Meccanica (2008) 43: 279–288DOI 10.1007/s11012-007-9090-6

Convex Fung-type potentials for biological tissues

Salvatore Federico · Alfio Grillo ·Gaetano Giaquinta · Walter Herzog

Received: 20 December 2006 / Accepted: 16 October 2007 / Published online: 28 November 2007© Springer Science+Business Media B.V. 2007

Abstract The anisotropic, non-linear elastic behaviorof biological soft tissue is typically accounted for bythe hypothesis of hyperelasticity, i.e., the existence ofan elastic potential. Fung-type potentials, based on theexponential of a quadratic form in the components ofthe Green-Lagrange strain, have been widely used insoft tissue modeling, and have inspired potentials inwhich the exponential was replaced by other monoton-ically increasing functions. It has been shown that sim-ple fitting of the parameters of a Fung-type potentialto experimental stress-strain curves may lead to non-convexity, with undesirable effects on the reliability ofthe algorithms used in Finite Element simulations. Inthis paper, we prove that the necessary and sufficientcondition for the strict convexity of a Fung-type po-tential is that the quadratic form in the exponential is

S. Federico · W. Herzog (�)Human Performance Laboratory, Faculty of Kinesiology,The University of Calgary, 2500 University Drive NW,Calgary, AB, T2N 1N4, Canadae-mail: walter@kin.ucalgary.ca

A. Grillo · G. GiaquintaDipartimento di Metodologie Fisiche e Chimiche perl’Ingegneria, Facoltà di Ingegneria, Università di Catania,Viale Andrea Doria 6, 98125 Catania, Italy

A. Grillo · G. GiaquintaConsorzio Nazionale Interuniversitario per le ScienzeFisiche della Materia, Sezione di Catania, Viale AndreaDoria 6, 98125 Catania, Italy

positive definite. This result provides a clear physicalmeaning for the parameters featuring in the quadraticform, and their relationship with the small-strain elas-tic moduli. This consistency relationship must be re-spected in order to guarantee that the Fung-type po-tential correctly reduces to the quadratic potential ofclassic linear elasticity in the small-strain approxi-mation. Furthermore, we show that, when the condi-tions of convexity and consistency with the linear the-ory are respected, Fung-type potentials become a one-parameter family, and we discuss the consequences ofthis result for when fitting experimental data.

Keywords Elasticity · Anisotropy · Convexity ·Biological tissue · Continuum mechanics

1 Introduction

Biological tissues can be divided into hard and soft,and they can be considered as a mixture comprisedof an anisotropic solid phase and a fluid phase whichfills the pores of the solid. The distinction betweenhard and soft tissues is made because of the extentto which tissues can undergo deformation before fail-ure. Hard tissues (e.g., bone) have brittle behavior andthey reach mechanical failure within the range of smallelastic deformations [6, 13]. Therefore, they are typi-cally treated as linearly elastic materials, in the realmof classical small-strain elasticity. On the contrary, soft

280 Meccanica (2008) 43: 279–288

tissues can experience non-linear elastic deformationsunder physiological conditions. For cases in whichtheir intrinsic viscoelasticity is negligible, the solidphase of soft tissues can be treated within the frame-work of non-linear finite elasticity [6, 13]. Virtuallyall biological tissues exhibit anisotropy, which, for softtissues, is caused by the presence and function-specificarrangement of collagen fibers. Therefore, under thecommon assumption of existence of an elastic poten-tial, soft tissues are modeled as anisotropic hyperelas-tic materials.

Among the many forms of elastic potentials thathave been proposed for the description of soft tissuebehavior, we would like to focus on the family of expo-nential potentials introduced by Y.C. Fung, that havebeen largely applied to blood vessels and skin, and thepotentials that were developed based on Fung’s idea.From uni-axial tensile test data from rabbit mesentery,Fung [3] discovered that a linear relationship exists be-tween the first Piola-Kirchhoff stress in the axial di-rection, T , and its derivative with respect to the axialstrain, E:

T ′(E) = αT (E) + β. (1.1)

By integration, and by imposing T (0) = 0, one obtains

T (E) = β

α(eαE − 1). (1.2)

Based on this result, Fung and his collaborators sug-gested several forms of an exponential elastic poten-tial, as a function of the Green-Lagrange strain, E

(e.g., Fung [4], Tong and Fung [20] and Fung et al.[5]), the most common of which can be described byan expression reported by Humphrey [9]:

