PLASTICITY THEORY pdf

Section 1.5 Constitutive Relations: Inelastic 591.5.3 Internal Variables: General Theory 65 Chapter 2: The Physics of Plasticity 2.1.1 Experimental Stress-Strain Relations 76 2.1.3 Tempe

When I first began to plan this book, I thought that I would begin the prefacewith the words “The purpose of this little book is ” While I never lost mybelief that small is beautiful, I discovered that it is impossible to put together

a treatment of a field as vast as plasticity theory between the covers of atruly “little” book and still hope that it will be reasonably comprehensive

I have long felt that a modern book on the subject — one that would beuseful as a primary reference and, more importantly, as a textbook in a grad-uate course (such as the one that my colleague Jim Kelly and I have beenteaching) — should incorporate modern treatments of constitutive theory(including thermodynamics and internal variables), large-deformation plas-ticity, and dynamic plasticity By no coincidence, it is precisely these topics

— rather than the traditional study of elastic-plastic boundary-value lems, slip-line theory and limit analysis — that have been the subject of

prob-my own research in plasticity theory I also feel that a basic treatment ofplasticity theory should contain at least introductions to the physical foun-dations of plasticity (and not only that of metals) and to numerical methods

— subjects in which I am not an expert

I found it quite frustrating that no book in print came even close toadequately covering all these topics Out of necessity, I began to prepareclass notes to supplement readings from various available sources Withthe aid of contemporary word-processing technology, the class notes came

to resemble book chapters, prompting some students and colleagues to ask,

“Why don’t you write a book?” It was these queries that gave me theidea of composing a “little” book that would discuss both the topics thatare omitted from most extant books and, for the sake of completeness, theconventional topics as well

Almost two years have passed, and some 1.2 megabytes of disk space havebeen filled, resulting in over 400 pages of print Naively perhaps, I still hopethat the reader approaches this overgrown volume as though it were a littlebook: it must not be expected, despite my efforts to make it comprehensive,

to be exhaustive, especially in the sections dealing with applications; I havepreferred to discuss just enough problems to highlight various facets of anytopic Some oft-treated topics, such as rotating disks, are not touched at

iii

all, nor are such general areas of current interest as micromechanics (except

on the elementary, qualitative level of dislocation theory), damage ics (except for a presentation of the general framework of internal-variablemodeling), or fracture mechanics I had to stop somewhere, didn’t I?The book is organized in eight chapters, covering major subject areas;the chapters are divided into sections, and the sections into topical subsec-tions Almost every section is followed by a number of exercises The order

mechan-of presentation mechan-of the areas is somewhat arbitrary It is based on the order inwhich I have chosen to teach the field, and may easily be criticized by thosepartial to a different order It may seem awkward, for example, that consti-tutive theory, both elastic and inelastic, is introduced in Chapter 1 (which

is a general introduction to continuum thermomechanics), interrupted for asurvey of the physics of plasticity as given in Chapter 2, and returned to withspecific attention to viscoplasticity and (finally!) rate-independent plasticity

in Chapter 3; this chapter contains the theory of yield criteria, flow rules,and hardening rules, as well as uniqueness theorems, extremum and varia-tional principles, and limit-analysis and shakedown theorems I believe thatthe book’s structure and style are sufficiently loose to permit some juggling

of the material; to continue the example, the material of Chapter 2 may betaken up at some other point, if at all

The book may also be criticized for devoting too many pages to cepts of physics and constitutive theory that are far more general than theconventional constitutive models that are actually used in the chapters pre-senting applications My defense against such criticisms is this: I believethat the physics of plasticity and constitutive modeling are in themselveshighly interesting topics on which a great deal of contemporary research isdone, and which deserve to be introduced for their own sake even if theirapplicability to the solution of problems (except by means of high-powerednumerical methods) is limited by their complexity

con-Another criticism that may, with some justification, be leveled is thatthe general formulation of continuum mechanics, valid for large as well assmall deformations and rotations, is presented as a separate topic in Chapter

8, at the end of the book rather than at the beginning It would indeed

be more elegant to begin with the most general presentation and then tospecialize The choice I finally made was motivated by two factors One isthat most of the theory and applications that form the bulk of the book can

be expressed quite adequately within the small-deformation framework Theother factor is pedagogical: it appears to me, on the basis of long experience,that most students feel overwhelmed if the new concepts appearing in large-deformation continuum mechanics were thrown at them too soon

Much of the material of Chapter 1 — including the mathematical damentals, in particular tensor algebra and analysis — would normally becovered in a basic course in continuum mechanics at the advanced under-

fun-graduate or first-year fun-graduate level of a North American university I haveincluded it in order to make the book more or less self-contained, and while

I might have relegated this material to an appendix (as many authors havedone), I chose to put it at the beginning, if only in order to establish a con-sistent set of notations at the outset For more sophisticated students, thismaterial may serve the purpose of review, and they may well study Section8.1 along with Sections 1.2 and 1.3, and Section 8.2 along with Sections 1.4and 1.5

The core of the book, consisting of Chapters 4, 5, and 6, is devoted toclassical quasi-static problems of rate-independent plasticity theory Chapter

4 contains a selection of problems in contained plastic deformation (or plastic problems) for which analytical solutions have been found: some ele-mentary problems, and those of torsion, the thick-walled sphere and cylinder,and bending The last section, 4.5, is an introduction to numerical methods(although the underlying concepts of discretization are already introduced

elastic-in Chapter 1) For the sake of completeness, numerical methods for bothviscoplastic and (rate-independent) plastic solids are discussed, since nu-merical schemes based on viscoplasticity have been found effective in solvingelastic-plastic problems Those who are already familiar with the material

of Sections 8.1 and 8.2 may study Section 8.3, which deals with numericalmethods in large-deformation plasticity, immediately following Section 4.5.Chapters 5 and 6 deal with problems in plastic flow and collapse Chap-ter 5 contains some theory and some “exact” solutions: Section 5.1 coversthe general theory of plane plastic flow and some of its applications, andSection 5.2 the general theory of plates and the collapse of axisymmetricallyloaded circular plates Section 5.3 deals with plastic buckling; its placement

in this chapter may well be considered arbitrary, but it seems appropriate,since buckling may be regarded as another form of collapse Chapter 6 con-tains applications of limit analysis to plane problems (including those of soilmechanics), beams and framed structures, and plates and shells

