An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

Journal of the Mechanics and Physics of Solids 48 (2000) 2445–2465 www.elsevier.com/locate/jmps

An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity John C. Criscione a,*, Jay D. Humphrey b, Andrew S. Douglas a, William C. Hunter a a

Departments of Biomedical and Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21205, USA b Biomedical Engineering Program, Texas A&M University, College Station, TX 77843, USA Received 20 March 1999; received in revised form 1 March 2000

Abstract A novel constitutive formulation is developed for finitely deforming hyperelastic materials that exhibit isotropic behavior with respect to a reference configuration. The strain energy per unit reference volume, W, is defined in terms of three natural strain invariants, K1–3, which respectively specify the amount-of-dilatation, the magnitude-of-distortion, and the mode-ofdistortion. Distortion is that part of the deformation that does not dilate. Moreover, pure dilatation (K2=0), pure shear (K3=0), uniaxial extension (K3=1), and uniaxial contraction (K3=⫺1) are tests which hold a strain invariant constant. Through an analysis of previously published data, it is shown for rubber that this new approach allows W to be easily determined with improved accuracy. Albeit useful for large and small strains, distinct advantage is shown for moderate strains (e.g. 2–25%). Central to this work is the orthogonal nature of the invariant basis. If h represents natural strain, then {K1,K2,K3} are such that the tensorial contraction of (∂Ki/∂h) with (∂Kj/∂h) vanishes when i⫽j. This result, in turn, allows the Cauchy stress t to be expressed as the sum of three response terms that are mutually orthogonal. In particular (summation implied) t=Ai∂W/∂Ki, where the ∂W/∂Ki are scalar response functions and the Ai are kinematic tensors that are mutually orthogonal.  2000 Elsevier Science Ltd. All rights reserved. Keywords: B. Finite strain; B. Elastic material; B. Rubber material; C. Energy methods; C. Mechanical testing

* Corresponding author. Fax: +1-858-534-0522. E-mail address: [email protected] (J.C. Criscione). 0022-5096/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 0 0 ) 0 0 0 2 3 - 5

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1. Introduction For hyperelastic materials that undergo finite deformation, the constitutive behavior is characterized by a strain energy density function W which depends on the local deformation gradient tensor F. It is advantageous to use, as arguments of W, scalars that depend on strain yet are invariant under coordinate transformation. For materials exhibiting behavior that is isotropic with respect to the reference configuration, three scalars are needed. Typically, W is defined by either the principal invariants {I1,I2,I3} of C(=FTF) as in Rivlin (1948) or the principal stretches {l1,l2,l3} of V (Ogden 1972, 1984, and 1986) where F=VR represents polar decomposition. With such an approach, the Cauchy (or true) stress t can be expressed as the sum of three response terms, each of which consists of a kinematic tensor multiplied by a scalar partial derivative of W with respect to an argument. With W(I1,I2,I3), for example (Atkin and Fox, 1980)

冘

∂W A, ∂Ii i i⫽1 3

t⫽

(1.1)

wherein, with B=FFT and I as the identity tensor, A1⫽2I−1/2 B, A2⫽2I−1/2 (I1B⫺B2), A3⫽2I31/2I. 3 3

(1.2)

Although it may depend on all three invariants, a particular ∂W/∂Ii is often referred to as the Ii response function. In general, the Ai in (1.2) are not mutually orthogonal because the tensorial contraction1 Ai:Aj is non-vanishing for i⫽j. An approach based on W(l1,l2,l3), on the other hand, does give rise to orthogonal response terms. For this case (see, for example, Ogden, 1984)

冘

∂W A, ∂li i i⫽1 3

t⫽

(1.3)

wherein, with the qi as respective principal directions of V, A1⫽l1q1丢q1, A2⫽l2q2丢q2, A3⫽l3q3丢q3.

(1.4)

Since the principal directions are orthonormal, it should be evident that the li response terms are mutually orthogonal with Ai:Aj vanishing for i⫽j. An important subclass of the W(l1,l2,l3) form is W=w(l1)+w(l2)+w(l3), where w(li) is the same function for each stretch (see Valanis and Landel, 1967). Ogden materials (see Ogden, 1984) are of this separable form. Although accurate for rubberlike materials over particular ranges of stretch, this separable form is restrictive (see Rivlin and Sawyers, 1976). Nonetheless, for incompressible materials such as rubber, a W(I1,I2,I3⬅1) form is often less accurate than the Ogden form and leads to an unacceptable propagation of measurement error in the moderate strain region (e.g. 1

Herein, ‘:’ is the tensorial contraction operator with A:B=tr(ABT).

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2–25%). In fact, Rivlin and Saunders (1951) restrict their analysis of rubber in biaxial stretch to I1ⱖ5 and I2ⱖ5. As an example of a deformation that is below their lowest strain, consider a uniaxial, isochoric stretch of 100% wherein I1=5 and I2=4.25. To improve constitutive descriptions of isotropic hyperelastic materials, we developed a constitutive formulation based on three invariants {K1,K2,K3} of natural strain that are physically meaningful. Respectively, K1, K2, and K3 define the amountof-dilatation, the magnitude-of-distortion, and the mode-of-distortion. Following Flory (1961) distortion is the part of the deformation without dilatation. These terse scalar strain measures, in turn, give rise to mutually orthogonal stress response terms. Based on a comparison of data from the literature for rubber-like materials, the approach herein appears to allow specific forms of W to be determined from test data with improved accuracy. With regard to the general form of W(K1,K2,K3), physical reasoning allows W to be refined a priori. Moreover, common material tests can directly determine terms in W because pure dilatation (K2=0), pure shear (K3=0), uniaxial extension (K3=1), and uniaxial contraction (K3=⫺1) are tests which hold an invariant constant. Invariants that give rise to mutually orthogonal stress response terms are advantageous in our opinion because orthogonal entities exhibit the minimum of covariance and the maximum of mutual independence. That is, the stress response due to each invariant is discrete or completely unlike (i.e. orthogonal to) that of the other invariants. In addition, the response functions can be isolated forthwith upon tensorial contraction of a kinematic tensor with t. To see this, consider a relation for t that is analogous to (1.1) and (1.3) but with W=W(K1,K2,K3) and Ai:Aj vanishing for i⫽j. Being invariant specific, the Ai for the Ki (derived in Section 4) are different than those of (1.2) and (1.4). Nonetheless (no summation implied), ∂W t:Ai ⫽ . ∂Ki Ai:Ai