W(E) = c

2[exp(Q(E)) − 1], (1.3)

where c is a positive constant with the physical dimen-sions of a strain energy, and Q is a quadratic form inthe components of E:

Q(E) = a1E211 + a2E

222 + a3E

233

+ 2a4E11E22 + 2a5E11E33 + 2a6E22E33

+ a7E212 + a8E

213 + a9E

223. (1.4)

The convexity of these Fung-type potentials with re-spect to E depends on the choice of the coefficientsof the quadratic form Q. As shown by Holzapfel et

al. [7], and later remarked by Humphrey [10], simplefitting of the parameters featuring in (1.3) and (1.4)to experimental stress-strain curves may lead to non-convexity of the potential. In several cases in the litera-ture, the parameters of Fung-type potentials have beenfitted to the experimental curves, without checking forconvexity.

Convexity with respect to E is an essential condi-tion for reliability of routines used in Finite Elementanalysis. Indeed, the stiffness matrices in Finite Ele-ment routines are typically based on the elasticity ten-sor, CCC [7, 8, 14, 15, 18] (which is the second derivativeof W with respect to E: see the definition in Sect. 2.4),and reliability of the simulations is guaranteed only ifthe stiffness matrices are positive definite, which hap-pens only when CCC is positive definite at every E, i.e.,if, W is convex at every E.

Sun and Sacks [18] attempted to formulate condi-tions for the convexity of Fung-type potentials. Basedon considerations made on the linearization of Fung’spotential (in the neighborhood of zero deformation),they reported some necessary conditions that the coef-ficients of their two-dimensional Fung-type potentialneeded to satisfy in order to obtain convexity.

In this study, we prove that the necessary and suf-ficient condition for a Fung-type potential (1.3) to beconvex with respect to E is that the quadratic formQ (see (1.4)) is positive definite. This result has sev-eral important consequences, which have not been dis-cussed in the literature.

The “necessary part” of our proof is based on therequirement of consistency of a non-linear potentialwith the quadratic potential representing its lineariza-tion. Indeed, a non-linear theory must reduce to its lin-earization in the limit of small strains, and this relatesthe parameters of the non-linear potential to the elasticconstants of the linear theory.

The first consequence of the condition of consis-tency with the linear theory is that the coefficients inthe quadratic form in a Fung-type potential are propor-tional to the linear elastic constants. This result pro-vides a precise physical meaning for these parame-ters, which has been lacking [6, 7]. The second con-sequence is that, once the parameters in the quadraticform are fixed by the condition of consistency withthe linear theory, Fung’s potential becomes a one-parameter functional family, with only one “free” pa-rameter available to fit experimental curves. Since oneparameter may not be enough for curve-fitting, we

Meccanica (2008) 43: 279–288 281

suggest a method to solve this problem while preserv-ing the physical requirements outlined above. Finally,we discuss the case of incompressibility, which is par-ticularly important for soft tissues.

2 Theoretical background

In this section, we recall some basics of the algebraof fourth-order tensors, functions of second-order ten-sors, the basic concepts of convexity, the definition ofthe elasticity tensor for finite strain and linear theory,and the decomposition of the strain into volumetricand distortional components. Finally, we introduce thenotation used in this paper for Fung-type potentials,and their generalization, which includes potentials inwhich the exponential is replaced by other monotoni-cally increasing functions, as in the potential proposedby Takamizawa and Hayashi [19].

2.1 Basic notation and conventions

The set of real numbers and the natural three-dimen-sional Euclidean space are denoted by R and E = R

3,respectively. The spaces of second- and fourth-ordertensors are denoted by E2 and E4, respectively. SinceCartesian coordinates are used throughout, no dis-tinction is made between covariant, contravariant, andmixed components of vectors and tensors. Also, Ein-stein summation convention for repeated indices is un-derstood.

Vectors and second-order tensors are indicated byboldface (a,b,A,B, . . .), whereas tensors with orderhigher than the second (fourth-order tensors, in par-ticular) are indicated by blackboard style (AAA,BBB, . . .).The tensor product is indicated by ⊗, and the colon“:” denotes double contraction of two tensors, i.e.,contraction of the last two indices of tensor AAA andthe first two indices of tensor BBB in the expressionAAA : BBB = Aij...rsBrs...kl . If A and B are second-ordertensors, then A : B = tr(ABT ) = tr(AT B) = AijBij .The second- and fourth-order null and identity tensorsare denoted O and I , and OOO and III, respectively.

