Sharp rank-one convexity conditions in planar isotropic elasticity for the additive volumetric-isochoric split

We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu}{2}\frac{\lVert F\rVert^2}{\det F}+f(\det F)$; such an energy is rank-one convex if and only if the function $f$ is convex.


Introduction
Undoubtedly, one of the most important constitutive requirements in nonlinear elasticity is the rank-one convexity of the elastic energy W : GL + (n) → R. For C 2 -smooth energy functions, rank-one convexity is nothing else than the Legendre-Hadamard ellipticity condition expressing the ellipticity of the Euler-Lagrange equations Div DW (∇ϕ) = 0 associated to the variational problem Ω W (∇ϕ)dx → min . (1.1) Thus, rank-one convexity for smooth energies amounts to the requirement for all F ∈ GL + (n). This condition is equivalent to the requirement that the acoustic tensor Q(F, η) ∈ Sym(n) with Q ik (F, η) = n j,l=1 ∂Fij ∂F kl · η j η l is positive semi-definite for all directions η ∈ R n , where For the dynamic problem, a non-negative acoustic tensor means that the system is hyperbolic [19,44]. When the dynamic equation is linearized at F ∈ GL + (n), then there exists a traveling wave solution with real wave speeds if and only if the problem is Legendre-Hadamard elliptic at F [68]. Moreover, the static response is metastable (stable against infinitesimal perturbations) under Legendre-Hadamard ellipticity (if c > 0 in (1.2)). The loss of ellipticity is generally related to instability phenomena (separation into arbitrary fine phase mixtures [68,26], shear banding) and possible discontinuous equilibrium solutions [59]. Checking rank-one convexity for a given nonlinear elastic material can be quite challenging, see e.g. [35,8,56,48] and Appendix D, although John Ball's polyconvexity [6] as an easy-to-verify sufficient condition can often be helpful [63,36]. A certain simplification occurs in the isotropic case. Indeed, any isotropic energy on GL + (n) admits a representation in terms of the singular values W (F ) = g(λ 1 , · · · , λ n ), where g : (0, ∞) n → R is permutation invariant. In planar isotropic hyperelasticity, Knowles and Sternberg have provided a criterion for rank-one convexity in terms of g, conclusively reducing the problem to the singular values representation [31,32,33,70,14], see also [11,15,73,75,55,67]. In the Knowles-Sternberg result, a family of inequality constraints in terms of first and second derivatives of g (including the Baker-Ericksen inequality [5]) has to be checked, which can still be daunting in practice.
The ellipticity condition in the planar and isotropic case has also been reformulated in terms of yet different representations, see e.g. [27,15,60], always in the form of a set of differential inequality constraints involving the used special representation. Generalizations to the case of functions defined not on GL + (2) but on R 2×2 have been obtained by Aubert [3] and Hamburger [27]. For some special families of energy functions more detailed information is available. For example, in the fluid-like case [12] W (F ) = f (det F ) with f : (0, ∞) → R , W is rank-one convex if and only if f is convex. (1.4) Moreover, for the Hadamard-material [12,7,Theorem 5.58ii)] (see also [13]) In a recent contribution [34,38], we have been able to classify all planar, isotropic rank-one convex functions W : GL + (2) → R which are isochoric (conformally invariant), i.e. satisfy W (AF B) = W (F ) for all A, B ∈ aR ∈ GL + (2) | a ∈ (0, ∞) , R ∈ SO(2) . (1.6) For these energies we have the simple characterization 1 W (F ) = h λ 1 λ 2 , h : (0, ∞) → R , then W is rank-one convex if and only if h is convex, (1.7) where λ 1 , λ 2 > 0 are the singular values of F . Similarly, rank-one convexity in isotropic planar incompressible elasticity W : SL(2) → R can be easily checked [20]: for an energy of the form W is rank-one convex if and only if φ is convex and nondecreasing. (1.8) These results also allow for an explicit calculation of the quasiconvex relaxation for conformally invariant and incompressible isotropic planar hyperelasticity [38,37].

