on compatibility conditions for the left cauchy–green deformation field in three dimensions

Journal of Elasticity56: 95–105, 1999.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

95

On Compatibility Conditions for the LeftCauchy–Green Deformation Field in ThreeDimensions

AMIT ACHARYACenter for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign,3315 DCL, 1304 W. Springfield Ave., Urbana, IL 61801, U.S.A. E-mail: [email protected]

Received 20 April 1999; in revised form 16 September 1999

Abstract. A fairly general sufficient condition for compatibility of the left Cauchy–Green deforma-tion field in three dimensions has been derived. A related necessary condition is also indicated. Thekinematical problem is phrased as a suitable problem in Riemannian geometry, whence the method ofsolution emerges naturally. The main result of the paper is general in scope and provides conditionsfor the existence of solutions to certain types of overdetermined systems of first-order, quasilinearpartial differential equations with algebraic constraints.

Key words: compatibility, left Cauchy–Green deformation, three dimensions or (3-D).

1. Introduction

The aim of this paper is to discuss the issue of necessary and sufficient conditionsfor the existence of a local deformation of a simply-connected reference configura-tion whose left Cauchy–Green deformation field matches a prescribed symmetric,positive-definite tensor field on the same reference. In this paper, the term deforma-tion refers to a mapf with an invertible deformation gradient (detF 6= 0; F = Df),and the left Cauchy–Green deformation tensor is generically represented by thesymbolB = FFT. To the best of the author’s knowledge, this problem remains openin three dimensions, having been solved for the plane case by Blume [1]. Relatedresults, for the plane case, are also presented in Duda and Martins [2]. A fairlygeneral sufficient condition for compatibility in three dimensions is provided inthis paper.

The problem dealt with in this paper is a fundamental issue in continuum kine-matics, apart from being of mathematical interest. As pointed out by Blume [1],the problem of deriving sufficient conditions for the existence of a deformationcompatible with a given left Cauchy–Green field is fairly difficult, especially whencompared with the same question for the right Cauchy–Green field (C = FTF)or the even simpler question of ‘deformation-gradient’ compatibility. While in thelatter cases the compatibility conditions come out neatly as the vanishing of the

96 A. ACHARYA

Riemann–Christoffel curvature tensor and the ‘curl’, respectively, of the prescribedfields, the conditions in the present case are not as explicit.

As in the case of the proof of compatibility for the right Cauchy–Green de-formation field, we phrase the compatibility question for the left Cauchy–Greendeformation field as a problem in Riemannian geometry. The problem is posedas a special case of the determination of conditions for which two pre-assignedpositive-definite, symmetric matrix fields may be considered as components of aRiemannian metric on a manifold. The speciality of the problem lies in the factthat the metric is known to be Euclidean (its components in agivenparametrizationare, uniformly, the identity matrix) – the question is to construct another coordinatesystem (parametrization) on which the other pre-assigned matrix field (rectangularCartesian components of theB tensor field in question) can be considered as con-travariant components of the same metric. It should be carefully noted that such aquestion is quite different in its details than the one asked in the case of the proofof right Cauchy–Green compatibility – a symmetric positive-definite matrix fieldon agivenparametrization is available (interpreted as the covariant components ofa metric tensor), and another coordinate system has to be determined such that onthat system the covariant metric components are, uniformly, the components of theidentity matrix.

The answer to the geometric version of theB-compatibility problem rests onbeing able to determine necessary and sufficient conditions for the existence of asolution to an overdetermined system of algebraic and partial differential equationswhich, typically, is not completely integrable. Such systems have been consid-ered in the differential geometry literature in connection with the determinationof various kinds of invariants of metric and affinely connected manifolds (Veblenand Thomas [9]; Thomas and Michal [8]; Schouten [5]) and, apparently, werefirst considered by Christoffel (Eisenhart [3, footnotes p. 17 and p. 77]) whoseconcern seems to be very closely related to the geometric version of the issue ofB-compatibility, as phrased in this paper.

In Section 1, we examine the issue ofB-compatibility and phrase the question interms of suitably defined coordinates and components. The coordinate-componentsform of the problem motivates the geometric formulation, which we discuss inSection 2. This approach differs significantly from the one in Blume [1], which isbased on the left polar decomposition of the deformation gradient. In Section 3 asolution to the problem posed in Section 2 is provided with a sketch of the proof ofthe main result.

