general problems in solid mechanics and non-linearity

1

General problems in solidmechanics and non-linearity

1.1 Introduction

Many introductory texts on the finite element method discuss the solution for linearproblems of elasticity and field equations.1–3 In practical applications the limitationof linear elasticity, or more generally of linear behaviour, often precludes obtainingan accurate assessment of the solution because of the presence of ‘non-linear’ effectsand/or because the geometry has a ‘thin’ dimension in one or more directions. In thisbook we describe extensions to the formulations introduced to solve linear problemsto permit solutions to both classes of problems.

Non-linear behaviour of solids takes two forms: material non-linearity and geometricnon-linearity. The simplest form of non-linear material behaviour is that of elasticityfor which the stress is not linearly proportional to the strain. More general situations arethose in which the loading and unloading response of the material is different. Typicalhere is the case of classical elastic–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. In thiscase it is no longer possible to write linear strain–displacement or equilibrium equationson the undeformed geometry. Even before finite deformation exists it is possibleto observe buckling or load bifurcations in some solids and non-linear equilibriumeffects need to be considered. The classical Euler column, where the equilibriumequation for buckling includes the effect of axial loading, is an example of this classof problem. When deformation is large the boundary conditions can also become non-linear. Examples are pressure loading that remains normal to the deformed body andalso the case where the deformed boundary interacts with another body. This latterexample defines a class known as contact problems and much research is currentlyperformed in this area. An example of a class of problems involving non-linear effectsin deformation measures, material behaviour and contact is the analysis of a rollingtyre. A typical mesh for a tyre analysis is shown in Fig. 1.1. The cross-section shownis able to model the layering of rubber and cords and the overall character of a tread.The full mesh is generated by sweeping the cross-section around the wheel axis with avariable spacing in the area which will be in contact. A formulation in which the meshis fixed and the material rotates is commonly used to perform the analysis.4–7

2 General problems in solid mechanics and non-linearity

(a) Tyre cross-section. (b) Full mesh.

Fig. 1.1 Finite element mesh for tyre analysis.

Generally the accurate solution of solid problems which have one (or more) smalldimension(s) compared to the others cannot be achieved efficiently using standardtwo- or three-dimensional finite element formulations. Traditionally separate theo-ries of structural mechanics are introduced to solve this class of problems. A plateis a flat structure with one thin (small) direction which is called the thickness. Ashell is a curved structure in space with one such small thickness direction. Struc-tures with two small dimensions are called beams, frames, or rods. A primary rea-son why use of standard two- or three-dimensional finite element formulations donot yield accurate solutions is the numerical ill-conditioning which results in theiralgebraic equations. In this book we combine the traditional approaches of struc-tural mechanics with a much stronger link to the full three-dimensional theory ofsolids to obtain formulations which are easily solved using standard finite elementapproaches.

This book considers both solid and structural mechanics problems and formulationswhich make practical finite element solutions feasible. We divide the volume into twomain parts. In the first part we consider problems in which continuum theory of solidscontinues to be used, whereas in the second part we focus attention on theories ofstructural mechanics to describe the behaviour of rods, plates and shells.

In the present chapter we review the general equations for analysis of solids inwhich deformations remain ‘small’ but material behaviour includes effects of a non-linear kind. We present the theory in both an indicial (or tensorial) form as well as in thematrix form commonly used in finite element developments. We also reformulate theequations of solids in a variational (Galerkin) form. In Chapter 2 we present a generalscheme based on the Galerkin method to construct a finite element approximate solutionto problems based on variational forms. In this chapter we consider both irreducible

Introduction 3

and mixed forms of finite element approximation and indicate where the mixed formshave distinct advantages. Here we also show how the linear problems of solids forsteady state and transient behaviour become non-linear when the material constitutivemodel is represented in a non-linear form. Some discussion on the solution of transientnon-linear finite element forms is included. Since the form of the inertial effectsis generally unaffected by non-linearity, in the remainder of this volume we shallprimarily confine our remarks to terms arising from non-linear material behaviour andfinite deformation effects.

