Hyperelastic Stability Landscape: A Check for Hill Stability of isotropic, incompressible Hyperelasticity depending on Material Parameters

Abstract In this paper, we describe a uniform and standardized approach for analytically verifying the stability of isotropic, incompressible hyperelastic material models. Here, we address stability as fulfillment of the Hill condition – i.e. the positive definiteness of the material modulus in the Kirchhoff stress – log–strain relation. For incompressible material behavior, all mathematically and mechanically possible deformations lie within a range bounded, on the one hand, by uniaxial states and, on the other hand, by biaxial states; shear deformation states lie in between. This becomes particularly clear when the possible states are represented in the invariant plane. This very representation is now also used to visualize the regions of unstable material behavior depending on the selected strain energy function and the respective data set of material parameters. This demonstrates how, for some constellations of energy functions, with appropriate selection or calibration of parameters, stable and unstable regions can be observed. If such cases occur, it is no longer legitimate to use them to initiate, for example, finite element simulations. This is particularly striking when, for example, a fit appears stable in uniaxial tension, but the same parameter set for shear states results in unstable behavior without this being specifically investigated. The presented approach can reveal simple indicators for this. Nevertheless, the simple shear deformation, where the principal axes lag behind the deformation $\gamma =\tan \alpha $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>tan</mml:mo> <mml:mi>α</mml:mi> </mml:math> of the shear angle $\alpha $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , i.e. the rotation tensor $\textbf{R} \neq \textbf{I}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>R</mml:mtext> <mml:mo>≠</mml:mo> <mml:mtext>I</mml:mtext> </mml:math> , still represents a special case that requires extra investigations. This is especially true given that all shear components of the logarithmic strains themselves exhibit a non–monotonic behavior with respect to the deformation angle.