(1.5)

Although natural strain is not typically used at present, it is becoming more common. In our opinion, natural strain is advantageous. (Appendix A discusses these advantages and briefly defines the natural logarithm of a tensor). For most material tests, the principal stretches and directions of V are known (see Ogden, 1984), and hence, natural strain (lnV) can be obtained. Furthermore, given the availability of efficient computational methods for any choice of strain measure, the primary challenge at present is the identification of robust constitutive relations. It is in this realm that the proposed W(K1,K2,K3) promises to offer its greatest utility. 2. Natural strain invariants Natural or Hencky strain (h⬅lnV) has the advantage that it additively separates dilatation from distortion (see Appendix A). In particular, the spherical and deviatoric parts of h can be expressed as, 1 3

tr(h)I⫽ln(J1/3I),

(2.1)

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dev(h)⫽ln(J−1/3V),

(2.2)

wherein J=det(V)=det(F) is the volume ratio, dev(h)=h⫺(tr(h)/3)I is the deviatoric part of h, J1/3I is the dilatation, and J⫺1/3V is the distortion (i.e. deformation without dilatation). Since tr(h) only depends on dilatation and dev(h) only depends on distortion, the magnitudes of the spherical and deviatoric parts of h are two terse invariants. We define the amount-of-dilatation (K1) and the magnitude-of-distortion (K2) as: K1⫽tr(h)⫽ln(J),

(2.3)

冑

(2.4)

K2⫽|dev(h)|⫽ dev(h):dev(h).

Note that K1苸(⫺⬁,⬁) gives magnitude and sign of dilatation (+ for expansion and ⫺ for contraction) whereas K2苸[0,⬁) only gives magnitude for the distortion. The magnitude roles of K1 and K2 can be appreciated when h is expressed as, h⫽13K1I⫹K2⌽,

(2.5)

where ⌽=dev(h)/K2. Due to this normalization, ⌽ has unit magnitude (i.e. ⌽:⌽=1), and since ⌽ is deviatoric (i.e. ⌽:I vanishes), the square of the magnitude of h is h:h=13K 12+K 22. With dev(h)=K2⌽ representing the distortion strain, ⌽ specifies how the distortion is done whereas K2 specifies how much. Moreover, the principal values of ⌽ are completely determined by det(⌽) because tr(⌽)=0 and tr(⌽2)=1. We submit that det(⌽) specifies a measure of the mode or type of distortion. In particular, we define the mode-of-distortion, as

冑

K3⫽3 6 det(⌽),

(2.6)

where the constant 3√6 was chosen so that K3苸[⫺1,1]. It is a straightforward exercise in kinematics to show that the following three distinct types of distortion have fixed modes (i.e. K3 is constant): uniaxial extension (K3=1), uniaxial contraction (K3=⫺1), and pure shear (K3=0). Fig. 1 displays these and other modes. That det(⌽) is bounded can be seen from its principal values, f1, f2, and f3, which are constrained such that f1+f2+f3=0 and f21+f22+f23=1. Hence,

冉

冑

冊

冉

冊

冑

f1⬅f1, f2⫽⫺12 f1⫹ 2−3f12 , f3⫽⫺12 f1⫺ 2−3f12 ,

(2.7)

with det(⌽)=f1(f12⫺1/2). However, since ⌽ is symmetric, f1 must be bounded such that f2 and f3 are real; and thus, ⫺1ⱕ3√6 det(⌽)ⱕ1. Furthermore, if K2⫽0 there is a one-to-one correspondence between the Kinvariants and the principal stretches (l1ⱖl2ⱖl3). The forward and inverse maps are, K1⫽ln(J)⫽ln(l1l2l3), K2⫽

(2.8a)

冪冉lnl − K 冊 +冉lnl − K 冊 +冉lnl − K 冊 , 2

1 1 3

1

2

1 2 3

1

2

1 3 3

1

(2.8b)

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Fig. 1. Constant K2 or K3 trajectories for in-plane biaxial stretches l1 and l2 with K1=0 (i.e. isochoric). Identical trajectories are displayed in the linear and log–log plots. The closed curves correspond to, K2=0.3, 0.6, 0.9, 1.2, 1.5, and 1.8. The rays correspond to, K3= ⫺1, ⫺0.5, 0, 0.5, and 1. Note that K2=0 when V=I, and note that all K3 contours intersect all K2 contours. Also, notice how K2 specifies the magnitudeof-distortion (i.e. how far out from the state with V=I) and K3 specifies the mode-of-distortion (i.e. which ray to follow from the state with V=I) with, for example, uniaxial stretch in l1 being an equivalent mode to uniaxial stretch in l2.

