Weakening of Compressive Strength of Granite by Piezoelectric Actuation of Quartz Using High-Frequency and High-Voltage Alternating Current: A 3D Numerical Study

The rock is described here as an elastic (up to failure) isotropic and heterogeneous material. This means that the rock constituent minerals are taken as isotropic, but each mineral phase has its specific material properties. However, the piezoelectric properties are taken, as they truly are (see Fig. 1), anisotropic. The failure model for rock consists of a viscoplastic part and a damage part. It should be noted that the 3D embedded discontinuity approach used by Saksala (2021) and Saksala et al. (2022) cannot be used here as it cannot adequately describe the compressive failure. In the present approach, the (perfectly) viscoplastic part indicates the stress states leading to rock failure and inelastic deformation while the damage part quantifies the failure by separate isotropic damage variables in compression and tension.

The stress states leading to failure and inelastic flow are indicated by a bi-surface yield criterion consisting of the Hokka power law criterion (Hokka et al. 2016) for compressive (shear) fracture, called here the MH criterion, and the modified Rankine (MR) criterion for the tensile fracture. The MH criterion matches the experimental data for Kuru granite under confined dynamic compression better than the Hoek–Brown criterion (Hokka et al. 2016). Perfect viscoplasticity is assumed, as the damage model accounts for stiffness and strength degradation. These criteria are written aswhere the symbols are as follows: σ_i is the ith principal stress of the stress tensor \({\varvec{\sigma}}\); σ_c, σ_t are the dynamic uniaxial compressive and tensile strengths of the material; \({\dot{\lambda }}_{\text{MH}} \: \mathrm{and} \: {\dot{\lambda }}_{\text{MR}}\) are the rates of the internal variables in compression and tension, respectively; and \({s}_{\text{MH}}{ \: \mathrm{and} \: s}_{\text{MR}}\) are the constant viscosity moduli in compression and tension, respectively. Moreover, the Macaulay brackets, i.e. the positive part operator, are used in the MR criterion while B and n in the MH criterion are experimentally calibrated parameters. The rate sensitivity of the model is provided by adding the product of viscosity and the rate of the internal variable to the static values, σ_c0, σ_t0, of the compressive and tensile strength. This method is reliable, as demonstrated by the experimental results by Zhao (2000) showing that the dynamic failure envelope of Bukit Timah granite can be obtained from the quasi-static one by translation. Equation (4) presents the loading–unloading conditions of Kuhn–Tucker form, which means that the present formulation is based on the viscoplastic consistency model by Wang et al. (1997).

The damage part is formulated with separate scalar damage variables in tension and compression due to the asymmetric behavior of rocks in these stress regions. Following Grassl and Jirásek (2006), the damage variables are driven by viscoplastic strain. Consequently, there is no need for additional damage loading functions, which simplifies the implementation considerably. Moreover, if damage was driven by total strain, spurious stress effects would occur in cyclic loading. The components of the damage model resulting in exponential softening can be written as

The symbol meanings here are: A_t and A_c control the final value of the damage variables; β_t and β_c, controlling the initial slope and the amount of damage dissipation, are defined by the fracture energies G_Ic and G_IIc; h_e is a characteristic length of a finite element; \({\varepsilon }_{\text{eqvt}}^{\text{vp}}\) is the equivalent viscoplastic strain in tension defined, in the rate form, as the trace of the viscoplastic strain rate tensor, \({\dot{{\varvec{\upvarepsilon}}}}_{\mathrm{vp}}\), using the Macauley brackets so that tensile damage evolution occurs only if the volumetric viscoplastic principal strain is positive; \({\varepsilon }_{\text{eqvc}}^{\text{vp}}\) is the equivalent viscoplastic strain in compression, being defined with the deviatoric part,\({\dot{\mathbf{e}}}_{\mathrm{vp}}\), of \({\dot{{\varvec{\upvarepsilon}}}}_{\mathrm{vp}}\). Moreover, Eq. (8) is the Koiter’s rule for bi-surface plasticity. The colon in (7) denotes the double contraction operator for tensors, i.e. \({\varvec{A}}:{\varvec{A}}={A}_{ij}{A}_{ij}\). Finally, the plastic potential \({g}_{\text{MH}}\) in (9) proposed by Saksala et al. (2017) with ψ is the dilation angle of the rock.

