Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

Considering the push-forward operations derived we obtain finally~ from: (8~84) with relation dv = Jdl/ and some rearranging, the linearized internal virtual-Work in the spatial description, i.e.

D~uJlVi11t(u,Ju) = /(gradJu: grad~uu+ gradJu: c: gradflu)dv , (8.89)

n

or, in index notation,

(8.90)

where + Cabcd represents the effective elasticity tensor in the spatial description. Relations (8.89) and (8.90) are .linear with respect to the terms 6u and Liu. They describe the fully nonlinear (finite) deformation case and have a similar symmetric

.structure to the linearized eqs. (8.8 l) and (8.82).

Formulations according to (8.81) and (.8 ..89) are in the literature sometimes called

.total-·Lagrangian and updated-Lagrangian, respectively. This really means that integrals are calculated over the respective regions of the reference and the current configuration. However, it is important to emphasize that the derived material representations (8.81) and (8.. 82) of the linearized virtual internal work are equivalent to the spatial versions (8.89) and (8.90). The two representations are bas.ed on the use of change of variables, and the resu Its are the same in both cases.

In order to recapture the small deformation (but nonlinear elastic) .case (in the "lit- erature often called the materia1ly nonlinear case) we fix the geometry, i.e. we do not distinguish between initial and current geometry. Further~ we do not account for the initial stress contribution Oaeabcl and ignore the quadratic terms in the Green-Lagrange strain tensor. In addition, for the fully linear case, the coefficients of the elasticity tensor are given and are not functions of strains anymore.

EXAMPLE 8.7 An alternative approach to derive the linearized .internal virtual work (8.89) in the spatial description is to use the formal equivalence of the material time derivative of scalar-valued functions in the spatial description with the directional derivative of these functions in the direction ~u (compare with Section 2.8).

Carry out the material time derivative of the internal virtual work <5H'int in the spatial description and use this property to obtain its linearization. This approach circum- vents the extensive pull-back and push-forward operations.

400 8 Variational Principles

.Solution. We again start with 8lVint in the spatial form (8.34) and change the domain

where the .Euler-Almansi strain tensor e depends on the displacement field u through the relationshi.P (2.90). The Cauchy stress tensor u also depends on u, here a function of the current position x.

Employin.g mate·rial time derivatives and the product rule we find from (8.9.lh that

where additionally the relationship between the symmetric Kirc.hho.ff-stress tensor r and u, i.e. ,,- = Ju, is to be used,. Recall that the superposed dot denotes the material time derivative, as usual.

Firstly, we derive the material time derivative of the virtual Eu1er~Almansi strain tensor cfo. By means of (8.7) and (2.14Jh_, we find from (8.18h that

with the spatial ve.1ocity gradient I = gradv (recall definition (2.135)).

Secondly, we focus attention on the crucial material time derivative of the Kirchhoff stress tensor T. We start with the relation for the (objective) Oldroyd stress rate of the spatial stress field T. Recall the Oldroyd stress rate of T which is identical to the Lie time derivative of 1, and with reference to eq. (5 ..59) expressed as .l\. (T) = 1- - Ir - rfr. By means of£,. (r) = Jc : d., i.e. eq. (6."161. ), we conclude that ..

where re and d = sym(J) are the spatial elasticity tensor and the rate of deformation tensor (compare with definitions (6.159) and (2..146), respectively).

Substituting relations (8.93)s, (8.1.8) and (8.94) into (8.92) and using the symmetry

of T and the minor symmetries of ·c,i.e. CatJcd = ClJacli = CabcJc, we obtain

cH-Vint(u,Ou) = j[r: (-gradOul) +gradc5u: Jc: d

By use of property (1.95) the sum of the first and last terms in eq. {8.95) vanishes and we obtain

App.lying now the formal equivalence of the material time derivatives and the·di~ rectional derivatives, and replacing the spatial velocity field v by the linear incr.eme.nt ~u, we may rewrite (8.96) as

again we arrive finally at the linearized internal virtual work in the spatial description,

EXERCISES

1.Take the part Oucabd of the effective elasticity tensor due to the -current stresses in the spatial description.

(a)Show that this term possesses the major symmetly, that means that nothing chang-es by the manipulation

(b)Further, show that it does not possess the minor symm.etry, i.e.

However, for the case in which the term 6a.cCTbrl is absent, under the assu mp- tion of a hyperelastic material the minor and the major symmetry of C:ubccl hold. Discuss the consequences in regard to finite element discretizations.

2.Show that the linearization of the internal virtual work cH'Vint may also be written

as

D~uc5I·Vint(u,8u) = ./ GradOu : A: GradAudF , no

where A is a (mixed) fourth-order tensor useful in numerical implementations. It

.is known as the -(fi:rst) elasticity tensor with the definition

402 .8 Variational Principles

where P denotes the first Piola-Ki.rchhoff stress tensor depending on the deformation gradient F.