In Chapter 3 we describe various possible methods for solving non-linear alge-braic equations. This is followed in Chapter 4 by consideration of material non-linearbehaviour and completes the development of a general formulation from which a finiteelement computation can proceed.

In Chapter 5 we present a summary for the study of finite deformation of solids.Basic relations for defining deformation are presented and used to write variational(Galerkin) forms related to the undeformed configuration of the body and also to thedeformed configuration. It is shown that by relating the formulation to the deformedbody a result is obtained which is nearly identical to that for the small deformationproblem we considered in the small deformation theory treated in the early chapters ofthis volume. Essential differences arise only in the constitutive equations (stress–strainlaws) and the addition of a new stiffness term commonly called the geometric or initialstress stiffness. For constitutive modelling we summarize in Chapter 6 alternativeforms for elastic and inelastic materials. Contact problems are discussed in Chapter 7.Here we summarize methods commonly used to model the interaction of intermittentcontact between surfaces of bodies.

In Chapter 8 we show that analyses of rigid and so-called pseudo-rigid bodies8 maybe developed directly from the theory of deformable solids. This permits the inclusionin programs of options for multi-body dynamic simulations which combine deformablesolids with objects modelled as rigid bodies. In Chapter 9 we discuss specialization ofthe finite deformation problem to address situations in which a large number of smallbodies interact [multi-particle or granular bodies commonly referred to as discreteelement methods (DEM) or discrete deformation analysis (DDA)].

In the second part of this book we study the behaviour of problems of structuralmechanics. In Chapter 10 we present a summary of the behaviour of rods (beams)modelled by linear kinematic behaviour. We consider cases where deformation effectsinclude axial, bending and transverse shearing strains (Timoshenko beam theory9) aswell as the classical theory where transverse effects are neglected (Euler–Bernoullitheory). We then describe the solution of plate problems, considering first the problemof thin plates (Chapter 11) in which only bending deformations are included and,second, the problem in which both bending and shearing deformations are present(Chapter 12).

The problem of shell behaviour adds in-plane membrane deformations and curvedsurface modelling. Here we split the problem into three separate parts. The firstcombines simple flat elements which include bending and membrane behaviour to forma faceted approximation to the curved shell surface (Chapter 13). Next we involve theaddition of shearing deformation and use of curved elements to solve axisymmetricshell problems (Chapter 14). We conclude the presentation of shells with a generalform using curved isoparametric element shapes which include the effects of bending,


shearing, and membrane deformations (Chapter 15). Here a very close link with thefull three-dimensional analysis will be readily recognized.

In Chapter 16 we address a class of problems in which the solution in one coordi-nate direction is expressed as a series, for example a Fourier series. Here, for linearmaterial behaviour, very efficient solutions can be achieved for many problems. Someextensions to non-linear behaviour are also presented.

In Chapter 17 we specialize the finite deformation theory to that which results inlarge displacements but small strains. This class of problems permits use of all theconstitutive equations discussed for small deformation problems and can address clas-sical problems of instability. It also permits the construction of non-linear extensionsto plate and shell problems discussed in Chapters 11–15 of this volume.

We conclude the descriptions applied to solids in Chapter 18 with a presentation ofmulti-scale effects in solids.

In the final chapter we summarize the capabilities of a companion computer program(called FEAPpv ) that is available at the publisher’s web site. This program may be usedto address the class of non-linear solid and structural mechanics problems described inthis volume.

1.2 Small deformation solid mechanics problems

1.2.1 Strong form of equations – indicial notation

In this general section we shall describe how the various equations of solid mechanics∗

can become non-linear under certain circumstances. In particular this will occur forsolid mechanics problems when non-linear stress–strain relationships are used. Thechapter also presents the notation and the methodology which we shall adopt throughoutthis book. The reader will note how simply the transition between forms for linear andnon-linear problems occurs.