Chapter 7 is an introduction to dynamic plasticity It deals both withproblems in the dynamic loading of elastic–perfectly plastic structures treated

by an extension of limit analysis, and with wave-propagation problems, dimensional (with the significance of rate dependence explicitly discussed)and three-dimensional The content of Chapter 8 has already been men-tioned

one-As the knowledgeable reader may see from the foregoing survey, a ent course may be built in various ways by putting together selected portions

coher-of the book Any recommendation on my part would only betray my ownprejudices, and therefore I will refrain from making one My hope is thatthose whose orientation and interests are different from mine will nonethelessfind this would-be “little book” useful

In shaping the book I was greatly helped by comments from some

out-standing mechanicians who took the trouble to read the book in draft form,and to whom I owe a debt of thanks: Lallit Anand (M I T.), Satya Atluri(Georgia Tech), Maciej Bieniek (Columbia), Michael Ortiz (Brown), andGerald Wempner (Georgia Tech).

An immeasurable amount of help, as well as most of the inspiration towrite the book, came from my students, current and past There are toomany to cite by name — may they forgive me — but I cannot leave out Vas-silis Panoskaltsis, who was especially helpful in the writing of the sections

on numerical methods (including some sample computations) and who gested useful improvements throughout the book, even the correct spelling

sug-of the classical Greek verb from which the word “plasticity” is derived.Finally, I wish to acknowledge Barbara Zeiders, whose thoroughly pro-fessional copy editing helped unify the book’s style, and Rachel Lernerand Harry Sices, whose meticulous proofreading found some needles in thehaystack that might have stung the unwary Needless to say, the ultimateresponsibility for any remaining lapses is no one’s but mine

A note on cross-referencing: any reference to a number such as 3.2.1,without parentheses, is to a subsection; with parentheses, such as (4.3.4), it

is to an equation

Addendum: Revised Edition

Despite the proofreaders’ efforts and mine, the printed edition remainedplagued with numerous errors In the fifteen years that have passed I havemanaged to find lots of them, perhaps most if not all I have also found itnecessary to redo all the figures The result is this revised edition

Chapter 1: Introduction to Continuum chanics

vii

Section 1.5 Constitutive Relations: Inelastic 59

1.5.3 Internal Variables: General Theory 65

Chapter 2: The Physics of Plasticity

2.1.1 Experimental Stress-Strain Relations 76

2.1.3 Temperature and Rate Dependence 85

2.2.2 Dislocations and Crystal Plasticity 942.2.3 Dislocation Models of Plastic Phenomena 100

2.3.2 “Plasticity” of Rock and Concrete 108

Chapter 3: Constitutive Theory

3.1.1 Internal-Variable Theory of Viscoplasticity 1113.1.2 Transition to Rate-Independent Plasticity 1163.1.3 Viscoplasticity Without a Yield Surface 118

3.2.1 Flow Rule and Work-Hardening 1223.2.2 Maximum-Dissipation Postulate and Normality 127

3.3.2 Yield Criteria Independent of the Mean Stress 1373.3.3 Yield Criteria Dependent on the Mean Stress 1413.3.4 Yield Criteria Under Special States of Stress or Deformation 144

3.5.1 Standard Limit-Analysis Theorems 1623.5.2 Nonstandard Limit-Analysis Theorems 168

Chapter 4: Problems in Contained Plastic

Defor-mation

4.1.1 Introduction: Statically Determinate Problems 1774.1.2 Thin-Walled Circular Tube in Torsion and Extension 1784.1.3 Thin-Walled Cylinder Under Pressure and Axial Force 1814.1.4 Statically Indeterminate Problems 184

4.3.1 Elastic Hollow Sphere Under Internal and External Pressure 2064.3.2 Elastic–Plastic Hollow Sphere Under Internal Pressure 2084.3.3 Thermal Stresses in an Elastic–Plastic Hollow Sphere 2134.3.4 Hollow Cylinder: Elastic Solution and Initial Yield Pressure 2164.3.5 Elastic–Plastic Hollow Cylinder 220

Section 4.4 Elastic–Plastic Bending 229

4.4.1 Pure Bending of Prismatic Beams 2294.4.2 Rectangular Beams Under Transverse Loads 2394.4.3 Plane-Strain Pure Bending of Wide Beams or Plates 245

4.5.1 Integration of Rate Equations 251

4.5.3 Finite-Element Methods for Nonlinear Continua 262

Chapter 5: Problems in Plastic Flow and Collapse

I: Theories and “Exact” Solutions

5.2.1 Introduction to Plate Theory 299

5.3.1 Introduction to Stability Theory 3145.3.2 Theories of the Effective Modulus 3195.3.3 Plastic Buckling of Plates and Shells 326

