Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

390 .8 Variational Prindple~

Here, J = J(u) = (detC) 112 defines the volume ratio and C = C(u) = J-2/:sc the corresponding modified right Cauchy-Green tensor, as .introduced in eq,. (6.79h. The strictly convex function Wvol describes the volumetric .elastic response while ·wiso .is associated with the isoclzoric elastic response of the hyperelastic material. We require '11 vol ( J) and '11 iso ( C) to be zero if and only if J = 1 and C = I, ensuring that the reference configuration is stress-free.

According to eq. (8.55h .the volumetric contribution '1ivol is characterized by a (positive) penalty parameter ft, > 0 which is independent of the deformation. The parameter ff, may be viewed as the bulk modulus. The function g is motivated mathematically.

It is known as the penalty function and may adopt the simpJe form

often used in numerical computations. Consequently, the meaning of the function as used in eq. (8.55) 1 differs s·ignificantly from its meaning in eq. (6.85), in which

..is of physical relevance.

We now derive the stationary position of Ilp with respect to the displacement field, which is basically a procedure according to Example 8.4. Starting with the fundamental condition (8.49) and following the steps which have led to eq. (8.54) we find, using

decomposition (8.55), that

where D,iuC denotes the directional derivative of the right Cauchy-Green tensor Cat ·u in the direction of 8u, which is 2<5E (see Section 8.1 ).

A specification of the integral in eq. (8.58) implies, by means of the chain rule and

relation (6.82)1 , that

nu

(the arguments of the functions have been omitted for simplicity)..

With reference to eq. (6.91}1 the term d'11v01 /dJ defines the hydrostatic pressure p. Hence, recalling definitions (6.8.9h and (6.90}1t we conclude that the terms ·in parenLheses of (8.59b are associated with the volumetric and isochoric stress contributions

Svol and SiscH respectively.

Using the second Piola-Kirchho:ff stress tensor S, which is based on the additive

392 ·s Vari.ational .Princip:Jes

with Ju denoting the variation of the displacement vector field, which is a function of space and time. The functional L (or in the literature sometimes introduced as -.L) is integrated with respect to time t over the closed time interval t E [t0 , ti]. The two points t0 and t 1 denote arbitrary .instants of time at which c5u is assumed to be zero at all points of the body, i.e.

valid for a region n. The scalar-valued functionals II and K denote the total potential energy (4.114) and the kinetic -energy (4.83) of the moving body, as usual.. The potential energy of the loads -exists and is given by eq. (8.48h.

Show that the vanishing variation of the functional L with the imposed restrictions (8.62) gives the principle of virtual work (8.42) for aIJ t5u which are zero

throughout the entire closed ti.me interval t E [t:o, ti.).

2.Consider a constant normal pressure p applied to a boundary surface enclosing a certain region. Show that there exists the associated potential

Ilext(u) = 7J f dv = ]J f J(X, t)dV , n no

whose variation gives the external virtual work JH".°"ext.• as defined in eq. (8.36), i.e& D6ullcxt (u) = JH~ext(u, 6u) = p ./~n(T n · ouds.

8.4 Linearization of the Principle of Virtual Work

such as the principle of virtual work in the forms of (8 ..33) or (8.42) are generally non1inear .in the unknown displacement vector field u. Typ"icalJy, the nonlinearities are due to geometric and material contributions_, i.e. the kinematics of the body and the constitutive equation of the material, respectively&

As mentioned above, one main objective of engineering analysis is to find the un- known field u which is the sofotion of the associated nonlinear boundary-value problem. Usually, (exact) closed-form mathematical solutions of a set of nonlinear partial" differential equations are only available for some special engineering problems; they are rather complicated and often unusable.

In order to keep the complexities of engineering problems intact, approximate numerica] solutions, based on, for -example, the finite e:Jement method, are required. A very common and simple numerical technique to solve nonlinear equations is to employ the reliable incremental/iterative solution technique of Newton's type. It is an efficient method with the feature of a quadratic convergence rate near the solution point. This

technique requires a consistent lin.earization of all the quantities associated with the

.considered nonlinear problem generating efficient recurrence update formulas. The honlinear problem then .is replaced by a sequence of linear problems which are easy to solve at each iteration.

Linearization is a systematic process which is based on the concept of directional derivatives, see the pioneering work of HUGHES and PISTER [1978]; for the more generalized concept see the book by ·MARSDEN and HUGHES [1994, Chapter 4], and °for".an application to rods and plates the work of WRIGGERS Il98"8] among others. For the concepts of linearization and directional derivative and their applications in nonlinear continuum mechanics see also the textbook by BONET and WOOD [1997].

The following part of this section deals with the Iin- ·earization of a nonlinear and smooth (possibly time-dependent) .function :F = :F(u) in the material description which is either scalar-valued, vector-valued or tensor-valued. tile single argu·ment of :F .is the dis.placement vector variable u.

Consider u, then the fundamental .relationship for the lineari.z.ation of the nonlinear function :F -is based ·On the first-order (Tay"lor's) expansion, which is expressed as

is also linear and the usu.a] properties of differentiation are valid. The quantity ~u denotes the .increment of the displacement field u, here expressed in the reference configuration.

The remainder o(~u), characterized by the Landau order symbol o( •),:is a small ·error that tends to zero faster than Liu -1- o, i.e. o(~u)/l.6.ul = o.

