Mechanics of solids and materials

The book blends both innovative topics e.g., large strain, strain rate, temperature, time-dependent deformation and localized plastic deformation in crystalline solids, and deformation o

MECHANICS OF SOLIDS AND MATERIALS

Mechanics of Solids and Materials intends to provide a modern and integrated

treat-ment of the foundations of solid mechanics as applied to the mathematical description

of material behavior The book blends both innovative topics (e.g., large strain, strain

rate, temperature, time-dependent deformation and localized plastic deformation in

crystalline solids, and deformation of biological networks) and traditional topics (e.g.,

elastic theory of torsion, elastic beam and plate theories, and contact mechanics) in a

coherent theoretical framework This, and the extensive use of transform methods to

generate solutions, makes the book of interest to structural, mechanical, materials, and

aerospace engineers Plasticity theories, micromechanics, crystal plasticity, thin films,

energetics of elastic systems, and an overall review of continuum mechanics and

ther-modynamics are also covered in the book

Robert J Asaro was awarded his PhD in materials science with distinction from Stanford

University in 1972 He was a professor of engineering at Brown University from 1975

to 1989, and has been a professor of engineering at the University of California, San

Diego since 1989 Dr Asaro has led programs involved with the design, fabrication, and

full-scale structural testing of large composite structures, including high-performance

ships and marine civil structures His list of publications includes more than 170 research

papers in the leading professional journals and conference proceedings He received the

NSF Special Creativity Award for his research in 1983 and 1987 Dr Asaro also received

the TMS Champion H Mathewson Gold Medal in 1991 He has made fundamental

contributions to the theory of crystal plasticity, the analysis of surface instabilities, and

dislocation theory He served as a founding member of the Advisory Committee for

NSF’s Office of Advanced Computing that founded the Supercomputer Program in the

United States He has also served on the NSF Materials Advisory Committee He has

been an affiliate with Los Alamos National Laboratory for more than 20 years and has

served as consultant to Sandia National Laboratory Dr Asaro has been recognized by

ISI as a highly cited author in materials science

Vlado A Lubarda received his PhD in mechanical engineering from Stanford

Univer-sity in 1980 He was a professor at the UniverUniver-sity of Montenegro from 1980 to 1989,

Fulbright fellow and a visiting associate professor at Brown University from 1989 to

1991, and a visiting professor at Arizona State University from 1992 to 1997 Since 1998

he has been an adjunct professor of applied mechanics at the University of California,

San Diego Dr Lubarda has made significant contributions to phenomenological

the-ories of large deformation elastoplasticity, dislocation theory, damage mechanics, and

micromechanics He is the author of more than 100 journal and conference publications

and two books: Strength of Materials (1985) and Elastoplasticity Theory (2002) He has

served as a research panelist for NSF and as a reviewer to numerous international

journals of mechanics, materials science, and applied mathematics In 2000 Dr Lubarda

was elected to the Montenegrin Academy of Sciences and Arts He is also recipient of

the 2004 Distinguished Teaching Award from the University of California

i

ii

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© Robert Asaro and Vlado Lubarda 2006

Information on this title: www.cambridge.org/9780521859790

This publication is in copyright Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press

isbn-10 0-511-14707-4

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hardback

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PART 1: MATHEMATICAL PRELIMINARIES

