An experimental database for the development, calibration and verification of constitutive models for sand with focus to cyclic loading: part I—tests with monotonic loading and stress cycles


For numerical studies of geotechnical structures under earthquake loading, aiming to examine a possible failure due to liquefaction, using a sophisticated constitutive model for the soil is indispensable. Such a model must adequately describe the material response to a cyclic loading under constant volume (undrained) conditions, amongst others the relaxation of effective stress (pore pressure accumulation) or the effective stress loops repeatedly passed through after a sufficiently large number of cycles (cyclic mobility, stress attractors). The soil behaviour under undrained cyclic loading is manifold, depending on the initial conditions (e.g. density, fabric, effective mean pressure, stress ratio) and the load characteristics (e.g. amplitude of the cycles, application of stress or strain cycles). In order to develop, calibrate and verify a constitutive model with focus to undrained cyclic loading, the data from high-quality laboratory tests comprising a variety of initial conditions and load characteristics are necessary. The purpose of these two companion papers was to provide such database collected for a fine sand. The database consists of numerous undrained cyclic triaxial tests with stress or strain cycles applied to samples consolidated isotropically or anisotropically. Monotonic triaxial tests with drained or undrained conditions have also been performed. Furthermore, drained triaxial, oedometric or isotropic compression tests with several un- and reloading cycles are presented. Part I concentrates on the triaxial tests with monotonic loading or stress cycles. All test data presented herein will be available from the homepage of the first author. As an example of the examination of an existing constitutive model, the experimental data are compared to element test simulations using hypoplasticity with intergranular strain.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24


  1. 1.

    Arangelovski G, Towhata I (2004) Accumulated deformation of sand with initial shear stress and effective stress state lying near failure conditions. Soils Found 44(6):1–16

    Article  Google Scholar 

  2. 2.

    Baki AL, Rahman MM, Lo SR, Gnanendran CT (2012) Linkage between static and cyclic liquefaction of loose sand with a range of fines contents. Can Geotech J 49:891–906

    Article  Google Scholar 

  3. 3.

    Baki MAL, Rahman MM, Lo SR (2014) Predicting onset of cyclic instability of loose sand with fines using instability curves. Soil Dynamics and Earthquake Engineering 61–62(3):140–151

    Article  Google Scholar 

  4. 4.

    Bauer E (1996) Calibration of a comprehensive constitutive equation for granular materials. Soils Found 36:13–26

    Article  Google Scholar 

  5. 5.

    Been K, Jefferies MG (1985) A state parameter for sands. Géotechnique 35(1):99–112

    Article  Google Scholar 

  6. 6.

    Bishop AW, Green GE (1965) The influence of end restraint on the compression strength of a cohesionless soil. Géotechnique 15(3):243–266

    Article  Google Scholar 

  7. 7.

    Bouckovalas GD, Andrianopoulos KI, Papadimitriou AG (2003) A critical state interpretation for the cyclic liquefaction of silty sands. Soil Dynamics and Earthquake Engineering 23:115–125

    Article  Google Scholar 

  8. 8.

    Boulanger RW (2003) High overburden stress effects in liquefaction analyses. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 129(12):1071–1082

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chen YC, Liao TS (1999) Studies of the state parameter and liquefaction resistance of sand. In: Proceedings of the 2nd International Conference on Earthquake Geotechnical Engineering, Lisbon, Portugal, pp 513–518

  10. 10.

    Chiaro G, Koseki J, Sato T (2012) Effects of initial static shear on liquefaction and large deformation properties of loose saturated Toyoura sand in undrained cyclic torsional shear tests. Soils Found 52(3):498–510

    Article  Google Scholar 

  11. 11.

    Dafalias Y, Papadimitriou A, Li X (2004) Sand plasticity model accounting for inherent fabric anisotropy. Journal of Engineering Mechanics 130(11):1319–1333

    Article  Google Scholar 

  12. 12.

    Dafalias Y, Manzari M (2004) Simple plasticity sand model accounting for fabric change effects. Journal of Engineering Mechanics 130(6):622–634

    Article  Google Scholar 

  13. 13.

    Dash HK, Sitharam TG, Baudet BA (2010) Influence of non-plastic fines on the response of a silty sand to cyclic loading. Soils Found 50(5):695–704

    Article  Google Scholar 

  14. 14.

    DIN 18126: Bestimmung der Dichte nichtbindiger Böden bei lockerster und dichtester Lagerung, 1996

  15. 15.

