On the Development of Volumetric Strain Energy Functions

Transcription

1 Institut für Mechanik On the Development of Volumetric Strain Energy Functions S. Doll, K. Schweizerhof Universität Karlsruhe, Institut für Mechanik 1999 Institut für Mechanik Kaiserstr. 12, Geb Karlsruhe Tel.: +49 (0) 721/ Fax: +49 (0) 721/

2 On the Development of Volumetric Strain Energy Functions (Dedicated to Professor Franz Ziegler on the occasion of his 60th birthday) S. Doll and K. Schweizerhof Institute for Mechanics, University of Karlsruhe, Karlsruhe, Germany Abstract To describe elastic material behaviour the starting point is the isochoric-volumetric decoupling of the strain energy function. The volumetric part is the central subject of this contribution. First some volumetric functions given in the literature are discussed with respect to physical conditions, then three new volumetric functions are developed which fulfill all imposed conditions. One proposed function which contains two material parameters in addition to the compressibility parameter is treated in detail. Some parameter fits are carried out on the basis of well-known volumetric strain energy functions and experimental data. A generalization of the proposed function permits an unlimited number of additional material parameters. 1 Introduction The success achieved in the application of finite element techniques during the recent years has the consequence that nowadays nonlinear material laws at finite strains are frequently applied 1

3 in structural analysis. In the case of finite elasticity strain energy functions for compressible (or nearly compressible) materials are preferable because displacement based finite elements can be used. As a special assumption the isochoric-volumetric decoupling of the energy function is frequently applied in this context. As an advantage appears that the isochoric and the volumetric material behaviour can be treated as completely independent which permits their decoupled treatment in the development of finite elements, e.g. using different integration schemes to avoid locking phenomena. A disadvantage of the split is the increase of computational effort due to the product formula that must be taken into account deriving the stresses and the elasticity tangent from the strain energy function. The outline of this contribution is as follows: After reviewing the isochoric-volumetric decoupling of the strain energy function the conditions imposed on the decoupled energy function are motivated and a complete representation is given. Then the focus is on the volumetric part, first discussing known functions and then developing new functions. 2 Decoupling of the Strain Energy Function As is well known (e.g. Ogden (1984)) under the assumption of isotropy the strain energy function depends only on the left (or right) Cauchy-Green tensor b through the invariants I b, II b, III b or the related principal stretches λ 1, λ 2, λ 3 : W = W(b) = W(I b,ii b,iii b ) = W(λ 1,λ 2,λ 3 ). In the compressible case all invariants and principal stretches are independent and no constraint exists. The determinant of the deformation gradient which allows to measure the local change 2

4 of volume during the deformation is given by J = III b = λ 1 λ 2 λ 3 with 0 < J <. For compressible materials a totally decoupled isochoric and volumetric material behaviour is commonly assumed (see citations in section 4). This leads to the definition of the isochoric left Cauchy-Green tensor b with the invariants I b = J 2 3 Ib, II b = J 4 3 IIb, III b = 1 (1) and the isochoric principal stretches λ i = J 1/3 λ i (2) which date back to Flory (1961). Because of the isochoric incompressibility the classical constraint (1) 3 holds, thus λ 1 λ2 λ3 = 1. Now only two of three isochoric principal stretches are independent. Including the isochoric-volumetric decoupling into the strain energy function leads to W = Ŵ(I b,ii b ) + U(J), W = Ŵ( λ 1, λ 2, λ 3 = 1 1 λ 1 λ 2 ) + U(J) (3) where Ŵ is the isochoric part and U(J) is the volumetric part. The question for which materials or in which range such a decoupled strain energy function holds is not discussed here (see e.g. Penn (1970) for some criticism of the additive split or van den Bogert & de Borst (1990) for the investigation of coupling terms). In the following we assume that the additive split (3) is valid for the materials considered. 3