The field equations for solid mechanics are given by equilibrium behaviour (bal-ance of momentum), strain-displacement relations, constitutive equations, boundaryconditions, and initial conditions.10–15

In the treatment given here we will use two notational forms. The first is a cartesiantensor indicial form and the second is a matrix form (see reference 1 for additionaldetails on both approaches). 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 shallfind that it is convenient to use a matrix form. The requirements for transformationsbetween the two will also be indicated.

In the sequel, when we use indicial notation an index appearing once in any termis called a free index and a repeated index is called a dummy index. A dummy indexmay only appear twice in any term and implies summation over the range of the index.

∗ More general theories for solid mechanics problems exist that involve higher order micro-polar or couple stresseffects; however, we do not consider these in this volume.

Small deformation solid mechanics problems 5

Thus if two vectors ai and bi each have three terms the form aibi implies

aibi = a1b1 + a2b2 + a3b3

Note that a dummy index may be replaced by any other index without changing themeaning, accordingly

aibi ≡ ajbj

Coordinates and displacementsFor a fixed Cartesian coordinate system we denote coordinates as x, y, z or in indexform as x1, x2, x3. Thus the vector of coordinates is given by

x = x1e1 + x2e2 + x3e3 = xi ei

in which ei are unit base vectors of the Cartesian system and the summation conventiondescribed above is adopted.

Similarly, the displacements will be denoted as u, v, w or u1, u2, u3 and the vectorof displacements by

u = u1e1 + u2e2 + u3e3 = ui ei

Generally, we will denote all quantities by their components and where possiblethe coordinates and displacements will be denoted as xi and ui , respectively, in whichthe range of the index i is 1, 2, 3 for three-dimensional applications (or 1, 2 for two-dimensional problems).

Strain--displacement relationsThe strains may be expressed in Cartesian tensor form as

εij = 1

2

(∂ui

∂xj

+ ∂uj

∂xi

)(1.1)

and are valid measures provided deformations are small. By a small deformationproblem we mean that

|εij | << 1 and |ω2ij | << ‖εij‖

where | · | denotes absolute value and ‖ · ‖ a suitable norm. In the above ωij denotes asmall rotation given by

ωij = 1

2

(∂ui

∂xj

− ∂uj

∂xi

)(1.2)

and thus the displacement gradient may be expressed as

∂ui

∂xj

= εij + ωij (1.3)


Equilibrium equations -- balance of momentumThe equilibrium equations (balance of linear momentum) are given in index form as

σji,j + bi = ρ ui , i, j = 1, 2, 3 (1.4)

where σij are components of (Cauchy) stress, ρ is mass density, and bi are body forcecomponents. In the above, and in the sequel, we use the convention that the partialderivatives are denoted by

f,i = ∂f

∂xi

and f = ∂f

∂t

for coordinates and time, respectively.Similarly, moment equilibrium (balance of angular momentum) yields symmetry of

stress given in indicial form asσij = σji (1.5)

Equations (1.4) and (1.5) hold at all points xi in the domain of the problem �.

Boundary conditionsStress boundary conditions are given by the traction condition

ti = σjinj = ti (1.6)

for all points which lie on the part of the boundary denoted as �t . A quantity with a‘bar’ denotes a specified function.

Similarly, displacement boundary conditions are given by

ui = ui (1.7)

and apply for all points which lie on the part of the boundary denoted as �u.Many additional forms of boundary conditions exist in non-linear problems. Con-

ditions where the boundary of one part interacts with another part, so-called contactconditions, will be taken up in Chapter 7. Similarly, it is necessary to describe howloading behaves when deformations become large. Follower pressure loads are oneexample of this class and we consider this further in Sec. 5.7.

Initial conditionsFinally, for transient problems in which the inertia term ρ ui is important, initial con-ditions are required. These are given for an initial time denoted as ‘zero’ by

ui(xj , 0) = d i(xj ) and ui(xj , 0) = vi(xj ) in � (1.8)

It is also necessary in some problems to specify the state of stress at the initial time.