    Flora A, Lirer S, Silvestri F (2012) Undrained cyclic resistance of undisturbed gravelly soils. Soil Dynamics and Earthquake Engineering 43:366–379

    Article  Google Scholar 

  16. 16.

    Fuentes W (2014) Contributions in mechanical modelling of fill material. Dissertation, Veröffentlichungen des Institutes für Bodenmechanik und Felsmechanik am Karlsruher Institut für Technologie, Heft 179

  17. 17.

    Georgiannou VN, Konstadinou M (2014) Effects of density on cyclic behaviour of anisotropically consolidated Ottawa sand under undrained torsional loading. Géotechnique 64(4):287–302

    Article  Google Scholar 

  18. 18.

    Georgiannou VN, Tsomokos A (2008) Comparison of two fine sands under torsional loading. Can Geotech J 45:1659–1672

    Article  Google Scholar 

  19. 19.

    Ghionna VN, Porcino D (2006) Liquefaction resistance of undisturbed and reconstituted samples of a natural coarse sand from undrained triaxial tests. J Geotech Geoenvir Eng ASCE 132(2):194–201

    Article  Google Scholar 

  20. 20.

    Hatanaka M, Uchida A, Ohara J (1997) Liquefaction characteristics of a gravelly fill liquefied during the 1995 Hyogo-Ken Nanbu earthquake. Soils Found 37(3):107–115

    Article  Google Scholar 

  21. 21.

    Hatanaka M, Feng L, Matsumura N, Yasu H (2008) A study on the engineering properties of sand improved by the sand compaction pile method. Soils Found 48(1):73–85

    Article  Google Scholar 

  22. 22.

    Herle I (1997) Hypoplastizität und Granulometrie einfacher Korngerüste. Promotion, Institut für Bodenmechanik und Felsmechanik der Universität Fridericiana in Karlsruhe, Heft Nr. 142

  23. 23.

    Huang A-B, Chuang S-Y (2011) Correlating cyclic strength with fines contents through state parameters. Soils Found 51(6):991–1001

    Article  Google Scholar 

  24. 24.

    Hyodo M, Murata H, Yasufuku N, Fujii T (1991) Undrained cyclic shear strength and residual shear strain of saturated sand by cyclic triaxial tests. Soils Found 31(3):60–76

    Article  Google Scholar 

  25. 25.

    Hyodo M, Tanimizu H, Yasufuku N, Murata H (1994) Undrained cyclic and monotonic triaxial behaviour of saturated loose sand. Soils Found 34(1):19–32

    Article  Google Scholar 

  26. 26.

    Hyodo M, Hyde AFL, Aramaki N (1998) Liquefaction of crushable soils. Géotechnique 48(4):527–543

    Article  Google Scholar 

  27. 27.

    Hyodo M, Hyde AFL, Aramaki N, Nakata Y (2002) Undrained monotonic and cyclic shear behaviour of sand under low and high confining stresses. Soils Found 42(3):63–76

    Article  Google Scholar 

  28. 28.

    Kiyota T, Sato T, Koseki J, Abadimarand M (2008) Behavior of liquefied sands under extremely large strain levels in cyclic torsional shear tests. Soils Found 48(5):727–739

    Article  Google Scholar 

  29. 29.

    Kiyota T, Koseki J, Sato T (2013) Relationship between limiting shear strain and reduction of shear moduli due to liquefaction in large strain torsional shear tests. Soil Dyn Earthq Eng 49:122–134

    Article  Google Scholar 

  30. 30.

    Kokusho T, Hara T, Hiraoka R (2004) Undrained shear strength of granular soils with different particle gradations. J Geotech Geoenviron Eng ASCE 130(6):621–629

    Article  Google Scholar 

  31. 31.

    Konstadinou M, Georgiannou VN (2013) Cyclic behaviour of loose anisotropically consolidated Ottawa sand under undrained torsional loading. Géotechnique 63(13):1144–1158

    Article  Google Scholar 

  32. 32.

    Matsuoka H, Nakai T (1982) A new failure criterion for soils in three-dimensional stresses. In: Deformation and failure of granular materials. Proc. IUTAM Symp. in Delft. pp. 253–263

  33. 33.

    Niemunis A (2003) Extended hypoplastic models for soils. Habilitation, Veröffentlichungen des Institutes für Grundbau und Bodenmechanik, Ruhr-Universität Bochum, Heft Nr. 34. Available from

  34. 34.