5 3 Requirements for the Strain Energy Function The strain energy function has to satisfy some physical conditions. For completeness the conditions are listed for the isochoric part as well as for the volumetric part separately. In each case a short motivation is given. For simplicity the considerations are based on the representation (3) 2. Some typical references are Ogden (1982), Ogden (1984) or Ciarlet (1988). It should be noted that all of the conditions imposed on the isochoric strain energy Ŵ in this contribution coincide exactly with those imposed on the strain energy function for incompressible materials. The only difference is that the isochoric principal stretches λ i replace the principal stretches of the incompressible case. Isotropy requires that the principal stretches can be arbitrarily ordered in the strain energy function: Ŵ( λ i, λ j, λ k ) = Ŵ( λ l, λ m, λ n ) for i j k i and l m n l. The determinant J and therefore the volumetric part fulfill this requirement automatically. For isotropic materials the derivatives must fulfill the following conditions: λ i Ŵ( λ 1, λ 2, λ 3 ) λ m=1 = λ Ŵ( λ k 1, λ 2, λ 3 ) λ m=1, 2 λ Ŵ( λ i λ 1, λ 2, λ 3 ) = i λ 2 m=1 λ Ŵ( λ j λ 1, λ 2, λ 3 ) j 2 λ Ŵ( λ i λ 1, λ 2, λ 3 ) = j λ 2 m=1 λ Ŵ( λ k λ 1, λ 2, λ 3 ) l λ m=1 λ m=1, for i j, k l. (4) Herein denotes the first partial derivative with respect to the indicated variable and 2 denotes the second partial derivative. These conditions hold only in the case of identical principal stretches. Because of definition (2) identical principal stretches lead always to λ m = 1. 4

6 The corresponding conditions on the derivatives of the volumetric part are then automatically fulfilled. In the strainless initial state no strain energy Ŵ( λ 1 = 1, λ 2 = 1, λ 3 = 1) = 0 and U(J = 1) = 0 (5) is stored. If the strainless state is assumed to be stressfree the condition J U J=1 = 0 (6) must hold, where p(j) = J U represents the volumetric stress (= hydrostatic pressure). Due to (4) 1 no similar statement for λ i Ŵ can be obtained. If strains are present, i.e. λ i 1, the stored energy must be always positive. Ŵ( λ 1 1, λ 2 1, λ 3 = λ λ 2 ) > 0, U(J 1) > 0 (7) In the case of infinitesimal strains the strain energy function leads in the limit to the classical Saint-Venant-Kirchhoff (SVK) material law Ŵ( λ 1 1, λ 2 1, λ 3 = 1 1 λ 1 λ 2 ) + U(J 1) W SVK. Considering the tangent of the stress-strain relation in the initial state the conditions [ λ i Ŵ + 2 λ Ŵ 2 i λ i λ Ŵ ] i λ j λ 1 1, λ 2 1, λ 3 = λ 1 1 λ 1 2µ for i j 2 (8) and JJ 2 U J 1 K occur, where µ is the shear modulus and K is the bulk modulus of the infinitesimal theory. In the limit case when the continuum degenerates to a single point, the strain energy tends to positive infinity and the volumetric stress to negative infinity: U(J +0) + and J U J +0. (9) 5

7 Accordingly a infinitely stretched continuum results in a positive infinite strain energy and a positive infinite volumetric stress U(J + ) + and J U J + +. (10) These two limit cases lead to undeterminable isochoric stretches (2) due to a product zero times infinity. Therefore no conditions for Ŵ and λ i Ŵ are available. With respect to the requirement of polyconvexity of the strain energy function the volumetric part has to satisfy the convexity condition 2 JJ U 0 (11) which appears in conjunction with the existence of solutions (see e.g. Ciarlet (1988)). In the following the attention is focused on the volumetric strain energy function U exclusively. 4 Discussion of Volumetric Strain Energy Functions The first part of this section extends the considerations of Liu & Mang (1996) and sets the motivation for the second part which deals with the design of alternative volumetric strain energy functions. 4.1 Functions Suggested in the Literature In Table 1 some volumetric strain energy functions suggested in the literature are summarized. Some characteristic references are: for U 1 Sussman & Bathe (1987), Simo (1988), van den 6

8 Bogert & de Borst (1990), Chang & Saleeb & Li (1991), van den Bogert & de Borst & Luiten & Zeilmaker (1991); for U 2 Hencky (1933), Valanis & Landel (1967), Simo & Taylor & Pfister (1985), Simo (1992), Roehl & Ramm (1996); for U 3 Simo & Taylor (1982), van den Bogert & de Borst (1990), Liu & Mang (1996); for U 4 Ogden (1972), Simo & Taylor (1991), Miehe (1994), Kaliske & Rothert (1997) and for U 5 Liu & Hofstetter & Mang (1992), Liu & Hofstetter & Mang (1994). The cited references show that the isochoric-volumetric decoupling of the strain energy function is very common, especially in the treatment of nonlinear elasticity using the finite element method. In some references the extension of incompressible materials to nearly incompressible materials is discussed. These proposed extensions are closely related to the volumetric strain energy function and can be interpreted in a similar fashion. Table 1: U(J) suggested in the literature (see references). Table 2: Fulfillment of the volumetric conditions for the given and proposed U(J). For the given volumetric strain energy functions the fulfillment of the conditions (5) (11) in section 3 is given in Table 2. A denotes the fulfillment of the corresponding condition. In the case of violation the limit value is listed instead. It is obvious that only the behaviour of the functions U 3 and U 4 is correct. The functions U 1, U 2 and U 5 show some deficiencies. In particular U 1 and U 5 should not be used in applications with large compression while U 2 does not make sense in cases where large volumetric expansions occur. In Figures 1, 2 and 3 the functions U 3, U 4a (index a stands for θ = 2) and their derivatives J U, JJ 2 U are given (the newly proposed functions U 6 8 are discussed later). All curves are plotted using K to scale. 7