冑

K3⫽3 6

冢

冣冢

lnl1−13K1 K2

冑

冉 冊

ln(l1)⫽13K1⫹K2 23sin q⫹23p ,

冑

ln(l2)⫽13K1⫹K2 23sin(q),

冑

冣冢

lnl2−13K1 K2

冉 冊

ln(l3)⫽13K1⫹K2 23sin q⫺23p ,

冣

lnl3−13K1 , K2

(2.8c)

(2.9a) (2.9b) (2.9c)

wherein q=⫺arcsin(K3)/3苸[⫺p/6, p/6]. Although K3 is indeterminate when K2=0, note that l1=l2=l3=exp(K1/3). Whenever K2 vanishes, the deformation is a pure dilatation regardless of the value of K3. In other words, the mode-of-distortion, K3, is irrelevant when there is no distortion (K2=0). To summarize, the li are of the form

再

li ⫽

li(K1,K2,K3) K2⫽0 . li(K1) K2=0

(2.10)

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3. Mutually orthogonal derivatives We have already described how K1–3 specify different aspects of deformation (recall K1 as amount-of-dilatation, K2 as magnitude-of-distortion, and K3 as modeof-distortion). In this section, we show that their partial derivatives with respect to h are mutually orthogonal. That is, (∂Ki/∂h):(∂Kj/∂h) vanishes when i⫽j. To find these derivatives, we utilize the definition (see Gurtin, 1981, pp. 19–21) that the partial derivative is the linear part of the mapping from one incremental space to another. In particular, dKi=(∂Ki/∂h):dh. For example, note from (2.3) that K1=I:h. Upon incrementing we obtain dK1=I:dh, and thus ∂K1 ⫽I. ∂h

(3.1)

Similarly, note that K2=⌽:h (recall 2.5). Incrementing yields dK2=⌽:dh+d⌽:h. However, with h expressed as (2.5) it is evident that d⌽:h=0 because d⌽:⌽=0 and d⌽:I=0 (obtained from incrementing ⌽:⌽=1 and ⌽:I=0, respectively). Therefore, dK2=⌽:dh and ∂K2 ⫽⌽. ∂h

(3.2)

The K3 derivative is more involved. We begin by incrementing (2.5) and substituting dK1=I:dh and dK2=⌽:dh. After rearrangement, d⌽⫽

1 [dh⫺13(I:dh)I⫺(⌽:dh)⌽]. K2

(3.3)

Now the Cayley–Hamilton theorem for ⌽ gives, ⌽3⫺12⌽⫺det(⌽)I⫽0,

(3.4)

and upon taking the trace, it yields 3det(⌽)=tr(⌽3). Hence, (2.6) becomes

冑

冑

K3⫽ 6tr(⌽3)⫽ 6⌽3:I,

(3.5)

and dK3=3√6⌽2:d⌽. Upon substitution of d⌽ with (3.3) and recalling that ⌽2:I=1 and ⌽2:⌽=K3/√6, we obtain, after rearrangement, dK3⫽

冉冑

冊

冑

1 3 6⌽2⫺ 6I⫺3K3⌽ :dh. K2

(3.6)

Therefore, ∂K3 1 ⫽ Y, where Y⫽3 6⌽2⫺ 6I⫺3K3⌽. ∂h K2

冑

冑

(3.7)

Note that Y is orthogonal to I (i.e. Y is deviatoric) because ⌽:I=0 and ⌽:⌽=⌽2:I=1.

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Moreover, Y is orthogonal to ⌽ since ⌽:⌽=1, (3.5), and (3.7) yield ⌽:Y=0. To summarize, I:⌽⫽0, I:Y⫽0, ⌽:Y⫽0.

(3.8)

Though orthogonal, Y and ⌽ share many similarities. In particular, they are coaxial and deviatoric, and their principal values depend solely on K3. Albeit bounded by [0,3], however, the magnitude of Y is variable whereas the magnitude of ⌽ is unity. To derive this, ⌽ is contracted onto (3.4) to yield tr(⌽4)=1/2, and together with (3.5), ⌽:⌽=1, and ⌽:I=0, we obtain Y:Y⫽9(1⫺K 32).

(3.9)

Hence, for uniaxial deformation (i.e. K3=±1) Y is the null tensor.

4. Constitutive relations based on W(K1,K2,K3) Central to the theory of finite strain hyperelasticity is the existence of a conservative material that allows all of the work done by boundary tractions to be stored as internal strain energy. For materials that exhibit an isotropic response with respect to their reference configuration, this principle requires (see, for example, Hill, 1978 or Atluri, 1984) Jt⫽∂W/∂h,

(4.1)

where W is the strain energy per unit reference volume, t is the Cauchy (or true) stress, and J=det(F) is the volume ratio. Because W(l1,l2,l3) is an admissible form for W (Ogden, 1972) and since (2.10) shows that the li can be expressed in terms of the Ki for all F, then,

再

W⫽

W(K1,K2,K3) K2⫽0 , W(K1) K2=0

(4.2)

is admissible. Using the ∂Ki/∂h from (3.1), (3.2), and (3.7), the chain rule for ∂W/∂h yields, t⫽

1 ∂W 1 ∂W 1 ∂W 1 I⫹ ⌽⫹ Y, J ∂K1 J ∂K2 J ∂K3 K2

(4.3)

where ∂W/∂Ki is the Ki material response function. Although it appears that t might become singular as K2→0, K2=0 is not excluded because we show in Section 5 that the last two terms vanish as K2 goes to zero. That is, ∂W/∂K2→0 as K2→0 and ∂W/∂K3→0 faster than K2→0. Since I, ⌽, and Y are mutually orthogonal, t is composed of three mutually orthogonal response terms, each dependent on a different response function. Also, the response functions can be isolated forthwith upon contraction of I, ⌽, and Y with t. Indeed (recall 3.8 and 3.9)

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1 3

1 ∂W t:I⫽⫺p⫽ , J ∂K1

(4.4a)

1 ∂W t:⌽⬅dev(t):⌽⫽ , J ∂K2

(4.4b)

9(1−K 23) 1 ∂W t:Y⬅dev(t):Y⫽ , K2 J ∂K3

(4.4c)

where p is the pressure and dev(t) may be substituted for t because ⌽ and Y are deviatoric. Physically, the K1, K2, and K3 response terms in (4.3) can be characterized respectively as: (1) the pressure, (2) the part of dev(t) that is proportional to the distortion strain dev(h), and (3) the part of dev(t) that is orthogonal to dev(h).