    Niemunis A (2008) Incremental Driver User’s manual. Available from

  35. 35.

    Niemunis A, Wichtmann T, Triantafyllidis T (2005) A high-cycle accumulation model for sand. Comput Geotech 32(4):245–263

    Article  MATH  Google Scholar 

  36. 36.

    Niemunis A, Herle I (1997) Hypoplastic model for cohesionless soils with elastic strain range. Mech Cohes-Frict Mater 2:279–299

    Article  Google Scholar 

  37. 37.

    Oda M, Kawamoto K, Suzuki K, Fujimori H, Sato M (2001) Microstructural interpretation on reliquefaction of saturated granular soils under cyclic loading. J Geotech Geoenviron Eng ASCE 127(5):416–423

    Article  Google Scholar 

  38. 38.

    Oka F, Yashima A, Tateishi A, Taguchi Y, Yamashita S (1999) A cyclic elasto-plastic constitutive model for sand considering a plastic-strain dependence of the shear modulus. Géotechnique 49(5):661–680

    Article  Google Scholar 

  39. 39.

    Porcino D, Caridi G, Ghionna VN (2008) Undrained monotonic and cyclic simple shear behaviour of carbonate sand. Géotechnique 58(8):635–644

    Article  Google Scholar 

  40. 40.

    Sharma SS, Ismail M (2006) Monotonic and cyclic behavior of two calcareous soils of different origins. J Geotech Geoenviron Eng ASCE 132(12):1581–1591

    Article  Google Scholar 

  41. 41.

    Sivathayalan S, Ha D (2011) Effect of static shear stress on the cyclic resistance of sands in simple shear loading. Can Geotech J 48:1471–1484

    Article  Google Scholar 

  42. 42.

    Stamatopoulos CA, Stamatopoulos AC, Balla LN (2004) Cyclic strength of sands in terms of the state parameter. In: The 11th international conference on soil dynamics and earthquake engineering (11th ICSD) and the 3rd international conference on geotechnical earthquake engineering

  43. 43.

    Stamatopoulos CA (2010) An experimental study of the liquefaction strength of silty sands in terms of the state parameter. Soil Dyn Earthq Eng 30(2):662–678

    Article  Google Scholar 

  44. 44.

    Sze HY, Yang J (2014) Failure modes of sand in undrained cyclic loading: impact of sample preparation. J Geotech Geoenviron Eng ASCE 140(1):152–169

    Article  Google Scholar 

  45. 45.

    Tatsuoka F, Maeda S, Ochi K, Fujii S (1986) Prediction of cyclic undrained strength of sand subjected to irregular loadings. Soils Found 26(2):73–89

    Article  Google Scholar 

  46. 46.

    Tatsuoka F, Toki S, Miura S, Kato H, Okamoto M, Yamada S-I, Yasuda S, Tanizawa F (1986) Some factors affecting cyclic undrained triaxial strength of sand. Soils Found 26(3):99–116

    Article  Google Scholar 

  47. 47.

    Vaid YP, Chung EKF, Kuerbis RH (1989) Preshearing and undrained response of sands. Soils Found 29(4):49–61

    Article  Google Scholar 

  48. 48.

    Vaid YP, Chern JC (1983) Effect of static shear on resistance to liquefaction. Soils Found 28(1):47–60

    Article  Google Scholar 

  49. 49.

    Vaid YP, Sivathayalan S (2000) Fundamental factors affecting liquefaction susceptibility of sands. Can Geotech J 37:592–606

    Article  Google Scholar 

  50. 50.

    Verdugo R, Ishihara K (1996) The steady state of sandy soils. Soils Found 36(2):81–91

    Article  Google Scholar 

  51. 51.

    von Wolffersdorff P-A (1996) A hypoplastic relation for granular materials with a predefined limit state surface. Mech Cohes-Frict Mater 1:251–271

    Article  Google Scholar 

  52. 52.

    Wichtmann T (2015). Homepage

  53. 53.

    Wichtmann T, Niemunis A, Triantafyllidis T, Poblete M (2005) Correlation of cyclic preloading with the liquefaction resistance. Soil Dyn Earthq Eng 25(12):923–932

    Article  Google Scholar 

  54. 54.