9 The fulfillment of the conditions (5) (11) can be checked now very easily. It appears also that the functions U 3 and U 4a lead to very similar shapes. Figure 1: Curves U(J)/K. Figure 2: Curves of the first derivative J U(J)/K. Figure 3: Curves of the second derivative 2 JJ U(J)/K. The compressibility parameter K only scales the functions but does not change their shapes. In this context K can be interpreted as a penalty parameter that enforces incompressibility if large values are chosen. The function U 4 seems to be superior compared to the other functions given in Table 1 because it contains one additional parameter θ which permits to fit the shape of the function to experimental data. However, only values of θ < 1 guarantee the fulfillment of all conditions. In the literature the choice θ = 2 is very popular. As found by Ogden (1972) the value θ = 9 1 fits experimental data for nearly incompressible rubber rather well. In this respect the violation of the conditions (10) 2 and (11) seems to be acceptable because for nearly incompressible materials a large K has to be taken and J remains close to 1. Applications with J > 1 should be handled with care due to the small volumetric stress obtained in this range (see also Figure 4). 8

10 4.2 Alternatively Proposed Functions Before developing alternative volumetric functions their desirable properties must be defined: First the functions must be conform with the physical conditions (5) (11) and second the functions should be as general as possible. The fulfillment of the second requirement has the advantage that a wide range of experimental data can be fitted with only one general function. Then a wide range of elastic volumetric material behaviour can be described with little effort which can be seen as an advantage in conjunction with finite element codes. This means not each special material should have its special volumetric function but its special parameters within a general function. Generality of a volumetric strain energy function is only achieved, if some additional material parameters are incorporated. In the following a class of volumetric functions with two additional parameters is proposed. The design process of such functions is described in detail with respect to the fulfillment of the physical conditions Function with Two Additional Parameters The proposed starting point of the development is the following equation p 6 (J) = J U 6 (J) = K(J α J β )(α + β) 1 with α > 0 and β > 1 (12) for the volumetric stress. The index 6 indicates a new volumetric strain energy function. Condition (6) is obviously fulfilled. The first term with the positive exponent (α > 0) vanishes as J +0. The second term with the negative exponent (β > 1) vanishes as J +. Consequently the conditions (9) 2 and (10) 2 hold. The differentiation of equation (12) with regard to J shows that conditions (8) 2 and (11) are fulfilled. It has to be noted, that starting with an undetermined constant in (12) the constant K(α+β) 1 follows directly from condition 9

11 (8) 2. The integration of equation (12) with regard to J results in the following according volumetric strain energy function: U 6 (J) = K[(α + 1) 1 J α+1 + (β 1) 1 J (β 1) ](α + β) 1 K(α + 1) 1 (β 1) 1. (13) The integration constant is chosen such that condition (5) 2 is satisfied. Conditions (7) 2, (9) 1 and (10) 1 are directly fulfilled. Differentiating or integrating (12) the negative exponent of the second term is preserved whereas the sign changes. This property holds as β > 1 and ensures that the volumetric strain energy function as well as the second derivative remain always positive. Thus in the limit U 6 tends to infinity as J +0 or J +, as it should. In equations (12) and (13) two special cases are contained. Setting β = α the relation p 6 (J) = p 6 (J 1 ) holds. That means in a homogeneously compressed brick with the stretch factor γ 1 (γ > 1) and in a homogeneously expanded brick with the stretch factor γ volumetric stresses act with identical absolute values but different signs. Once as compressive and once as tensile stress. Using β = α+2 the relation U 6 (J) = U 6 (J 1 ) holds, i.e. two identical bricks, the first compressed homogeneously by γ 1 and the second expanded homogeneously by γ, store the same volumetric strain energy. Figures 1, 2 and 3 contain the curves for function U 6a (index a means α = β = 2) and its derivatives. The identical constants refer to the first one of the special cases considered above. The fulfillment of the conditions (5) (11) is obvious. The major task now is to assess the new volumetric function U Assessment: The limit process α 0 in equation (12) leads directly to the pressure formula p(j) = K(1 J β )β 1 (14) 10