1.2 Coordinate Transformation: Rotation

1.22 Identities and Relations Involving

v

2 Basic Integral Theorems 26

2.2 Vector and Tensor Fields: Physical Approach 27

Contents vii

4.22 Additional Connections Between Current and Reference

4.24 Material Derivatives of Volume, Area, and Surface Integrals:

5.3 Balance of Angular Momentum: Symmetry ofσ 95

6.1 First Law of Thermodynamics: Energy Equation 113

6.2 Second Law of Thermodynamics: Clausius–Duhem

6.4 Thermodynamic Relationships with p, V, T, and s 120

6.4.2 Coefficients of Thermal Expansion and

6.5 Theoretical Calculations of Heat Capacity 123

6.8 Gibbs Conditions of Thermodynamic Equilibrium 129

6.12 Thermodynamics of Open Systems: Chemical

7.3 Constitutive Equations in Terms of B 151

7.4 Constitutive Equations in Terms of Principal Stretches 152

7.5 Incompressible Isotropic Elastic Materials 153

7.9 Elastic Moduli of Isotropic Elasticity 156

PART 3: LINEAR ELASTICITY

8.1 Elementary Theory of Isotropic Linear Elasticity 161

8.3 Restrictions on the Elastic Constants 164

8.5 Compatibility Conditions: Ces `aro Integrals 170

8.6 Beltrami–Michell Compatibility Equations 172

8.8 Uniqueness of Solution to Linear Elastic Boundary Value

8.9 Potential Energy and Variational Principle 175

Contents ix

9.4 Beam Problems with Body Force Potentials 188

9.6 Complete Boundary Value Problems for Beams 193

11.5 Torsion of a Rod with Multiply Connected Cross Sections 222

12.6.1 Displacement Fields in Half-Spaces 238

Contents xi

20.3 Calculation of the Constrained Fields: uc, ec, andσc 338

20.4 Components of the Eshelby Tensor for Ellipsoidal Inclusion 341

20.6 Inhomogeneous Inclusion: Uniform Transformation Strain 343

20.8 Inclusions in Isotropic Media 350

21.3.1 Interaction Between Dislocations and

21.7.2 Application of the Interface Force to

22.11.3.Energy due to Internal and External Sources of

Contents xiii

22.16 Conservation Laws for Plane Strain Micropolar Elasticity 404

PART 5: THIN FILMS AND INTERFACES

23.5 Strain Energy of a Dislocation Near a Bimaterial Interface 423

23.5.1 Strain Energy of a Dislocation Near a

25.3 Surface Diffusion and Interface Stability 450

PART 6: PLASTICITY AND VISCOPLASTICITY

26.10.Constitutive Equations for Pressure-Dependent Plasticity 478

27.2.1 Some Basic Properties of Dislocations in Crystals 511

27.2.2 Strain Hardening, Dislocation Interactions, and

27.5 Observations of Slip in Single Crystals and Polycrystals at

Contents xv

29 The Nature of Crystalline Deformation: Localized Plastic

29.1 Perspectives on Nonuniform and Localized Plastic Flow 557

29.1.1 Coarse Slip Bands and Macroscopic Shear Bands in

29.1.2 Coarse Slip Bands and Macroscopic Shear Bands in

29.2.5 Perturbations about the Slip and Kink Plane

30.4 Model Calculational Procedure 592

32.7.1 Thermodynamic Potentials per Unit Initial Mass 620

32.8 Multiplicative Decomposition of Deformation Gradient 622

Contents xvii

PART 8: SOLVED PROBLEMS

xviii

This book is written for graduate students in solid mechanics and materials science and

should also be useful to researchers in these fields The book consists of eight parts Part 1

covers the mathematical preliminaries used in later chapters It includes an introduction to

vectors and tensors, basic integral theorems, and Fourier series and integrals The second

part is an introduction to nonlinear continuum mechanics This incorporates kinematics,

kinetics, and thermodynamics of a continuum and an application to nonlinear elasticity

Part 3 is devoted to linear elasticity The governing equations of the three-dimensional

elasticity with appropriate specifications for the two-dimensional plane stress and plane

strain problems are given The applications include the analyses of bending of beams

and plates, torsion of prismatic rods, contact problems, semi-infinite media, and

three-dimensional isotropic and anisotropic elastic problems Part 4 is concerned with

microme-chanics, which includes the analyses of dislocations and cracks in isotropic and anisotropic

media, the well-known Eshelby elastic inclusion problem, energy analyses of imperfections

and configurational forces, and micropolar elasticity In Part 5 we analyze dislocations in

bimaterials and thin films, with an application to the study of strain relaxation in thin films

and stability of planar interfaces Part 6 is devoted to mathematical and physical theories

of plasticity and viscoplasticity The phenomenological or continuum theory of plasticity,

single crystal, polycrystalline, and laminate plasticity are presented The micromechanics

of crystallographic slip is addressed in detail, with an analysis of the nature of crystalline

deformation, embedded in its tendency toward localized plastic deformation Part 7 is an

introduction to biomechanics, particularly the formulation of governing equations of the

mechanics of solids with a growing mass and constitutive relations for biological

mem-branes Part 8 is a collection of 180 solved problems covering all chapters of the book This

is included to provide additional development of the basic theory and to further illustrate

its application

The book is transcribed from lecture notes we have used for various courses in solidmechanics and materials science, as well as from our own published work We have also

consulted and used major contributions by other authors, their research work and written

books, as cited in the various sections As such, this book can be used as a textbook for a

sequence of solid mechanics courses at the graduate level within mechanical, structural,

aerospace, and materials science engineering programs In particular, it can be used for

xix

the introduction to continuum mechanics, linear and nonlinear elasticities, theory of

dis-locations, fracture mechanics, theory of plasticity, and selected topics from thin films and

biomechanics At the end of each chapter we offer a list of recommended references for

additional reading, which aid further study and mastering of the particular subject

