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Numerical Modelling of the Dynamic Behaviour of a Soil in True Triaxial Tests with Bender Elements by Ana Rita Silva

The study of soil's small strain dynamic behaviour was the main goal of this dissertation, for the case of a residual soil from Porto granite, a geotechnical material that possess a complex mechanic behaviour. To that end, a detailed experimental research was developed, in true triaxial conditions, considering the measurement of seismic wave velocities through bender elements.
A three-dimensional numerical model was produced that characterized the behaviour of this particular residual soil using a finite difference program, FLAC3D. Its main advantages reside in its simplicity, versatility and the possibility of directly measuring seismic wave velocities, not only in the three principal directions, but also in inclined directions.
This study requires the implementation of bender elements in the platens of the true triaxial apparatus, in order to assess the influence of its cubical geometry and boundary conditions (rigid or flexible platens, or even reflective and absorbent), as well as the validation of stiffness parameters attained from the measurement of seismic waves.
For this purpose, a series of parametric and sensitivity studies were developed, considering the linear elastic constitutive model, with isotropic loading, in the previously mentioned software, to particularly evaluate the influence of each parameter in the numerical modelling of the true triaxial apparatus, and which values are better suited for its correct representation. These parameters are: time step, amplitude, frequency, damping, Poisson’s ratio and finally, boundary conditions and a cross-anisotropic constitutive model. With these studies completed, a comparison and validation between the numerical results attained and the laboratory results previously done by Ferreira (2009), regarding residual soil specimens from Porto granite was in order. Due to time limitations, only the dry specimens (w ≈ 0%), namely, R8D-TT and R4D-K0TT were used in this comparative study.

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Numerical Modelling of the Dynamic Behaviour of a Soil in True Triaxial Tests with Bender Elements by Ana Rita Silva

  1. 1. Numerical Modelling of the Dynamic Behaviour of a Soil in True Triaxial Tests with Bender Elements Ana Rita Silva - 201005317 Mestrado Integrado em Engenharia Civil – Especialização em Geotecnia Porto, 17 de Julho de 2014
  2. 2. • Understanding of the characteristics and dynamic properties of soils • Non-linear behaviour of soils Measurement of seismic waves In situ and Laboratory methods Bender Elements (BE) True Triaxial Apparatus (TT) • This numerical study through FLAC3D is a validation of the elastic and small strain stiffness parameters from seismic wave measurements on a residual soil from Porto granite in a TT - Ferreira (2009) Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions
  3. 3. Volumetric Seismic Waves P-wave S-wave Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Seismic Waves
  4. 4. • Non-linearity • Stress and strain • Stiffness degradation curve • “Small is beautiful” (Burland 1989) Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Dynamic Behaviour of Soils
  5. 5. Laboratory Methods • Piezoelectric Transducers • Piezoelectricity: ability of converting electrical energy into mechanical energy or vice-versa • Bender Elements: - Consists of two piezoceramic transdurcers composed of two piezoceramic plates rigidly attached - Electrical connection ensures an accurate flexural movement, in order to propagate shear waves during its deformation - Single transducer or T-shaped pair of transducers Advantages: - Simplicity (results interpretation and procedures) - Versatility and portability - Can be used in the small-strain domain - Immediate registration of the results - Quick, simple and low cost implementation of the support equipment Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Seismic Wave Testing Methods
  6. 6. Laboratory Methods • True Triaxial Apparatus • Types of boundaries: Rigid, flexible, mixed • Rigid boundary: six square platens assembled in a cubical frame of anodized aluminum • Example: collaborative project between FEUP and University of Western Australia Advantages: - Versatility (uniform shear stresses and normal stresses can be applied) - Provdes much information - Allow controlled gradual rotations of the principal axes and strain Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Seismic Wave Testing Methods