With respect to notation used in the paper, we use standard notational conven-tions of the tensor calculus when dealing with coordinates and components. ThesymbolD represents a derivative operator, and the same symbol with a subscriptwill represent a partial derivative on a suitably defined product space. The symbol∇ will represent the gradient of a scalar field. All tensorial quantities from thefirst order onwards will be represented in boldface and scalar quantities will berepresented in lower case italics. In representing a list as function arguments, we

COMPATIBILITY CONDITIONS FOR THE LEFT CAUCHY–GREEN DEFORMATION FIELD 97

shall often use just the kernel letter to represent the entire list, e.g.,x to representthe n-tuple (x1, x2, . . . , xn). Also, associated tensors will be freely used in thefollowing.

2. The problem of B-Compatibility

The compatibility problem for the Left Cauchy–Green deformation field may bestated as follows:

Given a pre-assigned positive-definite symmetric second order tensor fieldBon a reference configurationR, find necessary and sufficient conditions for theexistence of a regular deformationy of the reference such that

Dy(x)(Dy(x)

)T = B(x) for all x ∈ R. (1)

Consider a rectangular Cartesian parametrization ofE3, the ambient three di-mensional Euclidean point space. Let the reference configurationR be representedby coordinates{xα} in this coordinate system. It is now easy to see that the afore-mentionedB-compatibility question is equivalent to the following problem:

Given the rectangular Cartesian components of the tensorB (Bkm) as functionsof {xα}, find necessary and sufficient conditions on the field(Bkm) for the existenceof functionsyi of {xα} satisfying

∂yi

∂xρ

∂yj

∂xρ= Bij . (2)

The equivalence of the two questions follows from the equivalence of the exis-tence assertions defined by Equations (1) and (2). In fact, if functionsyi existsuch that (2) holds, then, choosing the region ofE3, sayC, that corresponds to{yi(x)} for all {xα} corresponding toR (where the{yi} are coordinates in the samerectangular Cartesian parametrization ofE3 chosen to define{xα}), as a deformedimage ofR, we see that we have defined a regular deformationy: R→ C throughthe association, represented symbolically as

x(x)→ y(y(x)

)for all x ∈ R

with

di ·Dy(x)eρ = ∂yi

∂xρ,

where

di = ∇yi(y); eρ = Dρx(x) [1].

Since we are working with a rectangular Cartesian parametrization,di = di =ei = ei and, hence,

Bij = ∂yi

∂xρ

∂yj

∂xρ= ei · [Dy(Dy)T

]ej ⇒ Dy(Dy)T = B.

The converse assertion, of course, is even simpler.

98 A. ACHARYA

3. Geometric Formulation

The issue of existence posed in (2) suggests the following question in Riemanniangeometry:

Given a real, positive-definite symmetric matrix fieldBij on a coordinate patch{xα} of some manifold, what are the necessary and sufficient conditions for theexistence of another coordinate patch{yi} such that, ifBij are viewed as the com-ponents of a metric tensor on{yi}, then the corresponding components on the{xα}system are, uniformly, the components of the identity matrix. That is, we seek tofind conditions for the existence of the coordinate patch{yi} such thatδαβ andBij

may be viewed as the contravariant components of a metric tensor on the coordinatepatches{xα} and{yi}, respectively. The corresponding covariant components arerepresented by the symbolsδαβ andBij .

We also note that the aforementioned geometric problem is equivalent to a localversion of the compatibility problem for the left Cauchy–Green deformation fieldas represented by the existence problem (1).

In order to solve the geometric problem, we first consider necessity. Supposethere exists{yi} such that (2) holds. This implies that the mapping{xα} → {yi} islocally invertible which further implies that∂Brp/∂ys is well defined and given by

∂Brp

∂ys= ∂Brp

∂xα

∂xα

∂ys. (3)

Now, if {zi} is some coordinate patch, then the Christoffel symbols of the secondkind with respect to{zi} and their transformation rules are given by

(z)0irs =

Bip

2

[∂Brp

∂zs+ ∂Bsp

∂zr− ∂Brs∂zp

](4)

and

∂yi

∂xα∂xβ= (x)0

ραβ

∂yi

∂xρ− (y)0

irs

∂yr

∂xα

∂ys

∂xβ, (5)

respectively (Sokolnikoff [6]). Since the metric components are constant on{xi},(x)0

ραβ = 0, and hence, (3)–(5) imply

∂yi

∂xα∂xβ= −B

im

2

[∂Brm

∂xβ

∂yr

∂xα+ ∂Bsm∂xα

∂ys

∂xβ− ∂Brs∂xρ

∂xρ

∂ym

∂yr

∂xα

∂ys

∂xβ

].