Standard notations and conventions are used throughout the text Symbols in bold, both

Latin and Greek, denote tensors or vectors, the order of which is indicated by the context

Typically the magnitude of a vector will be indicated by the name symbol unbolded Thus,

for example, a or b indicate two vectors or tensors If a and b are vectors, then the scalar

product, i.e., the dot product between them is indicated by a single dot, as a· b Since a

and b are vectors in this context, the scalar product is also ab cos θ, where θ is the angle

between them If A is a higher order tensor, say second-order, then the dot product of

A and a produces another vector, viz., A· a = b In the index notation this is expressed

as A i j a j = b i Unless explicitly stated otherwise, the summation convention is adopted

whereby a repeated index implies summation over its full range This means, accordingly,

that the scalar product of two vectors as written above can also be expressed as a j b j = φ,

whereφ is the scalar result Two additional operations are introduced and defined in the

text involving double dot products For example, if A and B are two second-rank tensors,

then A : B= A i j B i jand A· · B = A i j B ji For higher order tensors, similar principles apply

If C is a fourth-rank tensor, then C : e⇒ C i j kl e kl = { } i j

In finite vector spaces we assume the existence of a convenient set of basis vectors Most

commonly these are taken to be orthogonal and such that an arbitrary vector, say a, can be

expressedwrt its components along these base vectors as a = a1e1+ a2e2+ a3e3, where

{e1, e2, e3} are the orthogonal set of base vectors in question Other more or less standard

notations are used, e.g., the left- or right-hand side of an equation is referred to as the lhs,

or r hs, respectively The commonly used phrase with respect is abbreviated as wrt, and so

on

We are grateful to many colleagues and students who have influenced and contributed

to our work in solid mechanics and materials science over a long period of time and thus

directly or indirectly contributed to our writing of this book Specifically our experiences at

Stanford University, Brown University, UCSD, Ford Motor Company (RJA), Ohio State

University (RJA), University of Montenegro (VAL), and Arizona State University (VAL)

have involved collaborations that have been of great professional value to us Research

funding by NSF, the U.S Army, the U.S Air Force, the U.S Navy, DARPA, the U.S DOE,

Alcoa Corp., and Ford Motor Co over the past several decades has greatly facilitated our

research in solid mechanics and materials science We are also most grateful to our families

and friends for their support during the writing of this book

PART 1: MATHEMATICAL PRELIMINARIES

This chapter and the next are concerned with establishing some basic properties of vectors

and tensors in real spaces The first of these is specifically concerned with vector algebra and

introduces the notion of tensors; the next chapter continues the discussion of tensor algebra

and introduces Gauss and Stokes’s integral theorems The discussion in both chapters is

focused on laying out the algebraic methods needed in developing the concepts that follow

throughout the book It is, therefore, selective and thus far from inclusive of all vector

and tensor algebra Selected reading is recommended for additional study as it is for all

subsequent chapters Chapter 3 is an introduction to Fourier series and Fourier integrals,

added to facilitate the derivation of certain elasticity solutions in later chapters of the book

1.1 Vector Algebra

We consider three-dimensional Euclidean vector spaces, E, for which to each vector

such as a or b there exists a scalar formed by a scalar product a· b such that a · b =

a r eal number i n R and a vector product that is another vector such that a × b = c Note

the definitions via the operations of the symbols, · and ×, respectively Connections to

common geometric interpretations will be considered shortly

Withα and β being scalars, the properties of these operations are as follows

a b

c

Figure 1.1 Geometric meaning of a vector triple product The triple product

is equal to the volume of the parallelepiped formed from the three defining

From the above expressions it follows that if a× b = 0, then a and b are linearly dependent,

i.e., a = αb where α is any scalar.

