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General prob lems in sol idmechanics and non-l in earit y1 I Introduct ion

In the first volume we discussed quite generally linear problems of elasticity and offield equations. In many practical applications the limitation of linear elasticity ormore generally of linear behaviour precludes obtaining an accurate assessment ofthe solution because of the presence of non-linear effects and/or because of thegeometry having a thin dimension in one or more directions. In this volume wedescribe extensions to the formulations previously introduced which permit solutionsto both classes of problems.Non-linear behaviour of solids takes two forms: material non-linearity and geo-metric non-linearity. The simplest form of a non-linear material behaviour is thatof elasticity for which the stress is not linearly proportional to the strain. More gen-eral situations are those in which the loading and unloading response of the materialis different. Typical here is the case of classical elasto-plastic behaviour.When the deformation of a solid reaches a state for which the undeformed anddeformed shapes are substantially different a state of finite deformation occurs. Inthis case i t is no longer possible to write linear strain-displacement or equilibriumequations on the undeformed geometry. Even before finite deformation exists it ispossible to observe buckling or load bifurcations in some solids and non-linear equilib-rium effects need to be considered. The classical Euler column where the equilibriumequation for buckling includes the effect of axial loading is an example of this class ofproblem.Structures in which one dimension is very small compared with the other twodefine plate and shell problems. A plate is a flat structure with one thin directionwhich is called the thickness, and a shell is a curved structure in space with onesuch small thickness direction. Structures with two small dimensions are calledbeams, frames, or rods. Generally the accurate solution of linear elastic problemswith one (or more) small dimension(s) cannot be achieved efficiently by using thethree-dimensional finite element formulations described in Chapter 6 of Volume 1and conventionally in the past separate theories have been introduced. A primaryreason is the numerical ill-conditioning which results in the algebraic equationsmaking their accurate solution difficult to achieve. In this book we depart frompast tradition and build a much stronger link to the full three-dimensional theory.

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2 General problems in soli d mechanics and n on -lineari tyThis volume will consider each of the above types of problems and formulationswhich m ak e practical finite element solutions feasible. W e establish in the present c hap -ter the general formulation fo r both static and transient problems o f a non-linear kind.Here we show how the linear problems of steady state behaviour a nd transient beha-viour discussed in Volume 1 become non-linear. Some general discussion of transient

non-linearity will be given here, an d in the rem ainder of this volum e we shall primarilyconfine o ur remarks to quasi-static (i.e. no inertia effects) an d static problems only.In C hap te r 2 we describe various possible me tho ds for solving non-linear a lgebraicequations. This is followed in Chapter 3 by consideration of material non-linearbehaviour an d the development of a general fo rmu lation from which a finite elementcom putat ion c an proceed.W e then describe the solution of plate problems, considering first the pro blem of thinplates (Chapter 4) in which only bending deformations are included and, second, theproblem in which both bending and shearing deformations are present (Chapter 5).The problem of shell behaviour a dd s in-plane me mb rane deform ation s an d curvedsurface modelling. H ere we split the problem in to three sep ara te pa rts. Th e first, com -bines simple flat elements which include bending a nd me mb rane behaviour to fo rm afaceted a ppro xim ation t o the curved shell surface (Ch apte r 6). Next we involve theadd ition of shearing defo rma tion a nd use of curved elements to solve axisymmetricshell problems (Chapter 7). We conclude the presentation of shells with a generalfor m using curved isoparam etric element shap es which include the effects of bending,shearing, and mem brane deformations (Chapter 8). He re a very close link w ith the fullthree-dimensional analysis of Volume 1 will be readily recognized.In Chapter 9 we address a class of problems in which the solution in on e co ord ina tedirection is expressed as a series, for example a Fourier series. Here, for linearmaterial behavior, very efficient solutions can be achieved for many problems.Some extensions to non-linear behaviour are also presented.In the last pa rt of this volume we address the general problem of finite defo rma tionas well as specializations which perm it large displacem ents but ha ve small strains. InC hap te r 10 we present a su m m ary for the finite de for m atio n of solids. Basic relationsfor defining deform ation ar e presented an d used to w rite variational fo rms related tothe undeformed configuration of the body a nd also to the deformed configuration. Itis shown that by relating the formulation to the deformed body a result is obtainwhich is nearly identical to that for the small deformation problem we consideredin Volume 1 an d which we expan d upo n in the early chap ters of this volume. Essentialdifferences arise only in the constitutive eq ua tion s (stress-strain laws) an d theaddition of a new stiffness term commonly called the geometric o r initial stressstiffness. For constitutive modelling we summarize alternative forms for elastic andinelastic m aterials. In this ch apte r con tact problems are also discussed.In Chapter 11 we specialize the geom etric behaviour to th at which results in largedisplacem ents but small strains. This class of problems permits use of all the consti-tutive eq uatio ns discussed for small defo rm ation problems an d c an address classicalproblems of instability. It also permits the construction of non-linear extensions toplate and shell problems discussed in Chapters 4-8 of this volume.In Chapter 12 we discuss specialization of the finite deformation problem toaddress situations in which a large number of small bodies interact (multiparticleo r granu lar bodies) o r individual parts of the problem are treated a s rigid bodies.