Constitutive relationsAll of the above equations apply to any material provided the deformations remainsmall. The specific behaviour of a material is described by constitutive equationswhich relate the stresses to imposed strains and, often, other sources which causedeformation (e.g. temperature).


The simplest material model is that of linear elasticity where quite generally

σij = Cijkl(εkl − ε(0)kl ) (1.9a)

in which Cijkl are elastic moduli and ε(0)kl are strains arising from sources other than dis-

placement. For example, in thermal problems strains result from change in temperatureand these may be given by

ε(0)kl = αkl[T − T0] (1.9b)

in which αkl are coefficients of linear expansion and T is temperature with T0 a referencetemperature for which thermal strains are zero.

For linear isotropic materials these relations simplify to

σij = λδij (εkk − ε(0)kk ) + 2 µ (εij − ε(0)

ij ) (1.10a)

andε(0)

kl = δijα [T − T0] (1.10b)

where λ and µ are Lame elastic parameters and α is a scalar coefficient of linearexpansion.10,11 In addition, δij is the Kronecker delta function given by

δij ={ 1; for i = j

0; for i �= j

Many materials are not linear nor are they elastic. The construction of appropriateconstitutive models to represent experimentally observed behaviour is extremely com-plex. In this book we will illustrate a few classical models of behaviour and indicatehow they can be included in a general solution framework. Here we only wish toindicate how a non-linear material behaviour affects our formulation. To do this weconsider non-linear elastic behaviour represented by a strain–energy density functionW in which stress is computed as11

σij = ∂W

∂εij

(1.11)

Materials based on this form are called hyperelastic. When the strain–energy is givenby the quadratic form

W = 12εijCijklεkl − εijCijklε

(0)kl (1.12)

we obtain the linear elastic model given by Eq. (1.9a). More general forms are permit-ted, however, including those leading to non-linear elastic behaviour.

1.2.2 Matrix notation

In this book we will often use a matrix form to write the equations. In this case wedenote the coordinates as

x =⎧⎨⎩

x

y

z

⎫⎬⎭ =

⎧⎨⎩

x1

x2

x3

⎫⎬⎭ (1.13)


and displacements as

u =⎧⎨⎩

u

v

w

⎫⎬⎭ =

⎧⎨⎩

u1

u2

u3

⎫⎬⎭ (1.14)

For two-dimensional forms we often ignore the third component.The transformation to matrix form for stresses is given in the order

σ = [σ11 σ22 σ33 σ12 σ23 σ31

]T

= [σxx σyy σzz σxy σyz σzx

]T(1.15)

and strains by

ε = [ε11 ε22 ε33 γ12 γ23 γ31

]T

= [εxx εyy εzz γxy γyz γzx

]T(1.16)

where symmetry of the tensors is assumed and ‘engineering’shear strains are introducedas

γij = 2εij , i �= j (1.17)

to make writing of subsequent matrix relations in a concise manner.The transformation to the six independent components of stress and strain is per-

formed by using the index order given in Table 1.1. This ordering will apply tomany subsequent developments also. The order is chosen to permit reduction to two-dimensional applications by merely deleting the last two entries and treating the thirdentry as appropriate for plane or axisymmetric applications.

The strain–displacement equations are expressed in matrix form as

ε = Su (1.18)

with the three-dimensional strain operator given by

ST =

⎡⎢⎢⎢⎢⎢⎣

∂

∂x10 0

∂

∂x20

∂

∂x3

0∂

∂x20

∂

∂x1

∂

∂x30

0 0∂

∂x30

∂

∂x2

∂

∂x1

⎤⎥⎥⎥⎥⎥⎦

Table 1.1 Index relation between tensor and matrix forms

Form Index value

Matrix 1 2 3 4 5 6

Tensor (1, 2, 3) 11 22 33 12 23 3121 32 13

Cartesian (x, y, z) xx yy zz xy yz zx

yx zy xz

Cylindrical (r, z, θ) rr zz θθ rz zθ θrzr θz rθ


The same operator may be used to write the equilibrium equations (1.4) as

ST σ + b = ρ u (1.19)

The boundary conditions for displacement and traction are given by

u = u on �u and t = GT σ = t on �t (1.20)

where

GT =[n1 0 0 n2 0 n3

0 n2 0 n1 n3 00 0 n3 0 n2 n1

]

in which n = (n1, n2, n3) are direction cosines of the normal to the boundary �. Wenote further that the non-zero structure of S and G are the same.