2.2 Second- and fourth-order tensors

Tensors are multi-linear applications on E × · · · × E

into R. In particular, a fourth-order tensor TTT is a four-linear form on E × E × E × E into R. Alternatively,

and this is of interest here, it can be seen either as alinear operator on E2

TTT : E2 → E2 : A �→ TTTA = (TijklAkl)3i,j=1, (2.1)

or as a bilinear form on E2 × E2 into R,

TTT : E2 × E2 → R : (A,B) �→ A : (TTTB) = TijklAijBkl.

(2.2)

Whereas the components of second-order tensors canonly possess diagonal symmetry, i.e., A ∈ E2 is sym-metric if A = AT ⇒ Aij = Aji , the components offourth-order tensors can have two symmetries [1, 23].A fourth-order tensor, TTT ∈ E4, is diagonally symmet-ric if its components are invariant under interchange ofthe first and second pair of indices, i.e., Tijkl = Tklij ,and it is pair-symmetric if its components are in-variant under interchange of indices within the samepair, i.e., Tijkl = Tjikl = Tijlk = Tjilk . Finally, TTT isfull-symmetric if it is diagonally and pair symmetric.The full-symmetric identity tensor, III, has componentsIijkl = 1

2 (δikδjl + δilδjk) [23].Any full-symmetric fourth-order tensor TTT defines a

quadratic form

T (A) = A : (TTTA). (2.3)

The quadratic form T and the tensor TTT are said to bepositive semi-definite if, and only if, for every second-order tensor A ∈ E2, different from the null tensor O ,

T (A) = A : (TTTA) = TijklAijAkl ≥ 0. (2.4)

When the inequality is strict, T and TTT are said to bepositive definite.

If f : E2 → R : E �→ f (E) is a twice differentiablefunction of a second-order tensor, its first and secondderivatives are a second- and a fourth-order tensor, re-spectively, and are denoted by

∂Ef : E2 → E2 : E �→ ∂Ef (E)

=[

∂f

∂Eij

(E)

]3

i,j=1, (2.5)

∂2E2f : E2 → E4 : E �→ ∂2

E2f (E)

=[

∂2f

∂Eij ∂Ekl

(E)

]3

i,j,k,l=1. (2.6)

282 Meccanica (2008) 43: 279–288

Full-symmetric isotropic fourth-order tensors consti-tute a two-dimensional subspace of E4, a basis ofwhich can be found, e.g., in the work of Walpole [23]:

KKK = 1

3I ⊗ I ,

MMM = III − 1

3I ⊗ I .

(2.7)

It is easy to prove that KKK and MMM are “orthogonal”, inthe sense that KKK : MMM = MMM : KKK = OOO, and idempotent,i.e., KKK : KKK = KKK,MMM : MMM = MMM [23]. A full-symmetrictensor TTT can be written as the linear combination of KKK

and MMM:

TTT = 3κKKK + 2μMMM. (2.8)

If TTT is the elasticity tensor for isotropic linear elastic-ity, then κ and μ take the physical meaning of bulkmodulus and shear modulus, respectively.

Note that in Walpole’s work [23] KKK and MMM are indi-cated by J and K , respectively. Here we use a differentnotation to have a “visual” correspondence between κ

and KKK, and μ and MMM.

2.3 Convexity

Let V be a vector space on R, and D an open subset ofV . D is said to be convex if, for every x,y ∈ D, withx = y, the segment Sxy connecting x and y,

Sxy = {z ∈ V |z = (1 − λ)x + λy, λ ∈]0,1[}, (2.9)

is all included in D. A function f : D ⊆ V → R isconvex on D if, for every λ ∈]0,1[ and for everyx,y ∈ D, with x = y (see, e.g., [12]),

f ((1 − λ)x + λy) ≤ (1 − λ)f (x) + λf (y). (2.10)

If the inequality (2.10) is strict, function f is said tobe strictly convex. If f is twice differentiable on D, itcan be shown that f is convex on D if, and only if, forevery x ∈ D, its Hessian matrix, defined by

H f (x) = [∂2ij f (x)]3

i,j=1, (2.11)

is positive semi-definite. If f is strictly convex on D,the Hessian is positive definite on D, except, at most,at the point of absolute minimum of f , at which theHessian might be positive semi-definite. Note that, ifthe vector space V is the space E2 of second-ordertensors, then the Hessian is the fourth-order tensorHHHf (E) = ∂2

E2f (E), defined as in (2.6).