  7. 7. • Three-dimensional finite difference program for geotechnical engineering calculations • Possesses a lower processing capacity and models more complex behaviours than the finite element method • Explicit calculation: time domain – suited for the simulation of BE and the dynamic behaviour of soil • Possesses several constitutive models • Suitable especification of boundary conditions • Two types of hysteretic damping: local and Rayleigh • Graphical output in a variety of formats Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Program FLAC3D
  8. 8. Brick shaped mesh Radially graded mesh around brick Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Mesh Generation
  9. 9. Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Mesh Generation
  10. 10. - Mesh Generation Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions
  11. 11. Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Mesh Generation
  12. 12. Elastic: - linear stress-strain behaviour - deformation increases with applied forces - homogeneous and continuous materials Soil: residual soil from Porto granite Soil Bender Element Shear modulus (G)=80.00 MPa Shear Modulus (G)=1000 MPa Poisson’s ratio (ν)=0.10 Poisson’s ratio (ν)=0.25 Young’s modulus (E)=176.00 MPa Young’s modulus (E)=2500 MPa Bulk modulus (K)=73.33 MPa Bulk modulus (K)=1700 MPa Density (ρ)=2000 kg/m3 Density (ρ)=3000 kg/m3 Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Properties
  13. 13. -3.00E-06 -2.00E-06 -1.00E-06 0.00E+00 1.00E-06 2.00E-06 3.00E-06 -1.5E-04 -1.0E-04 -5.0E-05 0.0E+00 5.0E-05 1.0E-04 1.5E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 Y-displacement(m) Time (s) Input 10.0 kHz 1.00E-04 1.78E-05 (automatic) • Automatic time step: self-analysis • Lack of data/points to correctly translate the shape of the output signal Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Time Step
  14. 14. • Higher degrees of magnitude: more continuous signal and an improvement in quality and number of data -3.00E-06 -2.00E-06 -1.00E-06 0.00E+00 1.00E-06 2.00E-06 3.00E-06 -1.5E-04 -1.0E-04 -5.0E-05 0.0E+00 5.0E-05 1.0E-04 1.5E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 Y-displacement(m) Time (s) Input 10.0 kHz 1.00E-04 1.78E-05 (automatic) 5.00E-06 Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Time Step
  15. 15. -3.00E-06 -2.00E-06 -1.00E-06 0.00E+00 1.00E-06 2.00E-06 3.00E-06 -1.5E-04 -1.0E-04 -5.0E-05 0.0E+00 5.0E-05 1.0E-04 1.5E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 Y-displacement(m) Time (s) Input 10.0 kHz 1.00E-04 1.78E-05 (automatic) 5.00E-06 2.50E-07 • Chosen time step • Perfect balance: data, quality and processing capacity Varies from model to model Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Time Step
  16. 16. Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Frequency: Time Domain (TD) -1,6E-03 -1,4E-03 -1,2E-03 -1,0E-03 -8,0E-04 -6,0E-04 -4,0E-04 -2,0E-04 0,0E+00 2,0E-04 0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03 Y-displacement(m) Time (s) Input 0.5 kHz 0.5 kHz Input 1.0 kHz 1.0 kHz Input 2.0 kHz 2.0 kHz Input 4.0 kHz 4.0 kHz Input 5.0 kHz 5.0 kHz Input 10.0 kHz 10.0 kHz Input 25.0 kHz 25.0 kHz
  17. 17. Fmax: 1.63 kHz 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 Output(V) Input(V) Frequency (kHz) Input 0.5 kHz Output 0.5 kHz Fmax: 1.75 kHz 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 Output(V) Input(V) Frequency (kHz) Input 1.0 kHz Output 1.0 kHz Fmax: 1.88 kHz 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 Output(V) Input(V) Frequency (kHz) Input 2.0 kHz Output 2.0 kHz Fmax:… 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 Output(V) Input(V) Frequency (kHz) Input 4.0 kHz Output 4.0 kHz Fmax: 1.63 kHz 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 Output(V) Input(V) Frequency (kHz) Input 5.0 kHz Output 5.0 kHz Fmax: 2.13… 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 Output(V) Input(V) Frequency (kHz) Input 10.0 kHz Output 10.0 kHz Fmax: 1.88… 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 Output(V) Input(V) Frequency (kHz) Input 25.0 kHz Output 25.0 kHz Fmax: 1.88 kHz 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 Output(V) Input(V) Frequency (kHz) Input 50.0 kHz Output 50.0 kHz a) b) c) d) e) f) g) h) Variation of frequency: a) 0.5 kHz; b) 1.0 kHz; c) 2.0 kHz; d) 4.0 kHz; e) 5.0 kHz; f) 10.0 kHz; g) 25.0 kHz; h)50.0 kHz Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Frequency: Frequency Domain (FD)
  18. 18. LOCAL Damping -1.2E-04 -7.0E-05 -2.0E-05 3.0E-05 8.0E-05 1.3E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03 Y-displacement(m) Time (s) Input 10 kHz 0.01 0.05 0.10 0.50 0.90 -1.2E-04 -7.0E-05 -2.0E-05 3.0E-05 8.0E-05 1.3E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03 Y-displacement(m) Time (s) Input 10 kHz L: 0.05 Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Damping