For transient problems, initial conditions are denoted by

u(x, 0) = d(x) and u(x, 0) = v(x) in � (1.21)

The constitutive equations for a linear elastic material are given in matrix form by

σ = D(ε − ε0) (1.22)

where in Eq. (1.9a) the index pairs ij and kl for Cijkl are transformed to the 6 × 6matrix D terms using Table 1.1. For a general hyperelastic material we use

σ = ∂W

∂ε(1.23)

1.2.3 Two-dimensional problems

There are several classes of two-dimensional problems which may be considered. Thesimplest are plane stress in which the plane of deformation (e.g. x1 − x2) is thin andstresses σ33 = τ13 = τ23 = 0; and plane strain in which the plane of deformation (e.g.x1 − x2) is one for which ε33 = γ13 = γ23 = 0. Another class is called axisymmetricwhere the analysis domain is a three-dimensional body of revolution defined in cylin-drical coordinates (r, θ, z) but deformations and stresses are two-dimensional functionsof r, z only.

Plane stress and plane strainFor plane stress and plane strain problems which have x1 − x2 as the plane of defor-mation, the displacements are assumed in the form

u ={

u1(x1, x2, t)

u2(x1, x2, t)

}(1.24)


and thus the strains may be defined by:11

ε =

⎧⎪⎨⎪⎩

ε11

ε22

ε33

γ12

⎫⎪⎬⎪⎭ = S u + ε3 =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

∂

∂x10

0∂

∂x20 0

∂

∂x2

∂

∂x1

⎤⎥⎥⎥⎥⎥⎥⎥⎦{

u1

u2

}+

⎧⎪⎨⎪⎩

00

ε33

0

⎫⎪⎬⎪⎭ (1.25)

Here the ε33 is either zero (plane strain) or determined from the material constitutionby assuming σ33 is zero (plane stress). The components of stress are taken in the matrixform

σT = {σ11 σ22 σ33 τ12

}(1.26)

where σ33 is determined from material constitution (plane strain) or taken as zero (planestress).

We note that the local ‘energy’ term

E = σT ε (1.27)

does not involve ε33 for either plane stress or plane strain. Indeed, it is not necessaryto compute the σ33 (or ε33) until after a problem solution is obtained.

The traction vector for plane problems is given by

t = GT σ where GT =[n1 0 0 n2

0 n2 0 n1

](1.28)

and once again we note that S and G have the same non-zero structure.

Axisymmetric problemsIn an axisymmetric problem we use the cylindrical coordinate system

x =⎧⎨⎩

x1

x2

x3

⎫⎬⎭ =

⎧⎨⎩

r

z

θ

⎫⎬⎭ (1.29)

This ordering permits the two-dimensional axisymmetric and plane problems to bewritten in a very similar manner. The body is three dimensional but defined by asurface of revolution such that properties and boundaries are independent of the θcoordinate. For this case the displacement field may be taken as

u ={

u1(x1, x2, t)

u2(x1, x2, t)

u3(x1, x2, t)

}=

{ur(r, z, t)

uz(r, z, t)

uθ(r, z, t)

}(1.30)

and, thus, also is taken as independent of θ.


The strains for the axisymmetric case are given by:11

ε =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

εrr

εzz

εθθ

γrz

γzθ

γθr

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ε11

ε22

ε33

γ12γ23γ31

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

= S u =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂

∂x10 0

0∂

∂x20

1

x10 0

∂

∂x2

∂

∂x10

0 0∂

∂x2

0 0

(∂

∂x1− 1

x1

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

{u1

u2

u3

}(1.31)

The stresses are written in the same order as

σT = {σ11 σ22 σ33 τ12 τ23 τ31

}(1.32)

Similar to the three-dimensional problem the traction is given by

t = GT σ where GT =[n1 0 0 n2 0 00 n2 0 n1 0 00 0 0 0 n2 n1

](1.33)

where we note that n3 cannot exist for a complete body of revolution. Once again wenote that S and G have the same non-zero structure.