    Wichtmann T, Niemunis A, Triantafyllidis T (2013) On the “elastic stiffness” in a high-cycle accumulation model—continued investigations. Can Geotech J 50(12):1260–1272

    Article  Google Scholar 

  55. 55.

    Wichtmann T, Triantafyllidis T (2014) An experimental data base for the development, calibration and verification of constitutive models for sand with focus to cyclic loading. Part II: tests with strain cycles and combined loading. Acta Geotechnica (submitted)

  56. 56.

    Wichtmann T, Triantafyllidis T (2014) Stress attractors predicted by a high-cycle accumulation model confirmed by undrained cyclic triaxial tests. Soil Dyn Earthq Eng 69(2):125–137

    Google Scholar 

  57. 57.

    Wijewickreme D, Sriskandakumar S, Byrne P (2005) Cyclic loading response of loose air-pluviated Fraser River sand for validation of numerical models simulating centrifuge tests. Can Geotech J 42:550–561

    Article  Google Scholar 

  58. 58.

    Wong RT, Seed HB, Chan CK (1975) Cyclic loading liquefaction of gravelly soils. J Geotech Eng Div ASCE 101(6):571–583

    Google Scholar 

  59. 59.

    Xenaki VC, Athanasopoulos GA (2003) Liquefaction resistance of sand–silt mixtures: an experimental investigation of the effects of fines. Soil Dyn Earthq Eng 23:183–194

    Article  Google Scholar 

  60. 60.

    Yamada S, Takamori T, Sato K (2010) Effects on reliquefaction resistance produced by changes in anisotropy during liquefaction. Soils Found 50(1):9–25

    Article  Google Scholar 

  61. 61.

    Yamashita S, Toki S (1993) Effects of fabric anisotropy of sand on cyclic undrained triaxial and torsional strengths. Soils Found 33(3):92–104

    Article  Google Scholar 

  62. 62.

    Yang J, Sze HY (2011) Cyclic behaviour and resistance of saturated sand under non-symmetrical loading conditions. Géotechnique 61(1):59–73

    Article  Google Scholar 

  63. 63.

    Yang J, Sze HY (2011) Cyclic strength of sand under sustained shear stress. J Geotech Geoenviron Eng ASCE 137(12):1275–1285

    Article  Google Scholar 

  64. 64.

    Yasuda S, Soga M (1984) Effects of frequency on undrained strength of sands (in Japanese). Proc 19th Nat Conf Soil Mech Found Eng, pp 549–550

  65. 65.

    Yoshimi Y, Oh-Oka H (1975) Influence of degree of shear stress reversal on the liquefaction potential of saturated sand. Soils Found 15(3):27–40

    Article  Google Scholar 

  66. 66.

    Zhang J-M, Shamoto Y, Tokimatsu K (1997) Moving critical and phase-transformation stress state lines of saturated sand during undrained cyclic shear. Soils Found 37(2):51–59

    Article  Google Scholar 

  67. 67.

    Zhang J-M, Wang G (2012) Large post-liquefaction deformation of sand, part I: physical mechanism, constitutive description and numerical algorithm. Acta Geotechnica 7:69–113

    Article  Google Scholar 

Download references


Parts of the presented study have been performed within the framework of the project “Geotechnical robustness and self-healing of foundations of offshore wind power plants” funded by the German Federal Ministry for the Environment, Nature Conservation and Nuclear Savety (BMU, project No. 0327618). Other parts were conducted within the framework of the project “Improvement of an accumulation model for high-cyclic loading” funded by German Research Council (DFG, Project No. TR218/18-1 / WI3180/3-1). The authors are grateful to BMU and DFG for the financial support. All tests have been performed by the technicians H. Borowski, P. Gölz and N. Demiral in the IBF soil mechanics laboratory.

Author information



Corresponding author

Correspondence to Torsten Wichtmann.