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Small deformation non-linear solid mechanics problems 3In the final chapter we discuss extensions to the computer program described inChapter 20 of Volume 1 necessary to address the non-linear material, the plate andshell, and the finite deformation problems presented in this volume. Here the discus-sion is directed primarily to the manner in which non-linear problems are solved. Wealso briefly discuss the manner in which elements are developed to permit analysis of

either quasi-static (no inertia effects) or transient applications.1.2 Small deformation non-linear solid mechanicsproblems

1.2.1 Introduct ion and notat ionIn this general section we shall discuss how the various equations which we havederived for linear problems in Volume 1 can become non-linear under certain circum-stances. In particular this will occur for structural problems when non-linear stress-strain relationships are used. But the chapter in essence recalls here the notation andthe methodology which we shall adopt throughout this volume. This repeats matterswhich we have already dealt with in some detail. The reader will note how simply thetransition between linear and non-linear problems occurs.The field equations for solid mechanics are given by equilibrium (balance ofmomentum), strain-displacement relations, constitutive equations, boundary condi-tions, and initial condition^.^^'

In the treatment given here we will use two notational forms. The first is a Cartesiantensor indicial form (e.g. see Appendix B, Volume 1 ) and the second is a matrix formas used extensively in Volume 1 . In general, we shall find that both are useful to describeparticular parts of formulations. For example, when we describe large strain problemsthe development of the so-called geometric or initial stress stiffness is most easilydescribed by using an indicial form. However, in much of the remainder, we shall findthat it is convenient to use the matrix form. In order to make steps clear we shall herereview the equations for small strain in both the indicial and the matrix forms. Therequirements for transformations between the two will also be again indicated.For the small strain applications and fixed Cartesian systems we denote coordinates asx , z or in index form as x I x2,x3.Similarly, the displacements will be denoted as u v wor u l u2,u3.Where possible the coordinates and displacements will be denoted asx, and

u respectively, where the range of the index is I , 2 , 3 for three-dimensional applications(or 1,2 for two-dimensional problems). In matrix form we write the coordinates as

and displacements as

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4 General problems in solid mechanics and no n-linearit y1.2.2 Weak form for equil ibrium - fin ite element discretizationThe equilibrium equations (balance of linear momentum) are given in index form as+ b, = piii, i , = 1,2,3 (1.3)where aii are components of (Cauchy) stress, p is mass density, b j are body forcecomponents and (') denotes partial differentiation with respect to time. In theabove, and in the sequel, we always use the convention that repeated indices in aterm are summed over the range of the index. In addition, a partial derivative withrespect to the coordinate x i is indicated by a comma, and a superposed dot denotespartial differentiation with respect to time. Similarly, moment equilibrium (balanceof angular momentum) yields symmetry of stress given indicially as

(1.4)f = f . .I .IEquations (1.3) and (1.4) hold at all points x i in the domain of the problem R. Stressboundary conditions are given by the traction condition

- ( 13 ). = f f . . n .= t .J Jfor all points which lie on the part of the boundary denoted as r r .A variational (weak) form of the equations may be written by using the proceduresdescribed in Chapter 3 of Volume 1 and yield the virtual work equations given by'.8.9

In the above Cartesian tensor form, virtual strains are related to virtual displacementsas= + J 1 (1.7)

In this book we will often use a transformation to matrix form where stresses aregiven in the order

= [ f f l l f f 22 O33 f f12 f f23 f f31 1T (1.8)

= [ G x 0 . g z ? O Y J ff ; f f z , ]and strains by

where symmetry of the tensors is assumed and 'engineering' shear strains areintroduced as*

Ti = 2 E i I (1.10)to make writing of subsequent matrix relations in a consistent manner.