The volumetric-isochoric split
In isotropic linear elasticity, the quadratic elastic energy W lin and the linear Cauchy stress tensor σ lin can be uniquely represented as where µ, λ are the Lamé-constants, κ is the bulk-modulus, ∇u denotes the displacement gradient and dev n X = X − 1 n tr(X) · 1 is the deviatoric part of X ∈ R n×n , cf. Appendix A. The right hand side of the energy is automatically additively separated into pure infinitesimal shape change and infinitesimal volume change, respectively, with a similar additive split of the Cauchy stress tensor into a deviatoric part and a spherical part, depending only on the shape change and volumetric change, respectively.
A natural finite strain extension of (1.9) is given by the additive volumetric-isochoric split [30], i.e. by assuming the energy potential W to have the form for all s > 0. Here, again, the energy contribution is additively split into the isochoric part taking only the shape change into account (depending only on F n √ det F ), and a part penalizing only the volumetric extension in det F . Note the multiplicative decomposition of the deformation gradient F itself into volume preserving and shape changing part [57,23,18,66,72,54,17,43]. Such type of energy functions are often used when modeling slightly compressible material behavior [10,28,16,54,9] or when otherwise no detailed information on the actual response of the material is available [72,19,45]. In the nonlinear regime, this split is not a fundamental law of nature for isotropic bodies (as in the linear case) but rather introduces a convenient form of the energy. Formally, this split can also be generalized to the anisotropic case, where it shows, however, some severe deficiencies [17,43] from a modeling point of view. 2 It has been first shown by Richter [57,23,58,51] (see also Flory [18] and Sansour [61]) that the accompanying symmetric nonlinear Kirchhoff stress tensor τ = det F · σ (but not the Cauchy stress as repeatedly claimed in [44,45]) admits a similar additive structure in the sense that in which the deviatoric part of the Kirchhoff stress only depends on W iso and the spherical part only depends on W vol . A typical example of the foregoing volumetric-isochoric format is the geometrically nonlinear quadratic Hencky energy [47,29,65,36] as well as its physically nonlinear extension, the exponentiated Hencky-model [50,49] which has been used for the stable computation of the inversion of tubes [46], the modeling of tire-derived material [42] or applications in bio-mechanics [62]. While it is well known that (1.13) and (1.14) are overall not rank-one convex [50], there does not exist any elastic energy depending on dev n log V 2 , n ≥ 3 that is rank-one convex [21]. The situation is surprisingly completely different for the planar case. Indeed, (1.14) for n = 2 is not only rank-one convex, but also polyconvex [52,22] if k ≥ 1 8 . Because of this counter-intuitive result when descending from three-dimensions to two-dimensions, we became interested in the precise qualitative nature of the volumetric-isochoric split in the planar case with respect to rank-one convexity. 3 In the planar isotropic case (exclusively), the isochoric energy part W iso admits a number of equivalent representations, e.g.
In particular, for planar additively volumetric-isochoric split energies we can uniquely write without loss of generality [34, Lemma 3.1] where λ 1 , λ 2 > 0 denote the singular values of F and h, f are given functions. It is now tempting to believe that the rank-one convexity conditions on ψ and f , respectively on W iso and W vol also allow for a sort of split (like for the Kirchhoff stress tensor in (1.12)) and can be reduced to separate statements on h and f . However, this is not even the case in planar linear elasticity, where W lin from (1.9) is rank-one convex in the displacement gradient ∇u if and only if µ ≥ 0 and µ + κ ≥ 0, see Appendix A. This means that rank-one convexity of W lin implies that W lin iso is rank-one convex, whereas W lin vol might not be rank-one convex. We therefore need to expect some coupling in the conditions for h and f .
Our main result in this respect (Theorem 2.6) can be summarized as follows: If W is altogether rank-one convex then, either W iso or W vol must be rank-one convex (i.e. h or f are convex), but contrary to isotropic linear elasticity, W iso may also be non rank-one convex (see the example in Section 2.8). Although the conditions for h and f do not decouple completely, it is possible to reduce rank-one convexity to a family of coupled one-dimensional conditions.
The paper is now structured as follows. After a short introduction and visualization of rank-one convexity on GL + (2), we start with criteria by Knowles and Sternberg for the arbitrary objective-isotropic case on the two-dimensional set of the singular values λ 1 , λ 2 > 0 of F . Lemma 2.4 gives the corresponding conditions for the volumetric-isochoric split. The main Theorem 2.6 shows that it is indeed possible to reduce the conditions to a coupled family of one-dimensional differential inequalities which makes testing for rank-one convexity much more convenient in the volumetric-isochoric split case. Apart from a list of additional necessary conditions, we derive a simple classification for generalized Hadamard energies and give two examples of non-trivial rank-one convex energies for which W iso and W vol , respectively, are not rank-one convex. Finally, we show how criteria for the invertibility of the Cauchy stress response are also simplified in the volumetric-isochoric split case and highlight the connection of this invertibility property to rank-one convexity.
2 Rank-one convexity conditions on GL + (2) We first recall the following basic definitions.
is convex in t for all F ∈ R n×n and ξ, η ∈ R n . (2.1) which is equivalent to rank-one convexity.