Appendix: Equations of the constitutive model used for the element test simulations

Appendix: Equations of the constitutive model used for the element test simulations


Scalar variables are denoted by characters with normal letters (e.g. e), second-order tensors are formatted fat (e.g. \({\varvec{\sigma }}\), \({\mathbf{h}}\)), and fourth-order tensors are given in sans-serif font (e.g. \({\mathsf{L}}, {\text{I}}\)). A dyadic product is denoted by \({\mathbf{a}}\otimes {\mathbf{b}}\) (i.e. \(a_{ij} \, b_{kl}\) in index notation), a single contraction by \({\mathbf{a}}\cdot {\mathbf{b}}\,\hat{=}\, a_{ik} \, b_{kj}\) and a double contraction by \({\mathbf{a}}: {\mathbf{b}}\,\hat{=}\, a_{kl} \, b_{kl}\). The Euclidean norm is defined as \(\Vert {\mathbf{a}}\Vert = \sqrt{\mathbf{a}: \mathbf{a}}\), the trace of a tensor as \(\text{tr}\,{({\mathbf{a}})} \,\hat{=}\, a_{kk}\) and the deviator as \({\mathbf{a}}^* = {\mathbf{a}}- \text{tr}\,{({\mathbf{a}})}/3 \, {\mathbf {1}}\) with the second-order identity tensor \({\mathbf {1}} \,\hat{=} \delta _{ij}\). The Kronecker symbol \(\delta _{ij}\) is equal to 1 for \(i = j\) and 0 for \(i \ne j\). A normalization is denoted by an arrow above the respective symbol \(\overrightarrow{\mathbf{a}} = {\mathbf{a}}/\Vert {\mathbf{a}}\Vert\) , and a division by the trace of the tensor is identified by a roof \(\hat{\mathbf{a}} = {\mathbf{a}}/\text{tr}\,({\mathbf{a}})\).

Basic hypoplastic model after von Wolffersdorff [51]

The basic equation of the hypoplastic model proposed by von Wolffersdorff [51] reads:

$$\dot{{\varvec{\sigma }}}={\mathsf{L}}: \dot{{\varvec{\varepsilon }}} + {\mathbf{N}}\Vert \dot{{\varvec{\varepsilon }}}\Vert = \underbrace{\left( {\mathsf{L}}+ {\mathbf{N}}\frac{\dot{{\varvec{\varepsilon }}}}{\Vert \dot{{\varvec{\varepsilon }}}\Vert }\right) }_{{\mathsf{M}}} : \dot{{\varvec{\varepsilon}}}$$

with Jaumann stress rate \(\dot{{\varvec{\sigma }}}\), strain rate \(\dot{{\varvec{\varepsilon }}}\) and the linear and nonlinear stiffness tensors \({\mathsf{L}}\) and \({\mathbf{N}}\):

$${\mathsf{L}}=f_b \, f_e \, \frac{1}{\hat{{\varvec{\sigma }}} : \hat{{\varvec{\sigma }}}} \, \left( F^2 \, {\text{I}}\,+\, a^2 \, \hat{{\varvec{\sigma }}} \otimes \hat{{\varvec{\sigma }}}\right)$$
$${\mathbf{N}}=f_b \, f_e \, f_d \, \frac{F \, a}{\hat{{\varvec{\sigma }}} : \hat{{\varvec{\sigma }}}} \, \left( \hat{{\varvec{\sigma }}} + \hat{{\varvec{\sigma }}}^*\right)$$

Therein, \(I_{ijkl} = 0.5 (\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk})\) is a fourth-order identity tensor. The parameters a and F in Eqs. (8) and (9) describe the failure criterion of Matusoka and Nakai [32]:

$$a=\frac{\sqrt{3} \, (3 - \sin {\varphi _c})}{2 \, \sqrt{2} \, \sin {\varphi _c}}$$
$$F=\sqrt{\frac{1}{8} \, \tan ^2{\psi } + \frac{2 - \tan ^2{\psi }}{2 + \sqrt{2} \tan {\psi } \cos {(3 \theta )}} } \,-\, \frac{\tan {\psi }}{2 \, \sqrt{2}}$$
$$\tan {\psi }=\sqrt{3} \, \Vert \hat{{\varvec{\sigma }}}^*\Vert$$
$$\cos {(3 \theta )}=-\sqrt{6} \, \frac{\text{tr}\,{\left( \hat{{\varvec{\sigma }}}^* \cdot \hat{{\varvec{\sigma }}}^* \cdot \hat{{\varvec{\sigma }}}^*\right) }}{\left[ \hat{{\varvec{\sigma }}}^* : \hat{{\varvec{\sigma }}}^*\right] ^\frac{3}{2}}$$