2.4 Hyperelasticity: stress and elasticity tensor

The fundamental measure of the deformation, fromwhich all measures of strain derive, is the deformationgradient, F . From Cauchy’s polar decomposition the-orem follows the definition of the right Cauchy stretchtensor, C = F T F , which in turn features in the defin-ition of the Green-Lagrange strain, E = 1

2 (C − I ).For the case of a hyperelastic material, a given mea-

sure of stress can be obtained by differentiation of astrain energy potential with respect to the conjugatedstrain. If the potential, W , is written as a function ofthe Green-Lagrange strain, E, then the first derivativeof W is the second Piola-Kirchhoff stress, S,

S = ∂EW(E), (2.12)

and the second derivative defines the elasticity ten-sor, CCC,

CCC(E) = ∂2E2W(E). (2.13)

The elasticity tensor of the classical linear elasticitytheory, LLL, is obtained by Taylor series expansion of W

in the neighborhood of the undeformed configuration,E = O , and truncation to the quadratic order:

W(E) � W(O) + [∂EW(O)] : E+ 1

2E : [(∂2

E2W(O))E]. (2.14)

Since W(O) is an inessential additive constant and∂EW(O) is the second Piola-Kirchhoff stress (see(2.12)), which vanishes in a stress-free undeformedconfiguration, (2.14) reduces to the quadratic potentialof the linear theory:

W(E) � Wlin(E) = 1

2E : (LLLE), (2.15)

where

LLL = CCC(O) = ∂2E2W(O). (2.16)

It is important to note that a linearly elastic material isconvex (strictly convex) if, and only if, the elasticitytensor LLL is positive semi-definite (positive definite).

2.5 Volumetric and distortional strain

It is often convenient to separate the volumetric andthe distortional part of the deformation. Considering

Meccanica (2008) 43: 279–288 283

that J = detF represents the volumetric strain, Flory[2] devised a multiplicative decomposition of F , intoits volumetric and distortional components

F = (J 1/3I )F = J 1/3F . (2.17)

From (2.17) it is possible to derive the correspondingdecomposition for C

C = F T F = J 2/3FTF = J 2/3C, (2.18)

and for E

E = 1

2(C − I ) = J 2/3E + 1

2(J 2/3 − 1)I , (2.19)

E = 1

2(C − I ). (2.20)

It is useful to calculate the derivative of E with respectto E [7]:

∂EE = J−2/3(

III − 1

3C ⊗ C

−T)

. (2.21)

We note that, in the undeformed state, C = I ,E =O, ∂EE reduces to tensor MMM defined in (2.7)2.

2.6 Generalized Fung-type potentials in tensornotation

We write the Fung-type potential in (1.3) in the form

W(E) = a[exp(Q(E)) − 1], (2.22)

where the positive factor c/2 was rescaled to the newpositive constant a, with the same physical dimen-sions (energy density). Following the original idea ofFung [4] and similarly as done by Humphrey [10], wewrite the quadratic form in (1.4) by means of a full-symmetric fourth-order tensor QQQ:

Q(E) = 1

2E : (QQQE) = 1

2QijklEijEkl. (2.23)

Note that we prefer having a 1/2 factor in the quadraticform because in this way, apart from the physicaldimensions (the coefficients of QQQ, and therefore Q,are non-dimensional), Q resembles the quadratic po-tential of linear elasticity (see (2.15)). The use of afourth-order tensor in place of second-order polyno-mials has the advantage of enabling the exploitation

of tensor algebraic properties to study material sym-metry and convexity. The material symmetry (trans-verse isotropy, orthotropy, etc.) of the potential W isimmediately recognizable as that of tensor QQQ.

We also note that function exp(·) − 1, on whichFung-type potentials are based, is twice differentiable,strictly monotonically increasing and strictly convexon R. Therefore a generalized form of the poten-tial (2.22) can be obtained by substituting functionexp(·) − 1 by any other function g, which is si-multaneously twice differentiable, strictly monotoni-cally increasing and strictly convex on an open setD ⊆ R [24]:

W(E) = ag(Q(E)) = ag

(1

2E : (QQQE)

). (2.24)

This generalization of Fung-type potentials includes,for example, the Takamizawa-Hayashi potential [19],which is obtained if g = − ln[1 − (·)].

3 Convexity of generalized Fung-type potentials

In this section, we prove that the necessary and suffi-cient condition for a generalized Fung-type potential(2.24) to be strictly convex with respect to the Green-Lagrange strain, E, is that the fourth-order tensor QQQ inthe quadratic form Q is positive definite. For the proofof the “necessary part”, we shall exploit the physicalrequirement of consistency of a non-linear elastic po-tential with its linearization, i.e., the fact that an elasticpotential must reduce to the quadratic potential of thelinear theory, in the small-strain approximation.