Chapter 6: Problems in Plastic Flow and Collapse

II: Applications of Limit Analysis

Section 6.1 Limit Analysis of Plane Problems 338

6.1.1 Blocks and Slabs with Grooves or Cutouts 338

6.2.3 Combined Extension, Bending and Torsion 364

6.4.2 Limit Analysis of Shells: Theory 4046.4.3 Limit Analysis of Shells: Examples 407

Chapter 7: Dynamic Problems

7.1.3 Dynamic Loading of Plates and Shells 425

7.2.1 Theory of One-Dimensional Waves 4347.2.2 Waves in Elastic–Plastic bars 438

7.2.4 Application of the Method of Characteristics 448

Section 7.3 Three-Dimensional Waves 452

7.3.1 Theory of Acceleration Waves 453

Chapter 8: Large-Deformation Plasticity

8.1.2 Continuum Mechanics and Objectivity 473

8.2.3 Inelasticity: Thermomechanics 4858.2.4 Yield Condition and Flow Rule 487

Section 8.3 Numerical Methods in Large-Deformation

Introduction to Continuum Thermomechanics

Solid mechanics, which includes the theories of elasticity and plasticity, is

a broad discipline, with experimental, theoretical, and computational pects, and with a twofold aim: on the one hand, it seeks to describe themechanical behavior of solids under conditions as general as possible, re-gardless of shape, interaction with other bodies, field of application, or thelike; on the other hand, it attempts to provide solutions to specific problemsinvolving stressed solid bodies that arise in civil and mechanical engineering,geophysics, physiology, and other applied disciplines These aims are not inconflict, but complementary: some important results in the general theoryhave been obtained in the course of solving specific problems, and practicalsolution methods have resulted from fundamental theoretical work Thereare, however, differences in approach between workers who focus on one orthe other of the two goals, and one of the most readily apparent differences

as-is in the notation used

Most of the physical concepts used in solid mechanics are modeled bymathematical entities known as tensors Tensors have representationsthrough components with respect to specific frames or coordinate systems(a vector is a kind of tensor), but a great deal can be said about them with-out reference to any particular frame Workers who are chiefly interested inthe solution of specific problems — including, notably, engineers — generallyuse a system of notation in which the various components of tensors appearexplicitly This system, which will here be called “engineering” notation,has as one of its advantages familiarity, since it is the one that is gener-

1

ally used in undergraduate “strength of materials” courses, but it is oftencumbersome, requiring several lines of long equations where other notationspermit one short line, and it sometimes obscures the mathematical nature ofthe objects and processes involved Workers in constitutive theory tend touse either one of several systems of “direct” notation that in general use noindices (subscripts and superscripts), such as Gibbs’ dyadic notation, matrixnotation, and a combination of the two, or the so-called indicial notation inwhich the use of indices is basic The indices are used to label components

of tensors, but with respect to an arbitrary rather than a specific frame.Indicial notation is the principal system used in this book, althoughother systems are used occasionally as seems appropriate In particular,

“engineering” notation is used when the solutions to certain specific problemsare discussed, and the matrix-based direct notation is used in connectionwith the study of large deformation, in which matrix multiplication plays animportant part

Assuming the reader to be familiar with vectors as commonly taught inundergraduate engineering schools, we introduce indicial notation as follows:for Cartesian coordinates (x, y, z) we write (x1, x2, x3); for unit vectors(i, j, k) we write (e1, e2, e3); for the components (ux, uy, uz) of a vector u

There is a relation between the “e” tensor and the Kronecker delta known

as the e-delta identity:

eijkelmk= δilδjm− δimδjl.The fundamental operations of three-dimensional vector algebra, pre-sented in indicial and, where appropriate, in direct notation, are as follows

Decomposition: u = uiei (The position vector in x1x2x3-space is denoted

The notation for matrices is as follows A matrix with entries αij, where

i is the row index and j is the column index, is denoted [αij] or α Thetranspose of α is the matrix [αji], also denoted αT The determinant of α isdenoted det α, and the inverse of α is α−1, so that α α−1= α−1α = I, where

I = [δij] is the unit matrix or identity matrix

x∗i-axis and the xj-axis, then

e∗i · ej = βij.According to this equation, βij is both the xj-component of e∗i and the

x∗i-component of ej, so that

e∗i = βijejand

ei= βjie∗j.For any vector u = uiei= u∗ie∗i,

u∗i = βikuk, ui = βjiu∗j

If the free index i in the second equation is replaced by k and its right-handside is substituted for uk in the first equation, then

u∗i = βikβjku∗j.Similarly,

ui = βkiβkjuj.Since u∗i = δiju∗j and ui = δijuj, and since the vector u is arbitrary, it followsthat

βikβjk = βkiβkj = δij,that is, the matrix β = [βij] is orthogonal In matrix notation, β βT = βTβ =

I The determinant of a matrix equals the determinant of its transpose,that is, det α = det αT, and the determinant of a product of matrices equalsthe product of the determinants, so that det(αβ) = det α det β For anorthogonal matrix β, therefore, (det β)2 = det I = 1, or det β = ±1 If thebasis (e∗i) is obtained from (ei) by a pure rotation, then β is called properorthogonal , and det β = 1

An example of a proper orthogonal matrix is the matrix describing terclockwise rotation by an angle θ about the x3-axis, as shown in Figure1.1.1

The preceding definitions are independent of any decomposition of thevectors involved If the vectors are to be represented with respect to a basis(ei), then the linear operator must also be so represented The Cartesiancomponents of a linear operator are defined as follows: If v = λ(u), then itfollows from the definition of linearity that

vi= ei· λ(ejuj) = ei· λ(ej)uj = λijuj,where λijujdef= ei· λ(ej) Thus λ = [λij] is the component matrix of λ withrespect to the basis (ei) In matrix notation we may write v = λ u, where

u (v) is the column matrix whose entries are the components ui (vi) of thevector u (v) with respect to the basis (ei) In direct tensor notations it

is also customary to omit parentheses: in Gibbs’ notation we would write

v = λ · u, and in the matrix-based direct notation, v = λu

In a different basis (e∗i), where e∗i = βijej, the component matrix of λ isdefined by

If the component matrix of a linear operator has a property which isnot changed by transformation to a different basis, that is, if the property

is shared by λ and λ∗ (for any β), then the property is called invariant Aninvariant property may be said to be a property of the linear operator λitself rather than of its component matrix in a particular basis An example

is transposition: if λ∗ = β λ βT, then λ∗T = β λTβT Consequently we mayspeak of the transpose λT of the linear operator λ, and we may define itssymmetric and antisymmetric parts:

