Mathematics and Mechanics of Solids (2007) 12, 526–542. doi: 10.1177/1081286506064719 Published 7 April 2006 Page 1 of 16 The incremental bulk modulus, Young’s modulus and Poisson’s ratio in nonlinear isotropic elasticity: physically reasonable response N.H. Scott∗ School of Mathematics, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, U.K. 12 May 2020 [Dedicated to Michael Hayes with esteem and gratitude] Abstract An incremental (or tangent) bulk modulus for finite isotropic elasticity is de- fined which compares an increment in hydrostatic pressure with the corresponding increment in relative volume. Its positivity provides a stringent criterion for phys- ically reasonable response involving the second derivatives of the strain energy function. Also, an average (or secant) bulk modulus is defined by comparing the current stress with the relative volume change. The positivity of this bulk modulus provides a physically reasonable response criterion less stringent than the former. The concept of incremental bulk modulus is extended to anisotropic elasticity. For states of uniaxial tension an incremental Poisson’s ratio and an incremental Young’s modulus are similarly defined for nonlinear isotropic elasticity and have properties similar to those of the incremental bulk modulus. The incremental Pois- son’s ratios for the isotropic constraints of incompressibility, Bell, Ericksen, and constant area are considered. The incremental moduli are all evaluated for a spe- cific example of the compressible neo-Hookean solid. Bounds on the ground state Lam´eelastic moduli, assumed positive, are given which are sufficient to guarantee the positivity of the incremental bulk and Young’s moduli for all strains. However, although the ground state Poisson’s ratio is positive we find that the incremental arXiv:2005.04212v1 [physics.class-ph] 9 May 2020 Poisson’s ratio becomes negative for large enough axial extensions. Keywords Nonlinear elasticity, incremental elastic moduli, compressible neo- Hookean material, constant-area constraint, superficial incompressibility MSC (2010) 74B20 ∗Email: [email protected] Page 2 of 16 N. H. Scott 1 Basic nonlinear and linear isotropic elasticity In this section we recount the basic theory of nonlinear and linear elasticity which we shall need before defining in Section 2 the incremental bulk and Young’s moduli and the incremental Poisson’s ratio and deriving some of their properties. In Section 3 we consider these moduli in detail for a specific example of the compressible neo-Hookean material. In terms of the deformation gradient F the left and right Cauchy-Green tensors are B = FFT and C = FTF, respectively. In terms of the three invariants −1 J1 = tr C, J2 = tr C , J = √det C (1.1) the Cauchy stress of an isotropic elastic material may be written −1 σ = β0I + β1B + β−1B (1.2) in which I denotes the unit tensor and βΓ (Γ=0, 1, 1) are functions only of the three − invariants J1, J2, J defined at (1.1). The response functions βΓ are often assumed to satisfy the empirical inequalities [1, (51.27)] β0 0, β1 > 0, β−1 0. (1.3) ≤ ≤ It seems reasonable to assume that these inequalities are satisfied for small enough strains of a non-linearly elastic material. In the theoretical discussion of Section 2 the inequalities (1.3) are assumed at the beginning of Sections 2.1 and 2.2 in order to show that equal principal stresses imply equal corresponding principal stretches. In fact, all that is required is the invertibility of the stress-strain law (1.2). For the specific example of a certain compressible neo-Hookean material considered in Section 3 it is shown that the empirical inequalities (1.3) are indeed satisfied for deformations for which the invariant J lies within a certain range of values, see (3.3) below. Other possible restrictions upon the response functions are considered in [1]. If the isotropic elastic material is hyperelastic there exists a strain-energy function W (J1,J2,J) such that the Cauchy stress (1.2) reduces to −1 −1 −1 σ = W I +2J W1B 2J W2B (1.4) J − in which W1 := ∂W/∂J1, W2 := ∂W/∂J2 and WJ := ∂W/∂J. Comparing with (1.2) we see that −1 −1 β0 = W , β1 =2J W1, β−1 = 2J W2 (1.5) J − and the empirical inequalities (1.3) become W 0, W1 > 0, W2 0. (1.6) J ≤ ≥ From (1.4) the principal Cauchy stresses σii (i = 1, 2, 3) are given in terms of the principal stretches λi (i =1, 2, 3) by −1 2 −1 −2 σ = W +2J W1λ 2J W2λ (i =1, 2, 3). (1.7) ii J i − i Incremental