5. Behavior in the neighborhood of K2=0 Recall from Section 2 that K2 vanishes for all pure dilatations. For this case, K3 is both indeterminate and insignificant, because the principal stretches are entirely determined by K1 alone. In this section it is shown that the constitutive behavior is well behaved when K2 vanishes and K3 is indeterminate, because isotropic material symmetry requires that both K2 and K3 dependence in W vanish faster than K2. Toward this end, note that t must be a pressure when an isotropic material is purely dilatated (e.g. F=J1/3I) because the deformation and the material are both spherically symmetric. Thus, isotropic material symmetry requires that dev(t)→0 as K2→0. Whereby, lim dev(t):⌽⫽0, lim dev(t):Y⫽0,

K2→0

K2→0

(5.1)

because 兩⌽兩=1 and 兩Y兩苸[0,3] are bounded. Furthermore, since (4.4b) and (4.4c) are valid for all K1 and K3 in the neighborhood of K2=0, lim

K2→0

∂W 1 ∂W ⫽0, lim ⫽0. ∂K2 K2→0 K2 ∂K3

(5.2)

As K2→0, therefore, symmetry requires that K2 and K3 dependence in W go to zero as order K 22 or higher. With (5.2), the constitutive relation in (4.3) can be evaluated for pure dilatation and lim

K2→0

冉

冊

1 1 ∂W 1 ∂W 1 ∂W 1 ∂W I⫹ ⌽⫹ Y ⫽ I. J ∂K1 J ∂K2 J K2 ∂K3 J ∂K1

(5.3)

In other words, the distortional response must vanish as the distortion goes away. In addition to satisfying symmetry, constitutive relations for finite elasticity must be compatible with linearized elasticity laws whenever the strain is infinitesimal and the rotation is negligible. In particular, linearized elasticity laws are valid when

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兩K1兩 and K2 are small and R=I. In terms of the Lame´ constants2 for isotropic linearized elasticity, t=ltr(h)I+2mh because natural strain is equal to engineering strain for this case. Equivalently, t⫽(l⫹2m/3)K1I⫹2mK2⌽,

(5.4)

where (l+2m/3) and m are the bulk and shear moduli, respectively. Also, with a stressfree and strain–energy free reference, W=(t:h)/2 for linearized elasticity. Therefore, W⫽(l/2⫹m/3)K 21⫹mK 22,

(5.5)

and neither W nor t depends on K3 despite K3苸[⫺1,1] being finite. Indeed, any dependence on K3 is inherently nonlinear. To see this, note from (5.4) that dev(t)=2mdev(h). Hence, linearized elasticity is valid only if dev(t) is proportional to dev(h). However, the K3 response (or Y) term, though deviatoric, is always orthogonal to dev(h) and can never be proportional to it. Therefore, dev(t) is proportional to dev(h) if and only if the K3 response term is negligible. Moreover, we assert that infinitesimal shearing (K2¿1) should exhibit linear shearing behavior with dev(t) proportional to dev(h) even if the shear is superimposed on a finite dilatation (i.e. finite K1). In particular, a pure dilatation of an isotropic body yields an isotropic configuration that could be considered as an alternate reference configuration. A subsequent small shear from this new reference would be an infinitesimal deformation with behavior that is likely linear. For example, compare the shearing behavior of a hyperelastic foam block at the bottom of the ocean versus on the surface. In both cases, the shear stress should be linearly dependent on small shear strains. The shear modulus m likely would be a function of block’s volume such that m=m(K1) and dev(t)=2m(K1)K2⌽. Recall that dev(t) cannot be proportional to dev(h) if the Y term in (4.3) is nonnegligible. Hence, the constraint “dev(t) must be proportional to dev(h) for K2¿1” requires that the Y term in (4.3) go to zero faster than 2m(K1)K2⌽ when K2→0. 2 Equivalently, K −1 2 ∂W/∂K3 must be of order K 2 . Therefore, K3 dependence in W must 3 go to zero as order K 2 when K2→0. 6. Functional form of W(K1,K2,K3) In the previous section, we showed that physical reasoning allows the functional form of W to be refined a priori. To summarize, symmetry requires K2 dependence in W to go to zero as order K 22 or higher as K2→0; and if infinitesimal shears are to yield a linear response, K3 dependence must go to zero as order K 23 or higher as K2→0. Hence, if W can be expressed as a power series (i.e. all analytic strain energy functions), then W can be written as,

2

Elsewhere l denotes stretch; its use here as a Lame´ constant is common.

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冘

冉 冘 冊 冘冘冘

⬁

W⫽

⬁

aiK 1i⫹K 22 m⫹

i⫽0

i⫽1

⬁

⬁

⬁

biK 1i ⫹K 32

zijkK 1iK 2j K 3k,

(6.1)

i⫽0 j⫽0 k⫽0

wherein: a0, though typically taken as zero, is the strain energy of the reference configuration; ⫺a1 is the pressure in the reference configuration; 2a2 is the infinitesimal bulk modulus (l+2m/3); and m is the infinitesimal shear modulus. To verify that (6.1) is valid, note the K2 factors preceding the second and third terms. The first term represents zero order K2 dependence, the second term represents second order K2 dependence, and the third term represents third order and higher K2 dependence. Since K2 dependence in W must go to zero as order K 22 or higher as K2→0, linear K2 factors are not included in (6.1). Since K3 dependence must go to zero as order K 23 or higher as K2→0, K3 factors are only allowed in the third term. The separate terms in (6.1) can be isolated by specific material tests. First of all, a pure dilatation (K2=0) will determine the ai constants by fitting the resulting pressure versus K1 curve. Infinitesimal shears at small K1 will determine m. Furthermore, if infinitesimal shear responses can be measured at finite dilatation, then the bi can be fit by plotting the effect of dilatation, K1, on the shear modulus. Finite shears with and without finite dilatation (i.e. triaxial tests) are necessary to find the zijk constants. Fortunately, experimentation is simplified (relative to other theories) because uniaxial extension (K3=1) and contraction (K3=⫺1) are tests with an invariant held constant. In essence, if volume (or K1) can be controlled during uniaxial testing, only one invariant, K2, varies because K3 is held at +1 or ⫺1 (the extremes of the K3 domain). 7. Incompressible materials For the remainder of this paper, we focus on finite distortions of materials whose behavior can be considered as incompressible. The incompressibility constraint, J⬅1 or K1⬅0, reduces the number of variables in W from three to two such that W=W(K2,K3). Since the incompressibility constraint is discussed in many finite elasticity texts (see, for example, Ogden, 1984) we merely state the familiar result that ⫺qI must be added to (4.1) with q as a Lagrange multiplier. Albeit dependent on boundary conditions and equilibrium, q is arbitrary in the constitutive law. Following the approach of Section 4 and with p as the pressure, the constitutive law becomes ∂W ∂W 1 ⫺pI⫽⫺qI, dev(t)⫽ ⌽⫹ Y. ∂K2 ∂K3 K2