The transformation to the six independent components of stress and strain isperformed by using the index order given in Table 1.1. This ordering will apply to

* This form is necessary to allow the internal work always to be written as cTs

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Small deformation non-linear solid mechanics problems 5Table 1.1 Index relat ion between tensor and m atr ix form sFo rm Index valueM atrix 1 2 3 4 5 6T e n s o r ( l . 2 , 3 ) 1 1 22 33 12 23 3121 32 13

V Y 3) .Y:Tensor (.Y.J'.z) .Y.Y yy :z XY? J'r ?.Y

many subsequent developm ents also . Th e order is chosen to permit reduction t o two-dimensional app licati ons by merely deleting the last two entries and treating the thirdentry a s app ropr iate for plane o r axisymmetric applications.In matrix fo rm, the virtual w ork eq uation is written as (see Chapter 3 of Volume 1)

Finite element approximations to displacements and virtual displacements aredenoted byu(x, t ) = N(x)u( t ) and Su(x) = N(x)SU (1.12)

or in isoparametr ic form asu(6, t) = N(C)U(t); Su(6) = N(C)SU with x(S) = N(6)x (1 .13)

an d may be used t o com put e virtual strains asSE = SSU= (SN)SU = BSU (1.14)

in which the three-dimensional strain-displacement matrix is given by [see Eq. (6.1 l ) ,Volume 11

(1.15)

In the above, U denotes time-dependent nodal displacement parameters and 6Urepresents arbitrary virtual displacement parameters.Noting that the virtual parameters 6U are arbitrary we obtain for the discreteproblem*(1.16)

whereMU + P(o) = f

M = JI N T p N d R (1.17)f = j n N T b d f l+ .I N T t d r (1.18)

r r

* Fo r s implici ty we omit direct damp ing which leads to the term Cu (see Ch apte r 17, Volume I )

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Small deformation non-linear solid mechanics problems 7

(1.25)Using the GN22 formulae, the discrete displacements, velocities, and accelerationsar e linked by [see Eq . (18.62), Volum e 11

u n + l = u , + a t U , + ; ( l - / 3 2 ) A t 2 i i n + 9 2 A t 2 i i n + , (1.26)U 1 = U + (1 P I )Alii, + PI Alii, + 1 (1.27)

where At = tn i l , .Eq ua tion s (1.26) an d (1.27) ar e simple, vector, linear relationships as the coefficientPI a n d P2 are assigned a priori and it is possible to take the basic unknown in Eq.(1.24) as any one of th e three variables at t ime step n + 1 (Le. u , , + ~ , n + l o r + I ) .In such schemes wetake the con s tant P2 as zero and note that this allows un+ to be evaluated directlyfrom the initial values at time t , without solving any simultaneo us equat ions.Immediately, therefore, Eq. (1.24) will yield the values of ii, I by simple inversionof matr ix M .If the M ma trix is diagonalized by an y on e of the meth ods which we have discussedin Volume 1, the solution fo r ii, is trivial a nd the prob lem c an be considered solved.However, such explicit schemes are only conditionally stable as we have shown inChapter 18 of Volume 1 and may require many time steps to reach a steady statesolu tion. The refore for transient problems an d indeed for all static (steady state)problems, it is often more efficient to deal with implicit methods. Here, most con-veniently, u,+ can be taken a s the basic variable from w hich U,+ a n d ii, can becalculated by using Eq s (1.26) and (1.27). T he eq uatio n system (1.24) can thereforebe written as

*(u,,+1) = * , , + I = 0 (1.28)T he so luti on of this set of eq ua tion s will require a n iterative process if the relationsare non-linear. We shall discuss various non-linear calculation processes in somedetai l in Ch apte r 2; however, the New ton-Rap hson metho d form s the basis ofmo st practical schemes. In this meth od an ite ration is as given below

A very convenient choice for explicit schemes is that of ii,

( 1.29)where du: is an increment to the so lut ion* such that

(1.30)Fo r prob lem s in which pa th dependence is involved it is necessary to keep trac k of thetotal increment during the i terat ion and write

Uk+, 1 = U, + Aut'' (1.31)Th us the total increment ca n be accumulated by using the same solution increments as

(1.32)

U k + l k kn+ 1 UP,+ 1 + dun

Au: = u:: u,, = Auk + du:Note that an italic d s used for a solution increment and an upright 'd' for a differential.