If λA= 0 (i.e., λij = λji), then λ is a symmetric operator If λS = 0 (i.e.,

λij = −λji), then λ is an antisymmetric operator

Tensors

A linear operator, as just defined, is also called a tensor More generally,

a tensor of rank n is a quantity T represented in a basis (ei) by a componentarray Ti1 in (i1, , in= 1, 2, 3) and in another basis (e∗i) by the componentarray Ti∗1 in, where

Ti∗ i = βi k βi k Tk k

Thus a scalar quantity is a tensor of rank 0, a vector is a tensor of rank 1,and a linear operator is a tensor of rank 2 “Tensor” with rank unspecified

is often used to mean a tensor or rank 2 The tensor whose component array

is eijk is an isotropic tensor of rank 3 Tensors of rank 4 are found first inSection 1.4

An array whose elements are products of tensor components of rank

m and n, respectively, represents a tensor of rank m + n An importantexample is furnished by the tensor product of two vectors u and v (a dyad inthe terminology of Gibbs), the tensor of rank 2 represented by u vT = [uivj]and denoted u ⊗ v, or more simply uv in the Gibbs notation Thus anarbitrary tensor λ of rank two, whose components with respect to a basis(ei) are λij, satisfies the equation

λ = λijei⊗ ej.Clearly, (u⊗v)w = u(v·w); in the Gibbs notation both sides of this equationmay be written as uv · w

An operation known as a contraction may be performed on a tensor ofrank n ≥ 2 It consists of setting any two indices in its component arrayequal to each other (with summation implied) The resulting array, indexed

by the remaining indices, if any (the “free indices”), represents a tensor ofrank n − 2 For a tensor λ of rank 2, λii= tr λ is a scalar known as the trace

of λ A standard example is u · v = uivi = tr (u ⊗ v) Note that if n > 2then more than one contraction of the same tensor is possible, resulting

in different contracted tensors; and, if n ≥ 4, then we can have multiplecontractions For example, if n = 4 then we can have a double contractionresulting in a scalar, and three different scalars are possible: Tiijj, Tijij, and

Tijji

If u and v are vectors that are related by the equation

ui = αijvj,then α necessarily represents a tensor α of rank 2 Similarly, if α and β aretensors of rank 2 related by

αij = ρijklβkl,then the array ρijkl represents a tensor ρ of rank four The generalization

of these results is known as the quotient rule

A tensor field of rank n is a function (usually assumed continuously entiable) whose values are tensors of rank n and whose domain is a region

differ-R in x1x2x3 space The boundary of R is a closed surface denoted ∂R, and

the unit outward normal vector on ∂R will be denoted n The partial ative operator ∂/∂xi will be written more simply as ∂i A very commonalternative notation, which is used extensively here, is ∂iφ = φ,i.

deriv-If φ is a tensor field of rank n, then the array of the partial derivatives

of its components represents a tensor field of rank n + 1

The del operator is defined as ∇ = ei∂i, and the Laplacian operator as

∇2 = ∂i∂i =P

i∂2/∂x2i For a scalar field φ, the gradient of φ is the vectorfield defined by

∇φ = gradφ = eiφ,i.For a vector field v, we use ∇v to denote (∇ ⊗ v)T, that is,

∇v = vi,jei⊗ ej,but this notation is not universal: many writers would call this (∇v)T There

is no ambiguity, however, when only the symmetric part of ∇v is used, orwhen the divergence of v is defined as the trace of ∇v:

div v = ∇ · v = vi,i.Similarly, the curl of v is defined unambiguously as

curl v = ∇ × v = eieijkvk,j.For a tensor field φ of rank 2, we define ∇φ as represented by φjk,i, and

∇ · φ = div φ = φjk,jek.These definitions are, again, not universal

The three-dimensional equivalent of the fundamental theorem of calculus

as the divergence theorem This is the case we use most often

The two-dimensional Gauss’s theorem refers to fields defined in an area

A in the x1x2-plane, bounded by a closed curve C on which an infinitesimalelement of arc length is ds (positive when it is oriented counterclockwise)

It is conventional to use Greek letters for indices whose range is 1, 2; thusthe theorem reads

Figure 1.1.2 Normal vector to a plane curve

Now suppose that the curve C is described parametrically by x1 = x1(s),

x2 = x2(s) Then, as can be seen from Figure 1.1.2,

n1 = dx2

ds , n2 = −

dx1

ds .Thus, for any two functions uα(x1, x2) (α = 1, 2),

no holes); otherwise additional conditions are required

The preceding result is known as the two-dimensional integrabilitytheorem and will be used repeatedly

There exists an extension of Green’s lemma to a curved surface S in

x1x2x3 space, bounded by a (not necessarily plane) closed curve C metrized by xi = xi(s), i = 1, 2, 3 This extension (derived from Gauss’s

para-theorem) is known as Stokes’ theorem and takes the form

n = e3 From Stokes’ theorem follows the three-dimensional integrabilitytheorem: a field φ(x) such that u = ∇φ in a region R exists only if ∇×u = 0,

or, equivalently, uj,i= ui,j As in the two-dimensional version, this lastcondition is also sufficient if R is simply connected

The study of tensor fields in curvilinear coordinates is intimately tied todifferential geometry, and in many books and courses of study dealing withcontinuum mechanics it is undertaken at the outset The traditional method-ology is as follows: with a set of curvilinear coordinates ξi (i = 1, 2, 3)such that the position of a point in three-dimensional space is defined byx(ξ1, ξ2, ξ3), the natural basis is defined as the ordered triple of vectors

gi = ∂x/∂ξi, so that dx = gidξi; the summation convention here applieswhenever the pair of repeated indices consists of one subscript and one su-perscript The basis vectors are not, in general, unit vectors, nor are theynecessarily mutually perpendicular, although it is usual for them to havethe latter property One can find, however, the dual basis (gi) such that

gi · gj = δji A vector v may be represented as vigi or as vigi, where

vi = v · gi and vi = v · gi are respectively the covariant and contravariantcomponents of v For tensors of higher rank one can similarly define covari-ant, contravariant, and several kinds of mixed components The gradient of

a tensor field is defined in terms of the so-called covariant derivatives of itscomponents, which, except in the case of a scalar field, differ from the partialderivatives with respect to the ξi because the basis vectors themselves vary