Note that the linearization of dlVint contains one term only (compare with rela- tion (8.81 )).

3.Carry out the linearization of the external virtual work c5lVexb assuming pressure boundary loading, as derived in eq. (8..38). Consider a constant .pressure load v

and show by means of the product rule and relation (2.5) that

n~

Note that the two terms in eq. (8.98) are not symmetric in c5u .and .~u. "In fi- nite element discretizations the associated tangent stiffness matrix -is also not, in general, symmetric.

8.5Two-field Varia.tion.al Principles

So far we have considered single-field variational princip.les such as the principle of virtual work. It is not always the best principle to choose, particularly when constraint conditions are imposed on the deformation. In finite element analyses of problems whkh a.re associated with ·Constraint conditions, significant numerical difficulties must be expected within the context of a i.e. a standard dfa~placemen.t­

based method in which only the displacement field is dis.cretized. This ·method exhibits rather poor numerical performances such as penalty sensitivity and ill-conditioning of the stiffness matrix., as is well-known from

(i)the numerical analysis of rubber, which is frequently mode.led as a nearly incompressible or incompressible material,

(ii)bending dominated (plate and shell) problems,

(iii)e.Jastoplastic problems that are based on lrft.ow .theory (the plastic flow is isochoric), and

(iv)Stokes' ftow, which malhematicaHy yields a problem .identical to that for isotropic incompressible elastic.ity.

In the ·computational literature ·these devastating numerical difficulties are referred to .as locking phenomena. Essentially, these locking difficulties arise from the over-

stiffening of the systetn and are associated with a significant loss of accuracy observed,

in particular, with low-order finite elements (for fundamental studies and for more references see the :books by HUGHES [1.987] and ZIENKIEW"JCZ and TAYLOR [1989,

1991]).

To eliminate these difficulties inherent in the conventional single-field variational approach a great deal of research effort by engineers and mathematicians bas been de- ·voted to the developments of efficient so-called mixed .finite element methods (see

HUGHES (1987, and references therein]). For a more mathematically oriented presentation see lhe work of BREZZI and FORTIN [1991 ].

For these types of methods the constraints imposed on the deformation are dealt with within a variational sense resulting in effective multi-field variational principles.

This approach has attracted considerable attention in the computational mechanics lit- erature. Besides the usual displacement field, a mixed {finite element) method incorporates one or more additional fields (typically the internal pressure field, the volume ratio field ...) which are treated as independent variables. The basic idea within the ·mixed finite -element method is to discretize these additional variables independently with the aim of achieving nonlocking .and stable numerical solutions in the incompressible limit..

Lagrange-multiplier .method. In the subsequent part we focus attention on a suitable variational approach which captures nearly incompressible and incompressible hyperelastic material .response.

Rubber or rubber-like materials may show a very high resistance to volumetric changes compared with that to isochoric changes. Typically, the ratio of the bulk modulus to the .shear modulus is roughly of four orders of magnitude (exhibiting an almost incompressible response), and a ·very careful numerical treatment is needed. A standard disp"lacement-based method cannot be applied directly to these types of .problems., be- ·cause locking or =instabilities will occur. Hence, almost al.I hyperelastic materials which show a nearly incompressib1e or incompressible deformation behavior are treated with a mixed finite element formufation.

The Lagrange-multi-plier method, in which a constraint is introduced using a scalar parameter called the Lagrange multiplier, is often used to prevent volumetric locking,. Utilizing .P as the Lagrange multiplier to enforce the incompressibility constraint .l = 1, we .may formulate a functional HL in the -decoupled representation

no

where the term p(J(u)-1) denotes the Lagrange-multiplier term, with the volume ratio

J = J (u) = (detC) 1/ 2 • The function \JI iso = \]_i iso ( C) c·haracterizes the isochoric elastic response of the hyperelastic material with .the corres_ponding modified right Cauc-hy-

Green tensor C = C(u) = J-2/:ic. The loads do not depend on the motion of the body,

404 8 Variational Principles

so that the external potential energy IIext .is given by the standard relation (8.48)2 • Note that the Lagrange-multiplier term vanishes for the case of incompressible

finite elasticity. In addition, it is important to emphasize that the solution of the L<l:grange-multiplier method can be recovered from the penalty method just by taking the restriction on the value "' --+ ·oo (compare with the -outline given on p. 389). Hence, the penalty method may also be seen as an approximation to the Lagrange-multiplier method.

The Lagrange :multiplier p plays the role of the (physical) hydrostatic pressure. The described formulation gives rise to a two-field mixed finite element implementation involving u and pas the independent field variables. Since we consider extra numbers

of unknowns th.is technique requires additional computational effort.

The main objective is now to derive a two-field variational principle for finite elasticity by finding the stationary point of functional TI1.,· In other words, we must compute and set to zero the directional derivatives of IIL with respect to both the displacement

field u and the hydrostatic pressure field p.