12 given in Murnaghan (1951,page 73). Note, that in this contribution the compressive volumetric stress has a negative sign in contrast to the positive pressure in the cited papers. The advantage of the pressure formula (14) is its excellent adjustment to experimental data for Sodium (see references in Murnaghan (1951) for information concerning the experiments). Setting β = 3.79 the experimental values p(j) are approximated at pressures up to 10GPa within the accuracy of measurement (3 per cent). Therefore the practical applicability of U 6 is proofed. On the other hand the volumetric strain energy function based on Murnaghan s pressure formula (14) is a limit case (α 0) of the well-behaved more general function (13) which fulfills all conditions (5) (11). In Figure 4 the volumetric stress curve derived from U 6b (index b means α = 0.001, β = 3.79) is given. The curve for α = 0 is omitted here, because no difference compared to J U 6b is visible. Due to experimental considerations the fit of the volumetric stress (= pressure) seems to be superior over the fit of the energy function itself or the second derivative. Therefore the plots of the U-curves and JJ 2 U-curves are omitted. Figure 4: Fitted curves of the first derivative J U(J)/K. 2. Assessment: The task now is to fit the constants α and β in U 6 to obtain similar curves as given by the two frequently used and well-behaved functions U 3 and U 4a (index a means θ = 2). Because both functions are similar, the attention is restricted here to function U 4a. Looking at the first derivative of U 4a the comparison with equation (12) would lead directly to the constants α = β = 1, which in turn would violate the second restriction in relation (12). But the choice α = 1, β 1 (especially β = referenced as U 6c ) satisfies the second restriction and a perfect parameter fit with respect to U 4a can be observed. In Figure 4 the 11

13 volumetric stress curve J U 6c is plotted. The curve for J U 4a (see Figure 2) is omitted here because it is indistinguishable from J U 6c. Because for α = 1, β 1 the function U 6 becomes identical to the well-known function U 4a the applicability of U 6 is confirmed once more. 3. Assessment: As mentioned the volumetric function U 4b (index b means θ = 9) was found to fit experimental data rather well in the compression range for nearly incompressible rubber. But for this choice of θ the violation of two physical conditions must be accepted. The first derivative of U 4b in comparison with equation (12) would suggest to use the values α = 1 and β = 10 which would violate the restriction for parameter α in (12). Choosing α = and β = 10 (referenced as U 6d ) the restrictions for α, β and all physical conditions (5) (11) are fulfilled. The two volumetric stress graphs derived from U 4b, U 6d are given in Figure 4. In the compression range (J < 1) both graphs are nearly indistinguishable. Thus once more a good parameter fit to experimental data (see Ogden (1972) and references) is obtained and the general usability and versatility of U 6 is confirmed. As a drawback the physically non-reasonable small stresses for J > 1 should be mentioned. Due to the fulfillment of all conditions the performance of U 6d appears to be somewhat more reasonable in this range Generalization of the Two-Parameter-Function A further generalization of the volumetric strain energy function U 6 is possible by the additive composition of single functions of the proposed type (12). The volumetric stress of the generalized function is then given by n n p 6gen (J) = J U 6gen (J) = K [ (J α i J β i )][ (α i + β i )] 1 i=1 i=1 (15) with α i > 0 and β i > 1. 12

14 Integration of equation (15) results in the generalized function U 6gen (J). The integration constant has to be determined from (5) 2. It is straightforward to verify that U 6gen (J) fulfills all conditions (5) (11). Now more general strain energy functions with an unlimited number of additional material parameters α i, β i can be derived. It should be mentioned that every additive composition of the single volumetric strain functions U 1 6 which fulfills the physical conditions is an admissible generalization of the single functions. E.g. the function (U 1 + U 4 )/2 leads to an admissible new function which overcomes the deficiencies of the single function U Further Functions In addition to function U 6 two further functions U 7 (J) = K (exp(j 1) ln J 1) /2 U 8 (J) = K (J 1)ln J /2 (16) are newly proposed here in the context of volumetric strain energy functions. As reported in Table 2 these two functions fulfill all conditions (5) (11) as well. However, they do not contain some constants to influence their shapes, which reduces the possibilities to fit experimental data in general. But both functions (16) seem to be superior over U 1, U 2 and U 5 because they do not violate any condition. For further comparison the functions U 7 and U 8 respectively their derivatives are also plotted in Figures 1, 2 and 3. Two additional parameter fits of the single function U 6 are performed now with respect to the functions U 7 and U 8. This should give an idea of the possibilities and limits in the approximation of volumetric stress data using U 6. With α = 2.3, β = 1.4 referenced as U 6e the 13