A central role is played by the metric tensor with components gij = gi· gj,having the property that dx · dx = gijdξidξj

An alternative approach is based on the theory of differentiable manifolds(see, e.g., Marsden and Hughes [1983])

Curvilinear tensor analysis is especially useful for studying the mechanics

of curved surfaces, such as shells; when this topic does not play an tant part, a simpler approach is available, based on the so-called “physical”components of the tensors involved In this approach mutually perpendic-ular unit vectors (forming an orthonormal basis) are used, rather than thenatural and dual bases We conclude this section by examining cylindricaland spherical coordinates in the light of this methodology

impor-Cylindrical Coordinates

In the cylindrical coordinates (r, θ, z), where r =qx21+ x22, θ = tan−1(x2/x1),

and z = x3, the unit vectors are

er = e1cos θ + e2sin θ, eθ = −e1sin θ + e2cos θ, ez = e3,

be given by

∇ = er ∂

∂r + eθ

1r

∇2u = ∂

2

∂r2 +1r

Spherical Coordinates

The spherical coordinates (r, θ, φ) are defined by r = qx21+ x22+ x23,

θ = tan−1(x2/x1), and φ = cot−1

The unit vectors are

er= (e1cos θ+e2sin θ) sin φ+e3cos φ, eφ= (e1cos θ+e2sin θ) cos φ−e3sin φ,

eθ = −e1sin θ + e2cos θ,

∂vr

∂φ −vφr



+ eφ

1r

∇ · λ = er

∂λrr

∂r +

1r

(a) aii,

(b) aijaij,

(c) aijajk when i = 1, k = 1 and when i = 1, k = 2

4 Show that the matrix

453

5 −

4

5 016

25

1225

35

is proper orthogonal, that is, β βT = βTβ = I, and det β = 1

5 Find the rotation matrix β describing the transformation composed

of, first, a 90◦ rotation about the x1-axis, and second, a 45◦ rotationabout the rotated x3-axis

6 Two Cartesian bases, (ei), and (e∗i) are given, with e∗1 = (2e1+ 2e2+

e3)/3 and e∗2 = (e1− e2)/√2

(a) Express e∗3 in terms of the ei

(b) Express the ei in terms of the e∗i

(c) If v = 6e1− 6e2+ 12e3, find the vi∗.

7 The following table shows the angles between the original axes xi andthe transformed axes x∗i

x1 x2 x3

x∗1 135◦ 60◦ 120◦

x∗2 90◦ 45◦ 45◦

x∗3 45◦ 60◦ 120◦(a) Find the transformation matrix β, and verify that it describes arotation

(b) If a second-rank tensor λ has the following component matrixwith the respect to the original axes,

8 (a) Use the chain rule of calculus to prove that if φ is a scalar field,

then ∇φ is a vector field

(b) Use the quotient rule to prove the same result

9 Using indicial notation, prove that (a) ∇×∇φ = 0 and (b) ∇·∇×v = 0

10 If x = xiei and r = |x|, prove that

16 Starting with the expression in Cartesian coordinates for the ent operator ∇ and using the chain rule for partial derivatives, deriveEquation (1.1.5).

The first application of the mathematical concepts introduced in Section1.1 will now be to the description of the deformation of bodies that can bemodeled as continua A body is said to be modeled as a continuum if to anyconfiguration of the body there corresponds a region R in three-dimensionalspace such that every point of the region is occupied by a particle (materialpoint) of the body

Any one configuration may be taken as the reference configuration sider a particle that in this configuration occupies the point defined by thevector r = xiei When the body is displaced, the same particle will occupythe point r∗ = x∗iei (Note that here the x∗i no longer mean the coordinates

Con-of the same point with respect to a rotated basis, as in the Section 1.1, butthe coordinates of a different point with respect to the same basis.) The dif-ference r∗− r is called the displacement of the particle and will be denoted

u The reference position vector r will be used to label the given particle;the coordinates xi are then called Lagrangian coordinates Consequently thedisplacement may be given as a function of r, u(r), and it forms a vector fielddefined in the region occupied by the body in the reference configuration.Now consider a neighboring particle labeled by r + ∆r In the displacedconfiguration, the position of this point will be

r∗+ ∆r∗ = r + ∆r + u(r + ∆r)(see Figure 1.2.1), so that

C P

P P P P

j

r∗r

r + ∆r

6

A A A A

r∗+ ∆r∗

∆r∗

∆ru(r + ∆r)

of the particle labeled by r; the deformation of the neighborhood may bemeasured by the extent to which the lengths of the infinitesimal vectors dremanating from r change in the course of the displacement The square ofthe length of dr∗ is

sor 1 Clearly, E(r) describes the deformation of the infinitesimal hood of r, and the tensor field E that of the whole body; E(r) = 0 for all r

neighbor-in R if and only if the displacement is a rigid-body one

The deformation of a region R is called homogeneous if E is constant It

is obvious that a necessary and sufficient condition for the deformation to

be homogeneous is that the ui,j are constant, or equivalently, that u varieslinearly with r,

Infinitesimal Strain and Rotation

We further define the tensor ε and ω, respectively symmetric and symmetric, by

θi = 1

2eijkωkj, ωik = eijkθj.Since, moreover, eijkεjk = 0 because of the symmetry of ε, the first relationimplies that

defor-In a rigid-body displacement, then, dui= eijkθjdxk, or du = θ ×dr That is,

θ is the infinitesimal rotation vector : its magnitude is the angle of rotationand its direction gives the axis of rotation Note that ∇ · θ = 0; a vectorfield with this property is called solenoidal