Within the classical solution technique of Ne1vton 's method, Taylor's expansion is

.truncated after the first derivative of :F. Hence, the first term in (8.63) is a constant ·part, i.e. an approx.imate solution for a given state u. The second term il:F is the linear change in :F due to ~u at u. It is the directional derivative of :F at given u (fixed) in the direct.ion of the .incremental displacement field ~u, i.e.

where the linear Gateaux operator D( •) is with respe·ct to the incremental displacement

.~eld ~u.. We say that AF{u, ~u) is the Iinearization (or linear approximation) of ·;:at u.

Note that in .regard to eq. (8.8) the fi.rst variation D6µ.:F(u) of a vector function F(u) and the linearization .DA0 :F(u) of that vector function are based on the same ,concept of directional derivatives. By taking notice of this equivalence of variation and 'tinearization, all relations derived in the previous Section 8.1 can be adopted here.; we just use the symbol ~(•) instead of

which are analogous to eqs. (8.9), (8.10), (8.14).

EXAMPLE 8.5 Show th.at the linearizat.ion A6E of the virtual Green-Lagrange strain tensor <>E = sym(FTGrad6u), .as derived in eq. (8 .. 14h, may be expressed as

Solution. According to the rule (8.64), we compute the directional derivative of c)'E

in the direction of ~u at u, i.e.

~trE = Do118E = ell 6E(u + ci).u)lc=O

Since the virtual displacement field c5u is independent of the disp.lacement~ the term Grad.Su is not affected by the linearization. Knowing that d/de: F(u + e~u)lc==U =

Grad~u (see eq. (8.67)),. we find the desired result (.8.70). a

In order to linearize a nonlinear smooth vector function f = f (u) in the spatial description we adopt the concept for the first variation off introduced on p. 375. By analogy with relation (8.. 16), we may write

(8..72)

for the linear-ization (or Hnear approxi·mation) of .f. Since D 611F(u) = ~F(u, ~u) according to (8.64) we obtain

(8.73) which is analogous to eq..(8.1.7). For notational simplicity, the linearization operator~

which describes the fully non.linear (finite) deformation case. The terms 6abSnfJ. and Fc1A.F1icCAncv represent the effective elasticity tensor, which has the nature of the

(tangent) st~ffness matrix.

Relations (8.81) and (8.82) are linear with respect to Ju and ~u depending on X.

These relations show a clear mathematical structure in the sense that du and Liu can be interchanged without altering the result of the .integral; thus D il·urHVint(u, c5.u) = DJu{HVint(u, ~u). Relations .(8.81) and (8.82) lead to a symmetric (tangent) stiffness matrix upon discretization. Note that, for example, the set of nonlinear equations associated with nonlinear heat conduction results in a different mathematical structure leading to a non-symmetric stiffness matrix.

The first term in eq. (8.81.) comes from the current state of stress and represents the so-called geometrical stress contribution (in the literature sometim·es called the .initial stress contribution) to the linearization. Since SAB is not the initial stress (it is in fact the current stress}, the terminology is misleading. Within an incremental/iterative solution. technique we can think of Siw as the initial stress at every .increment, so the term initial stress contribution has some meaning. The second term in eq. (8.81) repre-

sents the so-called material contribution to the Iinearization.

The linearized principle of virtual work (8.8]) constitutes the starting point for approximation techniques such as the finite element method, typically leading to the

_geometrical (or initial stress) stiffness matrix and to the material s"tiffness matrix.

Note that for some cases .it is more convenient to discretize the nonlinear variational equation as a first step and to linearize the result with respect to the positions of the nodal points as a second step.

Linearization of "the principle of v.irtual work in spatial description. In order to

linearize the principle of virtual work in the spatial description we recall the nonlinear variational equation (8.33).

As above we consider the static case (ii = o) and assum.e the loads band t to be independent of the motion of the body. Only the linearization of the internal virtual work c)Hlint in the spatial des-cription remains. We adopt 6lVint in the spatial {or Eulerian) form (8..34). The idea is first to pull-back the spatial quantities to the reference configuration, so they correspond with the internal virtual work in the material description. Then they are linearized, as above, and as a last step it is necessary to push-forward the linearized terms.

Starting with the equivalence

(the .arguments have been omitted).

Hence, the push-forward operation on the second Piola-Kirchhoff stress tensor S yields, according to (3.-64), the Kirchhoff-stress tensor T, which is related to the Cauchy stress tensor by T = Ju. Pushing forward the linearized variation of the Green-

Lagrange strain tensor, ~8E, yields the linearized variation of the Euler-Almansi strain tensor, Li.c5e, as discussed in Example 8.6. Computing the push-forward of JE results in 8e'f as introduced. in Section 8.1.

Finally, we -derive the push-forward of the linearized second Piola-Kirchhoff stress tensor, i.e. the last term in (8.84), which will yield the linearized Kirchhoff stress tensor

~T. We write

(8.85)

with .DauS given by (8.79), i.e..DAuS = C : D~uE. By use of (8.69) and (8.·66), the term F(DuuS)FT in-eq.. (8.85h may be written as

(8.86)

where we have also employed the minor symmetries of C~ In order to proceed it is more instructive to employ index notation. With the definition (6.159) of the spatial elasticity t~nsor (c)aiu~tl == Cabcd' eq. (8."86) is equivalent to

cording to the concept of directional derivative introduced in (8.72). Replacing the associated direction ~u, used in the directional derivative, by the velocity vector v, Ar and grad~u result in the Lie ti.me derivative .tv ( r) of T and the spatial velocity gradient I (defined by (2.139)4), respectively. By using the symmetries of c relation (8.88) reads l~v(r) =Jc: d, which proves (6.16l).