We note that the strain–displacement relations between the u1, u2 and u3 componentsare uncoupled. If the material constitution is also uncoupled between the first four andthe last two components of strain (i.e. the first four stresses are related only to thefirst four strains) we may separate the axisymmetric problem into two parts: (a) a partwhich depends only on the first four strains which are expressed in u1, u2; and (b) aproblem which depends only on the last two shear strains and u3. The first problem issometimes referred to as torsionless and the second as a torsion problem. However,when the constitution couples the effects, as in classical elastic–plastic solution of abar which is stretched and twisted, it is necessary to consider the general case.

The torsionless axisymmetric problem is given by

ε =

⎧⎪⎨⎪⎩

εrr

εzz

εθθ

γrz

⎫⎪⎬⎪⎭ =

⎧⎪⎨⎪⎩

ε11

ε22

ε33

γ12

⎫⎪⎬⎪⎭ = S u =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂

∂x10

0∂

∂x21

x10

∂

∂x2

∂

∂x1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

{u1

u2

}(1.34)

with stresses given by Eq. (1.26) and tractions by Eq. (1.28). Thus the only differencein these two classes of problems is the presence of the u1/x1 for the third strain in


the axisymmetric case (of course the two differ also in the domain description of theproblem as we shall point out later).

1.3 Variational forms for non-linear elasticity

For an elastic material as specified by Eq. (1.23), the above equations may be givenin a variational form when no inertial effects are included. The simplest form is thepotential energy principle where

�PE =∫

�

W(Su) d� −∫

�

uT b d� −∫

�t

uT t d� (1.35)

The first variation yields the governing equation of the functional as16

δ�PE =∫

�

δ(Su)T ∂W

∂Sud� −

∫�

δuT b d� −∫

�t

δuT t d� = 0 (1.36)

After integration by parts and collecting terms we obtain

δ�PE = −∫

�

δuT(ST σ + b

)d�

+∫

�t

δuT(

GT σ − t)

d� = 0(1.37)

where

σ = ∂W

∂SuWhen W is given by the quadratic form (1.12) we recover the linear problem given byEq. (1.22). In this case the form becomes the principle of minimum potential energyand the displacement field which renders W an absolute minimum is an exact solutionto the problem.11

We note that the potential energy principle includes the strain–displacement equa-tions and the elastic model expressed in terms of displacement-based strains. It alsorequires the displacement boundary condition to be stated in addition to the theorem. Itis, however, the simplest variational form and only requires knowledge of the displace-ment field to be valid. This form is a basis for irreducible (or displacement) methodsof approximate solution.

A general variational theorem, which includes all the equations and boundary con-ditions, is given by the Hu–Washizu variational theorem.17 This theorem is given by

�HW(u, ε, σ) =∫

�

[W(ε) + σT (Su − ε)

]d�

−∫

�

uT b d� −∫

�t

uT t d� −∫

�u

tT (u − u) d�

(1.38)

in which t = GT σ. The proof that the theorem contains all the governing equations isobtained by taking the variation of Eq. (1.38) with respect to u, ε and σ. Accordingly,

Variational forms for non-linear elasticity 13

taking the variation of (1.38) and performing an integration by parts on δ(Su) we obtain

δ�HW =∫

�

δεT

[∂W

∂ε− σ

]d�

+∫

�

δσT [Su − ε] d� −∫

�u

δtT (u − u) d�

−∫

�

δuT(ST σ + b

)d� +

∫�t

δuT(t − t

)d� = 0

(1.39)

and it is evident that the Hu–Washizu variational theorem yields all the equations forthe non-linear elastostatic problem.

