9. Verification of the consistency of homogenization: tangent op-
14.2 Fitting the macroscopic material response
14.2 Fitting the macroscopic material response
With the grain geometry fitted and the inclusions assigned to several grains, the macroscale response to biaxial compression needs to be fitted against results obtained from drained triaxial compression tests. Therefore the elastic properties of carbonates (calcite), tectosilicates (quartz) and heavy minerals (pyrite) are assigned to the corresponding grains according to Table 14.2. The mechanical characteristics for the calibration of the remaining grains representing the clay matrix and the interfaces are obtained from triaxial tests [Andra, 2013, Armand et al., 2013]. For the numerical modelling under plane strain conditions to be comparable with the results of triaxial data, the different appearance of stiffness and Poisson’s ratio need to be taken into account. An apparent plane-strain modulus Epscan be defined to compare with the Young’s modulus derived from the triaxial tests;
Eps=δσa
δεa = E
1 − ν2 (14.1)
The same can be done for the Poisson’s ratio:
νps= −δεlat
δεa = ν
1 − ν (14.2)
The plane strain modulus Eps is the ratio between variations of axial stress σa and axial strain εa, the plane strain Poisson’s ratio νps is the ratio between variations of axial strain and lateral strain εlat . When applied to the reference parameters for the COx claystone (E ≈ 4.0 GP a and ν ≈ 0.3, [Armand et al., 2013]), the plane strain equivalent parameters become:
Eps= 4.4 GP a, νps= 0.428 (14.3)
This difference in elastic properties has to be taken into account when comparing the numerical biaxial and experimental triaxial results. The material strength (nominal peak stress) derived from triaxial tests can not be translated directly to plane strain conditions as it requires information on the dependency of the the peak strength (the ’failure’) on the Lode angle (defined by the dependency on the intermediate principal stress). This information can generally be obtained from the failure criterion, which is unknown. Moreover, in the doublescale modelling approach, out-of-plane stress is not defined, which means that the Lode angle can not be obtained for making the comparison. Only a qualitative prediction can be made, based on the stress paths of biaxial- and triaxial tests and the properties of general failure criteria. This would predict a lower (or at least an equal) peak response in case of triaxial compression compared to the biaxial compression test (plane strain conditions) when constant confinement stress is applied.
To avoid the quantification of the difference in peak response, the peak re-sponse of the biaxial compression test is fitted to be equal to the triaxial com-pression test.
solids: SimOn −CO3 F eS2 clay
E [GPa] 95 84 305 2.3
ν [-] 0.074 0.317 0.154 0.11
interfaces:
δct/n 0.05
Dt/n0 0.002 Tnmax[MPa] 1.0 Ttmax[MPa] 5.5
Fig. 14.3: Microstructure ’8’ with carbonate (gray), tectosilicate (red) and pyrite (deep blue) inclusions in a matrix of clay (yellow). Grain selection for material allocation has been performed randomly.
The remaining micromechanical parameters (the clay matrix elasticity and the interface cohesion parameters) are adjusted in order to fit the numerical material point response to the biaxial compression to the post-peak part of the experi-mental results. This is done following the following procedure:
• The initial state of softening for normal and tangential components of co-hesion D0n and Dt0 are chosen high, such that the initial stiffness of the interfaces is high with respect to the stiffness of the grains. The initial response of the REV is therefore mainly controlled by the stiffness of the grains.
• The stiffness parameters E and µ to be used for the grains that make up the clay matrix are fitted such that the initial response to compressive loading represents the correct macroscale Young’s modulus and Poisson’s ratio (see (14.1) and (14.2).
• The normal and tangential components of the maximum cohesive forces Tnmax and Ttmax are used for fitting the macroscale peak response. The ratio between Ttmax and Tnmax is increased in order to reduce the pressure-dependency of the response.
• The critical opening is chosen such that the macroscale deformation at mac-roscale peak strength corresponds well with the experimental observations.
With this procedure in mind, the parameters are fitted in a manually-iterative (trial-and error) procedure until a reasonable fit with the experimental results is obtained.