Ellipticity domains for some nonlinear energy functions
For an isotropic energy function W : GL + (2) → R, it is possible to visualize the ellipticity domain in terms of the singular values λ 1 , λ 2 > 0 of F , as shown in Figure 1. The isotropy of W ensures the symmetry at the bisector, i.e. W (λ 1 , λ 2 ) = W (λ 2 , λ 1 ). For purely isochoric energies W (F ) = W iso , the ellipticity domain consists only of cones and can therefore be reduced to a one-dimensional problem at det F = 1 or any other fixed determinant. Of course, this is not the case for energies with an arbitrary volumetric part f (λ 1 · λ 2 ).   [55,11,71,69,4,3]). Let W : GL + (2) → R be an objective-isotropic function of class C 2 with the representation in terms of the singular values of the deformation gradient F given by W (F ) = g(λ 1 , λ 2 ), where g ∈ C 2 ((0, ∞) 2 , R). Then W is rank-one convex if and only if the following five conditions are satisfied: i) g xx ≥ 0 and g yy ≥ 0

The classical Knowles and Sternberg ellipticity criterion
"TE-inequalities" (separate convexity) for all x, y ∈ (0, ∞) , ii) Furthermore, if all the above inequalities are strict, then W is strictly rank-one convex.
This criterion has recently been generalized to plane strain orthotropic materials [1], see also [74,39]. Hamburger [27] gives an extension to R 2×2 instead of GL + (2) in the same format. In this paper, we restrict ourselves to the class of two-dimensional isotropic energy functions W : GL + (2) → R with additive volumetricisochoric split 4 Note that the case W (F ) = +∞ is excluded by definition. 4 The latter requirement is due to the invariance of the singular value representation g(x, y) = h x y +f (x·y) under permutation of the arguments, which implies h x y = h x y for all x, y > 0.
Starting with a rank-one convex energy function W with additive volumetric-isochoric split, it is not clear whether the separate parts W iso and W vol need to be rank-one convex, too. Indeed, subsequently we will see that this is not always the case.

