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Closed-form solution of the Ogden–Hill’s compressible hyperelastic model for ramp loading


This article deals with the visco-hyperelastic modelling approach for compressible polymer foam materials. Polymer foams can exhibit large elastic strains and displacements in case of volumetric compression. In addition, they often show significant rate-dependent properties. This material behaviour can be accurately modelled using the visco-hyperelastic approach, in which the large strain viscoelastic description is combined with the rate-independent hyperelastic material model. In case of polymer foams, the most widely used compressible hyperelastic material model, the so-called Ogden–Hill’s model, was applied, which is implemented in the commercial finite element (FE) software Abaqus. The visco-hyperelastic model is defined in hereditary integral form, therefore, obtaining a closed-form solution for the stress is not a trivial task. However, the parameter-fitting procedure could be much faster and accurate if closed-form solution exists. In this contribution, exact stress solutions are derived in case of uniaxial, biaxial and volumetric compression loading cases using ramp-loading history. The analytical stress solutions are compared with the stress results in Abaqus using FE analysis. In order to highlight the benefits of the analytical closed-form solution during the parameter-fitting process experimental work has been carried out on a particular open-cell memory foam material. The results of the material identification process shows significant accuracy improvement in the fitting procedure by applying the derived analytical solutions compared to the so-called separated approach applied in the engineering practice.

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This research has been supported by the Hungarian Scientific Research Fund, Hungary (Project Identifier: PD 108691) and the National Talent Programme of the Hungarian Government (Contract Identifier: NTP-EFO-P-15-0085). The research leading to these results has received funding from the Hungarian–American Enterprise Scholarship Fund’s (HAESF). These supports are gratefully acknowledged.

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Correspondence to Szabolcs Berezvai.

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Berezvai, S., Kossa, A. Closed-form solution of the Ogden–Hill’s compressible hyperelastic model for ramp loading. Mech Time-Depend Mater 21, 263–286 (2017).

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  • Visco-hyperelasticity
  • Constitutive model
  • Polymer foams
  • Finite element analysis
  • Parameter-fitting