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8 General problems in soli d mechanics and non -linearityin which a quantity without the superscript k denotes a converged value from aprevious time step. The initial iterate may be taken as zero or, more appropriately,as the converged solution from the last time step. Accordingly,

u,,+~= u,, giving also Aui = 0 (1.33)A solution increment is now computed from Eq. (1.29) as

(1.34)- I k'uf: (KT) * l 7 + Iwhere the tangent nmtrix is computed as

K; = ~dun+ IFrom expressions (1.24) and (1.26) we note that the above equations can be rewrittenas

We note that the above relation is similar but not identical to that of linear elasti-city. Here D is the tangent modulus matrix for the stress-strain relation (whichmay or may not be unique but generally is related to deformations in a non-linear manner).Iteration continues until a convergence criterion of the form(1.35)

or similar is satisfied for some small tolerance E . A good practice is to assume thetolerance at half machine precision. Thus, if the machine can compute to about 16digits of accuracy, selection of E = lo-' is appropriate. Additional discussion onselection of appropriate convergence criteria is presented in Chapter 2.Various forms of non-linear elasticity have in fact been used in the present contextand here we present a simple approach in which we define a strain energy W as afunction of EW = W E) = W(E,)

and we note that this definition gives us immediatelyddec = - (1.36)

If the nature of the function W is known, we note that the tangent modulus D:becomes

The algebraic non-linear solution in every time step can now be obtained by theprocess already discussed. In the general procedure during the time step, we haveto take an initial value for u , ~ + ~ ,or example, u , + ~= u, (and similarly for u , + land i i 1 7 +] ) nd then calculate at step 2 the value of at k = 1, and obtain1

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Small deformation non-linear solid mechanics problems 9duf, upd ating the value of u t+ by Eq. (1.30). Th is of course necessitates calculationof stresses at t , + , to obtain the necessary forces. It is worthwhile noting that thesolu tion for steady s tate problems proceeds o n identical lines with so lution variablechosen as u, ,+ but now we simply say u, ? = t i f 7+ = 0 as well as the co rrespondingterms in the governing equations.

1.2.4 Mixed or irreduc ible formsThe previous formulation was cast entirely in terms of the so-called displacementformulation which indeed was extensively used in the first volume. However, as wementioned there, on some occasions it is convenient to use mixed finite elementforms and these are especially necessary when constraints such as incompressibilityarise. It has been frequently noted that certain constitutive laws, such as those ofviscoelasticity an d asso ciative plasticity th at w e will discuss in C ha pt er 3, the ma terialbehaves in a nearly incompressible manner. For such problems a reformulationfollowing the procedures given in C ha pt er 12 of Volum e 1 is necessary. We remindthe reader that on such occasions we have two choices of formulation. We canhave the variables u a n d p (where p is the mean stress) as a two-field formulation(see Sec. 12.3 or 12.7 of Volume 1) or we can have the variables u, p and E (whereE is the volume change) as a three-field formulation (see Sec. 12.4, Volume I ) . A nalternative three-field for m is the enha nce d strain ap pro ac h presented in Sec. 11.5.3of Volume 1 . The matter of which we use depends on the form of the constitutiveequ ation s. F o r situatio ns where changes in volume affect only the pressure the two-field for m ca n be easily used. H owe ver, for problem s in which the response is coupledbetween the deviatoric and mean components of stress and strain the three-fieldformulations lead to much simpler forms from which to develop a finite elementmodel. To illustrate this point we present again the mixed formulation of Sec. 12.4in Volume 1 and show in detail how such coupled effects can be easily includedwithout any change to the previous discussion on solving non-linear problems. Thedevelopment also serves as a basis fo r the development of an extended fo rm whichpermits the tre atm en t of finite de for m atio n problems. Th is extension will be presentedin Sec. 10.4 of Chapter 10.A three-f ie ld m ixed m ethod for general const i tu tive modelsIn orde r to develop a mixed form for use with constitutive models in which m ean a nddeviatoric effects can be coupled we recall (Chapter 12 of Volume 1) that mean anddeviatoric m atrix op era tor s are given by

(1.37)

where I is the identity matrix.