  19. 19. RAYLEIGH Damping -1.2E-04 -7.0E-05 -2.0E-05 3.0E-05 8.0E-05 1.3E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03 Y-displacement(m) Time (s) Input 10 kHz R: 10 kHz -1.2E-04 -7.0E-05 -2.0E-05 3.0E-05 8.0E-05 1.3E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 8.0E-03 Y-displacement(m) Time (s) Input 10.0 kHz 2.5 kHz 5.0 kHz 8.0 kHz 10.0 kHz 12.0 kHz Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Damping
  20. 20. • Face C is the zone most influenced by ν, given the higher reflection of the signal, due to the large dimensions of the model • P waves (vary – E ≠ constant) • : Model’s first arrival (first inflection) • : Theoretical first arrival • S waves (constant – G = constant) • : Theoretical first arrival Poisson’s ratio chosen = 0,10 -1.2E-03 -1.0E-03 -8.0E-04 -6.0E-04 -4.0E-04 -2.0E-04 0.0E+00 2.0E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 Y-displacement(m) Time (s) input 5.0 kHz ν = 0.00 tS tP 0.00 input ν = 0.10 tS tP 0.10 input ν = 0.20 tS tP 0.20 input ν = 0.30 tS tP 0.30 input ν = 0.40 tS tP 0.40 input ν = 0.48 tS tP 0.48 Variation of ν and the first arrival of both P and S waves in face C Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Poisson’s Ratio (ν)
  21. 21. ABSORBENT -6.00E-04 -5.00E-04 -4.00E-04 -3.00E-04 -2.00E-04 -1.00E-04 0.00E+00 1.00E-04 0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03 Y-displacement(m) Time (s) A: Transmitter A: Midpoint A: Receiver Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Boundary Conditions
  22. 22. • Transmitter: No difference • Midpoint and Receiver: - reflection, energy, absorbency (midpoint behaves as if it was in an infinite medium) • The further from the transmitter, the more difference between first arrival of P and S waves • P waves • : Model’s first arrival • : Theoretical first arrival • S waves • : Theoretical first arrivalREFLECTED VERSUS ABSORBENT Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Boundary Conditions -6.00E-04 -5.00E-04 -4.00E-04 -3.00E-04 -2.00E-04 -1.00E-04 0.00E+00 1.00E-04 0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03 Y-displacement(m) Time (s) R: Transmitter A: Transmitter R: Midpoint A: Midpoint R: Receiver A: Receiver tP tS tS: Reflection tS
  23. 23. Displacements (example: EV = 0.7 EH) VERTICAL DisplacementHORIZONTAL Displacement Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Anisotropy
  24. 24. Displacements (example: EV = 0.7 EH) 7.E-05 0.0E+00 3.0E-05 6.0E-05 9.0E-05 1.2E-04 1.5E-04 1.8E-04 Z XY Isotropic: EV = EH Anisotropic: EV = 0.7xEH HORIZONTAL and Vertical Displacements for isotropic and anisotropic models Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Anisotropy • Isotropic behaviour: great similarity between the values of the displacements • Anisotropic behaviour: higher V displacement, given the lower stiffness (enough to verify the presence of anisotropy)
  25. 25. Anisotropic and Isotropic behaviour from face B to D Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Anisotropy Waves (for EV = 0.7 EH; EH = EH isotropic) Horizontal Polarization -1,5E-04 -1,0E-04 -5,0E-05 0,0E+00 5,0E-05 1,0E-04 1,5E-04 0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03 Displacement(m) Time (s) Input 5.0 kHz B to D: Anisotropic B to D: Isotropic Horizontal Displacement
  26. 26. Anisotropic and Isotropic behaviour from face B to D and A to C Vertical polarization Horizontal polarization Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Anisotropy -3,5E-04 -3,0E-04 -2,5E-04 -2,0E-04 -1,5E-04 -1,0E-04 -5,0E-05 0,0E+00 5,0E-05 1,0E-04 1,5E-04 0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03 Displacement(m) Time (s) Input 5.0 kHz A to C: Anisotropic A to C: Isotropic B to D: Anisotropic B to D: Isotropic tP: EV = 0.70EH tP: EV = EH tS: EV = 0.70EH tS: EV = EH Vertical Displacement Horizontal Displacement Waves (for EV = 0.7 EH; EH = EH isotropic)
  27. 27. Specimen γ [kN/m3] w0 [%] e0 R2W-TT 19.1 28.4 0.770 R3W-K0TT 18.8 30.7 0.850 R4D-K0TT 14.1 1.2 0.883 R8D-TT 12.7 1.0 1.067 • Comparison already considering the validity of the simulation model • The tests made in the TT by Ferreira (2009) consisted in dry (w ≈ 0%) and wet (w ≈ 30%) reconstituted residual soil specimens from Porto granite Physical properties of the reconstituted specimens tested in the TT Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Comparison with Laboratory Tests