Necessary and sufficient conditions for the planar volumetric-isochoric split
The following Lemma expresses the Knowles-Sternberg criterion (Lemma 2.3) in terms of the functions h and f for the specific form g(x, y) = h x y + f (x · y) of g in the case of an additive volumetric-isochoric split.
Lemma 2.4. Let W : GL + (2) → R be an objective-isotropic function of class C 2 with additive volumetricisochoric split, with the representation in terms of the singular values of the deformation gradient F given by Then W is rank-one convex if and only if the following four conditions are satisfied: (equivalent to Knowles and Sternberg ii) ) , (equivalent to Knowles and Sternberg iv) ) , Remark 2.5. Note that Knowles and Sternberg condition iii) is redundant in this case, since it is already implied by condition B), cf. [11].
Proof. We start by computing the partial derivatives The isotropy of the energy implies the symmetry under permutation of the two singular values λ 1 and λ 2 , thus of the energy function with respect to the singular values. Next, we utilize the specific form of the additive split by introducing the transformation of variables which allows us to decouple the conditions for g(x, y) into conditions for f (z) and h(t). Moreover, the invariance property h(t) = h 1 t of the isochoric part h yields Note that for t = 1, equation (2.7) already implies 6 In the following, we transform the conditions i)-v) of Knowles and Sternberg Lemma 2.3 one by one.
Knowles and Sternberg i) g xx ≥ 0 and g yy ≥ 0 for all x, y, ∈ (0, ∞): We compute and The first condition (2.8) already has the desired form. For the second condition (2.9) we use the symmetry property (2.7) for a further transformation, using the equality which is equivalent to condition (2.8). Thus Knowles and Sternberg ii) we obtain the equivalence Knowles and Sternberg iii) We compute and the latter condition being already implied by condition B), which in turn follows from Knowles and Sternberg ii).
Knowles and Sternberg iv)+v) We compute and substitute For further computations, we need to express the Knowles-Sternberg conditions in a form not involving square roots 7 in the term √ g xx g yy . However, we will need to pay close attention to the sign of occurring terms.
Knowles and Sternberg iv) √ g xx g yy + g xy + g x − g y x − y ≥ 0 for all x, y ∈ (0, ∞) , x = y: It holds For the first case in equation (2.22) we compute For the second case in equation (2.22) we start with Thus, with the definitions we obtain the equivalence 7 The square root expression does not allow for a proper decoupling of h(t) and z(z), since Knowles and Sternberg v) √ g xx g yy − g xy + g x + g y x + y ≥ 0 for all x, y ∈ (0, ∞): It holds √ g xx g yy ≥ g xy − g x + g y x + y ⇐⇒ −g xy + g x + g y x + y ≥ 0 or g xx g yy ≥ −g xy + g x + g y x + y 2 . (2.26) For the first case in equation (2.26) we compute For the second case in equation (2.26) we start with and compute Thus with the previous definitions thereby establishing the last equivalence and completing the proof of Lemma 2.4.

Reduction to a family of coupled one-dimensional differential inequalities
In the following theorem we show that, surprisingly, it is possible to reduce the two-dimensional problem of rank-one convexity for planar objective-isotropic energies with additive volumetric-isochoric split to only coupled one-dimensional conditions. Theorem 2.6. Let W : GL + (2) → R be an objective-isotropic function of class C 2 with additive volumetricisochoric split with the representation in terms of the singular values of the deformation gradient F via W (F ) = g(λ 1 , λ 2 ) = h λ1 λ2 + f (λ 1 λ 2 ), where f, h ∈ C 2 ((0, ∞), R), and let h 0 = inf t∈(0,∞) t 2 h (t) and f 0 = inf z∈(0,∞) z 2 f (z). Then W is rank-one convex if and only if 1) h Proof.
thus f is convex. Finally, W can be decomposed into the sum of the rank-one convex energy W and the F → f (det F ), which is rank-one convex due to the convexity of f . Therefore, the energy W is rank-one convex. Now, let W be rank-one convex. We observe that for given h and arbitrary fixed t > 0, each of the conditions A)-D) of Lemma 2.4 can be written as an inequality P t,h (z 2 f (z)) ≥ 0 for some continuous function P t,h . Furthermore, using the same functions, each of the conditions 1)-4) can be expressed in the form P t,h (f 0 ) ≥ 0. If W is rank-one convex, i.e. if the conditions A)-D) are satisfied for all z ∈ (0, ∞), then due to continuity, therefore conditions 1)-4) hold as well.

Further necessary conditions for rank-one convexity
We continue the discussion with a list of necessary conditions for rank-one convexity of an energy with volumetric-isochoric split in terms of the isochoric part h and volumetric part f . Proof. Condition a) The separate convexity condition 1) in Theorem 2.6 states

