Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

We write

where the overbars (ii) denote ·pres·cribed (given) functio.ns on the .boundaries an(•) c an of a continuum body occupying the region n. The unit..exterior vector normal to the boundary surface 80.u is characterized by n. The prescribed displacement field u and the prescribed Cauchy traction vector i (force ·measured per unit current. surface area) are specified on a portion Dnu c Dn and on the remainder Dnu, respectively. Note that in the previous section the symbol u also stands for the modified displacement fie:ld ..

We call the .prescribed body force b and surface traction i loads. We say that the continuum body -is subjected to holonomic external co.nstraints if u ·= uon the boundary surface Dnu. External constraints are nonholonomic if they are given by an inequality.

The second-order differential equations (8.22) themselves require additional data in the form of initial condi.tio.ns. The displacement field ult=o and the velocity field

where (• )0 denotes a prescribed function in 110 • Since we agreed to consider a stressfree reference configuration at t == 0, the initial values (• )0 are assumed to be zero in our case. However, in dynamics -the configuration at t = 0 is sometimes not chosen as a reference configuration.

In order to achieve compatibility of the boundary and .initial conditions we require

Now, the ·problem is to find a motion th.at satisfies eq. (.8.22) with :the prescribed boundary and initial ·conditions (8.24),. (.8.25) and co.m=patibility conditions (8.26).

This leads to the formulation in the strong form (or dassical form) of the ini...

·tial boundary-value problem (IBV.P). Given the body force, and the boundary and initial conditions. find the displace·ment field u so that (considering only mechanical variables}

ti(x, tHt=o = uo(X) .

Note that the S·et (8 ..27) of equations generally defines a nonlinear initial boundaryvalue problem for the unknown displacement field u. In addition, we need a constitutive equation for the stress u which is, in general, .a nonlinear function of the displacement

field u.

If data depend on time and the acceleration is assumed to vanish, i.e. ii = o,

the considered problem is called quasi..static. For this ·case the equation of motion is subjected to the conditions (8.24), (8.25)i, and the requirement that eq. (8.26) 1 holds for compatibility.

If the data are independent of time the proble.m is referred to as static. For this case we consider a body in static equilibrium for which the set (8.27) of equations reduces to the associated nonlinear boundary.value problem (BV~) of elastostatics, i.e.

Thus, the solution of a static problem at a point of a continuum body .depends only on the data of the boundary and not on initial conditions (there ·is no need for initial

conditions)..

Principle of virtual work in spatial description. An ana.lytical solution of the

nonlinear initial boundary-va]ue problem des.cribed is on.ly possible for so.me special cases. Therefore, on the basis of variational principles, solution strategies such as the finite element method are often used in order to achieve approximate solutions.

In order to develop the principle -of virtual work we start with Cauchy's first equa- tion of ·motion (.4.53) which we multiply with an arbitrary vector-valued function T/ =

TJ(X), defined on the current .configuration n of the body. Integration over the region n of the body yields the scalar-valued function

For the first argument of function .f we conveniently introduce the displacement vector field u rather then the motion x for a given time t,. The s·econd argument off is a

function T/ (at ajixed instant of time t), called a test function (or wei.ghting function). It is a smooth function with 77 ·=o on the boundary surface .OOu.

Eq. (8.2.9) is known as the weak form of the equation of motion with respect "to the spatial configuration. Equations in the weak form often remain valid for discontinuous problems such as shocks where most of the variables undergo a discontinuous variation. For this type of problem differential equations are not necessarily appropriate.

Since T/ is arbitrary, the vector equation divu + b = pii on fl is equivalent to the weak form (8.29). The method used to prove th.is important property goes along with the fundamental lemma of the calculus of variations. The solution of the problem in the strong form is identical to the solution in the weak form. For further details see the books by HUGHES [1987] and MARSDEN and HUGHES [1994].

Subsequently, applying the product rule ( l .289) to the term divu · 71, i.e..

.and using the divergence theorem in the form of (.l.299), eq. (8.29) may be written as

Since 1J vanishes on the part of the boundary surface 8f111 where u is prescribed, the surface integral only nee-ds to be integrated over the portion 80a- c an. By use

of boundary conditions (8.24) and by formulating the initial conditions (8 ..25) in the weak form, we obtain the foil owing set .of scalar equations known as the variational problem:

This set of equations characterizes the weak form (or variational form) of the initial boundary-value problem. It is the equivalent counterpart .in the strong form

(827) which is satisfied when (8.32) is satisfied. Note that the stress boundary condi- tions on the portion an(J are part in the weak form (8.32h, so they are often referred to as natural boundary conditions. However, the conditions u = =ii which are prescribed over the boundary surface anu are called essential boundary conditions of the variational problem.