A triple product is defined as

It is evident from simple geometry that the triple product is equal to the volume enclosed

by the parallelepiped constructed from the vectors a, b, c This is depicted in Fig 1.1 Here,

again, the listed vector properties allow us to write

iff a, b, c are linearly dependent.

Because of the first of the properties (1.3), we can establish an orthonormal basis (Fig 1.2)

that we designate as{e1, e2, e3}, such that

1.1 Vector Algebra 3

e

e

e 2

3

υ

Figure 1.2 A vectorυ in an orthonormal basis.

where the repeated index i implies summation, i.e.,

+1, if i, j, k are an even permutation of 1, 2, 3,

−1, if i, j, k are an odd permutation of 1, 2, 3,

0, if any of i, j, k are the same.

(1.21)

Some useful results follow Let a= a pepand b= brer Then,

a· b = (apep)· (b rer)= a p b r(ep· er)= ap b r δ pr (1.22)Thus, the scalar product is

Similarly, the vector product is

a× b = apep × brer = a p b rep× er = ap b r  priei = i pr(a p b r)ei. (1.24)Finally, the component form of the triple product,

is

c· (i pr a p b rei)= i pr a p b r c i = pri a p b r c i = i j k a i b j c k (1.26)

e e

e

e' e'

e'

1 1

2 2

3

3

Figure 1.3 Transformation via rotation of basis.

1.2 Coordinate Transformation: Rotation of Axes

Letυ be a vector referred to two sets of basis vectors, {e i} and {e

i }, i.e.,

υ = υ iei = υ

We seek to relationship of theυ i to the υ

i Let the transformation between two bases(Fig 1.3) be given by

For example, in the two-dimensional case, we have

e 1= cos θe1+ sin θe2,

with the corresponding transformation matrix

α =

cosθ sinθ

1.4 Symmetric and Antisymmetric Tensors 5

1.3 Second-Rank Tensors

A vector assigns to each direction a scalar, viz., the magnitude of the vector A second-rank

tensor assigns to each vector another (unique) vector, via the operation

More generally,

Second-rank tensors obey the following additional rules

1.4 Symmetric and Antisymmetric Tensors

We call the tensor A symmetric if A = AT A is said to be antisymmetric if A = −AT

An arbitrary tensor, A, can be expressed (or decomposed) in terms of its symmetric and

antisymmetric parts, via

1.5 Prelude to Invariants of Tensors

reverses sign if p , q, r undergo an odd permutation, and is equal to 0 if any of p, q, r are

made equal Thus set p = 1, q = 2, r = 3, and multiply the result by pqr to take care of

the changes in sign or the null results just described The full expression becomes

is invariant to changes of the basis{f, g, h}.

Given the validity of the expressions for χ1, χ2, and χ3, we thereby discover three

invariants of the tensor A, viz.,

We deferred formal proofs of several lemmas until now in the interest of presentation We

provide the proofs at this time

LEMMA 1.1: If a and b are two vectors, a × b = 0 iff a and b are linearly dependent.

Proof: If a and b are linearly dependent then there is a scalar such that b= αa In

this case, if we express the vector product a × b = c in component form, we find that

c i = i j k a j αa k = αi j k a j a k But the summations over the indices j and k will produce pairs

of multiples of a β a γ , and then again a γ a β, for which the permutator tensor alternates

algebraic sign, thus causing such pairs to cancel Thus, in this case a × b = 0.

Conversely, if a× b = 0, we find from (1.3) to (1.8) that a × b = ±|a||b| If the plus signs

holds, we have from the second of (1.3)

(|b|a − |a|b) · (|b|a − |a|b) = 2|a|2|b|2− 2|a||b|a · b = 0. (1.52)Because of the third property in (1.3) this means that |b|a = |a|b When the minus sign

holds, we find that|b|a = −|a|b In either case this leads to the conclusion that b = αa.

Next we examine the relations defining properties of the triple product when pairs

of the vectors are interchanged Use (1.26) to calculate the triple product This yields

[a, b, c] =  i j k a i b j c k Next imagine interchanging, say a with b; we obtain [b , a, c] =

 i j k b i a j c k = i j k a j b i c k = −ji k a j b i c k = −i j k a i b j c k, where the last term involved merely

a reassignment of summation indices Thus [a, b, c] = −[b, a, c] Proceeding this way all

members of (1.11) are generated.

We now examine the triple product property expressed in (1.12).