It must be remembered that a finite rotation is described by an onal, not an antisymmetric, matrix Since the orthogonality conditions aresix in number, such a matrix is likewise determined by only three indepen-dent numbers, but it is not equivalent to a vector, since the relations amongthe matrix elements are not linear

orthog-Significance of Infinitesimal Strain Components

The study of finite deformation is postponed until Chapter 8 For now,let us explore the meaning of the components of the infinitesimal straintensor ε Consider, first, the unit vector n such that dr = ndr, where

dr = |dr|; we find that

dr∗2= dr∗· dr∗ = (1 + 2Eijninj)dr2= (1 + 2ε. ijninj) r2

But for α small, √1 + 2α= 1 + α, so that dr. ∗= (1 + ε. ijninj) r Hence

dr∗− drdr

= εijninj

is the longitudinal strain along the direction n (note that the left-hand side

is just the “engineering” definition of strain)

Next, consider two infinitesimal vectors, dr(1) = e1dr(1) and dr(2) =

e2dr(2) In indicial notation, we have dx(1)i = δi1dr(1) and dx(2)i = δi2dr(2).Obviously, dr(1)· dr(2) = 0 The displacement changes dr(1) to dr(1)∗ and

2ε12= (1 + ε11)(1 + ε22)γ12

= γ12

for infinitesimal strains

x 1

x 2

6

Figure 1.2.2 Shear angle

Since the labeling of the axes is arbitrary, we may say in general that,for i 6= j, εij = 1

2γij Both the εij and γij, for i 6= j, are referred to as shearstrains; more specifically, the former are the tensorial and the latter are theconventional shear strains A state of strain that can, with respect to someaxes, be represented by the matrix

ap-if both ε and ω (or θ) are small compared to unity If the rotation is finite,then the strain must be described by E (or by some equivalent finite defor-mation tensor, discussed further in Chapter 8) even if the deformation per

se is infinitesimal

As an illustration, we consider a homogeneous deformation in which the

x1x3-plane is rotated counterclockwise about the x3-axis by a finite angle θ,while the the x2x3-plane is rotated counterclockwise about the x3-axis by theslightly different angle θ − γ, with |γ|  1 Since all planes perpendicular tothe x3 axis deform in the same way, it is sufficient to study the deformation

of the x1x2-plane, as shown in Figure 1.2.3 It is clear that, with respect

to axes rotated by the angle θ, the deformation is just one of simple shear,

C C C C C C C C C C

M

?

9

-θ

θ−γ

C C

Figure 1.2.3 Infinitesimal shear strain with finite rotation

and can be described by the infinitesimal strain matrix given above Withrespect to the reference axes, we determine first the displacement of pointsoriginally on the x1-axis,

u1(x1, 0, x3) = −(1 − cos θ)x1, u2(x1, 0, x3) = sin θx1,

and that of points originally on the x2-axis,

u1(0, x2, x3) = − sin(θ − γ)x2, u2(0, x2, x3) = −[1 − cos(θ − γ)]x2.The latter can be linearized with respect to γ:

u1(0, x2, x3) = −(sin θ − γ cos θ)x2, u2(0, x2, x3) = −(1 − cos θ − γ sin θ)x2.Since the deformation is homogeneous, the displacement must be linear in

x1 and x2 and therefore can be obtained by superposition:

u1(x1, x2, x3) = −(1 − cos θ)x1− (sin θ − γ cos θ)x2,

u2(x1, x2, x3) = sin θx1− (1 − cos θ − γ sin θ)x2

Knowing that u3= 0, we can now determine the Green–Saint-Venant straintensor, and find, with terms of order γ2 neglected,

is that E measures strain with respect to axes that are, in effect, fixed in thebody.

Further discussion of the description of finite deformation is postponeduntil Chapter 8

Alternative Notations and Coordinate Systems

The “engineering” notations for the Cartesian strain components are εx,

εy, and εz for ε11, ε22, and ε33, respectively, and γxy for γ12, and so on

In cylindrical coordinates we find the strain components by taking thesymmetric part of ∇ ⊗ u as given by Equation (1.1.6):

∂uz

∂θ +

∂uθ

∂z .(1.2.1)

In spherical coordinates we similarly find, from Equation (1.1.8),

εr = ∂ur

∂r , εφ=

1r

Volumetric and Deviatoric Strain

The trace of the strain tensor, εkk= ∇ · u, has a special geometric icance: it is the (infinitesimal) volumetric strain, defined as ∆V /V0, where

signif-∆V is the volume change and V0 the initial volume of a small neighborhood

An easy way to show this is to look at a unit cube (V0 = 1) whose edgesparallel the coordinate axes When the cube is infinitesimally deformed, thelengths of the edges change to 1 + ε11, 1 + ε22, and 1 + ε33, respectively,making the volume (1 + ε11)(1 + ε22)(1 + ε33) = 1 + ε. kk The volumetricstrain is also known as the dilatation

The total strain tensor may now be decomposed as

εij = 1εkkδij + eij.The deviatoric strain or strain deviator tensor e is defined by this equation.Its significance is that it describes distortion, that is, deformation withoutvolume change A state of strain with e = 0 is called spherical or hydrostatic

1.2.3 Principal Strains

It is possible to describe any state of infinitesimal strain as a superposition

of three uniaxial extensions or contractions along mutually perpendicularaxes, that is, to find a set of axes x∗i such that, with respect to these axes,the strain tensor is described by

εni, or

εijnj− εni= (εij− εδij)nj = 0

Then, in order that n 6= 0, it is necessary that

det(ε− εI) = −ε3+ K1ε2+ K2ε + K3= 0, (1.2.3)where K1 = εkk, K2 = 1

2(εijεij − εiiεkk), and K3 = det ε are the so-calledprincipal invariants of the tensor ε Since Equation (1.2.3) is a cubic equa-tion, it has three roots, which are the values of ε for which the assumptionholds, namely ε1, ε2, and ε3 Such roots are known in general as the eigen-values of the matrix ε, and in the particular case of strain as the principalstrains

The principal invariants have simple expressions in terms of the principalstrains:

— are real (if they were not real, their physical meaning would be dubious)

We can also show that the principal axes are mutually perpendicular.Theorem 1 If ε is symmetric then the εI are real

Proof Let ε1= ε, ε2 = ¯ε, where the bar denotes the complex conjugate.Now, if n(1)i = ni, then n(2)i = ¯ni Since εijnj = εni, we have ¯niεijnj =

ε¯nini; similarly, εijn¯j = ¯ε¯ni, so that niεijn¯j = ¯ε¯nini However, εij = εji;consequently niεijn¯j = ¯niεijnj, so that (ε − ¯ε)¯nini = 0 Since ¯nini is, forany nonzero vector n, a positive real number, it follows that ε = ¯ε (i.e., ε isreal).