3.1 Sufficient condition

Here we prove that if QQQ is positive definite, then W isstrictly convex.

First, we note that the quadratic form Q is convex(strictly convex) with respect to E if, and only if, ten-sor QQQ is positive semi-definite (positive definite). In-deed, the second derivative of Q is

∂2E2Q(E) = QQQ, ∀E ∈ E2, (3.1)

and thus, if QQQ is positive semi-definite (positive defi-nite), ∂2

E2Q is everywhere positive semi-definite, andQ is everywhere convex (strictly convex).

284 Meccanica (2008) 43: 279–288

Then, our thesis is proved if we show that the com-position of a twice differentiable, strictly monoton-ically increasing, strictly convex function g with atwice differentiable, strictly convex function h isstrictly convex. For functions h defined on R, this is astandard result of basic calculus. In the following, weprove the theorem for functions h defined on the spaceE2 of second-order tensors (the case of functions de-fined on E = R

3 or Rn is analogous).

Theorem Let D ⊆ R be an open set, and g : D ⊆R → R and h : E2 → R be twice differentiable. If g

is strictly monotonically increasing and strictly convexon D, and h is strictly convex on E2, then f = g ◦ h isstrictly convex on E2.

Proof The first derivative of f is given by

∂Ef = ∂E[g ◦ h] = [g′ ◦ h]∂Eh, (3.2)

and the second derivative is

∂2E2f = [g′′ ◦ h]∂Eh ⊗ ∂Eh + [g′ ◦ h]∂2

E2h. (3.3)

Multiplication on the left and on the right by any non-null tensor A ∈ E2\{O}, yields

A : [(∂2E2f )A] = [g′′ ◦ h]A : [(∂Eh ⊗ ∂Eh)A]

+ [g′ ◦ h]A : [(∂2E2h)A], (3.4)

which reduces to

A : [(∂2E2f )A] = [g′′ ◦ h](A : ∂Eh)2

+ [g′ ◦ h]A : [(∂2E2h)A]. (3.5)

Because g is strictly convex and strictly monotonicallyincreasing on D, then [g′′ ◦ h] > 0 and [g′ ◦ h] > 0.Thus, the right hand side of (3.5) is the positive lin-ear combination of (A : ∂Eh)2 ≥ 0 (the equality holdswhen A and ∂Eh are orthogonal) and of the quadraticform A : [(∂2

E2h)A], which is positive definite becauseof the strict convexity of h. Thus

A : [(∂2E2f )A] > 0,A ∈ E2\{O}, (3.6)

which proves the strict convexity of f on E2. �

By using this result, and replacing f by a−1W andh by Q, we deduce that the positive definiteness of thefourth-order tensor QQQ implies strict convexity of thegeneralized Fung-type potential (2.24).

3.2 Necessary condition

Here we prove that if W is strictly convex, then QQQ ispositive definite.

Taylor expansion of the generalized Fung-type po-tential in (2.24), in the neighborhood of the unde-formed configuration (E = O), and up to the quadraticorder, yields (see (2.15))

W(E) � Wlin(E) = 1

2ag′(0)E : (QQQE). (3.7)

Comparison with (2.15) gives

QQQ = [ag′(0)]−1LLL. (3.8)

Now, the strict convexity of W implies that of Wlin

and hence the positive definiteness of tensor LLL. Fur-thermore, a must be a strictly positive number andg′(0) is strictly positive, as g is strictly monotonicallyincreasing by hypothesis. Therefore, the positive defi-niteness of LLL implies that of QQQ.

We note that, for Fung-type and Takamizawa-Hayashi potentials, we have, respectively

g(x) = exp(x) − 1 ⇒ g′(0) = 1, (3.9)

g(x) = − ln(1 − x) ⇒ g′(0) = 1, (3.10)

and, in both cases, (3.8) reduces to

QQQ = a−1LLL. (3.11)

4 Consequences of convexity

In the previous section, we have proved that a general-ized Fung-type potential (see (2.24))

W(E) = ag(Q(E)) = ag

(1

2E : (QQQE)

)(4.1)

is strictly convex if, and only if, its quadratic form Q(and thus tensor QQQ) is positive definite. This result isfairly evident, but its proof offers interesting new in-sights.

