On the Incompressible Behavior in Weakly Nonlinear Elasticity

This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses.


Introduction
The weakly nonlinear theory of elasticity applied to incompressible solids has renewed interest with growth in dynamic characterization of soft materials such as tissue. A recent article by Saccomandi and Vergori [1] recaps the historical developments and places the weakly nonlinear and nonlinear theories in an appropriate context. Thus, readers interested in a historical basis should consult Saccomandi and Vergori and the references therein [1]. The benefits of the weakly nonlinear theory are appropriate to highlight and restate. Namely, the weakly nonlinear theory of incompressible solids contains only three independent stiffness constants, including exactly one second-, third-, and fourth-order stiffness. Thus, weakly nonlinear elastodynamic models involving the three stiffnesses provides a straightforward bridge to experiments whereas the full nonlinear theory of elasticity often deals with functional dependencies requiring empirical fits of experimental data.
With this motivation, several researchers have investigated the behavior of elastic stiffnesses when constraints of incompressibility are applied within the weakly nonlinear theory. Kostek, Sinha, and Norris [2] deduced connections between the third-order stiffness constants c 111 , c 112 , and c 123 and the constants A and B present in the adiabatic equation C.M. Kube kube@psu.edu 1 Engineering Science and Mechanics, Penn State University, University Park, PA 16802, USA of state for an inviscid fluid. In a similar comparison with liquids, Hamilton, Ilinskii, and Zabolotskaya obtained the behavior for the fourth-order stiffnesses E, F , G, and H in terms of A , B , and C in addition to a fourth-order constant D [3]. However, the comparison to a fluid does not lead to the possibility of shear deformation, which is a shortcoming if a connection to elastodynamic shear deformation in soft solids is desired. The existence of linear and nonlinear shear deformation was known theoretically for many years [4] and modern techniques involving the use of shear wave elastography are commonplace [5,6]. Destrade and Ogden uncovered the behavior of the linear and nonlinear stiffnesses under the constraints of incompressibility applied to the weakly nonlinear theory of compressible solids [7]. They were able to predict the magnitude of each of the stiffnesses to within the order of the Lamé parameter λ [7]. With the behavior of the stiffnesses known, the link between the strain energy of general compressible solids to that of an incompressible solid with three independent parameters was established as predicted decades before [4]. Perhaps most importantly, other theoretical models developed for static and dynamic problems involving compressible solids are able to apply the limiting behavior of the elastic constants to extend predictions to incompressible solids. Destrade and Ogden [7] provided several examples of extending previously developed wave propagation results to incompressible solids.
In Destrade and Ogden [7], the constraints stemming from incompressibility were applied to the relationship between logarithmic strain and the conjugate stress tensor (rather than the more common stress-strain relationship). In doing so, an arbitrary Lagrange multiplier is not needed in this case as stated in Destrade and Ogden, "...a strain-stress relation is more amenable to the imposition of incompressibility, because it leads to unambiguous, finite limit(s) for one (or several) compliance(s)" [7]. Thus, the analysis of the compliances are fundamental to determining the behavior in the incompressible limit. However, unlike in [8], which analyzed the behavior of the second-order compliance constants, the compliance constants and their behavior were not explicitly provided [7]. More recently, Saccomandi and Vergori [1] followed a similar approach of eliminating the Green-Lagrange strain invariants in the incompressibility constraint in favor of invariants of the second Piola-Kirchhoff stress tensor. They obtained improved estimates of the Landau constants by then analyzing the stress-strain relationship including the determination of the Lagrange multiplier p. Once again, while the behavior of the compliance constants are integral to the method, their explicit behavior was not provided.
This article reports, for the first time, the explicit behavior of the second-, third-, and fourth-order compliance constants as the solid tends toward incompressibility. Additionally, the usual stiffness/compliance relationships are shown to recover the behavior of the stiffness constants and are closely connected to previous results [1,7]. Understanding the behavior of both stiffnesses and compliances allows pure stress formulations in elasticity problems to be extended to incompressibility. For example, problems in elastodynamics have recently been developed using the so-called stress equations, which find compliance constants in their solutions [9][10][11].

