Pure elastic stiffness of sand represented by response envelopes derived from cyclic triaxial tests with local strain measurements

  • Lukas Knittel
  • Torsten WichtmannEmail author
  • Andrzej Niemunis
  • Gerhard Huber
  • Edgar Espino
  • Theodoros Triantafyllidis
Research Paper


The elastic stiffness of a fine sand at small to moderate strains (\(\varepsilon \le 2 \times 10^{-4}\)) has been studied based on cyclic triaxial tests on cube-shaped samples with local strain measurements. Three samples with different densities (between very loose and very dense) were tested at up to 21 average stresses in succession (variation of mean pressure and stress ratio). At each average stress cycles were applied in six different directions, as a result of a simultaneous sinusoidal oscillation of the axial and lateral effective stress. Ten preconditioning cycles with larger strain amplitudes \(\varepsilon ^{\mathrm{ampl}}= 3 \times 10^{-4}\) per polarization were intended to induce a shakedown. An almost purely elastic response was then observed during the 5 subsequent cycles per polarization with lower strain amplitudes \(\varepsilon ^{\mathrm{ampl}}= 2 \times 10^{-4}\). The stress and strain paths measured during the latter ones were used to derive the components of the incremental elastic stiffness tensor \({\mathsf{E}}= \{E_{PP}, E_{PQ}, E_{QP}, E_{QQ}\}\) at a given average stress. The analysis of \({\mathsf{E}}\) is based on all 60 reversal points and the corresponding stress and strain points lying in a certain distance \(\Delta \varepsilon\) (e.g. \(10^{-4}\)) from the reversals. A graphical presentation of \({\mathsf{E}}\) by means of response envelopes is shown. They are used to calibrate the hyperelastic stiffness incorporated into a new hypoplastic constitutive model.


Cyclic triaxial tests Local strain measurements Response envelopes Sand Small-strain stiffness 



The cyclic tests have been performed by the technician H. Borowski in the IBF soil mechanics laboratory.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  • Lukas Knittel
    • 1
  • Torsten Wichtmann
    • 2
    Email author
  • Andrzej Niemunis
    • 1
  • Gerhard Huber
    • 1
  • Edgar Espino
    • 3
  • Theodoros Triantafyllidis
    • 4
  1. 1.Karlsruhe Institute of Technology (KIT), Institute of Soil Mechanics and Rock Mechanics (IBF)KarlsruheGermany
  2. 2.Chair of Soil Mechanics, Foundation Engineering and Environmental GeotechnicsRuhr-University BochumBochumGermany
  3. 3.Keller Grundbau GmbHBochumGermany
  4. 4.Department of Civil EngineeringUniversity of PatrasPatrasGreece

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