4.1 Consistency with the linear theory

When proving the “necessary” part of our theorem(Sect. 3.2), we showed that, for W to reduce to the

Meccanica (2008) 43: 279–288 285

quadratic potential of the linear theory in the small-strain approximation, QQQ must be proportional to theelasticity tensor of the linear theory, LLL (see (3.8)or (3.11)).

Now, let us imagine to fit a generalized Fung-typepotential to an experimental curve, and to obtain afourth-order tensor for the quadratic form Q, whichwe call QQQ

∗, and a positive multiplicative constant,which we call a∗. Also, let us assume that we putconstraints on our optimization algorithm in order toobtain a positive definite QQQ

∗, and therefore a con-vex potential. The problem is that there is no guar-antee that the QQQ

∗ found in this way obeys condi-tion (3.8). Therefore, by linearizing the potential con-structed with tensor QQQ

∗, we obtain a small-strain elas-ticity tensor LLL

∗ = a∗g′(0)QQQ∗, which in general differsfrom the real small-strain elasticity tensor, LLL, obtainedby small-strain experiments.

It is evident that from the requirement of con-sistency with the linear theory follows another con-straint on the components of QQQ, aside from that follow-ing from convexity. This constraint implies a physicalmeaning for QQQ and its components, which has beenmissing so far [6, 7].

By using (3.8), the potential (2.24) can be written as

W(E) = ag

(1

2[ag′(0)]−1E : (LLLE)

). (4.2)

Equations (3.8) and (4.2) show that, in order to re-spect the condition that an elastic potential should re-duce to the quadratic potential of the linear theoryin small strains, all coefficients in the quadratic formQ(E) = 1

2 [ag′(0)]−1E : (LLLE) are fixed, up to the con-stant term ag′(0). Therefore, for a given LLL, the onlyway to fit experimental non-linear stress-strain curvesis by varying the constant a, which features in the ex-pression of the stress as

S = [ag′(0)]−1[ag′

(1

2[ag′(0)]−1E : (LLLE)

)]LLLE.

(4.3)

This means that once LLL is fixed, say, by experimentalmeasurement of the linear elastic moduli, W becomesa one-parameter family of potentials. The parameter a

is the only residual “degree of freedom” to fit large-strain experimental curves, which may not be enoughfor the description of complex biological materials.

4.2 Fitting experimental curves

On the one hand, we have two essential requirements:convexity and consistency with the linear theory in thelimit of small strains; on the other hand, we need morethan one parameter to fit experimental curves. Severalattempts have been made to increase the number of the“free” parameters, and here we briefly recall some.

Tong and Fung [20] added third-order terms in theargument of the exponential function, and added aquadratic function:

W(E) = α1E211 + α2E

222 + 2α4E11E22

+ c exp(a1E211 + a2E

222 + a3E

212

+ 2a4E11E22 + γ1E311 + γ2E

322

+ γ4E211E22 + γ5E11E

211). (4.4)

This idea probably came from the potential proposedby Vaishnav et al. [22], a third-order polynomial in thediagonal components of E:

W(E) = c1E211 + c2E11E22 + c3E

222 + c4E

311

+ c5E211E22 + c6E11E

222 + c7E

322. (4.5)

However, as pointed out by Holzapfel et al. [7], be-cause of the cubic nature of the polynomial potential(4.5), there is no set of constants that ensures its con-vexity, except for some limited range of strains, or forthe trivial case in which all the coefficients of the cubicterms vanish. The exponential potential (4.4) of Tongand Fung [20] has the same limitation. Another gen-eralization was proposed by Fung [4] and expressedby Tong and Fung [20] as a function of the Green-Lagrange strain, E:

W(E) = 1

2E : (AAAE) + β0E : (BBBE)

× exp

(M : E + 1

2E : (GGGE) + · · ·

). (4.6)

The fourth-order tensors AAA,BBB,GGG, the second-ordertensor M , and the constant β0 must be determined ex-perimentally. Jemioło and Telega [9] remarked that thepotential in (4.6) contains too many parameters withsimilar physical meaning. This is true because the elas-ticity tensor LLL of the small strain theory is given by acombination of the tensors featuring in (4.6). There-fore, after fitting the potential to experimental stress-strain curves, some of these tensors may not be posi-tive definite.

286 Meccanica (2008) 43: 279–288

In the following, we propose a method to overcomethe difficulties outlined above.