Theorem 2 If ε is symmetric then the n(I) are mutually perpendicular.Proof Assume that ε1 6= ε2; then εijn(1)j = ε1n(1)i and εijn(2)j = ε2n(2)i But n(2)i εijn(1)j − n(1)i εijn(2)j = 0 = (ε1− ε2)n(1)i n(2)i Hence n(1)· n(2)= 0

If ε1 = ε26= ε3, then any vector perpendicular to n(3)is an eigenvector, sothat we can choose two that are perpendicular to each other If ε1= ε2 = ε3(hydrostatic strain), then every nonzero vector is an eigenvector; hence wecan always find three mutually perpendicular eigenvectors Q.E.D

If the eigenvectors n(i) are normalized (i.e., if their magnitudes are fined as unity), then we can always choose from among them or their nega-tives a right-handed triad, say l(1), l(2), l(3), and we define Cartesian coor-dinates x∗i (i = 1, 2, 3) along them, then the direction cosines βij are given

de-by l(i)· ej, so that the strain components with respect to the new axes aregiven by

εij ∗= lk(i)l(j)l εkl.But from the definition of the n(i), we have

ll(i)εkl= εiδklll(i)= εil(i)k (no sum on i),

ε∗ = βεβT = LTεL = LTLΛ = Λ

If one of the basis vectors (ei), say e3, is already an eigenvector of ε,then ε13 = ε23 = 0, and ε33 is a principal strain, say ε3 The remainingprincipal strains, ε1 and ε2, are governed by the quadratic equation

ε2− (ε11+ ε22)ε + ε11ε22− ε212

This equation can be solved explicitly, yielding

ε1,2= 1

2(ε11+ ε22) ±

12

q

(ε11− ε22)2+ 4ε122

In the special case of simple shear, we have ε11= ε22= 0 and ε12= 1

2γ.Consequently ε1,2 = ±1

2γ With respect to principal axes, the strain tensor

If a second-rank tensor field ε(x1, x2, x3) is given, it does not automaticallyfollow that such a field is indeed a strain field, that is, that there exists adisplacement field u(x1, x2, x3) such that εij = 1

2(uj,i+ui,j); if it does, thenthe strain field is said to be compatible

The determination of a necessary condition for the compatibility of apresumed strain field is closely related to the integrability theorem Indeed,

if there were given a second-rank tensor field α such that αji = uj,i, thenthe condition would be just eikmαji,k= 0 Note, however, that if thereexists a displacement field u, then there also exists a rotation field θ suchthat εij + eijlθl = uj,i Consequently, the condition may also be written as

eikm(εij+ eijlθl),k= 0 But

eikmeijlθl,k= (δjkδlm− δjmδkl)θl,k= θm,j,since θk,k= 0 Therefore the condition reduces to

eikmεij,k= −θm,j.The condition for a θ field to exist such that the last equation is satisfiedmay be found by again invoking the integrability theorem, namely,

eikmejlnεij,kl= 0 (1.2.4)The left-hand side of Equation (1.2.4) represents a symmetric second-rank tensor, called the incompatibility tensor , and therefore the equationrepresents six distinct component equations, known as the compatibilityconditions If the region R is simply connected, then the compatibility

conditions are also sufficient for the existence of a displacement field fromwhich the strain field can be derived In a multiply connected region (i.e., aregion with holes), additional conditions along the boundaries of the holesare required.

Other methods of derivation of the compatibility conditions lead to thefourth-rank tensor equation

εij,kl+εkl,ij−εik,jl−εjl,ik= 0;

the sufficiency proof due to Cesaro (see, e.g., Sokolnikoff [1956]) is based onthis form It can easily be shown, however, that only six of the 81 equationsare algebraically independent, and that these six are equivalent to (1.2.4)

A sufficiency proof based directly on (1.2.4) is due to Tran-Cong [1985].The algebraic independence of the six equations does not imply that theyrepresent six independent conditions Let the incompatibility tensor, whosecomponents are defined by the left-hand side of (1.2.4), be denoted R Then

Rmn,n= eikmejlnεij,kln= 1

2eikmejln(ui,jkln+uj,ikln) = 0,regardless of whether (1.2.4) is satisfied, because eikmuj,ikln= ejlnui,jkln= 0.The identity Rmn,n= 0 is known as the Bianchi formula (see Washizu[1958] for a discussion)

Compatibility in Plane Strain

Plane strain in the x1x2-plane is defined by the conditions εi3 = 0 and

εij,3= 0 for all i, j The strain tensor is thus determined by the dimensional components εαβ(x1, x2) (α, β = 1, 2), and the only nontrivialcompatibility condition is the one corresponding to m = n = 3 in Equation(1.2.4), namely,

strain is one of plane displacement , in which u1 and u2 are functions of x1

and x2only, and u3vanishes identically The conditions εi3= 0 are, in terms

of displacement components,

u3,3= 0, u1,3+u3,1= 0, u2,3+u3,2= 0,leading to

u3 = w(x1, x2), uα = u0α(x1, x2) − x3w,α.The strain components are now

εαβ = ε0αβ(x1, x2) − x3w,αβ,where ε0 is the strain derived from the plane displacement field u0= u0αeα.The conditions εαβ,3= 0 require that w,αβ= 0, that is, w(x1, x2) = ax1+

bx2+ c, where a, b and c are constants The displacement field is thus thesuperposition of u0 and of −ax3e1− bx3e2 + (ax1+ bx2+ c)e3, the latterbeing obviously a rigid-body displacement In practice, “plane strain” issynonymous with plane displacement