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10 General prob lems in solid mechanics and n on -linearityAs in Volume 1 we introduce independent parameters E : and p describing volumetricchange and mean stress (pressure), respectively. The strains may now be expressed ina mixed form as

E = I,, (SU) + f mE, (1.38)and the stresses in a mixed form as

IS = Id6+ mp (1.39)where 6 is the set of stresses deduced directly from the strains, incremental strains, orstrain rates, depending on the particular constitutive model form. For the present weshall denote this stress by

6 = S( ) ( I .40)where we note it is not necessary to split the model into mean and deviatoric parts.

The Galerkin (variational) equations for the case including transients are nowgiven by

p ] d f l = O6p[mT(Su) ] dR = 0

Introducing finite element approximations to the variables asu u = N,U, p p = NPp and

and similar approximations to virtual quantities asE E = N,E,

6u Si = Nu , 6p = 6p = Np6p and 6 ~, 62, = N,6E,the strain and virtual strain in an element become

E = IdBU +fmN,E, ,6~= Id B 6U + f mN,SE,

(1.41)

( 1.42)in which B is the standard strain-displacement matrix given in Eq. (1.15). Similarly,the stresses in each element may be computed by using

IS = I d 8 + mN,p (1.43)where again 6 are stresses computed as in Eq. (1.40) in terms of the strains E .into Eq. (1.41) we obtain the set of finite element equationsSubstituting the element stress and strain expressions from Eqs (1.42) and (1.43)

P +MU= fPp cp = 0

-CTE, + EU = 0( 1.44)

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Small deformation non-linear solid mechanics problems 1 1where

P = J B T c d R ,n P, = J N:mTirdRnE = N , m B d RJn (1 .45)

If the pressure and volumetr ic s t ra in approximat ions are taken local ly in eachelement and N, = N, it is possib le to solve the secon d an d third eq ua tio n of (1.44)in each e lement individual ly . Not ing that the array C is now symmetric posit ivedefinite, we may always write these asp = c - P,

E = C - ' E u =W uThe mixed s t ra in in each e lement may now be com puted as

whereB, = N,W

(1.46)

( 1.47)

( 1.48)defines a mixed f o rm of the volumetr ic s t ra in-displacement equat ions.forms''.' 'From the above resul ts i t i s possible to wri te the vector P in the alternative = B T c d R

1

The computa t ion of P may then be represented in a m atr ix form as(1 .49)

(1 S O )in which we note the inclusion of the transpose of the matr ices appear ing in theexpression for the m ixed strain given in E q. (1.47). Based o n this result we observetha t i t is not necessary to compute the true mixed stress except when reportingfinal results where, fo r si tuations involving near incom pressible beha viour, it is crucialto compute explicit ly the mixed pressure to avoid any spurious volumetric stresseffects.

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1 2 General problems in solid mechanics and no n-li nearit yThe last step in the process is the com puta tion o f the tangent for the equ ation s. Thisis straightforward using form s given by E q. (1.40) where we obt ain

dir = DTdUse of Eq. (1.47) to express the incremental mixed strains then gives

(1.51)It should be noted that construction of a modified modulus term given by

requires very few o pera tions because of the sparsity and form of the array s In and m.Con sequently, the m ultiplications by the coefficient matrices B an d B, in this form isfar m ore efficient th an constructing a full B asB = I d B + f m B , ( 1 . 5 3 )

and operat ing o n DT directly.The above form for the mixed element generalizes the result in Volume 1 and isvalid for use with many different linear and non-linear constitutive models. InC ha pt er 3 we con sider stress-strain beh avio ur m odelled by viscoelasticity, classicalplasticity, and generalized plasticity formulations. Each of these forms can lead tositu atio ns in which a nearly incompressible response is required an d for manyexamples included in this volume we shall use the above mixed fo rm ula tion . Tw obasic forms are considered: four-noded qua drilater al or eight-noded brick isopara-metric elements with constan t interpolation in each element for one -term appro xim a-tions to N , and Nl, by unity; an d nine-noded quadrilateral o r 27-noded brickisop aram etric elements with linear interp olation fo r N, an d N, .* Accordingly, intwo dim ensions we use

N,, = N , = [ 1 < 771 o r [ 1 -Y 1.1and in three dimensions

N , = N , = [ l 77 < ] o r [ l .Y y z ]Th e elements created by this process may be used t o solve a wide range of problem s insolid mechanics, as we shall illustrate in later chapters of this volume.