Hence, variational problems are related to initial boundary-value problems which

.are described through differentia] equations .and i~.itial. and boundary conditions. The differential equation is usually called the Euler-Lagrange equation in the weak formulation which in our case is Cauchy·'s first equation of motion (8.27)i. Formulations in the weak form are mathematically helpful for jnvestigations of -existence, uniqueness or stability of solutions (see, for example, MARSDEN and HUGHES [1994, Sections 6.1-6.5]).

Note that the test function 'fJ in (8.32) is arbitra1y. If we look upon 'T/ as the vir-

382 8 Variation-a·) Prindples

tual displacement field t5u, defined on the current co11jiguration, then the formulation in the weak form of the initial boundary...value problem (8.32) leads to the fundamental principle of vlrtual work (or principle of virtual displacement). Considering the symmetry of .u and the variation of the Euler-Almansi strain tensor 8-e, as derived in eq. (8.1 Sh, we arrive at the principle of virtual work in the spatial description expressed in tenns of the virtual displacement, i.e.

the boundary surface 800 where the traction vector i is prescribed. We require that Ju vanishes on anu, where the displacement field u .is prescribed (see the boundary conditions (8.24) 1). The virtual displacement field is assumed to be infinit.esima/, which is not a requirement for an arbitrary test function. It is an imaginary (not .a ·'rear') change of the continuum which is subjected to the loadings.

The principle of virtual work is the simplest variational principle and it states: the virtual stress work u : c5e at fixed u is equal to the work done by the body force b and inertia force pii per unit current volume and the surface traction t pe:r unit current

are known as internal (mechanical) virtual work 61'Vint and external (mechanical) virtual work c5lVcx-t.

In the first case the stress u does internal work along the virtual strains cfo. .In the second case external work is done by the loads, which are the body Jorce b and the surface traction t, along the virtual displacement 6u about region n and its boundary surface DO, respectively.. For vanishing accelerations ii, the internal virtual work equals the external virtual work, i.e. 8lVint = <SH~!xt.

It is important to emphasize that the principle of virtual work does not necessitate the -existence of a potential. No statement in regard to a particular material is invoked. Therefore, the principle of virtual work is general in the sense that .it is applicable to any material including inelastic materials.

p = const

6

a"'Y

/e2

.Figure 8.3 Constant pressure boundary loading and parametrization of the pressure loaded surface an(1.

Principle of virtual work in ·material description. Now we are in a position to express the principle of virtual work in terms of material variables.

We assume a region .0.0 of the continuum body which is bounded by a reference boundary surface 8.fl0 • This boundary surface is partitioned into·disjoint parts (compare

As a point .of departure we recall the equation of motion in the material description (4.63). We use the form

tensor, the reference body force .and the inertia force per unit reference volume, respec- tively.

where the unit exterior vector normal to the boundary surface onou is characterized by

N. The prescribed displacement field u .and the prescribed first PioIa-Kirchhoff traction vector T (force measured per unit reference surface area) are specified on the disjoint parts 8rl0 u and 800 en respectively. The second-order differential equation (8.40) must be supplemented by initial conditions for the displacement field and the velocity field at the instant of time t = 0 (see eq,. (8.25)).

Using the above concept, we may show the princip:le of virtual work in the material

description expressed in tenns of the virtual displacement, i.e.

with the virtual displacement field 6u (here defined on the reference co1~figuration.) satisfying the condition OU = 0 on the part of the boundary surface anou where the displacement field u is prescribed. The surface traction Tacts on the portion 8fl0 er c

80.0 • .According to relation (3.1), T has the same direction as t, but T ¥ t. It is i~portant to note that the description of the variationa_l equation (8.42) is equivalent to that of (8.33). ·

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

EXAMPLE ·s.3 Show that the material form of the principle of virtual work, as given in (8A2), can be obtained alternatively by a pull-back operation of relation (8.31.) to the reference configuration.

Solution. In order to show (8.42) we must express the internal and external virtual work cn.v·int and JTVext in eqs. (8.34) and (8.35) and the contribution J~ pii · oudv in terms of material variables.