Hence, our hypothesis implies that there exists a solution to the system of equations

uiρujρ = Bij , (6)

∂yi

∂xα= uiα, (7)

∂uiα

∂xβ= −B

im

2

[∂Brm

∂xβurα +

∂Bsm

∂xαusβ −

∂Brs

∂xρf ρm(u)u

rαu

sβ

], (8)


wheref ρm represents the matrix inverse function. Conversely, suppose the fieldsBij are such that there exists a solution to (6)–(8). Then, it is clear from (6) and (7)that there exist functions{yi}, defined on the coordinate patch{xα}, that satisfy (2).Hence, our problem reduces to finding necessary and sufficient conditions for theexistence of solutions to the system (6)–(8).

We also note here that if an invertible matrix-valued solution to (8) exists, then asolution to (7) necessarily exists because of the symmetry, inα andβ, of the right-hand side of (8) (Thomas [7]), and the solution is locally invertible as a functionof {xα}.

In the next section we provide a sketch of the proof of the result that providesa sufficient condition for the existence of solutions to (6) and (8) and, conse-quently, (7).

4. Algebraic Conditions for the Existence of Solutions

Consider the system of differential equations

∂wi

∂xα(x) = ψi

α

(w(x), x

), i = 1,2, . . . , R;α = 1,2, . . . , n, (9)

in which thewi are functions of independent variables{xα} andψiα are differ-

entiable as many times as required. The domain ofψiα, also referred to as(z, x)

space in the following, is assumed to be an open, connected set ofRR ×Rn. Forapplication to the main problem of the paper, we identifywi with the functionsukµ,k,µ = 1,2,3, so thatR = 9 andn = 3. Also, we identify (9) with system (8). Weseek solutions

wi(x) (10)

of (9) which satisfy a system of equations

F (0)(w(x), x

) = 0, (11)

where the domain ofF (0) is the(z, x) space. For the compatibility problem at hand,we identify F (0)(z, x) = 0 as the nine equations appearing in (6), i.e.,uiαu

jα −

Bij (x) = 0, of which only six are independent.DefineF (1)(z, x) = 0 to be the set of equations consisting of the equations of

integrability of (9), i.e., the equations obtained by formally differentiating (9) withrespect toxα (assumingwi to be functions of{x}) and then eliminating the secondderivative ofwi using the relations

∂wi

∂xα∂xβ= ∂wi

∂xβ∂xα

and eliminating the first derivative ofwi using (9).In searching for solutions to (9), the equations in the setF (1)(z, x) = 0 are

either identically satisfied, or not. We first consider the former case.

100 A. ACHARYA

If the equations in the setF (1) = 0 are identically satisfied in the(z, x) space,then it is well known that there exists a unique solution to (9) (Thomas [7]) witharibitrarily assignable initial data at{x0}. Let the initial data forwi(x0) be chosensuch that the conditionsuiα(x0)u

jα(x0) = Bij (x0) are satisfied. We now show that

such a local solution of (9), which corresponds to a solution{uiα} to (8), is sufficientfor the existence of a solution to (6), (7).

First we note that since the solution to (8) is an invertible matrix at{x0}, it isinvertible in a neighborhood around{x0} due to the continuity of the solution. Also,as pointed out in the previous section, a solution to (7) necessarily exists, and thesolution is locally invertible as a function of{xα}.

We now assume that such a local solution to (7) has been constructed. With thesolutions of (8) and (7) in hand, we shall be able to prove that{uiα} that satisfy (8)also satisfy (6) by using a general property of the Riemannian geometry – that ofpreservation of angles between vector fields under parallel transport.