1.3 Non-linear quasi-harmonic field problemsIn subsequent chapters we shall touch upon non-linear problems in the context ofinelastic constitutive equations for solids, plates, and shells and in geometric effects

Form ulat ions using the e ight-noded quadri la tera l a nd twenty-noded br ick serendipity e le inents may alsobe constructed: however . we showed in Chapter 1 I of V o l u m e I that these elements do not fully satisfy themixed patch tes t.

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Non-linear quasi-harmonic fi eld problems 13arising from finite defo rma tion. In Volum e 3 non-linear effects will be considered forvarious fluid mechanics situations. However, non-linearity also may occur in manyother problems and in these the techniques described in this chapter are still univer-sally applicable. A n exam ple of such situa tions is the qu asi-h arm onic equatio n whichis encou ntered in man y fields of engineering. He re we consider a simple quas i-ha rm o-nic problem given by (e.g. heat co ndu ction )p ~ VTq Q(4) 0 (1.54)with suitable boundary conditions. Such a form may be used to solve problemsranging from temperature response in solids, seepage in porous media, magneticeffects in solids, an d p otential fluid flow. In t he ab ove, q is a flux an d generally thiscan be written as

q = q(4,V4 k ( A V4)V4or , after l inearization,

dq = kod 4 + k ' d(V4)where

The source term Q ( 4 ) also can in trodu ce non-linearity.the q term the problemA discretization based o n Ga lerkin procedures gives after integration by par ts of

(1.55)a n d is still valid if q a n d / o r Q (and indeed the boun dary conditions) are dependent o n4o r i ts derivat ives. Introducing the interpo lat ions

4 = N & a n d 64= NS (1.56)a discretized fo rm is given a s

l f( ) c P,($) = 0 (1.57)where

C = N T p c N d R.I,P ,= N T q d RI

f = 1 NT Q( ) dR NTq,l dTI Tu

(1.58)

Equat ion (1.57) may be solved following similar procedures described in Chapter18, Volume 1 . Fo r instance, just as we did with GN 22 we can now use G N 1 1 as@,I+ I = 4 + ( 1 h Q & + (1.59)

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14 General problems in soli d mechanics and no n-li nearit yOnce again we have the choice of using + I o r n+ as the primary solution vari-able. To this extent the process of solving transient problems follows absolutely thesame lines as those described in the previous section (and indeed in the previousvolume) and need not be further discussed. We note again that the use of +,+, asthe chosen variable will allow the solution me thod to be applied to static o r steadystate problems in which the first term of Eq. (1.54) becomes zero.

1.4 Some typical examples of transient non-linearcalculationsIn this section we report results of some transient problem s of structural mechanics a swell as field pro blem s. As we mentioned earlier, we usually will not consider transientbehaviour in latter parts of this book as the solution process for transients followessentially the path described in Volume 1.Transient heat conductionTh e governing eq ua tion f or this set of physical problem s is discussed in the previoussection, with 4 eing the temperature T now [Eq. (1.54)].Non-linearity clearly can arise from the specific heat, e thermal conductivity, k ,an d source, Q , being temperature-dependent o r from a radiat ion boun dary cond it ion

a(T T,)'=Tdn (1.60)with n 1. Here (Y is a convective heat transfer coefficient and T, is an ambientexternal tempe rature . We shall show two examples t o il lustrate the ab ove .The first concerns the freezing of ground in which the latent heat of freezing isrepresented by varying the material properties with temperature in a narrow zone,as shown in Fig. 1.1. Further, in the transition from the fluid to the frozen state avariation in conductivity occurs. We now th us have a problem in which bo th matricesC a n d P [Eq. (1 .58)] ar e variable, a nd solution in Fig. 1.2 illustrates the progression ofa f reez ing f ront which was derived by using the three-point (Lees) a l g ~ r i t h m l ~ . ' ~ithC = C,7and P = P,.

A computational feature of some significance arises in this problem as values ofthe specific heat become very high in the transition zone and, in time steppingcan be missed if the temperature step straddles the freezing point. To avoid thisdifficulty and keep the heat balance correct the concept of enthalpy is introduced,definingTH = s p c d T (1.61)

No w, whenever a change of tem pera ture is considered, an a ppro priate value of pc iscalculated that gives the correct change of H .The heat conduction problem involving phase change is of considerable impor-tance in welding and casting technology. Some very useful finite element solutionsof these problems have been obtained.14 Further elaboration of the proceduredescribed above is given in reference 15.