4.3 Proposed solution: zero-zero convex elasticpotentials

A possible way to increase the number of free para-meters in a generalized Fung-type potential, while ful-filling the requirement of convexity and consistencywith the linear theory, is to add one or more convex“zero-zero” terms, i.e., convex functions Φ with zerostiffness at zero strain, such that ∂2

E2Φ(O) = OOO. Sucha term does not contribute to the elasticity tensor of thelinear theory, LLL, as can be shown by differentiating thepotential

W(E) = ag

(1

2E : (QQQE)

)+ Φ(E), (4.7)

and evaluating it at E = O:

∂2E2W(O) = LLL = ag′(0)QQQ + ∂2

E2Φ(O)

= ag′(0)QQQ +OOO = ag′(0)QQQ. (4.8)

Therefore, the positive definiteness of QQQ and its rela-tion to LLL are preserved.

An example of a convex zero-zero term can be

Φ(E) = b[Z(E)]n = b

[1

2E : (ZZZE)

]n

, (4.9)

where b is a material constant with the physical dimen-sions of an energy density, Z is a quadratic form in E,ZZZ is its full-symmetric, positive semi-definite fourth-order tensor, and n ≥ 2 is a positive integer. Consider-ing that ∂EZ(E) = ZZZE and ∂2

EZ(E) = ZZZ, it followsthat the second derivative of Φ ,

∂2E2Φ(E) = bn(n − 1)[Z(E)]n−2(ZZZE) ⊗ (ZZZE)

+ bn[Z(E)]n−1ZZZ, (4.10)

vanishes identically at E = O , i.e., Φ is zero-zero.Furthermore, function Φ is convex by virtue of thetheorem proved in Sect. 3, as it is the compositionof a positive semi-definite quadratic form, and the n-power function, (·)n, is strictly convex and strictlymonotonically increasing on the set of positive realnumbers, R

+.This method prevents the drawback characteriz-

ing the potential of Tong and Fung [20] (see (4.6)),

in which too many parameters had a similar physi-cal meaning [11]. Indeed, by using the additive term(4.9), by virtue of the zero-zero property, tensor ZZZ hasno effect on the linear elastic constants, which featuresolely in the tensor QQQ of the original potential.

For the case of isotropic materials, tensor ZZZ can beone of the two positive semi-definite tensors KKK andMMM of the fourth-order tensor basis in the subspace ofisotropic tensors, defined in Sect. 2.5, or a positive lin-ear combination of them. For the case of transverseisotropy with respect to a given direction, w, tensor ZZZ

can be chosen, for example, as a positive linear com-bination of the first four of the positive semi-definitetensors of the fourth-order tensor basis relative to di-rection w [1, 23].

Finally, we note that, if we expand a generalizedFung-type potential in Taylor series in the quadraticform Q(E), in the neighborhood of zero, we obtain:

W(E) = ag(Q(E)) = a

∞∑m=0

g(m)(0)

m! [Q(E)]m. (4.11)

It is evident that adding a function like (4.9) to a Fung-type potential means “adjusting” the term with m = n

in the series expansion.

5 Incompressible Fung-type potentials

The solid phase of soft tissues is often assumed tobe incompressible. For the case of incompressibility,an elastic potential can be redefined as a function ofthe purely distortional strain E (see (2.19) and (2.20))[7], thus incorporating the incompressibility constraintJ = detF = 1. An incompressible generalized Fung-type potential reads then

W(E) = W (E) = a[exp(Q(E)) − 1]

= a

[exp

(1

2E : (QQQE)

)− 1

]. (5.1)

Let us now calculate the elasticity tensor, CCC, secondderivative of W with respect to E:

CCC = ∂2EW

= (∂EE)T : (∂2E

2W : ∂EE) + ∂EW : ∂2E2E, (5.2)

where [(∂EE)T ]ijpq = [∂EE]pqij . If (5.2) is evaluatedin the undeformed configuration (E = E = O), then

Meccanica (2008) 43: 279–288 287

∂EW vanishes identically, ∂EE reduces to tensor MMM

defined in (2.7)2,CCC = ∂2EW becomes the linear elas-

ticity tensor LLL, and ∂2E

2W becomes tensor ag′(0)QQQ.