We begin by considering the intental virtual work (8.34). With the help of identities (8~18.h, (2.50) and transformation (8.7) we obtain

where the symmetry of the Cauchy stress tensor u is to be used.

Applying property (l.95) .and Piola transformation (3.8.) we obtain the canonical

representation of the material version, i.e..

The external ·virtual wo.rk 8ll1~xt in the form of eq. (8.35) may be transformed by means of the relation for the body force b, i.e. b = J-1.u, and the change in volume given by d-v = Jdll. In addition, we must show the equivalence of the

386 8 Variational .Princi.ples

prescribed traction vectors. With relation (3."I) and boundary conditions (8.24h and (8.4.lh we deduce that ids= T.dS. Hence,

OIVext(u,Ou) = Jb · 15udv + JI· r5uds = JB · 15udV + / T · OudS . (8.45)

n ano no ano ct

The remaining term in eq. (8.42), i.e. the inertia force p0 ii per unit reference volume over the region H0 , may simply be established from the third tenn in the .associated eq. (8.31) by means of p0dV'"= pd11, i.e. conservation of mass in the local fonn (4.6). This result together with (8.44) .and (8.45) leads to the desired re"lation (8.42). II

EXERCISE

I. Starting at eq. (8.44h, show that the internal virtual work cHlint may also be expressed as the contraction of .the symmetric second Piola-Kirchhoff stress tensor

.S .and the virtual Green-Lagrange strain tensor ,YE de-fined in eq. (8.14).

ln addition,. show that <Hl'i-nt may also be given in terms of the Mandel stress tensor :E =CS .and the v~iriation of the symmetric Green-Lagrange strain tensor,

. .

8.3 Principle of Stationary Potential Energy

In the principle of virtual work, as derived in the last section, the stresses are considered without .their connections to the strains. We have not taken into account a particular material.

Io this section we assume a conservative mechanical system {compare with p. 159} requiring the existence of an energy functional n for both the stresses and the loads. The assumption of the existence of n is common in many fields .in solid ·mechanics. The loads .may depend on the motion, but they must emanate from a functional. A formulation based on .energy functionals is very useful, for ex.ample., for the development of robust nmnerical algorithms that are based on optimization techniques.

In the following we introduce a stationary energy principle in which the displacement vector field u is taken to be the only fundamental unknown.

energy ·princi.ple. From now on we assume that the loads do not depend ·6n the motion of the body, which is usually the case, for example, for body forces .

.~t. means that the directions of the loads remain paralle.I and their values unchanged ihroughout the deformation of the body. We say that such loads are 'dead'.

Instead of a vibrating body we consider a body in static equilibrium under the action

:of specified 'dead' loadings and boundary conditions on anu u according to

.e.q. (.8.41 ). Then the total potential energy IT of the system is .given as the sum of the ·internal and external potential energy, flint and Ilexb i.e.

... "._: .. Si.nee the deformation gradient F depends on the displacement vector fi.eld u by the relation according to eq. (2.45h, i.e. F = Gradu + I, we indicate explicitly the dependence of Fon u and write F = F(u). For.. nh1t and Ilext we will also indicate subsequently the dependence on u. Note that for a rigid body the term TIint is zero. The 'dead' loadings, given by the external forces Band T, are distributed over the volume of the continuum body and its von Neumann boundary, respectively.

One main objective of common engineering interests .is to find the state of .equilibrium (the de.formed configuration) for which the potential is stationary. The stationary Pt?Sition of the total potential energy II is obtained by requiring the directional deriva- tive with respect to the displacements .u to vanish in all directions '5u. Compute

which is known as the prindple of stationary ·potential energy, another fundamental variational principle in mechanics. In other words, we require that the first variation of the total potential energy, denoted 8Il, vanishes. The variation of fl clearly .is a function ofboth u and c5u. The arbitrary vector field Ju is consistent with the conditions imposed on the continuum body. Thus, ·C~U = 0 over anu, where .boundary displacements are prescribed.

In order to decide if the solution corresponds to a nuitimum, a mininmm or .a saddle point we must determine the second vadation of the total potential energy fl, denoted by cFIT(u, ,~u, ~u) = D~u,uull(u). Here, ~u .is the increment of the displacement field u which will be discussed later in Section 8.4. The quantity .D~u,ilun(u) is obtained from the directional derivative of variational equation (8.49) with respect to u in the direction ~u (i.-e. the second directional derivative of TI with respect to u), which .is