\(\varphi _c\) is the critical friction angle (material constant). The barotropy and pyknotropy factors read:

$$f_d=\left( \frac{e - e_d}{e_c - e_d}\right) ^\alpha$$
$$f_e=\left( \frac{e_c}{e}\right) ^\beta$$
$$f_b=\frac{\left( \frac{e_{i0}}{e_{c0}}\right) ^\beta \frac{h_s}{n} \frac{1 + e_i}{e_i} \left( \frac{3 p}{h_s}\right) ^{1-n}}{3 + a^2 - a \, \sqrt{3} \, \left( \frac{e_{i0} - e_{d0}}{e_{c0} - e_{d0}}\right) ^\alpha }$$

with material constants \(\alpha ,\, \beta ,\, h_s\) and n. The pressure dependence of the void ratios \(e_d,\, e_c\) and \(e_i\), corresponding to the densest, the critical and the loosest possible state, is described by (Bauer [4]):

$$\frac{e_i}{e_{i0}}=\frac{e_c}{e_{c0}} \,=\, \frac{e_d}{e_{d0}} \,=\, \exp {\left[ -\left( \frac{3 p}{h_s}\right) ^n\right] }$$

with the void ratios \(e_{i0},\, e_{c0}\) and \(e_{d0}\) (material constants) at pressure \(p = 0\). The material parameters \(e_{i0}, \,e_{c0},\, e_{d0},\, \varphi _c,\, h_s,\, n,\, \alpha\) and \(\beta\) used for the simulations are summarized in the first eight columns of Table 2.

Extension of hypoplastic model by intergranular strain according to Niemunis and Herle [36]

In order to eliminate an excessive accumulation of strain (ratcheting) of the original hypoplastic model proposed by von Wolffersdorff [51] in the case of a cyclic loading, Niemunis and Herle [36] introduced the additional state variable “intergranular strain” \(\mathbf{h}\), which memorizes the last part of the previous strain path. A measure of the mobilization of the intergranular strain is \(\rho = \Vert {\mathbf{h}}\Vert /R\) with a material constant R describing the range of an elastic locus. Depending on the actual strain rate \(\dot{{\varvec{\varepsilon }}}\) in relation to the direction of the intergranular strain \(\overrightarrow{\mathbf{h}}\), the stiffness \({\mathsf{M}}\) in Eq. (7) is increased according to:

$$\begin{aligned} {\mathsf{M}}=\,& {} \left[ \rho ^\chi \,m_T + (1-\rho ^\chi )\,m_R\right] \,{\mathsf{L}} \\&+\left\{ \begin{array}{ll} \rho ^\chi (1-m_T){\mathsf{L}}:\overrightarrow{\mathbf{h}} \otimes \overrightarrow{\mathbf{h}} + \rho ^\chi \,\mathbf{N}\otimes \overrightarrow{\mathbf{h}} &{}\text{for}\,\overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} > 0\\ \rho ^\chi (m_R-m_T){\mathsf{L}}:\overrightarrow{\mathbf{h}} \otimes \overrightarrow{\mathbf{h}} &{} \text{ for }\,\overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} \,\le\, 0 \end{array} \right. \end{aligned}$$

with material constants \(m_T,\, m_R\) and \(\chi\). The evolution of the rate \(\dot{\mathbf{h}}\) of intergranular strain obeys:

$$\begin{aligned} \dot{\mathbf{h}}= & {} \left\{ \begin{array}{ll} \left( {\text{I}}- \overrightarrow{\mathbf{h}} \otimes \overrightarrow{\mathbf{h}} \varrho ^{\beta _r}\right) : \dot{{\varvec{\varepsilon }}} &{} \text{for} \, \overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} > 0\\ \dot{{\varvec{\varepsilon }}} &{} \text{for} \,\overrightarrow{\mathbf{h}}:\dot{{\varvec{\varepsilon }}} \,\le\, 0\\ \end{array} \right. \end{aligned}$$

with another material constant \(\beta _r\). If a sufficiently large monotonic strain is applied after a change in the strain path direction, the comparatively low stiffness of the original hypoplastic model according to Eq. (7) is regained. The material parameters \(R,\, m_T,\, m_R,\, \chi\) and \(\beta _r\) used for the simulations are summarized in the last five columns of Table 2.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wichtmann, T., Triantafyllidis, T. An experimental database for the development, calibration and verification of constitutive models for sand with focus to cyclic loading: part I—tests with monotonic loading and stress cycles. Acta Geotech. 11, 739–761 (2016).

Download citation


  • Cyclic triaxial tests
  • Database
  • Fine sand
  • Monotonic triaxial tests
  • Stress cycles