We denote the solution of (7) based on the solution{uiα} of (8) as{yi}. Define

(y)0irs :=

Bip

2

[∂Brp

∂ys+ ∂Bsp∂yr− ∂Brs∂yp

].

Clearly, the following relations hold

∂uiα

∂xβ= −(y)0irsurαusβ .

We now consider the expression

∂

∂ym

(Bijuαi u

β

j

),

whereuαi represent the components of the inverse ofuiα. If it can be shown that theabove expression is identically zero in a local neighborhood of{y(x0)}, then thechain rule and the choice of the initial data imply that (6) is indeed satisfied locallyaround{x0}. Now,

Bij∂uαi

∂ymuβ

j = −Bijuαk∂ukγ

∂xρuρmu

γ

i uβ

j = Bkj (y)0ikmuαi uβj ,

whereuαi uiβ = δαβ and the fact thatuiα satisfy (8) have been used. Consequently,

∂

∂ym

(Bijuαi u

β

j

) = [∂Bij∂ym+ Bkj (y)0ikm + Bik(y)0jkm

]uαi u

β

j = 0,

which follows from Ricci’s theorem (Sokolnikoff [6, p. 77 and p. 86]) – the termin the square parenthesis vanishes (merely by the smoothness and definition of the(y)0 field) since it is the covariant derivative of the contravariant metric tensor andthe fundamental tensors are “covariantly constant”.


Next we consider the case where the equations of the setF (1)(z, x) = 0 arenot identically satisfied in the(z, x) space. In such a case, we define the equa-

tions F(1)(z, x) = 0 as a system of equations consisting ofF (1)(z, x) = 0 and

F (0)(z, x) = 0. We also defineF(j+1)

(z, x) = 0 (j > 1) to be the set of equa-

tions obtained by formally differentiatingF(j)(z, x) = 0 with respect toxα and

eliminating the derivative ofzi by means of (9).We now consider two integersN andM, with N > 1 and 1 6 M 6 R. We

assume that

(1) there existM equations in the setsF(1) = 0 through F

(N) = 0 denoted byGλ = 0, λ = 1 to M, andM of the variableszi (from the list zi, i =1,2, . . . , R) denoted byzi, i = 1,2, . . . ,M which satisfy

det

[∂Gλ

∂zi(z, x)

]6= 0, i, λ = 1, . . . ,M, (12)

in the(z, x) space.Let the remainingR − M =: P variableszi, obtained by ignoringzi, i =1,2, . . . ,M, from the listzi, i = 1,2, . . . , R, be denoted byzj , j = 1,2,. . . , P . We also assume that, given a point(z1

0, . . . , zP0 , x

10, . . . , x

n0)

(2) the unique local solutionzi = ϕi(z1, . . . , zP, x), i = 1, . . . ,M, of Gλ = 0,λ = 1, . . . ,M, (which exists because of(12)) satisfies all the equations of

the setsF(1) = 0 throughF

(N+1) = 0 identically in a local neighborhood of(z1

0, . . . , zP0 , x

10, . . . , x

n0).

Under these assumptions, we show that there exists a local solution to(9) and(11) of the form(10) that is determined byP constants.

We denote the functionsψiα, i = 1, . . . ,M by ψ i

α. Similarly, we denote the

functionsψjα , j = 1, . . . , P , by ˜ψj

α. For each of the functionsGλ, ψ iα, and˜ψj

α of(z, x) we define the corresponding functions of(z, z, x) by the rules given below.Let5 be a function that delivers the list(z, x) corresponding to the list(z, z, x).We now define

Gλ(z, z, x) = Gλ

(5(z, z, x)

), λ = 1, . . . ,M,

ψi

α(z, z, x) = ψ iα

(5(z, z, x)

), i = 1, . . . ,M, α = 1, . . . , n,

ψjα (z, z, x) = ˜ψj

α

(5(z, z, x)

), j = 1, . . . , P, α = 1, . . . , n.