  28. 28. 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03 Z XY X, Y, Z axial displacements (m) 5 10 25 50 75 100 150 200 300 400 • Deformation of the model concomitant with the applied stress • Differences between experimental and numerical results: compliance and bedding errors (in the laboratory) • Valid comparison, despite the difference between the degrees of magnitude of the displacement Three-dimensional view of the strains measured for isotropic loading 1.3E-04 2.6E-04 6.7E-04 0.0E+00 3.0E-04 6.0E-04 9.0E-04 Z XY 100 kPa 200 kPa 500 kPa Experimental results Numerical results Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Comparison with Laboratory Tests: R8D-TT Displacements
  29. 29. • Shape and configuration of the waves independent from the stresses applied – characteristics of the constitutive model (linear elastic with constant G) • Experimental: increase of stiffness and necessarily the seismic wave velocities with increasing loads (soil hardening due to particle rearrangement) Simulations of seismic wave propagation for several stress values Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Comparison with Laboratory Tests: R8D-TT Waves -3,0E-04 -2,5E-04 -2,0E-04 -1,5E-04 -1,0E-04 -5,0E-05 0,0E+00 5,0E-05 1,0E-04 1,5E-04 2,0E-04 0,0E+00 1,0E-03 2,0E-03 3,0E-03 4,0E-03 5,0E-03 6,0E-03 Z-displacement(m) Time (s) Input 5.0 kHz 100 kPa 200 kPa 500 kPa tP tS
  30. 30. • The higher the stress applied, the higher the displacement • Isotropic material: V displacements are higher than H, in agreement with the loading conditions • Anisotropic material: diferences betwen H and V displacements in agreement with loading conditions: even though the soil is more rigid in the V direction, it exhibits a higher strain in the V direction due to the V stress Horizontal and vertical displacements, correspondent to horizontal and vertical stresses: a) 35 kPa and 100 kPa; b) 70 kPa and 200 kPa Numerical results 2.E-05 0.0E+00 3.0E-05 6.0E-05 9.0E-05 1.2E-04 Z XY Isotropic EV = EH Anisotropic EV = 1.4xEH Anisotropic EV = 0.7xEH a) 4.E-05 0.0E+00 3.0E-05 6.0E-05 9.0E-05 1.2E-04 Z XY Isotropic: EV = EH Anisotropic: EV = 1.4xEH Anisotropic: EV = 0.7xEH b) Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Comparison with Laboratory Tests: R4D-K0TT Displacements
  31. 31. • Different displacements, but similar waves • There is virtually no change between both types of constitutive models: use of another model Stiffness (assumed constant) commands the behaviour of the model • Different from reality -2.0E-04 -1.5E-04 -1.0E-04 -5.0E-05 0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04 0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 Z-displacement(m) Time (s) Input 5.0 kHz Isotropic: Z-disp 175/500 kPa Anisotropic: Z-disp 175/500 kPa tP: EV = EH tP: EV = 1.4xEH tS Signal variation, considering a horizontal stress of 175 kPa and 500 kPa Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions - Comparison with Laboratory Tests: R4D-K0TT Waves
  32. 32. Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions • Even though there were no different properties that considered the presence of BE, a wave signal could still be successfully created and evaluated and in considerably less amount of time in the simpler model Parametric and sensitivity studies proved the validity of the model by correctly characterizing the soil • Comparison between experimental and numerical results was successful and the measurement of the stiffness parameters achieved in the laboratory verified, regardless of the differences between the constitutive models • Since this was the first numerical modelling approach to study the TT, a greater effort was made in the implementation and improvement of the model (Parametric and Sensitivity Studies) • Altogether, this was a pioneer work in a short amount of time and a learning experience for all involved!
  33. 33. Contents Scope and objectives State of the Art Numerical Modelling Parametric and Sensitivity Studies Results and Discussion Conclusions Further Works • Regarding the sensitivity studies: • Study of the influence of the water level; • Multi-parametric study regarding the relationship between the input frequency and the Rayleigh damping; • Include in the study a qualitative processing of the signal. • Regarding the numerical modelling in FLAC3D: • Investigation of all parametric and sensitivity studies in the frequency-domain; • The simulation of a cubical cell to compare with more accuracy the influence of flexible or absorbent boundaries; • The use of a different constituive model that would consider the evaluation of stiffness; • The evaluation of all the comparisons between numerical and experimental results with the more complex model.

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