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Some typical examples of transient non-linear calculations 1 5

Fig. 1.1 Estimation of thermophysical properties i n phase change problems. The latent heat effect is approxi-mated by a large capacity over a small temperature interval 2AT.

Th e second non -linear ex amp le concerns th e problem of spontaneous ignition. '6 W ewill discuss the steady state case of this pro blem in Ch ap ter 3 an d no w will be con-cerned only with transient cases. Here the heat generated d epe nds on the tempe rature(1.62)= Se

and the situation can become physically unstable with the computed temperaturerising contin uou sly to extreme values. In Fig . 1.3 we show a transient solution of asphere at an initial temp eratur e of T = 290 K immersed in a bath of 500 K . Th e solu-tion is given for two values of the para me ter b with k = pc = 1, a n d the non-linearitiesare now so severe that an iterative solutio n in each time increment is necessary. F o rthe larger value of b the tem pera ture increases to infinite value in a nite time an d thet ime interval for the com putat ion had to be chang ed continuously to acc ount for this.Th e finite time fo r this point to be reached is know n as the induction time an d is show nin Fig. 1.3 for various values of 8.The quest ion of changing the t ime interval during the co mpu tat ion h as not beendiscussed in detail, but clearly this must be done quite frequently to avoid largechanges of the unknown function which will result in inaccuracies.

- T

Structura l dynamicsHere the examples concern dyn am ic structural transients with material an d geometricnon-linearity. A highly non-linear geometrical and material non-linearity generallyoccurs. Neglecting damping forces, Eq. (1.16) can be explicitly solved in an efficientmanner .If the explicit com puta tion is pursued to the po int when steady state con dition s areapproached, that is, until ii = u 0 the solution to a static non-linear problem isobta ined . This type of technique is frequently efficient as an alternative to the me tho ds

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5.V-Ln._fNa .'*.

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Some typical examples of transient n on-linear calculations 17

Fig. 1.3 Reactive sphere. Transient temperature behaviour for ign ition ( 6 = 16) and non-ignition h = 2 )cases: (a) induct ion time versus Frank-Kamenetskii parameter; temperature profiles; (b) temperature profilesfor ignition (d = 16) and non-ignition (&= 2) transient behaviour of a reactive sphere.

descr ibed above an d in Cha pte r 2 an d h as been a pplied successful ly in the context off ini te d ifferences un der the n am e of dynamic re laxat ion for the solut ion of non-l inearsta t ic problems.T w o examples of explic it d yna m ic analysis will be given here . The first problem,illustrated in Plate 3, is a large three-dimensional problem and i ts solut ion wasobta in ed wi th the use of a n explic it dyna m ic scheme. In such a case implicit schemeswould be totally inapplicable and indeed the explicit code provides a very efficientsolut ion of the crash problem shown. I t must . however . be recognized that such

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18 General problems in sol id mechanics and no n-linearity

Fig. 1.4 Crash analysis: (a) mesh a t t = Oms; (b) mesh a t t = 20ms; c) mesh a t t = 40 ms.

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Some typ ical examples of transient n on-linear calculations 19final solutions are not necessarily unique. As a second example Figure 1.4 shows atypical crash analysis of a m ot or vehicle carried ou t by similar mean s.

arthquake response of soil - structuresWe have mentioned in Chapter 19, Volume 1, the essential problem involving inter-action of the soil skeleton or matrix with the water contained in the pores. Thisproblem is of extreme im portan ce in earthq uak e engineering an d here again solution

Fig. 1.5 Retaining wall subjected to earthquake excitation comparison of experiment (centrifuge) andcalculations l a

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20 General problems in solid mechanics and non -linearityof transient non-linear equations is necessary. As in the mixed problem which wereferred to earlier, the variables include displacement, and the pore pressure in thefluid p .In Chapter 19 of Volume 1, we have in fact shown a comparison between somecentrifuge results and computations showing the development of the pore pressurearising from a particular form of the constitutive relation assumed. Many suchexamples and indeed the full theory are given in a recent text, and in Fig. 1.5 weshow an example of comparison of calculations and a centrifuge model presentedat a 1993 workshop known as VELACS19. This figure shows the displacements ofa big retaining wall after the passage of an earthquake, which were measured in thecentrifuge and also calculated.