Since the equations of the setF(N+1) = 0 are satisfied, the relations

∂Gλ

∂zi

(ϕ(z, x

), z, x

)ψi

α

(ϕ(z, x

), z, x

)+ ∂Gλ

∂zj

(ϕ(z, x

), z, x

)ψjα

(ϕ(z, x

), z, x

)+ ∂Gλ

∂xα

(ϕ(z, x

), z, x

) = 0, i = 1, . . . ,M, j = 1, . . . , P (13)

102 A. ACHARYA

are satisfied in the variables(z1, . . . , zP , x). Choose an arbitrary pointxα0 and zj0,j = 1, . . . , P . Let f α(t) be a path in the{x} space withf α(0) = xα0 . Along thispath we seek the solution of the system

dgj

dt(t) = ψj

α

(ϕ(g(t), f (t)

), g(t), f (t)

)df αdt(t), j = 1, . . . , P,

(14)gj (0) = z

j

0.

From the existence theorem of ordinary differential equations, a local solution tosuch a system exists. With such a solution in hand, we note that

Gλ

(ϕ(g(t), f (t)

), g(t), f (t)

) = 0 (15)

is identically satisfied int , and differentiating the above expression yields

∂Gλ

∂zi

(ϕ(g(t), f (t)

), g(t), f (t)

){∂ϕi∂zj

(g(t), f (t)

)dgjdt(t)

+ ∂ϕi

∂xα

(g(t), f (t)

)df αdt(t)

}+ ∂Gλ

∂zj

(ϕ(g(t), f (t)

), g(t), f (t)

)dgjdt(t)

+ ∂Gλ

∂xα

(ϕ(g(t), f (t)

), g(t), f (t)

)df αdt(t) = 0. (16)

Combining (13) and (16) yields

∂Gλ

∂zi

(ϕ(g(t), f (t)

), g(t), f (t)

){ψi

α

(ϕ(g(t), f (t)

), g(t), f (t)

)df αdt(t)

− ∂ϕi

∂zj

(g(t), f (t)

)dgjdt(t)− ∂ϕi

∂xα

(g(t), f (t)

)df αdt(t)

}+ ∂Gλ

∂zj

(ϕ(g(t), f (t)

), g(t), f (t)

)×{ψjα

(ϕ(g(t), f (t)

), g(t), f (t)

)df αdt(t)− dgj

dt(t)

}= 0

and, noting the definition ofgj and the fact that the matrix(∂Gλ/∂zi)(z, x) is

invertible at all(z, x), we find that{ψi

α

(ϕ(g(0), f (0)

), g(0), f (0)

)− ∂ϕi∂zj

(g(0), f (0)

)× ψj

α

(ϕ(g(0), f (0)

), g(0), f (0)

)− ∂ϕi

∂xα

(g(0), f (0)

)}df α

dt(0) = 0 (17)

holds. Since{zj0, xα0 } and the path were chosen arbitrarily, (17) implies that

ψi

α

(ϕ(z, x

), z, x

)− ∂ϕi∂zj

(z, x

)ψjα

(ϕ(z, x

), z, x

)− ∂ϕi

∂xα

(z, x

) = 0 (18)


is satisfied identically in the variables(z1, . . . , zP , x).Now choose, arbitrarily, two distinct pointsxα0 andxα. Join these points by a

pathf α, parametrized by the variablet . Along any such path we seek the solu-tion of system (14) of ordinary differential equations with arbitrarily chosen initialconditions atxα0 , where the existence of a local solution is guaranteed.

According to the result of Thomas [7],gj defined by (14) are path-independentand satisfy

∂gj

∂xα(x) = ψj

α

(ϕ(g(x), x

)g(x), x

)(20)

if the conditions

∂ψkα

∂zi

(ϕ(z, x

), z, x

)(∂ϕi∂zj

(z, x

)ψj

β

(ϕ(z, x

), z, x

)+ ∂ϕi

∂xβ

(z, x

))+ ∂ψ

kα

∂xβ

(ϕ(z, x

), z, x

)+ ∂ψkα

∂zj

(ϕ(z, x

), z, x

)ψj

β

(ϕ(z, x

), z, x

)= ∂ψk

β

∂zi

(ϕ(z, x

), z, x

)(∂ϕi∂zj

(z, x

)ψjα

(ϕ(z, x

), z, x

)+ ∂ϕi

∂xα

(z, x

))+ ∂ψ

kβ

∂xα

(ϕ(z, x

), z, x

)+ ∂ψkβ

∂zj

(ϕ(z, x

), z, x

)ψjα

(ϕ(z, x

), z, x

), (21)

k = 1, . . . , P , are satisfied identically in the variables(z1, . . . , zP , x). Since theequations of the setF (1) = 0 are identically satisfied, we find that

