Large Isotropic Elastic Deformations: On a Comprehensive Model to Correlate the Theory and Experiments for Compressible Rubber-Like Materials

Abstract The comprehensive incompressible strain energy function devised in a preceding contribution (J. Elast. 153:219–244, 2023) is extended in this work for application to the finite deformation of isotropic compressible rubber-like materials. Based on the two established approaches in the literature for constructing a compressible strain energy function $W$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> from the incompressible counterpart, two models are developed and presented: one model is developed on using a $J\left (= \lambda _{1} \thinspace \lambda _{2} \thinspace \lambda _{3} \right )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mspace/> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mspace/> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> term added to the general functional form of the incompressible model; and the second model on using the isochoric, or modified, principal stretches $\left (\bar{\lambda }_{a}\right )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , $a = 1,2,3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:math> , in the functional form of the incompressible model, to account for the deviatoric contribution $W_{dev}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>e</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The volumetric input $W_{vol}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>v</mml:mi> <mml:mi>o</mml:mi> <mml:mi>l</mml:mi> </mml:mrow> </mml:msub> </mml:math> is considred as an additive part. Each model is then simultaneously fitted to extant multi-axial experimental datasets, and the favourable correlation between the models’ predictions and the experimental data is demonstrated. Exemplar challenging individual datasets including a shear-softening behaviour exhibited by an elastomeric foam are also considered, whereby the excellent predictions of the said behaviours by the models will be illustrated. The compatibility of both models with the kinematics of slight compressibility will also be discussed and presented.