1.5 Concluding remarksIn this chapter we have summarized the basic steps needed to solve a general small-strain solid mechanics problem as well as the quasi-harmonic field problem. Only astandard Newton-Raphson solution method has been mentioned to solve theresulting non-linear algebraic problem. For problems which include non-linearbehaviour there are many situations where additional solution strategies are required.In the next chapter we will consider some basic schemes for solving such non-linearalgebraic problems. In subsequent chapters we shall address some of these in thecontext of particular problems classes.The reader will note that, except in the example solutions, we have not discussedproblems in which large strains occur. We can note here, however, that the solutionstrategy described above remains valid. The parts which change are associated withthe effects of finite deformation on computing stresses and thus the stress-divergenceterm and resulting tangent moduli. As these aspects involve more advanced conceptswe have deferred the treatment of finite strain problems to the latter part of thevolume where we will address basic formulations and applications.

References1. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: The Basis, Volume 1.2. S.P. T imoshenko and J .N . Goodier . Theory of Elasticity, McGraw-Hill , New York, 3rd3. I.S. Sokolnikoff, The Mathenzatical Theory of Elasticity, McGraw-Hil l , New York, 2nd4. L.E. Malvern. Introduction to t h e Meclzanics o f a Continuous Medium , Prentice -Hall, Engle-5. A.P. Boresi and K.P. Chong. Elasticity in Engineering Mechonics, Elsevier, New York,6 . P .C . Cho u and N.J. Pagano . Ela sticity: Tensor , Dyadic und Engineering Appro aches, Dover7. I.H . Shames and F .A. Cozzarell i. Elastic and Inelastic Stress Analysis, Taylor Francis,

Arno ld, L ondo n, 5th edit ion, 2000.edition, 1969.edition, 1956.wood Cliffs, NJ, 1969.1987.Publications, Mineola, N Y , 1992; reprinted from the 1967 Van N ostrand edit ion.W ashingto n, D C , 1997; revised printing.

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References 2 18 . J .C. S imo, R.L . Tay lor and K.S. Pister. Variational an d projection methods for the volumeconstraint in finite deformation plasticity. Comp. Me th. Appl . Mech. Eng. , 51 177-208,1985.9. K . Washizu. Variational Methods in Elasticity and Plasticitv, Pergamon Press , New Y ork,3rd edit ion, 1982.10. T. J .R . Hughes . General ization of selective integ ration procedures to an isotropic an d non-

l inear media. In t . J . Nu m . Metlz. Eng. , 15 1413-18, 1980.11. J .C . Simo an d T.J .R. Hughes . O n the variat ional founda t ions of assumed s t rain methods.J . Appl . Mech. , 53 1), 51-4, 1986.12. M . Lees. A linear three level difference scheme f or quasilinear pa rab olic equ ation s. Maths .Comp. , 20, 516-622, 1966.13. G. Comini , S . Del Guidice, R .W . Lewis an d O .C. Zienkiewicz. F inite element solution ofnon-linear cond uction problems w ith special reference to pha se change . Int. J . Num . Me th .Eng., 8, 613-24, 1974.14. H.D. Hibbit t and P.V. Marcal. Numerical thermo-mechanical model for the welding andsubsequent loading of a fabricated structure. Com puters and Structures, 3, 1145-74, 1973.15. K . Morgan, R.W. Lewis and O.C. Zienkiewicz. An improved algori thm for heat con-vection problems with phase change. In t . J . Num . Meth . Eng. , 12 1191-95, 1978.16. C .A . Anderson an d O .C . Zienkiewicz. Spo ntaneou s ignit ion: finite element solutions fors teady a nd t ransient condi t ions . T ra ns A S M E , J . Heat Transfer, 398-404, 1974.17. J .R.H . Ot te r , E. Cassel and R.E. Hobbs. Dynamic relaxat ion. Proc. Inst. Civ. Eng., 3518. O.C . Zienkiewicz, A.H.C . C ha n, M . Pa sto r and B.A. Schrefler. Compu tational Geo-niechanics: W it h Special Reference to Earth qua ke Engineering, John Wiley, Chichester,

Sussex, 1999.19. K . Arulanandan and R.F. Sco t t (eds ) . Proceedings of V E L A C S S ym p os iu m , Balkema,Rotterdam. 1993.

633-56, 1966.