LEMMA 1.2: If a, b, c, and d are arbitrary vectors, and α and β arbitrary scalars, then

Proof: Begin with the property of scalar products between vectors expressed in (1.2)

and replace c with c × d Then,

(αa + βb) · (c × d) = αa · (c × d) + βb · (c × d) = α[a, c, d] + β[b, c, d]. (1.54)

Of course, the first term in the above is the triple product expressed on the lhs of the

v1 b1 c1

v2 b2 c2

v3 b3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

The expression just generated is zero as may be seen, for example, by letting x be equal to

e1, e2, e3, respectively Note that the third equation of (1.70) below has been used.

1.9 Coordinate Transformation of Tensors 9

LEMMA 1.5: Suppose that for any vector p, p · q = p · t, then we have

 ir s  i j k p s q j r k = qr ( ps r s) − rr ( ps q s) (1.65)

to complete the proof

A simple extension of the last lemma is that

The proof is left as an exercise

1.9 Coordinate Transformation of Tensors

Consider coordinate transformations prescribed by (1.28) A tensor A can be written

1.10 Some Identities with Indices

The following identities involving the Kronecker delta are useful and are easily verified

Let u and v be two vectors; then there is a tensor B= uv defined via its action on an

arbitrary vector a, such that

LEMMA 1.7: If u and v are arbitrary vectors, then

Proof: Replace A in (1.47) with uv, and use {a, b, c} as a basis of E Then the third

equation from (1.47) becomes

[(uv)· a, (uv) · b, (uv) · c] = I I Iu v[a, b, c],

Next, we note that the first equation from (1.47), with A = uv, leads to

[(uv)· a, b, c] + [a, (uv) · b, c] + [a, b, (uv) · c] = Iuv[a, b, c], (1.77)where

I uv = tr (uv).

But the lhs of the relation (1.77) can be rearranged as

v· a[u, b, c] + v · b[a, u, c] + v · c[a, b, u].

Since{a, b, c} is a basis of E, u can be expressed as

But, the first of (1.47) gives

(αa · v + βb · v + γ c · v) [a, b, c] = tr (uv)[a, b, c],

so that

1.12 Orthonormal Basis

Let us now refer the tensor A to an orthonormal basis, {e1, e2, e3} The eiare unit vectors

in this context Let A i jbe the components of A relative to this basis Then

A · ej = Apjep and A i j= ei · A · ej (1.81)

Now form A= Apqepeqand look at its operation on a vector a= arer We have

( A pqepeq) · arer = Apq a repeq· er = Apr a rep = ar A prep

e e

e

n n

n

1 1

2 2

Note if A= uv, then A i j = ui v j, because A= ui v jeiejand u= uiei , v = v jej.

By using an orthonormal basis{ei}, the invariants of a second-rank tensor A can be

expressed from (1.47) as follows First, consider

1.13 Eigenvectors and Eigenvalues

Letσ be a symmetric tensor, i.e., σ i j = σji in any orthonormal basis,{ei} Examine the

“normal components” ofσ, e.g., σ nn = n · σ · n = ni σ i j n j Look for extremum values for

1.13 Eigenvectors and Eigenvalues 13

σ nn wrt the orientation of n Let θ be the angle between n and e1 (Fig 1.4) We require

i = σi j n j Since∂n/∂θ is orthogonal to n, we conclude that

T(n) = σ · n must be codirectional with n Hence, T (n) = σ · n = λ (n)n This leads to the

homogeneous set of equations

Conditions need to be sought whereby (1.89) can have nontrivial solutions

LEMMA 1.8: Recall (1 .5) viz det A = [A · f, A · g, A · h], where {f, g, h} is an arbitrary

basis of E If [A · f, A · g, A · h] = 0, then {A · f, A · g, A · h} must be linearly dependent.

That is, [a, b, c] = 0 iff {a, b, c} are linearly dependent.

Proof: If one of{p, q, r} are zero, [p, q, r] = p · (q × r) = 0 Next, if {a, b, c} are linearly

dependent, there existα, β, γ (not all zero) such that

Return now to the possible solution of the equation A · n = 0 Suppose |A| = 0, then if

{f, g, h} form a basis of E, they are linearly dependent, i.e.,

λ3− IAλ2+ I IAλ − I I IA= 0. (1.96)Equation (1.96), referred to as a characteristic equation, has three solutions.