Weakly Nonlinear Elasticity
For a weakly nonlinear elastic solid, the second Piola-Kirchhoff stress tensor S can be written as a third-order polynomial in the Green-Lagrange strain E, where c ijkl , c ijklmn , and c ijklmnpq are tensors whose components are the second-, third-, and fourth-order elastic stiffnesses, respectively. Equation (1) is given in general anisotropic form to motivate possible extensions of this work to anisotropic materials. The current article considers isotropic solids only for which S can be written as where the strain invariants are The form of S in Eq. (2) is obtained through substituting the isotropic forms of c ijkl , c ijklmn , and c ijklmnpq given in Appendix A into Eq. (1) and evaluating terms involving Kronecker delta functions. Under a deformation, a compressible solid changes its differential volume according to dV = det F dV 0 where F is the deformation gradient. The quantity det F is related to the principal invariants of the Green-Lagrange strain [2,12], where the principal invariants, and their relation to J 1 , J 2 , and J 3 , are Thus, in terms of the strain invariants J 1 , J 2 , and J 3 , Eq. (4) becomes A solid is incompressible when it is capable of only isochoric deformations, i.e., det F = 1. This motivates forming the parameter v from Eq.
Thus, the vanishing of v based on the combinations of strain invariants seen in the righthand side of Eq. (7) provides the desired constraint to analyze incompressibility. Now, as described by [7], transforming the Green-Lagrange strain in Eq. (7) to the second Piola-Kirchhoff stress circumvents the need for an arbitrary Lagrange multiplier. The transformation is provided through the strain-stress relationship, where s ijkl , s ijklmn , and s ijklmnpq are tensors containing components of second-, third-, and fourth-order elastic compliances, respectively. Similar to Eq. (2), E for an isotropic solid takes the form where are invariants of the second Piola-Kirchhoff stress. The next step involves obtaining the strain invariants needed in Eq. (7) by determining the trace of E, E 2 , and E 3 using Eq. (9), which leads to Additionally, the following three quantities are needed in Eq. (7), Truncating Eqs. (11)- (12) to include up to third-order terms involving S and substituting into Eq. (7) gives An expression similar to Eq. (13), but not equivalent, was given by Saccomandi and Vergori [1]. In Eq. (14), the a i coefficients are functions of the second-, third-, and fourth-order compliance constants. Previous work [1,7] made use of inverting the strain invariants into stress invariants, but did not explicitly involve or make use of the compliance constants. The advantage of using compliance constants includes a logical path forward to investigate anisotropic materials and also provides new insights into the behavior of compliances when solids tend toward incompressibility.

Behavior of Nonlinear Compliance Constants in the Incompressible Limit
The behavior of the compliance constants seen in Eq. (14) Equations (15) provide the complete behavior of the second-, and third-order compliances in the incompressible limit. It is interesting to note that all of the compliances remain finite, whereas only the stiffnesses c 44 and c 456 remain finite.
To examine the behavior of the fourth-order compliances, begin by analyzing the expression involving a 6 to give The behavior of s 1123 is obtained from the a 5 Note that each term multiplying factors of a 1 in Eqs.

Behavior of Nonlinear Stiffness Constants in the Incompressible Limit
To investigate the behavior of the stiffness constants under constraints of incompressibility, begin by analyzing the linear elastic case involving a 1 in Eq. (14). Using stiffness/compliance relations, a 1 can be written as Thus, the vanishing of a 1 leads to c 12 ∼ ∞ and a 1 ∼ 1/(3c 12 ).
To find the behavior of the third-and fourth-order stiffnesses, extensive use is made of the stiffness/compliance relationships found in Appendix B, which were derived with no consideration of incompressibility.

Acoustoelasticity
Previous researchers [1,7] provide results for various nonlinear wave phenomenon using the obtained behavior of the stiffnesses in the incompressible limit. Thus, those results are not repeated here. However, it is worth noting an interesting connection between acoustoelasticity and the compliance constants. The shear phase velocity v T of a wave propagating in directionn and displacementû in an incompressible solid containing the static stress σ is governed by the expression [13] which is a generalized form for the specific cases reported in [7]. It is interesting that the fourth-order compliances s 1123 , s 1144 , s 1456 , and s 4455 are functions of only c 44 and c 456 . Thus, acoustoelastic measurements provide direct access to these fourth-order compliances.

Conclusion
In this article, the incompressible behavior of both compliances and stiffnesses are derived explicitly. The second-, third-, and fourth-order compliance constants are shown to be finite. Additionally, the third-and fourth-order compliances are found to be functions of the finite stiffnesses c 44 and c 456 , but not fourth-order stiffnesses. Thus, experiments sensitive to c 44 and c 456 can be connected to the fourth-order compliance constants. Lastly, the results are presented in both the Brugger convention [14] and Landau-Lifshitz notation [15]. Formulating the theory using the Brugger convention was necessary as the stiffness/compliance relationships given in Appendix B were obtained through tensorial operations. Furthermore, the Brugger convention is more commonly used in treating anisotropic problems, which provides a path forward to investigating the incompressible behavior of anisotropic materials.

Appendix B: Stiffness/Compliance Relationships
where I ijkl = δ ik δ jl + δ il δ jk /2 is the fourth-rank identity. The term ∂E mn /∂S kl is found by taking the derivative of E in Eq. (8) with respect to S, ∂E mn ∂S kl = s mnkl + s mnklpq S pq + 1 2 s mnklpqrs S pq S rs , which was obtained by using ∂S ij /∂S kl = I ijkl . Substituting Eqs. (8) and (39) into Eq. (38) and keeping terms up to second-order in S gives