moduli in nonlinear elasticity Page 3 of 16 In terms of the Lam´emoduli λ0, µ0 of linear isotropic elasticity, the bulk modulus, Poisson’s ratio and Young’s modulus are given by 2 λ0 µ0(3λ0 +2µ0) K0 = λ0 + 3 µ0, ν0 = , E0 = , (1.8) 2(λ0 + µ0) λ0 + µ0 see [2, Table 3]. The positive definiteness of the strain energy of linear isotropic elasticity requires the inequalities µ0 > 0, K0 > 0 (1.9) to hold. In turn these inequalities imply the further inequalities 1 2 1 < ν0 < , E0 > 0, λ0 > µ0. (1.10) − 2 − 3 Commonly in linear isotropic elasticity it is found that inequalities more restrictive than (1.10) hold: 1 0 < ν0 < 2 , E0 > 0, λ0 > 0. (1.11) 2 Incremental elastic moduli 2.1 The incremental bulk modulus Consider an initially stress-free isotropic elastic material subject to a hydrostatic pressure p = σ11 = σ22 = σ33 = σpp/3 given by (1.2) and (1.7), employing the summation convention− on− repeated− suffixes.− By subtraction of the diagonal components of (1.2) and use of (1.3)2,3 it can be shown that the principal stretches must be equal, so that λ1 = λ2 = λ3 = λ, say, and therefore the strain resulting from the pure hydrostatic pressure σ = pI is the pure dilatation F = λI. After the application of the hydrostatic − 3 pressure p, an initial volume V0 becomes V = V0J = V0λ . A further increment δp 2 in pressure results in a further volume change δV = 3V0λ δλ. The incremental bulk modulus is defined by change in pressure Incremental bulk modulus = − relative change in volume δp δσ11 Jδσ11 λδσ11 = − = 3 3 = = . (2.1) δV/V δ(V0λ )/(V0λ ) δJ 3δλ Thus the incremental (or tangent) bulk modulus is defined in the limit δλ 0 by either of the equivalent forms → λ ∂σ11 K(λ) := , (2.2) 3 ∂λ λ1=λ2=λ3=λ or J ∂σ K(λ) := pp . (2.3) 3 ∂J 1/3 λ1=λ2=λ3=J Page 4 of 16 N. H. Scott 2 −2 3 For a pure dilatation of stretch λ we have J1 = 3λ , J2 = 3λ , J = λ and (1.7) reduces to −1 −5 σ11 = W +2λ W1 2λ W2. (2.4) J − Use of (2.4) in (2.2) shows that the incremental bulk modulus of isotropic hyperelasticity may be expressed in the form 2 −1 10 −5 K(λ)= λ W1 + λ W2 (2.5) − 3 3 3 −7 2 −2 −2 + λ W +4λW11 +4λ W22 +4λ W 1 4λ W 2 8λ W12 JJ J − J − 2 in which W12 denotes ∂ W/∂λ1∂λ2, etc. This is similar to the incremental bulk moduli of Rivlin and Beatty [3] in their analysis of the stability of a compressible unit cube under dead loading. Rivlin and Beatty [3] also define incremental Young’s moduli and Poisson’s ratios similar to those defined below. These incremental elastic moduli are also related to the generalized Lam´emoduli of Beatty [4, Appendix]. On physical grounds we might expect δp and δV to have opposite signs, so that, for example, a further increase in pressure results in a further decrease in volume. Then physically reasonable response would require K(λ) > 0 for 0 <λ< , (2.6) ∞ a criterion which may be shown to be equivalent to the P-C inequality of Truesdell and Noll [1, (51.3)]. However, for an empirically determined material it might not be possible to insist that K(λ) be positive for all λ. We might find that K(λ) can be positive only for a smaller λ-interval, provided that this interval includes the stress-free ground state λ = 1. Then the material model is physically realistic only for strains within this λ- interval. We find in the limit λ 1 of (2.2) that K(1) = K0, so that the ground state incremental bulk modulus is→ equal to the usual bulk modulus of linear isotropic elasticity. We may define an average (or secant) bulk modulus in which the current stress is compared with the relative volume change: σ11 K(λ) := . (2.7) λ3 1 − We find that K(1) = K0 = K(1), so that this alternative definition of bulk modulus agrees with (2.2) in the ground state. Physically reasonable response would require K(λ) > 0 for 0 <λ< . (2.8) ∞ However, this condition is less restrictive than (2.6) as it requires only that σ11 and λ 1 − have the same sign, not that δσ11 and δλ should have the same sign. By integrating (2.2) we find that, since the reference configuration λ = 1 is stress free, (2.7) may be written λ ′ 3 1 K(λ ) ′ K(λ)= 2 ′ dλ , (2.9) λ + λ +1 · λ 1 1 λ − Z Incremental moduli in nonlinear elasticity Page 5 of 16 clarifying the sense in which K(λ) may be regarded as an average of the incremental bulk modulus. It is clear from (2.9) that criterion (2.6) implies (2.8) but that the converse does not hold: it is possible for K(λ) to be negative with K(λ) remaining positive, see, for example, (3.6) and (3.9).