(7.1)

Unlike other incompressibility constitutive relations in finite strain incompressible elasticity, the pressure-like Lagrange multiplier q is always equal to the pressure p since the K2 and K3 response terms are deviatoric. With K1⬅0 and a strain energy free reference configuration, W in (6.1) can be expressed as,

冘冘 ⬁

W⫽mK ⫹K 2 2

3 2

⬁

yjkK 3kK 2j ⫺qK1,

j⫽0 k⫽0

(7.2)

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where m is the infinitesimal shear modulus and the yjk are constants. Following the method of Rivlin and Saunders (1951), it should be evident that the coefficients within W in (7.2) can be determined from biaxial stretching tests with K2 and K3 separately held constant at multiple values. Such testing trajectories are illustrated in Fig. 1. The values of K2 and K3 can be obtained from the principle stretches with (2.9). Since the principle stretch directions are known and constant for biaxial stretching tests, h is easily evaluated and (2.5), (3.8) and (4.4b,c) can be used to calculate the response functions. A particularly interesting subclass of W(K2,K3) has only linear dependence on K3. Hence, let (7.2) with yjn=0 (nⱖ2) be expressed as,

冘 ⬁

W⫽

冘 ⬁

g K 2i⫹K3

1 i i−1

i⫽2

xj K 2j ,

(7.3)

j⫽3

where g1/2=m is the infinitesimal shear modulus. In this case, W can be determined from either pure shear or uniaxial testing. First, consider pure shear (K3=0) of a thin sheet with l1=l, l2=1, and l3=1/l. Thus K2=√2 ln(l), and the kinematic tensors ⌽ and Y are ⌽⫽

1

冑2

(q1丢q1⫺q3丢q3),

(7.4)

冉

冑

冊

Y⫽ 6 12q1丢q1⫺q2丢q2⫹12q3丢q3 ,

(7.5)

where the qi are the principal directions of h (or V) with q3 as the sheet normal. With the sheet free of tractions on the top and bottom surfaces, let t=t1q1丢q1+t2q2丢q2. Contraction of ⌽ and Y on t yield, t:⌽⫽

1 9

1

冑2

t1⫽

(t:Y)K2⫽

冑6 9

| 冘

∂W ∂K2

⬁

⫽

K3=0

冉 冊

giK 2i,

t ⫺t2 K2⫽

1 2 1

(7.6)

i⫽1

| 冘

∂W ∂K3

⬁

⫽

K3=0

xj K 2j .

(7.7)

j⫽3

Hence, fitting t1 and (t1/2⫺t2)K2 vs K2 provides a means of experimentally determining the values of gi and xj in Eq. (7.3). For uniaxial extension (K3=1), let l1=l⬎1, and l2=l3=l⫺1/2. Thus K2=√3/2 ln(l), Y vanishes, and ⌽ is

冑

⌽⫽

2 3

冉

冊

q1丢q1⫺12q2丢q2⫺12q3丢q3 .

(7.8)

With t=t1q1丢q1 (i.e. traction free lateral surfaces), contraction of ⌽ on t yields

冑

t⫽

2 3 1

| 冘

∂W ∂K2

⬁

⫽

K3=1

i⫽1

冘 ⬁

giK 2i⫹

j⫽2

(j⫹1)xj+1K 2j ⬅g(K2).

(7.9)

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Next consider uniaxial contraction (K3=⫺1) in the direction q3 (the sheet normal), or equivalently equibiaxial stretch with l1=l2=l⬎1 and l3=l⫺2. Then K2=√6 ln(l), Y vanishes again, and ⌽ is

冑

⌽⫽

冉

冊

q 丢q1⫹12q2丢q2⫺q3丢q3 .

2 1 3 2 1

(7.10)

With t=t0(q1丢q1+q2丢q2) contraction of ⌽ on t yields ∂W 冑 t ⫽∂K 2 3 0

2

| 冘 冘 ⬁

⫽

K3=−1

⬁

giK 2i⫺

i⫽1

(j⫹1)xj+1K 2j ⬅h(K2).

(7.11)

j⫽2

Fitting of g(K2)+h(K2) identifies the gi values; similarly, fitting of g(K2)⫺h(K2) provides the xj values. Thus, materials for which W is linear in K3 can be completely characterized by either a pure shear test (K3=0) or a comparison of uniaxial tests (K3=±1). 8. Analysis of rubber data of Jones and Treloar Fig. 2 displays our digitization of the data reported by Jones and Treloar (1975), who extensively tested rubber subjected to pure shear, general biaxial stretch, and uniaxial stretch. For purposes of illustration let W be of the form (7.3), that is, linear in K3. Hence, W can be found from the pure shear test (K3=0), and its predictive

Fig. 2. Digitization of Jones and Treloar (1975) biaxial stretch data from tests on natural rubber. Tests are from left to right: pure shear, general biaxial stretch, equibiaxial stretch, and uniaxial stretch. The symbols ‘䊊’ and ‘+’ respectively denote the t1 and t2 values for each digitized point. Data were digitized at regular abscissa intervals from scanned images of the smooth curves in Jones and Treloar as follows: pure shear, Fig. 2; uniaxial stretch, Fig. 1 curve A; general biaxial stretch, Fig. 3 with average of t1 curves; and equibiaxial stretch, Fig. 4 with one point from each curve.