∂ψkα

∂zi

(ϕ(z, x

), z, x

)ψi

β

(ϕ(z, x

), z, x

)+ ∂ψkα

∂xβ

(ϕ(z, x

), z, x

)+ ∂ψ

kα

∂zj

(ϕ(z, x

), z, x

)ψj

β

(ϕ(z, x

), z, x

)= ∂ψk

β

∂zi

(ϕ(z, x

), z, x

)ψi

α

(ϕ(z, x

), z, x

)+ ∂ψkβ

∂xα

(ϕ(z, x

), z, x

)+ ∂ψ

kβ

∂zj

(ϕ(z, x

), z, x

)ψjα

(ϕ(z, x

), z, x

), (22)

i = 1, . . . ,M; j, k = 1, . . . , P , holds. Consequently, (18) and (22) together implythat (21) holds.

We now consider the functions of{x} defined by

hi(x) = ϕi(g(x), x), i = 1, . . . ,M, (23)

and note that the relations

Gλ

(h(x), g(x), x

) = 0, λ = 1, . . . ,M, (24)

104 A. ACHARYA

are satisfied for all{x}. Differentiating (24) with respect toxα , we obtain

∂Gλ

∂zi

(h(x), g(x), x

) ∂hi∂xα

(x)+ ∂Gλ

∂zj

(h(x), g(x), x

) ∂gj∂xα

(x)

+ ∂Gλ

∂xα

(h(x), g(x), x

) = 0. (25)

Now, (13) yields

∂Gλ

∂zi

(h(x), g(x), x

)ψi

α

(h(x), g(x), x

)+ ∂Gλ

∂zj

(h(x), g(x), x

)ψjα

(h(x), g(x), x

) + ∂Gλ

∂xα

(h(x), g(x), x

) = 0. (26)

Subtracting (26) from (25), and noting (20) and the invertibility of

∂Gλ

∂zi

(h(x), g(x), x

),

we now find that the set of functionsh andg defined by (23) and the solution of (20)indeed satisfy (9). Noting the fact that (11) are some of the equationsF (1) = 0, itis clear that we have determined a solution to (9) and (11). The general solutionhasP arbitrary constants that enter as initial conditions in defining the functionsgj , j = 1, . . . , P .

The above proof is an expanded version of the proofs of sufficiency in Eisen-hart [3] and Veblen and Thomas [9]. The sufficient condition above, in the main,is claimed as necessary as well, in earlier works [3, 5, 8, 9]; their line of argumentcould not be followed to produce a proof of necessity of the sufficient conditionabove, for the existence of solutions. The main issue dealt with in this paper alsoappears to be related to Cartan’s Method of Equivalence (Gardner [4]).

Finally, an immediate necessary condition for the existence of a solution to thedifferential-algebraic system of Equations (9) and (11) is that there exist functionsof the form (10) which satisfy the purely algebraic system of equations

F(1)(w(x), x

) = 0, F(2)(w(x), x

) = 0, . . .

identically in the variables(x1, x2, . . . , xn).

Acknowledgement

I wish to thank Prof. D. E. Carlson for his suggestions on this work.

References

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2. F.P. Duda and L.C. Martins, Compatibility conditions for the Cauchy–Green strain fields:Solutions for the plane case.J. Elasticity39 (1995) 247–264.


3. L.P. Eisenhart,Non-Riemannian Geometry, Vol. VIII. Amer. Math. Soc. Colloquium, New York(1927).

4. R.B. Gardner, The method of equivalence and its applications. In:CBMS-NSF Regional Conf.Series in Appl. Math., Vol. 58. SIAM, Philadelphia (1989).

5. J.A. Schouten,Ricci-Calculus, 2nd edn. Springer, Berlin/Gottingen/Heidelberg (1954).6. I.S. Sokolnikoff,Tensor Analysis. Wiley, New York (1951).7. T.Y. Thomas, Systems of total differential equations defined over simply connected domains.

Ann. Math.35 (1934) 730–734.8. T.Y. Thomas and A.D. Michal, Differential invariants of relative quadratic differential forms.

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(1926) 279–296.

on compatibility conditions for the left cauchy–green deformation field in three dimensions

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