1.14 Symmetric Tensors

Symmetric tensors, e.g., S, possess real eigenvalues and corresponding eigenvectors,

{λ1, λ2, λ3} and {p1, p2, p3}, respectively We may write S in the various forms such as

S = S · I = S · (pr pr)= S · pr pr = λ (r )prpr Thus the spectral representation of S is

1.15 Positive Definiteness of a Tensor

If for any arbitrary vector a, a· A · a ≥ 0, the tensor A is said to be positive semidefinite If

a· A · a > 0, A is said to be positive definite.

Let S be a symmetric, positive semidefinite tensor with the associated eigenvectors and

eigenvalues, pi andλ i Then, as before,

a· (A · AT)· a = (a · A) · (AT · a) = (a · A) · (a · A) ≥ 0. (1.104)

1.16 Antisymmetric Tensors

If

the tensor W is said to be antisymmetric.

Let a and b be arbitrary vectors, then

Let{q, r, p} be a unit orthonormal basis; then

p= q × r, q= r × p,

Recall that if{i, j} is a pair of unit vectors from the set {q, r}, then

W i j = i · W · j, W = W i jij. (1.112)Thus

a= (a · p)p + (a · q)q + (a · r)r, (1.117)and form

The proper orthogonal tensor has one real eigenvalue, which is equal to 1 The

correspond-ing eigenvector, p, is parallel to the axis of rotation associated with Q, i.e.,

θ Figure 1.5 Geometric interpretation of an orthogonal tensor in terms of therotation of a material fiber.

Thus, the pairs (q, r) and (Q · q, Q · r) are orthogonal to p Because of this property, we

then can write

vector, x As shown, the effect of the tensor operation is to rotate x about the eigenvector

p by the angleθ That is, the result of operating with Q is to produce a rotated vector,

x = Q · x, as shown in the figure.

As expected, orthogonal tensors enter the discussion of material motion prominently

with respect to describing rotations of bodies and material fibers This will, for example,

appear explicitly in our consideration of the the polar decomposition theorem introduced

in the next section

1.18 Polar Decomposition 19

As an example, with respect to the basis e = {e1, e2, e3}, let Q11= Q22= cos θ, Q12 =

−Q21= − sin θ, Q33= 1, and Qi j = 0 otherwise Equations (1.129) then become

cosθp1+ sin θp2 = p1,

p3 = p3,

with p= piei The solution to this set is trivially p1 = p2= 0 and p3= 1; thus p = e3

Next choose q = e1and r = e2to satisfy the requirement of a right-handed triad basis,

{p, q, r} Clearly then Qqq = q · Q · q = cos θ = r · Q · r = Qrr, Qr q = r · Q · q = sin θ =

−(q · Q · r) = −Qqr, and hence

Q= pp + cos θ(qq + rr) − sin θ(qr − rq). (1.144)

But x= xses = (x · ξs)ξ s, whereξ s = {p, q, r} and where x1= cos α, x2= sin α and x3 =

0 The relationship x = Q · x leads to

x = [pp + cos θ(qq + rr) − sin θ(qr − rq)] · (cos αq + sin αr)

= cos(θ + α)q + sin(θ + α)r.

The result is exactly what was expected, namely that the fiber x inclined byθ to the e1base

vector is now rotated byα so that its total inclination is θ + α.

1.18 Polar Decomposition Theorem

Let A be an arbitrary tensor that possesses an inverse A−1 The following theorem, known

as the polar decomposition theorem, will be useful in the analysis of finite deformations

and the development of constitutive relations

THEOREM 1.1: An invertible second-order tensor A can be uniquely decomposed as

where Q is an orthogonal tensor and U and V are symmetric, positive definite tensors.

Proof: Recall that the forms AT· A and A · AT are positive semidefinite, symmetric

tensors If A is invertible, i.e., A−1exists, then det A

Let U be the square root of AT· A and V be the square root of A · AT But then U and

V have unique inverses, U−1and V−1 Consequently, if

and if

Thus, the pairs (q, r) and (Q · q, Q · r) are orthogonal to p Because of this... square root of AT· A and V be the square root of A · AT But then U and< /b>

V have unique inverses, U−1and V−1... p2= and p3= 1; thus p = e3

Next choose q = e1and r = e2to satisfy the requirement of a right-handed