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capability can be explored with the general biaxial and uniaxial data (i.e. K3苸[⫺1,1]). Utilizing equations (7.6) and (7.7), Fig. 3 plots ∂W/∂K2 and ∂W/∂K3 versus K2 from the digitized data. The solid lines show polynomial fits using constants obtained by inspection. Based on this one test, we estimate that W has the following functional form (units of MPa), W⫽0.48K 22⫺0.053K 23⫹0.088K 24⫹(0.065⫹0.039K2)K 23K3,

(8.1)

where m=0.48 MPa. To examine the predictive capability, Fig. 4 (right hand panel) plots predictions of t1 and t2 using (8.1) versus the digitized stress data. The line of identity represents perfect prediction, and the root mean squared (RMS) deviation from the line of identity is 0.0035 MPa or 0.28% of the mean stress value. Included in Fig. 4 are data from pure shear (K3=0), uniaxial extension (K3=1), equibiaxial stretch (K3=⫺1), and general biaxial stretch wherein K3 was varied from +1 (l1=0.66, l2=2.29, K2=1.01) to ⫺1 (l1=l2=2.29, K2=2.03). Thus, (8.1), derived from K3=0 alone, accurately predicts data for K3苸[⫺1,1] in the K2 ranges tested. For comparison, Fig. 4 (left hand panel) plots the prediction of t1 and t2 by equation 21 of Jones and Treloar (1975), versus our digitization of their data. Their relation is also accurate with RMS deviation from the line of identity of 0.0048 MPa or 0.37% of the mean stress value. Their constitutive relation is of the Valanis and Landel form and is (in units of MPa), lw⬘(l)⫺w⬘(1)⫽0.69(l1.3⫺1)⫹0.01(l4.0⫺1)⫺0.0122(l−2.0⫺1),

(8.2)

whereby, ti⫺tj ⫽0.69(li1.3⫺lj1.3)⫹0.01(li4.0⫺lj4.0)⫺0.0122(l−2.0 ⫺l−2.0 ). i j

(8.3)

Although slightly different, we consider the predictive capability of (8.1) and (8.2) to be equivalent.

Fig. 3. Response function plots for Jones and Treloar (1975) pure shear (K3=0) data. The ‘䊊’ denote values for the digitized data, and the solid lines are polynomial fits obtained by inspection. The parametric constants are (in MPa) x3=0.065, x4=0.039, g1=0.96, g2=⫺0.16, and g3=0.35.

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Fig. 4. Comparison of predictive capability between Jones and Treloar analysis (left) and K-invariant analysis (right). All lines are lines of identity representing perfect prediction of t1 and t2 values. For legibility, identity lines for different tests are offset vertically, and correspond to (top to bottom): all tests combined, pure shear, general biaxial stretch, uniaxial stretch, and equibiaxial stretch. The ‘䊊’ and ‘+’ respectively denote t1 and t2 values.

9. Analysis of rubber data of Rivlin and Saunders If a form of W, linear in K3, is assumed for the rubber specimens used in the biaxial tests of Rivlin and Saunders (1951)3, then values of the constants in (7.3) can be found from their data despite K2 or K3 not being held constant during their tests. This is possible because the K3 response function does not depend on K3 when W is linear in K3. Hence, using (4.4c) to evaluate ∂W/∂K3 from data, its resulting dependence on K2 is just xiK 2i (sum over i=3,4,…) which is plotted in Fig. 5. Note that Fig. 5 (left hand panel) is noisy since (4.4c) is sensitive to experimental error for uniaxial deformations (i.e. for (1⫺K 32)→0). The solid line represents our polynomial fit by inspection for the xi values. Using (4.4b) to evaluate the K2 response function, giK 2i (sum over i=1,2,…) can be obtained once jK3xj K 2j−1 (sum over j=3,4,…) is subtracted from ∂W/∂K2. Fig. 5 (right hand panel) plots the data and our polynomial fit of the gi values by inspection. Note the closeness of this polynomial fit with data. In units of MPa, we estimate 3

Rivlin and Saunders (1951) biaxial data are from their Tables 1 and 2.

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Fig. 5. Response function plots for Rivlin and Saunders (1951) biaxial stretch data. The ‘䊊’ and ‘+’ denote data points from constant I1 and I2 tests, respectively. The solid lines are fits obtained by inspection with parametric constants (in MPa): x3=0.045, x4=0.05, g1=0.94, g2=⫺0.24, and g3=0.38.

W⫽0.47K 22⫺0.08K 23⫹0.095K 24⫹(0.045⫹0.05K2)K 23K3,

(9.1)

with m=0.47 MPa. Fig. 6 (right hand panel) shows further that (9.1) accurately represents the data with an RMS deviation from the line of identity of 0.0084 MPa or 0.54% of the mean stress value; and note the lack of scatter. For comparison, Fig. 6 (left hand panel) shows the prediction by Rivlin and Saunders of their own data with an RMS deviation from the line of identity of 0.011 MPa

Fig. 6. Comparison of predictive capability between Rivlin and Saunders analysis (left) and K-invariant analysis (right). All lines are lines of identity representing perfect prediction of t1 and t2 values. For legibility, identity lines for different tests are offset vertically, and correspond to (top to bottom): both tests combined, I1 constant, and I2 constant. The ‘䊊’ and ‘+’ respectively denote t1 and t2 values. Note the decreased scatter on the right.