Lemma 2.8. Consider an arbitrary isotropic rank-one convex energy function
Condition b) For any a > 0, rank-one convexity of W implies the convexity of W along the straight line t → diag(a, 1) + t diag(1, 0) in rank-one direction diag(1, 0), i.e. the convexity of the mapping and thus the convexity of the mapping x → h(x) + f (x). Condition c) The monotonicity condition is already known as condition 2) in Theorem 2.6 (as well as Lemma 2.4).
Condition d) The inequality can be obtained by considering the restriction W SL = W | SL(2) of W to the special linear group {F ∈ GL + (2) | det F = 1}. Then rank-one convexity of W implies rank-one convexity [37] of W SL with respect to SL (2). Recall that any function W SL : SL(2) → R can be expressed in the form (1.8), i.e.
with φ : [0, ∞) → R, and W SL is rank-one convex on SL (2) if and only if φ is monotone and convex [37]. For for all F ∈ SL(2) with singular values λ max > λ min and thus Then for any t ≥ 1, and thus due to the rank-one convexity of W SL . For t ≤ 1, the inequality can be obtained analogously. If the energy is strictly rank-one convex, it holds φ (x) > 0 for all x > 0 and thus th (t) + h (t) > 0 for all t > 0. Condition e) The inequality follows directly from Lemma 2.7 where it is shown that rank-one convexity implies

Application to generalized Hadamard energies
In particular, our rank-one characterization applies to the family of planar isotropic functions Here, K ≥ 1 is the so-called distortion function, which plays an important role in the theory of quasiconformal mappings [2,38]. Note that K ≡ 1 if and only if ϕ is conformal. It is well known that the mapping F → K(F ) itself is rank-one convex and even polyconvex on GL + (2). Using our characterization for rank-one convexity of volumetric-isochoric split energies, we obtain the following lemma.
Lemma 2.9. The energy function W : and only if f is convex.
Proof. We first observe that Thus h is globally convex and (strictly) increasing for t > 1, hence the isochoric term µK is rank-one convex. If the volumetric term f is convex as well, the energy function W is rank-one convex as a sum of two rank-one convex functions. Now, in order to show the reverse implication, let W be rank-one convex. Then W is separately convex, which is expressed by condition 1) of Theorem 2.6 as h 0 + f 0 ≥ 0. We compute and thus Therefore, the function f must be convex (which implies that the volumetric part F → f (det F ) is rank-one convex).
The result of Lemma (2.9) should be compared with the particular planar and quadratic case [12, 7, 26, Theorem 5.58ii)] W : GL + (2) → R of the Hadamard-material, for which In an intriguing recent contribution [25,24], the quasiconvex envelope for the Hadamard material has been computed analytically for a non-convex function f and sufficiently large shear modulus µ. It would be interesting to obtain a similar relaxation result for W (F ) = µ 2 F 2 det F + f (det F ).

Idealized planar isotropic nonlinear energy function
From a modeling perspective, it is a useful idealization to assume that the isochoric and volumetric response are governed by the same functional form. Consider an energy function W : GL + (2) → R of the form where h ∈ C 2 ((0, ∞); R) with h(t) = h 1 t for all t ∈ (0, ∞) and µ, κ > 0. This class of energies includes, for example, the Hencky energy (1.13) and the exponentiated Hencky energy (1.14). Note that any energy W of this form is tension-compression symmetric, i.e. satisfies W (F ) = W (F −1 ) for all F ∈ GL + (2).
Theorem 2.6 immediately shows that an energy of the form (2.42) is rank-one convex if and only if h is convex. In this case, the volumetric and isochoric parts are rank-one convex, individually. Thus for this particular type of split energy function, rank-one convexity is indeed equivalent to rank-one convexity of both the volumetric and the isochoric part. Moreover, according to the results of Section 3, the corresponding Cauchy stress-stretch response is locally invertible if h is uniformly convex.