15 function U 7 is fitted and with α = 0.45, β = 1.05 referenced as U 6f the function U 8 is approximated. The derived volumetric stress curves are plotted in Figure 5. The comparison shows that both U 7 and U 8 can be nearly approximated by U 6. To expect a perfect approximation in the whole range of J would require more parameters, e.g. using the generalized function U 6gen. But the really good parameter fit over most important ranges has to be noted. Figure 5: Fitted curves of the first derivative J U(J)/K. 5 Concluding Remarks Well known and newly proposed volumetric strain energy functions have been discussed. The newly proposed functions are closely related to frequently used functions. The advantage of these new functions is the fulfillment of all known physical conditions. The more general function U 6 contains two material constants in addition to the compressibility parameter. Therefore a good fit to given data is possible. The good adaptability of the proposed volumetric strain energy function U 6 to well-known functions and experimental data is proofed. It must be noted that applications with small volumetric deformations do not require general volumetric strain energy functions. Because of the physical conditions all admissible functions are close together in this range of deformation. The proposed function U 6 with two additional constants and its generalization U 6gen with an arbitrary number of additional constants must be seen in the context of large volumetric deformations. A good adjustment of the volumetric function to given data in a wide range of volumetric deformation can only 14

16 be expected if sufficient material parameters are available, as in U 6, U 6gen. However, it has to be noted that the determination of the constants is difficult, because in the volumetric stress formula the constants appear simultaneously in the exponent and in the denominator. Furthermore, considering real materials, the modeling failure due to the assumed isochoricvolumetric split of the strain energy function should be taken into account to prevent an overprecise volumetric part. 6 References Bogert, Van den, P.A.J.; Borst, De, R. [1990], Constitutive Aspects and Finite Element Analysis of 3D Rubber Specimens in Compression and Shear, NUMETA 90: Numerical Methods in Engineering: Theory and Applications (Eds. G.N. Pande, J. Middleton), Elsevier Applied Science, Swansea, pp Bogert, Van den, P.A.J.; Borst, De, R.; Luiten, G.T.; Zeilmaker, J. [1991], Robust Finite Elements for 3D-Analysis of Rubber-Like Materials, Engineering Computations, 8, pp Chang, T.Y.; Saleeb, A.F.; Li, G. [1991], Large Strain Analysis of Rubber-Like Materials Based on a Perturbed Lagrangian Variational Principle, Computational Mechanics, 8, pp Ciarlet, P.G. [1988], Mathematical Elasticity. Volume 1: Three Dimensional Elasticity, Elsevier, Amsterdam. Flory, P.J. [1961], Thermodynamic Relations for High Elastic Materials, Transactions of the Faraday Society, 57, pp Hencky, H. [1933], The Elastic Behavior of Vulcanized Rubber, Applied Mechanics 15

17 (Journal of Applied Mechanics), 1, pp Kaliske, M.; Rothert, H. [1997], On the Finite Element Implementation of Rubber-Like Materials at Finite Strains, Engineering Computations, 14, pp Liu, C.H.; Hofstetter, G.; Mang, H.A. [1992], Evaluation of 3D FE-Formulations for Incompressible Hyperelastic Materials at Finite Strains, Proceedings of the First European Conference on Numerical Methods in Engineering (Eds. C. Hirsch, O.C. Zienkiewicz, E. Oñate), September 1992, Brussels, Belgium, pp Liu, C.H.; Hofstetter, G.; Mang, H.A. [1994], 3D Finite Element Analysis of Rubber-Like Materials at Finite Strains, Engineering Computations, 11, pp Liu, C.H.; Mang, H.A. [1996], A Critical Assessment of Volumetric Strain Energy Functions for Hyperelasticity at Large Strains, Zeitschrift fuer angewandte Mathematik und Mechanik, 76(S5), pp Miehe, C. [1994], Aspects of the Formulation and Finite Element Implementation of Large Strain Isotropic Elasticity, International Journal for Numerical Methods in Engineering, 37, pp Murnaghan, F.D. [1951], Finite Deformation of an Elastic Solid, John Wiley, New York. Ogden, R.W. [1972], Large Deformation Isotropic Elasticity: on the Correlation of Theory and Experiment for Compressible Rubberlike Solids, Proceedings of the Royal Society of London. Series A, 328, pp Ogden, R.W. [1982], Elastic Deformations of Rubberlike Solids, Mechanics of Solids, The Rodney Hill 60th Anniversary Volume (Eds. H.G. Hopkins and M.J. Sewell), Pergamon Press, pp