Therefore, we obtain:

LLL = ag′(0)MMM : QQQ : MMM. (5.3)

Now the question becomes: is the relationship betweenLLL and QQQ still univocal? For the sake of simplicity, weshall answer this question for the case of an isotropicmaterial. In this case, LLL and QQQ are given by the decom-position (2.8):

LLL = 3κKKK + 2μMMM, (5.4)

QQQ = 3kKKK + 2mMMM. (5.5)

However, because of the incompressibility constraint,the bulk modulus κ in (5.4) is indeterminate. This in-determinacy is transferred to the hydrostatic compo-nent of the stress, in which the pressure p features as aLagrange multiplier (see, e.g., [7]). Therefore, for in-compressible isotropic materials, the volumetric com-ponent 3κKKK of LLL drops, and LLL becomes simply:

LLL = 2μMMM. (5.6)

At this point, we note that the relationship (5.3) be-tween LLL and QQQ is “saved” by the action of tensor MMM onQQQ, in (5.5). Indeed, MMM filters out the volumetric com-ponent of QQQ, which is immediately seen from the factthat tensors KKK and MMM are “orthogonal” and idempotent(see Sect. 2.5):

MMM : QQQ : MMM = 3kMMM : KKK : MMM + 2mMMM : MMM : MMM= 2mMMM. (5.7)

By substituting (5.6) and (5.7) into (5.3) we find

2μ = ag′(0)2m, (5.8)

which shows that, also for incompressible materials, arelationship holds between the components of QQQ andthe components of LLL, and it is univocal if one setsk = 0 in the definition of QQQ (see (5.5)). This result, ob-tained for isotropic materials, holds in general. For anincompressible material, the volumetric component oftensor QQQ, which would be filtered out in the operationMMM : QQQ : MMM, must be set to zero.

6 Discussion

A rigorous proof of the conditions under which Fung-type potentials are convex with respect to the Green-Lagrange strain, E, was missing. Nonetheless, we notethat Wilber and Walton [24] studied the conditions forthe strong ellipticity with respect to the deformationgradient, F , of generalized Fung-type potentials, acondition which is related to the reality of elastic wavespeeds, and existence-uniqueness problems in linearelasticity (e.g., [12, 21]). Strong ellipticity and convex-ity in E in general do not imply each other (e.g., [16,24]). Indeed, there are cases in which a strongly ellip-tic potential is non-convex in E, thus implying that thefinite strain elasticity tensor CCCCCCCCC is non-definite. In thiswork, we are interested in convexity in E because itis the only condition that ensures the positive definite-ness of CCCCCCCCC, on which the stiffness matrices of Finite El-ement routines are typically based, e.g., [7, 8, 14, 15,18]. The positive definiteness of CCCCCCCCC implies the posi-tive definiteness of the stiffness matrices, and thus thereliability of numerical simulations.

We have shown that strict convexity of a gener-alized Fung-type potential is guaranteed if the quad-ratic form Q(E) = 1

2E : (QQQE), based on the full-symmetric fourth-order tensor QQQ, is positive definite.We also showed that, for potential (2.24) to correctlyreduce to the quadratic potential of linear elasticity(2.15) in the limit of small strains, QQQ must be pro-portional to the linear elasticity tensor LLL (see (3.8)or (3.11)), a relationship that, with suitable modifica-tions, remains valid for incompressible materials (see(5.3)). This result implies a clear physical meaningfor QQQ, which had not been known [6], and empha-sizes that the consistency with the linear theory mustalways be respected when fitting a non-linear potentialto large-strain experimental data [17].

The condition of consistency with the linear the-ory implies that, once LLL is fixed by small-strain ex-perimental measurements, only the parameter a in(2.24) remains “free” for fitting large-strain experi-mental stress-strain curves. Since, for the characteri-zation of a complex, anisotropic biological material,one parameter is not sufficient in general, it is essen-tial to find ways to increase the number of “degrees offreedom” for curve-fitting.

We proposed the addition of “zero-zero” potentials,i.e., potentials with zero stiffness at zero strain, whichpreserve the relationship between tensors QQQ and LLL,

288 Meccanica (2008) 43: 279–288

and thus preserve the consistency with the linear the-ory, while making more material parameters avail-able for curve-fitting. This approach to generalizedFung-type potentials (see (2.24)) provides a basis forsoft-tissue modeling that ensures convexity and con-sistency with the linear theory in the limit of smallstrains, which are essential requirements for numeri-cal applications and for the theoretical correctness of aconstitutive theory. Naturally, zero-zero potentials canbe used in a perfectly analogous way for other materialmodels.

Acknowledgements We gratefully acknowledge Dr. Ray-mond W. Ogden, for crucial references and discussions. TheCanada Research Chair Programme, the Alberta Ingenuity Fund(AIF, Canada), the Alberta Heritage Foundation for Medical Re-search (AHFMR, Canada), Consorzio Nazionale Interuniversi-tario per le Scienze Fisiche della Materia (CNISM—NationalInter-University Consortium for the Physical Sciences of Mat-ter, Italy).

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