2 For each of the displacement fields in the preceding exercise, determinethe matrices representing the finite (Green–Saint-Venant) and infini-tesimal strain tensors and the infinitesimal rotation tensor, as well asthe infinitesimal rotation vector

3 For the displacement field (a) of Exercise 1, determine the longitudinalstrain along the direction (e1+ e2)/√2

4 For the displacement field given in cylindrical coordinates by

u = ar er+ brz eθ+ c sin θ ez,where a, b and c are constants, determine the infinitesimal strain com-ponents as functions of position in cylindrical coordinates

5 Determine the infinitesimal strain and rotation fields for the ment field u = −w0(x1)x3e1+ w(x1)e3, where w is an arbitrary contin-uously differentiable function If w(x) = kx2, find a condition on k inorder that the deformation be infinitesimal in the region −h < x3 < h,

(a) the strain and rotation matrices,

(b) the volume strain and the deviatoric strain matrix,

(c) the principal strain invariants K1, K2, K3,

(d) the principal strains and their directions

7 For the displacement field u = α(−x2x3e1+x1x3e2), determine (a) thestrain and rotation fields, (b) the principal strains and their directions

as functions of position

8 For the plane strain field

εx = Bxy, εy = −νBxy, γxy = (1 + ν)B(h2− y2),

where B, ν and h are constants,

(a) check if the compatibility condition is satisfied;

(b) if it is, determine the displacement field u(x, y), v(x, y) in 0 <

x < L, −h < y < h such that u(L, 0) = 0, v(L, 0) = 0, and

Global Equations of Motion

Mechanics has been defined as the study of forces and motions It iseasy enough to define motion as the change in position of a body, in time,with respect to some frame of reference The definition of force is more

elusive, and has been the subject of much controversy among theoreticians,especially with regard to whether force can be defined independently of New-ton’s second law of motion An interesting method of definition is based on athought experiment due to Mach, in which two particles, A and B, are close

to each other but so far away from all other bodies that the motion of eachone can be influenced only by the other It is then found that there existnumbers mA, mB (the masses of the particles) such that the motions of theparticles obey the relation mAaA= −mBaB, where a denotes acceleration.The force exerted by A on B can now be defined as FAB= mBaB, and FBA

is defined analogously If B, rather than being a single particle, is a set ofseveral particles, then FAB is the sum of the forces exerted by A on all theparticles contained in B, and if A is also a set of particles, then FAB is thesum of the forces exerted on B by all the particles in A

The total force F on a body B is thus the vector sum of all the forces erted on it by all the other bodies in the universe In reality these forces are

ex-of two kinds: long-range and short-range If B is modeled as a continuumoccupying a region R, then the effect of the long-range forces is felt through-out R, while the short-range forces act as contact forces on the boundarysurface ∂R Any volume element dV experiences a long-range force ρb dV ,where ρ is the density (mass per unit volume) and b is a vector field (withdimensions of force per unit mass) called the body force Any oriented sur-face element dS = n dS experiences a contact force t(n) dS, where t(n) iscalled the surface traction; it is not a vector field because it depends notonly on position but also on the local orientation of the surface element asdefined by the local value (direction) of n

If a denotes the acceleration field, then the global force equation of tion (balance of linear momentum) is

Equations (1.3.1)–(1.3.2) are known as Euler’s equations of motion,applied by him to the study of the motion of rigid bodies If a body isrepresented as an assemblage of discrete particles, each governed by Newton’slaws of motion, then Euler’s equations can be derived from Newton’s laws.Euler’s equations can, however, be taken as axioms describing the laws ofmotion for extended bodies, independently of any particle structure They

are therefore the natural starting point for the mechanics of bodies modeled

as continua

Lagrangian and Eulerian Approaches

The existence of an acceleration field means, of course, that the ment field is time-dependent If we write u = u(x1, x2, x3, t) and interpretthe xi as Lagrangian coordinates, as defined in Section 1.2, then we havesimply a = ˙v = ¨u; here v = ˙u is the velocity field, and the superposed dotdenotes partial differentiation with respect to time at constant xi (calledmaterial time differentiation) With this interpretation, however, it must beagreed that R is the region occupied by B in the reference configuration,and similarly that dV and dS denote volume and surface elements measured

displace-in the reference configuration, ρ is the mass per unit reference volume, and t

is force per unit reference surface This convention constitutes the so-calledLagrangian approach (though Lagrange did not have much to do with it) tocontinuum mechanics, and the quantities associated with it are called La-grangian, referential, or material (since a point (x1, x2, x3) denotes a fixedparticle or material point) It is, by and large, the preferred approach in solidmechanics In problems of flow, however — not only fluid flow, but also plas-tic flow of solids — it is usually more instructive to describe the motion ofparticles with respect to coordinates that are fixed in space — Eulerian orspatial coordinates In this Eulerian approach the motion is described not

by the displacement field u but by the velocity field v If the xi are spatialcoordinates, then the material time derivative of a function φ(x1, x2, x3, t),defined as its time derivative with the Lagrangian coordinates held fixed,can be found by applying the chain rule to be

˙

φ = ∂φ

∂t + viφ,i.The material time derivative of φ is also known as its Eulerian derivativeand denoted D

Dtφ.

If the displacement field is infinitesimal, as defined in the preceding tion, then the distinction between Lagrangian and Eulerian coordinates canusually be neglected, and this will generally be done here until finite defor-mations are studied in Chapter 8 The fundamental approach is Lagrangian,except when problems of plastic flow are studied; but many of the equationsderived are not exact for the Lagrangian formulation Note, however, onepoint: because of the postulated constancy of mass of any fixed part of B,the product ρ dV is time-independent regardless of whether ρ and dV aregiven Lagrangian or Eulerian readings; thus the relation

sec-ddt