Four triaxial tests (1 at 2 M P a confinement, 3 at 12 M P a confinement [Andra, 2013]) are used for the calibration. Their stress-strain curves are given in Figure 14.4 together with the results of the calibrated numerical response to biaxial compression for different REV orientations. They show a good agreement in initial stiffness, peak stress and axial strain at peak stress. Lateral strain shows good agreement at low stress levels, but the numerical results show a stronger dilatancy at higher stress levels and when softening takes place. This high degree of dilatancy is strongly linked to the fact that relative displacement between grains (which is needed for interface softening) can only be obtained by the opening of
14.2 Fitting the macroscopic material response 159
adjacent interfaces. As the rearrangement of grains is not taken into account in the finite element formulation, this relative opening of the interfaces continues regardless the state of deformation.
−0.04
response biaxial compression [2 MPa]
−0.04
response biaxial compression [12 MPa]
Fig. 14.4: Deviatoric stress response to biaxial material point compression for orient-ations at an interval of 15◦. In black the experimental results, obtained by triaxial compression.
In Figure 14.5 the derivatives of axial stress with respect to axial strain are given, again showing good agreement with respect to the experimental results up to the point at which the peak stress is reached. It is important to note that strong fluctuations in the stiffness can be observed in the post-peak domain for several 2 M P a confinement tests, some of which indicate a snapback at the end of the numerical simulation. This effect can not be observed under 12 M P a confinement, where the response is much smoother.
−0.04
response biaxial compression [2 MPa]
−0.04
response biaxial compression [12 MPa]
Fig. 14.5: Structural stiffness. In red the responses of numerical simulations for θREV at an interval of 15◦. In black the experimental results.
A small variation in initial stiffness can be observed in the numerical results.
This variation in initial stiffness is the combined effect of the heterogeneity of the grain stiffness, the grain elongation with respect to grain shape and distribution of interface orientations. Figure 14.6 shows the initial stiffness of the material
under confining pressure as a function of the orientation of deformation.
−900 −60 −30 0 30 60 90
1000 2000 3000 4000 5000 6000 7000
orientation [°]
initial stiffness E ≈ ∆σa/∆εa [MPa]
12 MPa confinement 2 MPa confinement
Fig. 14.6: Anisotropy of the initial stiffness under different confining pressures. Ori-entation θREV = −10◦corresponds to loading perpendicular to the bedding plane, which was introduced as θbed= 10◦.
The anisotropy in the peak response is shown in Figure 14.7. Its profile is much more irregular than that of the initial stiffness, as the peak strength is mainly determined by the interfaces. Nevertheless, the numerical results show a very good agreement with the experimental results in general.
By determining the peak stress at different confining stresses, the apparent friction angle and cohesion is defined by adopting a Mohr-Coulomb failure cri-terion (Figure 14.7).
A)
−900 −45 0 45 90
5 10 15 20 25 30 35 40
Deviatoric]stress]σ22−σ11
2]MPa]confinement 12]MPa]confinement 12]MPa]− exp.]perpendicular 12]MPa]− exp.]parallel
2]MPa]− experimental
orientation][deg] B)
−90 −45 0 45 900
2 4 6 8 10
orientation [deg]
cohesion c [MPa]
friction angleφ[deg]
0 5 10 15 20 25
Fig. 14.7: A) Evolution of the deviatoric stress peak with respect to the REV orient-ation for the 2 M P a and 12 M P a lateral stress simulorient-ations. B) Friction angle φ and cohesion c interpreted from the Mohr-Coulomb failure criterion derived from the biaxial compression simulations at 2 M P a and 12 M P a.
Unloading and reloading during biaxial compression tests can be performed to study the residual stiffness of the material as an effect of the interface damage.
Two examples are given for the material response with REV orientations of 0◦and 50◦ respectively, for which unloading to the initial state is applied from several stages of compression. The results are shown in Figure 14.8. The unloading-reloading branches are perfectly elastic (no hysteresis is taken into account). The