Examples of non-trivial rank-one convex energies
In contrast to the class (2.40) of functions, it is possible to find rank-one convex energies with either a non rank-one convex volumetric part W vol or non rank-one convex isochoric part W iso . As an example, consider the energy The isochoric part W iso = e is not rank-one convex, cf. Figure 1. However, with formula (C.5) the infinitesimal shear modulus 8 µ = ψ (1) = 1 5 > 0 and the infinitesimal bulk modulus κ = f (1) = 2 15 > 0 imply rank-one convexity in the infinitesimal case. Similar to the non rank-one convex energy in Figure 1, numerical results for the Knowles-Sternberg criterion directly suggest that the energy (2.43) is everywhere rank-one convex on GL + (2). In order to prove rank-one convexity with Theorem 2.6, we compute and determine the constant f 0 : ≥ 0 for all t ≥ 1 .
The remaining conditions 3) and 4) of Theorem 2.6 are too complex to be solved analytically, but are numerically visualized in Figure 2. As a second example, consider the energy with a non rank-one convex volumetric part W vol in the form of a double-well potential. Again, with formula (C.5) the infinitesimal shear modulus 9 µ = ψ (1) = 48 5 > 0 and the infinitesimal bulk modulus κ = f (1) = −8 < 0 imply rank-one convexity in the infinitesimal case. Similar to the non rank-one convex energy in Figure 1, employing numerical calculations for the Knowles-Sternberg criterion directly suggests that the energy (2.46) is everywhere rank-one convex on GL + (2). Again, the rank-one convexity of W can be shown using Theorem 2.6. First, we compute Again, the remaining conditions 3) and 4) of Theorem 2.6 are too complex to allow for analytical solutions, but are easily verified numerically similar to the case shown in Figure 2.

Invertibility of the Cauchy stress tensor in planar elasticity
The volumetric-isochoric split allows for a simplified investigation of invertibility for the Cauchy stress-stretch law [64] as well. First, recall that in planar linear isotropic elasticity, the energy potential and the Cauchy stress response are given by
Now, consider the planar nonlinear isotropic case with an additive volumetric-isochoric split.
is strictly rank-one convex and f is uniformly convex with W vol (F ) = f (det F ). Then the Cauchy stress-stretch law is locally invertible.

Conclusion
Our investigation clearly demonstrates that for planar isotropic hyperelasticity, assuming a volumetricisochoric split of the elastic energy endows the theory with a lot of additional mathematical structure which can be consequently exploited. Indeed, we have shown how the classical Knowles-Sternberg planar ellipticity criterion -which is represented by a family of two-dimensional differential inequalities -can be reduced to a family of only one-dimensional coupled differential inequalities for split energies, which allows for a much more accessible rank-one convexity criterion.
By using these reduced inequalities, we could show that it is possible for a volumetric-isochorically split energy to be altogether rank-one convex even if the isochoric part is non-elliptic, in stark contrast to the linear case, where only the volumetric part of a rank-one convex energy might be non-elliptic. We also applied our method to the general class of Hadamard-type materials and obtained a universal classification of their rank-one convexity.
In future contributions, based on encouraging preliminary results, we aim to extend our investigation from rank-one convexity to polyconvexity. Unfortunately, our methods are at present strictly restricted to the planar isotropic case; it remains to be seen whether similar statements can be established for the three-dimensional case as well.

References
and observe that µ is the infinitesimal shear modulus, λ = f (1) − µ is the infinitesimal Lamé-parameter and κ = f (1) the infinitesimal bulk modulus. Overall, in terms of the infinitesimal strain ε = sym ∇u. Now, we consider the more general case for some arbitrary smooth ψ : [1, ∞) → R. Then the linearization of W is given by Therefore, the infinitesimal shear modulus and bulk modulus are given by µ = ψ (1) and κ = f (1), respectively.

D Algebraic characterization of rank-one convexity
In order to illustrate the extent to which our results simplify the test for rank-one convexity of energy functions with an additive volumetric-isochoric split compared to a more direct approach, we provide here some of our initial computations which did not utilize the specific singular value representation in (1.15). Again, consider an energy function of the form We compute the bilinear form of the second derivative: Beginning with the second derivative of the volumetric part, we first find For the second derivative of the volumetric part, we therefore find For a rank-one matrix H = ξ ⊗ η, using that det(ξ ⊗ η) = 0, we further obtain Now, we turn our attention to the derivatives of the isochoric part, finding We compute (D. 12) as well as Assuming again that H = ξ ⊗ η is a rank-one matrix with det(ξ ⊗ η) = 0, we find (D. 19) Thus, we have deduced In conclusion Finally, we consider the common representation of the isochoric part in terms of the distortion function K : GL + (2) → [1, ∞), cf. (2.40 At this point, it seems unfeasible to continue with the approach outlined above. In particular, methods based on direct computations in terms of matrix-valued arguments instead of a singular value based representation appear to be highly unsuited for drawing any further conclusions regarding the rank-one convexity of W iso and W vol , respectively.