We may also establish a direct link between the Hu–Washizu theorem and othervariational principles. If we express the strains ε in terms of the stresses using theLaurant transformation

U(σ) + W(ε) = σT ε (1.40)

we recover the Hellinger–Reissner variational principle given by18–20

�HR(u, σ) =∫

�

[σT Su − U(σ)

]d�

−∫

�

uT b d� −∫

�t

uT t d� −∫

�u

tT (u − u) d�

(1.41)

In the linear elastic case we have, ignoring initial strain and stress effects,

U(σ) = 12 σij Sijkl σkl (1.42)

where Sijkl are elastic compliances. While this form is also formally valid for generalelastic problems. We shall find that in the non-linear case it is not possible to findunique relations for the constitutive behaviour in terms of stress forms. Thus, we shalloften rely on use of the Hu–Washizu functional as the basis for a mixed formulation.

We may also establish a direct link to the minimum potential energy form and theHu–Washizu theorem. If we satisfy the displacement boundary condition (1.20) a priorithe integral term over �u is eliminated from Eq. (1.38). Generally, in our finite elementapproximations based on the Hu–Washizu theorem (or variants of the theorem) we shallsatisfy the displacement boundary conditions explicitly and thus avoid approximatingthe �u term.

If we then satisfy the strain-displacement relations a priori then the Hu–Washizutheorem is identical with the potential energy principle. In constructing finite elementapproximations, the potential energy principle is a basis for developing displacementmodels (also referred to as irreducible models1) whereas the Hu–Washizu form is abasis for developing mixed models.1 As we will show in Chapter 2 mixed methods havedistinct advantages in constructing robust finite element formulations. However, thereare also advantages in having a finite element formulation where the global problem isexpressed in a displacement form. Noting how the Hu–Washizu form reduces to thepotential energy principle provides a link on treating the reductions to their approximatecounterparts (see Sec. 2.6).


One advantage of a variational theorem is that symmetry conditions are automaticallyobtained; however, a distinct disadvantage is that only elastic behaviour and static formsmay be considered. In the next section we consider an alternative approach of weakforms which is valid for both elastic or inelastic material forms and directly admitsthe inertial effects. We shall observe that for the elastostatic problem a weak form isequivalent to the variation of a theorem.

1.4 Weak forms of governing equations

A variational (weak) form for any set of equations is a scalar relation and may beconstructed by multiplying the equation set by an appropriate arbitrary function whichhas the same free indices as in the set of governing equations (which then becomes adummy index and sums over its range), integrating over the domain of the problem andsetting the result to zero.1,17

1.4.1 Weak form for equilibrium equation

For example, in indicial form the equilibrium equation (1.4) has the free index i, thus toconstruct a weak form we multiply by an arbitrary vector with index i and integrate theresult over the domain �. Virtual work is a weak form in which the arbitrary functionis a virtual displacement δui , accordingly using this function we obtain the form

δ�eq =∫

�

δui

[ρui − σji,j − bi

]d� = 0

Generally stress will depend on strains which are derivatives of displacements. Thus,the above form will require computation of second derivatives of displacementto form the integrands. The need to compute second derivatives may be reduced(i.e. ‘weakened’) by performing an integration by parts and upon noting the symmetryof the stress we obtain

δ�eq =∫

�

δui ρ ui d� +∫

�

δεij (uk) σij d � −∫

�

δui bi d� −∫

�

δui ti d� = 0

(1.43)

where virtual strains are related to virtual displacements as

δεij (uk) = 12 (δui,j + δuj,i) (1.44)

This may be further simplified by splitting the boundary into parts where traction isspecified, �t , and parts where displacements are specified, �u. If we enforce pointwiseall the displacement boundary conditions∗ and impose a constraint that δui vanisheson �u, we obtain the final result

δ�eq =∫

�

δui ρ ui d� +∫

�

δεij (uk) σij d� −∫

�

δui bi d� −∫

�t

δui ti d� = 0

(1.45)∗ Alternatively, we can combine this term with another from the integration by parts of the weak form of thestrain–displacement equations.