In addition to the virtual displacement field c5'u we need an arbitrary smooth scalar function 8p(x) = 8p(x(X)) = Jp(X), which is interpreted as the ·virtual p.ressure

field! here defined in the reference configuration. We find the stationary conditions with respect to u and pas

for all 8u satisfying c5u = 0 on the boundary surface anou and all <5p.

Firstly,, we compute the directional derivative of (8.99) in the direction nf an arbitrary virtual displacement ~u. By means of the chain rule and the relations derived in Section 8.1, i.e. DsuJ = 8J = Jdivc5u and DduC = oC = 2oE, we obtain, with reference to (8.JOl)t, the weak form

flo

(.the arguments have been omitted). The contribution due to the loads is given by eq. (8.5lh as D.suilext = -(5ll'ext' with the external virtual work cHFext' i.e. (8.45).

In order to rewrite the term Jpdiv<5u of (8.102h we use the analogue.of eq. (l.279)2

and invoke relation (8.7) and property (l .95) to obtain

divc5u = I : gradJu = I : Grad8u F- 1

The :Jinearization of the principle of virtual work (8 .105) in the direction of the .increment D,.p .gives

while a linearization process in the direction of Au was already ·carried -out in detail within the last section., leading to

Diu,AuII1,(u,p) = / (GradJu : Grad.6.u S

no

+FTGrad6u : (<Cvol + C150 ) : FTGrad~u)dV . (8.109)

Sinc·e (8. l05) is based on the additive decomposition of the stresses S, we obtain the decoupled representation of the ·elasticity tensor C = Cvot + Cj80 , with the definitions

For an explicit treatment of these expressions recall Section 6.6, in particular relations (6.166)4 and -(64168). Note that for the considered case the scalar quantity p must be replaced by pin eq. (6..166)4 ..

Since the structure of the linearized princip.le of virtual work is symmetric and because of the symmetry .betwe.en eqs. (8.l07h and (8.108) a finite ele.ment implemen-

tation of this set of equations will lead to a symmetric (tangent) stiffness matrix.

In order to use these equations within a finite element regime so-called interpolation functions must be invoked separately for the displacement field u, the pressure field p and their variations .au and op, respectively (see SUSSMAN and BATHE [1987], ZIENKfEWl.CZ and TAYLOR [198.9] among others). A wen-considered choice of these functions is a crucial task in order to alleviate volumetric locking. It was observed that a discontinuous (constant) pressure and a continuous displacement interpolation over a typical finite element domain is computationally more efficient than with the choice of

suits in a stiffness matrix which is not positive definite for incompressible materials.

.In order to overcome the numerical difficulties associated wjth this fact and to avoid ill-conditioning .of the stiffness matrix .associated with the .penalty approach, regularization procedures such as the so-called perturbed Lagrange-multiplier method (in

the literature -often referred to as the .augmented method) have

been introduced successfully (see, for example, GLOWINSKI and LE TALLEC [1.982, 1984]).

"It may be viewed as a two-field variational princ.iple in which the functional (8.99)

no

(see the work of CHANG et al (1991]), where the third (penalty) tenn in functional (8.1 :I I) reguJarizes (relaxes) the incompressibility constraint J = 1 involved in the :first term of the integral. The Lagrange multjplier JJ which enforces the constraint no longer has the mean-ing of a pressure, in contrast to the Lagrange-multiplier method..

The positive penalty parameter n, may be viewed as a (constant) bulk modulus. Incompressible materials can be treated by repfacin_g .1/"" with zero, so the first and t~e third terms in (8.111) vanish. For this incompressible limit the penalty method, the Lagrange-multiplier method and the perturbed Lagrange-multiplier method lead to identical equations.

By taking for the term (.J(u) - 1) a more general, sufficiently smooth and strictly

convex function Q(J(u)) (so that ·Q(J{u)) = 0 if.and ouly if J = 1) the functional

(8.l 11) .is identical to that proposed by BRINK and STEIN [ 1996]. Functional (8. ll I) may also be identified with a special form of the two-fie"Jd variational principle given by ATLURI and REISSNER [ 1989]. This work deals with a general framework for -in-

corporating volume constraints into multi-field variational principles. In addition, note

that the formulation (8. I 11) also reduces from a mixed {finite element) formulation proposed by SUSSMAN and BATHE l1987l

EXERCISES

1.Consider the decoupled strain energy formulation proposed ·in (8.55), i.e. w(C) = WvoifJ) + Wiso(C), with '11vo1(J) = tt.9-(J), and treat the displacement u and the hydrostatic .pressure p as independent ·field variables (permitted to be varied).

Require that dQ(.J)/ dJ == 0 ifand only if .J = I.

(a)Derive the stationary condition with respect to u and incorporate the definition of the volumetric stress contribution, i.e. (6.89), with the constitutive equation for the hydrostatic pressure p according to (6."91 ).1• Show that the resulting variational equation is, in accordance with the principle of virtual work, in the form (8.105).

(b)Obtain the additional variational equation .in the form