Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

pies leads to unpleasant pressure oscillation. There are some remedies for this prob- -lem .in the computational literature, for example, the pressure smoothing technique by HUGHES et al. [1979]. However, a certain improvement is based on the idea of introducing additional independent field variables such as the volume ratio, leading to a more efficient three-field variational principle.

Simo...Taylor-Pister variational princip·l-e. A very efficient variational principle that takes account of nearly .incompressible response was origina11y proposed by SIMO et al. f1985] and is known as the mixed Jacobian-pressure formulation (for relevant applications to elasto.mers see SIMO [J.987] and SIMO and TAYLOR [199.la.D. :It emanates from a three-field variational principle of Hu "(1955] and WASHIZU [1.955]. Thereby, be.sides the displacement and pressure fields u and p, a third additional kinematic field variab1e1 which we denote by J, is treated independently within finite element dis~ cretizations. The principle is decomposed into volumetric, isochoric and external parts and is defined by the express-ion

flsTP{u,p, J) = /[\J!vo1(J) + p(J(u) - i) + W;so(C(u))]dV+ Ilcx~(u) . (8.113) no

Following Simo-Taylor-Pister the first two terms in the three-field variational principle are responsible for the nearly incompressible behavior of the material. They describe volume-chan_ging (dilational.) deformations and are expressed by J, p and the new variable J-. The kinematic variable J-enters the functional as a constraint which is ·enforced by the Lagrange multiplier p.. The Lagrange multiplier is an independent field variable which may be identified as the hydrostatic pressure..

In .addition to the virtual displacement and pressure fields -r5u and ,ov, we introduce an arbitrary smooth (vector) function oJ(x) = oJ(x(X)) = <5](X) for the constraint, which we call the virtual volume change (here defined on the reference confi_guration).. Jn equilibrium, functional (8.113) must be stationary. The necessary conditions for the stationarity of functional IlsTP with respect to the three field ·variables (u, p, i) are evaluated separately. We require

D5uilsTp(u,v, ]) = 0 ,

(8.114)

for all c5u satisfying ou == o on the part of the boundary surface DOnu where displacements u are prescribed and all c5v, 8].

Differentiating functional IlsTP with respect to changes in u gives the weak form of the elastic equilibrium~ i.e.. the principle of virtua1 work in the form of (8.. 105). For an explicit derivation recall the manipulations of the last se-ction.

A straightforward differentiation of fisTP with respect to changes in the field vari-

8.6 Three-field Variational Principles

ables p, .i gives the weak enforcement of the equivalence between .J and .i; and the. constitutive equation for the volumetric changes, Le. · ·

For arbitrary :c5p, the variational equation (8.115) results in th~ Euler-Lagrange equation J - } = 0.. It implies that the additional independent variable J equals J == (detC (u)) 1/.2 , i.e. the kinematic constraint associated with the volumetric behav-

·ior. For arbitrary -6J, eq. (8 ..l 16) results in the second constraint condition in the local form~ that is the Euler-Lagrange equation d'1Iv01 /dJ - p = 0. This is the standard

constitutive equation imp··lying the volumetric stresses to be equal to the hydrostatic pressure.

A finite element procedure in which the dilatation J and the pressure variab]es pare discretized by the same local .interpolations as for the displace-ment field u would not give any advantage. To prevent volumetric locking an appropriate choice of the i.nterpo-

Jation functions for the volumetric variables p, j and their variations OJJ, 8j is crucial.

A simple formulation arises by discretizi.ng the dilatation and pressure variables over a typica1 finite element domain with the same discontinuous (constant) function which need not be continuous across the finite element boundaries. This approach is known as the me.an dilatation method and is proposed in the notable work of NAGTEGAAL et al. [1974] who recognized _the effect of volumetric locking in elastoplastic Jrflow theory.

Since the interpolation functions are discontinuous, the volumetric variables p, J can be eliminated on the finite ele.ment level, a process known as static condensa- tion in the computationa] mechanics literature. Therefore, the variational equations (8. I 15) and (8.116) need not be solved on the global level leading back to a reduced displacement-based method.

The work of BRINK and STEIN [1996] is a comparative study of various multi- field variational principles. It emerges that under certain conditions the above three- field variational principle and some two-field principles yield the same discrete result in each step of the Newton method.

EXERCISES

l.Consider the functional (8,. 1l3) with the three independent field variables (u, p, }) and the associated variation equations (8."115), (8.l 16), (8.105), and show that for each step of the Newton type method the problem is completely described by the

and the linearized principle of virtual work, which has the .form of eqs.. {8.109)

and (8. l 08).

2.The described functional IlsTP (u, p, ]) .takes into account only the volumetric strain and stress components. Study a more general and very powerful type of a Hn-Washizu variational prindpl-e fundamental for various finite-element methods, i..e.

with the three independent variables u, F, P and the prescribed quantities Bon Oo, Ton 80.oCT and u on cK2o u· The loads Band Tare .assumed to be conservative and the first Piola-Kirchhoff traction vector Tis given in eq. (3.3h.

(a)With identity (1.289) show that the Hu-Washizu variational principle can be posed as a g-eneralization of the principle of virtual work, i.e.

where the total potential energy 11 is given in (8.47) and (8.48).

(b)Invoke the stationarity of Huw with respect to u, F and ·r. The vector- valued and tensor-valued functions 6u and c5F1 c5P are arbitrary with the

-conditions 80 = o over the boundary surface 8fl0 u and 8P = ·O on 800 u·

Show that the associated Euler-Lagrange equations for the functional Ilnw

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