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or 0.72% of the mean stress value. Although Rivlin and Saunders did not explicitly report a W, we inferred it from their curves drawn in the response function plots (Figure 6 of Rivlin and Saunders, 1951, with a form consistent with Rivlin and Sawyers, 1976). We estimate W⫽0.1673(I1⫺3)⫹0.0256(I2⫺3)⫺0.0003(I2⫺3)2,

(9.2)

in units of MPa. We consider the K-invariant description to be more accurate than (9.2), especially for small strain. In fact, Rivlin and Saunders intentionally did not test I1⬍5 or I2⬍5 because of unacceptable propagation of measurement error in their response functions. Although, the comparison in Fig. 6 may seem unfair since (9.1) has more parameters than (9.2), both were determined by inspection of response function plots from the same data set. Additional orders of K2 were included in (9.1) because the data suggested it. What the I1 and I2 response function plots of Rivlin and Saunders (1951) suggest is less clear.

10. Discussion Central to this theory for isotropic finite hyperelasticity is the utilization of invariants that tersely describe natural strain, h⬅lnV. In particular, K1苸(⫺⬁,⬁) specifies the amount-of-dilatation, K2苸[0,⬁) the magnitude-of-distortion, and K3苸[⫺1,1] the mode-of-distortion. Note that the domain of this invariant space is a rectangular infinite half slab. Since the ∂Ki/∂h are mutually orthogonal (see Section 3), t is expressed as the sum of three mutually orthogonal response terms (recall 4.3). Furthermore, each part of t is dependent on a different Ki response function that can be isolated forthwith upon contraction of t with a kinematic tensor (recall equation 4.4). Another advantage is the easy identification of symmetry conditions for isotropic materials. In particular, when K2=0, the deformation is a pure dilatation and t must be a pressure. When K3=±1 the deformation is uniaxial, and t must be axially symmetric. With regards to characterizing materials, W can be refined a priori (recall equations 6.1 and 7.2) and terms can be identified with typical experiments. Moreover, our analysis of rubber data suggests that linear K3 dependence in W may be sufficient for an accurate characterization. The analysis of Rivlin and Saunders (1951) is classic, and we have followed their philosophic approach in part. Specifically, we agree that tests which hold an invariant constant are crucial for determining the functional form of W directly from data. Though readily obtained from the characteristic equation, the principal invariants of C, unfortunately, do not give rise to orthogonal response terms in t. In contrast to the rectangular domain spaces of {K1,K2,K3} and {l1,l2,l3}, the {I1,I2,I3} domain space has boundaries that are non-planar; and worse yet, each boundary depends on all three invariants. There is considerable improvement if the invariants of Penn (1970) are used. With L1=I1/J2/3, L2=I2/J4/3, and L3=J2=I3, only L1 and L2 depend on distortion and L3苸(0,⬁) only depends on dilatation. Consequently, the L3 response term is orthogonal to that of L1 and L2. Also, one domain boundary in {L1,L2,L3} space becomes L3=0 with

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the other two boundaries being independent of L3. Now, the domain constraint for L1 and L2 can be obtained from the characteristic equation of J⫺2C by forcing it to yield real principal values. We obtain, 4L13⫹4L23⫺L12L22⫺18L1L2⫹27ⱕ0,

(10.1)

and this 2-D constraint is shown in Fig. 7 alongside the L1 and L2 (or I1 and I2 if J=1) invariant contours in a distortion stretch plane. As is evident from Fig. 7, the uniaxial deformation conditions are curvilinear functions of both invariants, and pure dilatation is degenerately described by L1=3 or L2=3 since L1 must be 3 if L2=3 and vice versa. Also shown in Fig. 7, the L1 and L2 contours practically overlap for small strains, and they are tortuous and intertwine for all strains (compare with the K2 and K3 contours in Fig. 1). Consequently, for biaxial testing of rubber, the I1 and I2 response functions are more difficult to isolate, and they are excessively sensitive to experi-

Fig. 7. Domain constraint (upper panels) and biaxial stretch contours (lower panels) of L1 and L2 (or I1 and I2 when I3=1). Left and right hand panels respectively depict the moderate and large stretch ranges. The domain is the area between the two non-linear curves. Note that uniaxial extension and contraction are the domain boundaries, and pure shear corresponds to the identity line. The L1 and L2 contours are for similar values with 3.05, 3.1, and 3.15 on the left and 3.5, 4.5, and 5.5 on the right. The rays in the lower plots indicate uniaxial and pure shear tests. For moderate stretches, note the covariance of L1 and L2 in the top panel and the overlapping contours in the bottom panel.