18 Ogden, R.W. [1984], Non-Linear Elastic Deformations, Ellis Horwood, Chichester. Penn, R.W. [1970], Volume Changes Accompanying the Extension of Rubber, Transactions of the Society of Rheology, 14(4), pp Roehl, D.; Ramm, E. [1996], Large Elasto-Plastic Finite Element Analysis of Solids and Shells with the Enhanced Assumed Strain Concept, International Journal of Solids and Structures, 33(20 22), pp Simo, J.C.; Taylor, R.L. [1982], Penalty Function Formulations for Incompressible Nonlinear Elastostatics, Computer Methods in Applied Mechanics and Engineering, 35, pp Simo, J.C.; Taylor, R.L.; Pfister, K.S. [1985], Variational and Projection Methods for Volume Constraint in Finite Deformation Elasto-Plasticity, Computer Methods in Applied Mechanics and Engineering, 51, pp Simo, J.C. [1988], A Framework for Finite Strain Elastoplasticity Based on Maximum Plastic Dissipation and the Multiplicative Decomposition: Part I. Continuum Formulation, Computer Methods in Applied Mechanics and Engineering, 66, pp Simo, J.C.; Taylor, R.L. [1991], Quasi-Incompressible Finite Elasticity in Principal Stretches. Continuum Basis and Numerical Algorithms, Computer Methods in Applied Mechanics and Engineering, 85, pp Simo, J.C. [1992], Algorithms for Static and Dynamic Multiplicative Plasticity that Preserve the Classical Return Mapping Schemes of the Infinitesimal Theory, Computer Methods in Applied Mechanics and Engineering, 99, pp Sussman, T.; Bathe, K.J. [1987], A Finite Element Formulation for Nonlinear Incompress- 17

19 ible Elastic and Inelastic Analysis, Computer & Structures, 26(1/2), pp Valanis, K.C.; Landel, R.F. [1967], The Strain-Energy Function of a Hyperelastic Material in Terms of the Extension Ratios, Journal of Applied Physics, 38(7), pp

20 List of Tables: Table 1: U(J) suggested in the literature (see references). Table 2: Fulfillment of the volumetric conditions for the given and proposed U(J). List of Figures: Figure 1: Curves U(J)/K. Figure 2: Curves of the first derivative J U(J)/K. Figure 3: Curves of the second derivative JJ 2 U(J)/K. Figure 4: Fitted curves of the first derivative J U(J)/K. Figure 5: Fitted curves of the first derivative J U(J)/K. 19

21 U 1 (J) = K (J 1) 2 /2 U 2 (J) = K (ln J) 2 /2 U 3 (J) = K [ (J 1) 2 + (ln J) 2 ] /4 U 4 (J) = Kθ 2 (θ ln J + J θ 1) for θ < 1 U 5 (J) = K (J ln J J + 1) Table 1: U(J) suggested in the literature (see references). 20

22 Literature Proposed Condition U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 (5) 2 (6) (7) 2 (8) 2 (9) 1 K/2 K (9) 2 K (10) 1 (10) 2 0 (11) 1 ln J Table 2: Fulfillment of the volumetric conditions for the given and proposed U(J). 21

23 U(J)/K 4 7 6a 3 4a J Figure 1: Curves U(J)/K. 22

24 J U(J)/K 4 7 6a 2 4a J -2-4 Figure 2: Curves of the first derivative J U(J)/K. 23

25 2 JJ U(J)/K a 1 4a J Figure 3: Curves of the second derivative 2 JJ U(J)/K. 24

26 J U(J)/K 2 6c 1 6b 6d b J Figure 4: Fitted curves of the first derivative J U(J)/K. 25

27 J U(J)/K 4 7 6e 2 6f J ,6f Figure 5: Fitted curves of the first derivative J U(J)/K. 26