References 15

or in matrix form as

δ�eq =∫

�

δuT ρ u d� +∫

�

δ(Su)T σ d� −∫

�

δuT b d� −∫

�t

δuT t d� = 0

(1.46)

The first term is the virtual work of internal inertial forces, the second the virtual workof the internal stresses and the last two the virtual work of body and traction forces,respectively.

The above weak form provides the basis from which a finite element formulationof equilibrium may be deduced for general applications. It is necessary to add appro-priate expressions for the strain–displacement and constitutive equations to completea problem formulation. Weak forms for these may be written immediately from thevariation of the Hu–Washizu principle given in Eq. (1.39).

We note that the form adopted to define the matrices of stress and strain permits theinternal work of stress and strain to be written as

εij σij = εT σ = σT ε (1.47)

Similarly, the internal virtual work per unit volume may be expressed by

δW = δεij σij = δεT σ (1.48)

In Chapter 4 we will discuss this in more detail and show that constructing constitutiveequations in terms of six components of stress and strain must be treated appropriatelyin reductions from the original nine tensor components.

1.5 Concluding remarks

In this chapter we have summarized the basic steps needed to formulate a generalsmall-strain solid mechanics problem. The formulation has been presented in a strongform in terms of partial differential equations and in a weak form in terms of integralexpressions. We have also indicated how the general problem can become non-linear.In the next chapter we describe the use of the finite element method to constructapproximate solutions to weak forms for non-linear transient solid mechanics problems.

References

1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Funda-mentals. Butterworth-Heinemann, Oxford, 6th edition, 2005.

2. T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Dover Publi-cations, New York, 2000.

3. R.D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt. Concepts and Applications of FiniteElement Analysis. John Wiley & Sons, New York, 4th edition, 2001.

4. F. de S. Lynch. A finite element method of viscoelastic stress analysis with applicationto rolling contact problems. International Journal for Numerical Methods in Engineering,1:379–394, 1969.


5. J.T. Oden and T.L. Lin. On the general rolling contact problem for finite deformations of aviscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 57:297–367,1986.

6. P. le Tallec and C. Rahier. Numerical models of steady rolling for non-linear viscoelasticstructures in finite deformation. International Journal for Numerical Methods in Engineering,37:1159–1186, 1994.

7. S. Govindjee and P.A. Mahalic. Viscoelastic constitutive relations for the steady spinning ofa cylinder. Technical Report UCB/SEMM Report 98/02, University of California at Berkeley,1998.

8. H. Cohen and R.G. Muncaster. The Theory of Pseudo-rigid Bodies. Springer, New York, 1988.9. S.P. Timoshenko and J.M. Gere. Theory of Elastic Stability. McGraw-Hill, New York, 1961.

10. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition,1969.

11. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition,1956.

12. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, EnglewoodCliffs, NJ, 1969.

13. A.P. Boresi and K.P. Chong. Elasticity in Engineering Mechanics. Elsevier, New York, 1987.14. P.C. Chou and N.J. Pagano. Elasticity: Tensor, Dyadic and Engineering Approaches. Dover

Publications, Mineola, NY, 1992. Reprinted from 1967 Van Nostrand edition.15. I.H. Shames and F.A. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis,

Washington, DC, 1997. (Revised printing.)16. F.B. Hildebrand. Methods of Applied Mathematics. Prentice-Hall (reprinted by Dover Publishers,

1992), 2nd edition, 1965.17. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd

edition, 1982.18. E. Hellinger. Die allgemeine Aussetze der Mechanik der Kontinua. In F. Klein and C. Muller,

editors, Encyclopedia der Mathematishen Wissnschaften, volume 4. Tebner, Leipzig, 1914.19. E. Reissner. On a variational theorem in elasticity. Journal of Mathematics and Physics, 29(2):

90–95, 1950.20. E. Reissner. A note on variational theorems in elasticity. International Journal of Solids and

Structures, 1:93–95, 1965.

general problems in solid mechanics and non-linearity

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