inverse of the deformation gradient

.F-1 is given by

or

Solution. With the definition (239) of the deformation gradient and rule (8.8), we may compute the directional derivative of F in the -direction of the virtual displacement field c)u at the position u(x) of the current configuration., i.e,.

.oF = D60 F = dd F(u + cOu)j0 ==a

Alternatively, knowing that .the operators 8(•) and 8(•) commute we obtain simp.ly result (8.9) by thinking of c~(•) as a linear operator. Applying re]ation (2.45)2 we conclude that

In order to show eq. (8..10) we start with the variation of the identity F- 1F = l which gives c5F- 1 = -F--i8FF- 1 (compare with eq. (2.143)i). Substituting (8,.9) and using transformation (8~7) we find that

<~F-J. = -F-1 (Gradc~u.F-1 ) = -F-1grad'5u ,

which is the desired expression (8.10). •

Finally we establish the variation-of the Green-Lagrange strain tensor E. By in_g the definition (2.67) of E, we obtain with :the _product rule that oE =

FT&F]/2. With relation (8.9) and ,property ( 1..84) we arrive at

cm = ~[(FTGradc5u)T+ F''Gradc5u] = sym(FTGradc5u) '

~

or, in index notation,

~hich is an important relation used in subsequent studies. The notation sy111(•) is used to indicate the symmetric part of a tensor (compare with -eq. ( 1..112)1). Since E = (C - J)/2, we find additionally that oE = (oC)/2.

variation of a function in spatial description. Let f = f (u) be a smooth (possibly time-dependent) vector function ·in the spatial description.. Note that the value of the function f (u) = x.(F(u))~ which we consider as the push-forward of :F, is _either a scalar, vector or tensor.

In order to -obtain the first variation off we formally apply the important concept of pull-back and push-fonvard ope.rations introduced in Sect.ion 2.5. The variation of f ..is obtained by the following three steps:

(i)compute the pull-back of .f to the reference configuration, which results -in the associated function .F{u) = x:-1 (.f (u) );

(ii)apply the concept of variation to F, as introduced in eq. (8.8}, and

(iii)carry out the push-fonvard of the result to the current configuration.

This concept is actually the same as for the computation .of the Lle time derivative introduced in Section 2.8. Instead of the direct.ion v used for the Lie time derivative we take here the virtual displacement field Ju.

Consequently, for the first variation of a vector function f given in the spatial description we merely write, with reference to eq. (2.187),

(8.16)

Since Dliu:F(u) = -6F(u, Ju) according to (8.8)i, we obtain

(8.17)

Therefore, the first variation of function f = f (u_) is the push-forward of the first variation of the associated function F(u) = x; 1(/(u)) in the direction of the virtual displacement field {}u. If .f = .f (u) is a scalar-valued function, then f = f (u) = F{u) and the variation of f coincides with the variation of its associated function F; thus, t5.f =6F.

Note that in our terminology the introduced operator c5 is used for the variation of a function in both the material and spatial descriptions.

EXAMPLE 8.2 Show that the first variation c5e of the Euler-Almansi strain tensor e may be expressed as

Solution. Compare Example 2.15 on p. 107. From rule (8.16) we obtain the variation of the spatial tensor e, i.e..

8.2P·rinc'iple of Virtual Work

which is the push-forward of the directional derivative of the associated Green-Lagrange strain tensor E = x; 1 ( e) in the direction of 6u.

By means of rule (8.8).i, relation (8..14) 1 and transfonnation (8.7) we find from

(8.20)2 that

According to eq. ( 1.112) the variation of the spatial tensor e is the symmetric part of the tensor ,grad8u, which gives the desired result (8.18)2~ •

EXERCISES

·· ·1. Show that the .first variations of the volume.ratio J and the inverse right Cauchy-

.. 2.. Show that the first variations of the spatial line, surface and volume elements are

8(dx) = gradc5udx ,

8(ds) = (cliv6u I - gracfr6u)ds , c}{dv) = cliv8udv ,

where I denotes the second-order unit tensor. Compare eqs. (2.173)3 , (2. I78)3 and (2.180)2.

8.2 · Principle of Virtual ·work

In· the following two sections we study variational principles with only one field of unknowns, called single•field variational .principles. In particular, we introduce work ;and stationary principles in which the displacen1ent vecfor u is the only unknown field .

... ::...J'hese principles are fundamental and will become essential in establishing finite

.·eiem~nt forinulations.

~nUial boundary-value problem.. The finite element method requires the formula- Jion·ar the balance Jaws in the form of variational principles.

378 8 Variat"ional Prlnciples

As one of the most fundamental balance laws we recall Cauchy's first equation of

motion (i.e. balance of mechanical en.ergy)discussed in Section 4.3. Knowing

spatial velocity field v .may be expressed as the time rate of change of the displacement field u, we may write Cauchy's first equation of motion, i.e. (4..53), as

From the fundamental standpoint adopted in Section 4.3, the Cauchy stress tensor is governed by the symmetry condition u = uT deriving from the balance of .angular momentum. The -spatial mass density of the material is p = J- which describes continuity ofmtis,s. The body force b per unit current volume which acts on a particle -in region n is considered to be a prescribed (given) force while the term pii characterizes the .inertfa force per unit current volume. Note that when we write ·eq. (8.22) we mean divu(x, n+ b(x, t) ·=p(x, t)ii.(x, l) at every point x E fl and for all times t.

In the following we c-onsider boundary conditions and initial conditions for the motion x -= required to satisfy the second-order -differential equation (8.22). We assume subsequently that the boundary surface an of a continuum body B occupying region n is decomposed ·into disjoint :parts so that

with anu n DOu = 0 . {8.23)

Figure 8.2 illustrates the decomposition of the boundary surface 8f2 nal space at time t!'

n

anu

"Figure 8.2 Partition of a boundary surface an.

We distinguish two classes of boundary conditions, namely the

ary conditions'f which correspond to a displacement field u = u(x, t)., and the von Neumann boundary condi.tions, which are identified physically with the surface traction -t = t(x, t, n).