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mental error for mild stretch, I1⬍5 or I2⬍5. Equibiaxial stretching is singular for all I1 and I2 (see Rivlin and Saunders, 1951). In our opinion, the limitations of the W(I1,I2) approach for moderate strains (and all strains to a lesser extent) is rooted in the L1 and L2 (or I1 and I2) covariance. For instance, W=C1(I1⫺3), W=C1(I2⫺3), and W=C1(α(I1⫺3)+(1⫺α)(I2⫺3)) all give the same stress response in the infinitesimal strain limit. Any attempt to differentiate I1 dependence from I2 dependence for small strain is fundamentally ill posed. For moderate strains there is only slight improvement. Of course, many forms of W(I1,I2) have been suggested for rubber with some good predictions of data (neo-Hookean and Mooney–Rivlin are classic; see also, Ling et al., 1993; James et al., 1975; Alexander, 1968; Hart-Smith, 1967; Gent and Thomas, 1958). It is well recognized that a W(I1,I2,I3) approach leads to an elegant formulation of the kinematics and analytically obtained solutions to many boundary value problems. With regard to the W(l1,l2,l3) approach of Ogden (1984) and the analysis of Jones and Treloar (1975), we consider the predictive capability of their approach (see Fig. 4) to be equal to ours for materials that satisfy the Valanis–Landel (VL) hypothesis (1967). Indeed, Ogden’s approach has some advantages that are similar to ours. In particular, ∂li/∂h:∂lj/∂h=0 when i⫽j and the li have a rectangular domain in invariant space (i.e. the infinite quadrant with li⬎0). However, the K-invariants are more concise in their description of strain and symmetry. Compare, for instance, a pure dilatation which is tersely described by K2=0; whereas for the li formulation, the functional representation l1=l2=l3 is needed. Similarly for uniaxial symmetry, K3=±1 is more concise than the following: li=lj, li⫽lk, i⫽j, j⫽k, and k⫽i. With regard to constitutive behavior, symmetry requires that K2 and K3 dependence in W go to zero as order K 22 or higher when K2 goes to zero (recall Sections 5 and 6). This result is more easily implemented than W(l1,l2,l3)=W(l2,l3,l1)=W(l3,l1,l2)=W(l2,l1,l3)=W(l1,l3,l2)=W(l3,l2,l1). The W(l1,l2,l3) symmetry restriction is typically satisfied by invoking the VL hypothesis. Although often accurate, the VL hypothesis is restrictive (Rivlin and Sawyers, 1976). Another potential advantage of the approach herein is the utilization of direct tensors rather than the spectral decomposition. With the W(l1,l2,l3) approach, solving the eigensystem is obligatory. In contrast, direct expressions for h are available (see Fitzgerald, 1980) as well as simple estimates (see Bazant, 1998; Hunter and Criscione, 2000) for moderate strains. Without the need to solve eigensystems, the computational efficiency of a finite element analysis may be enhanced. While the constitutive formulation presented here is novel, an analogous set of invariants for natural strain has been proposed in the Russian literature by Lurie (1990, pp. 189–192, 222–223). In this set, K1 and K2 appear, but Lurie chose the third invariant to be the arcsine of K3, which he termed the phase-of-strain (see our Eqs. (2.9a, 2.9b, 2.9c)). Despite the partial similarity in the invariants, Lurie’s constitutive formulation differs notably from the one presented here in Section 4. Instead of solving for two mutually orthogonal aspects of dev(t), he solves for the magnitude of dev(t) and the phase of dev(t). In both approaches the pressure is identified as the material response to K1. However, in contrast to the forthright

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method of (4.4) for response function isolation, Lurie’s approach requires the solution of a system of two equations to obtain the response functions in terms of 兩dev(t)兩 and the stress–strain phase shift. In both this approach and that of Lurie, pure shear is identified as a convenient test since K3 and the phase-of-strain are zero. Nonetheless, an analysis of uniaxial tests remains wanting for Lurie’s approach, particularly since the stress–strain phase shift must be zero for this case. This is so because when the strain is axis-symmetric so is t. With cosine dependence violating symmetry, a thorough analysis of uniaxial deformation leads to the result that W must only depend on the sine of the strain phase, which is K3 itself. A general functional form of W is not discussed by Lurie, nor is experimental application.

Acknowledgements The first author would like to acknowledge that numerous discussions with Dr Joseph C. Criscione, materials scientist, contributed to the foundation of this theory. Also, we acknowledge Dr Alexander Spector for pointing out/interpreting works of Lurie so that we could compare our analysis with his. Financial support from the NIH (grant HL 30552 to WCH) is gratefully acknowledged.

Appendix A This Appendix describes the natural logarithm of positive definite, symmetric tensors; shows how natural strain additively decouples dilatation and distortion; and highlights some advantages of natural strain for materials that undergo finite strain. In order to define the natural logarithm, consider a positive definite, symmetric tensor A with the spectral representation A⫽a1a1丢a1⫹a2a2丢a2⫹a3a3丢a3,

(A.1)

where ai are the principal values and ai are the principal directions. All positive definite, symmetric tensors have spectral representations with three real principal values ⬎0 and three orthogonal principal directions. The natural logarithm ln(A) has the spectral representation ln(A)⫽ln(a1)a1丢a1⫹ln(a2)a2丢a2⫹ln(a3)a3丢a3.

(A.2)

If a positive definite, symmetric tensor B is coaxial to A (i.e. they share the same principal directions), then with substitution of their respective spectral decompositions it can be shown that ln(AB)=ln(A)+ln(B). Also note that A and B commute such that AB=BA. Non-coaxial tensors do not commute and the log of their product is not necessarily the sum of their logs. These properties of the tensor logarithm are applied here to the natural strain tensor which is ln(V). If F=VR represents polar decomposition of the deformation

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gradient, then V is positive definite and symmetric, and FFT=VV. Since V is coaxial with itself, ln(V)=ln(VV)/2. This demonstrates that polar decomposition is not necessary to obtain natural strain. Now consider the decomposition of Flory (1961) with V=(J1/3I)(J⫺1/3V) where J=l1l2l3=det(V) is the volume ratio, the li are the principal stretches, and det(J⫺1/3V)=1. It should be evident that J1/3I is the dilatational part of the deformation whereas (J⫺1/3V) is the distortional part because it does not dilatate. Since I and V are coaxial, it follows that ln(V)⫽ln(J1/3I)⫹ln(J−1/3V).

(A.3)

Hence, dilatation and distortion additively decouple within ln(V). Moreover since ln(l1)+ln(l2)+ln(l3)=ln(J), it should be evident that the spherical part of ln(V) or (tr(ln(V))/3)I is dilatation strain whereas the deviatoric part is distortion strain. Upon noting that natural strain is approximately equal to engineering strain for small deformations, recall from basic solid mechanics for small deformations that the spherical part of engineering strain is dilatation strain whereas the deviatoric part is distortion strain. Hence, natural strain allows the additive decoupling of dilation and distortion for infinitesimal engineering strain to be generalized to all strains, large or small. With C=FTF and B=FFT, the Lagrangian E=(C⫺I)/2 and Almansi A=(I⫺B⫺1)/2 strain measures do not allow this decoupling for finite strain. Furthermore, natural strain has extension–contraction symmetry such that ln(V⫺1)=⫺ln(V). Lagrangian and Almansi strain do not. Also note that the principal values of Lagrangian strain are biased toward extension with the range (⫺1/2,⬁) whereas those of Almansi strain are biased toward contraction with the range (⫺⬁,1/2). Natural strain is unbiased, and the range of its principal values is (⫺⬁,⬁).

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