At this point, it is informative to illustrate the bi-surface yield criterion in the principal stress space. As the MH criterion does not depend on the intermediate stress space, the criteria are plotted in σ₁–σ₃ stress space in Fig. 2.

It can be observed in Fig. 2 that, in the present case, the MH criterion actually matches the tensile strength, 10 MPa, in uniaxial tension quite well but clearly overestimates the bi-axial tensile strength. Moreover, the flow direction of the MH criterion would be wrong in uniaxial tension, i.e. it would deviate from the Rankine flow direction (see Fig. 2b), which is the correct one. Still more, if the tensile strength was lower, say 7 MPa, the MH criterion would overpredict it by more than 40%. Therefore, the modified (rounded) Rankine criterion is needed as a tensile cutoff. Figure 2b also illustrates the corner plasticity direction at the intersection of the yield surfaces. In this case, the flow direction, upon stress return mapping, is calculated by the Koiter’s rule in Eq. (8), while the return mapping is performed by the standard cutting plane algorithm (Wang et al. 1997; Simo and Hughes 1998). The return mapping scheme is further elaborated in Appendix 1.

The final model components are the nominal-effective stress relation, which specifies how damage variables operate on the stress, and the constitutive law (within the small deformation framework) between the effective stress and the elastic strain:where \(\overline{{{\varvec{\upsigma}}}}\) is the nominal stress, i.e. the one returned onto the failure surface when the trial stress violates the criteria. Moreover, \({\mathbf{C}}_{\mathrm{e}}\) is the elasticity tensor, and \({{\varvec{\upvarepsilon}}}_{\mathrm{tot}}\) is the total strain. Finally, d is the third-order piezoelectric constants tensor (its 3 × 6 matrix form can be inferred from Fig. 1), and \(\mathbf{E}=-\nabla {\phi }_{e}\) is the electric field with \({\phi }_{e}\) being the scalar electric potential.

The present formulation combines the damage and viscoplasticity parts of the model in the effective stress space (Grassl and Jirásek 2006) allowing to separate the damage and viscoplasticity computations so that the stress return mapping is first performed in the effective principal stress space. Then, the damage variables are updated and, finally, the nominal stress is calculated by (11). This formulation requires that the damage variables are driven by the (visco)plastic strain.

The equations governing the HF-HV-AC pretreatment are the standard piezoelectro-mechanical equations, i.e. the balance of linear momentum and the piezoelectro-static balance, written in the finite element discretized form as (Allik and Hughes 1970)with

In these equations, symbols are as follows: t is time; A is the finite element assembly operator; \({\mathbf{B}}_{\upphi }^{\mathrm{e}}\) (\(\mathbf{E}={\mathbf{B}}_{\upphi }^{\mathrm{e}}{{\varvec{\upphi}}}_{e}\)) is the gradient of the electric potential interpolation matrix \({\mathbf{N}}_{\upphi }^{\mathrm{e}}\); \({\mathbf{B}}_{\mathrm{u}}^{\mathrm{e}}\) is the gradient of displacement interpolation matrix \({\mathbf{N}}_{\mathrm{u}}^{\mathrm{e}}\); M is the (lumped) mass matrix; C is the damping matrix with a constant α; \(\varrho\) is the electric charge; \({\mathbf{f}}_{t}^{\mathrm{int}}\) and \({\mathbf{f}}_{t}^{\mathrm{ext}}\) are the internal and external force vectors; and \({\mathbf{f}}_{t}^{\upphi }\) is the electric loading vector. The piezoelectric coupling and (diagonal) dielectric constants matrices depend on the damage variables bywhere I is the unit matrix and \({\epsilon }_{0}\) and \({\epsilon }_{\mathrm{r}}\) are, respectively, the permittivity of vacuum and the relative permittivity of the mineral phase in the rock. By these relations, the piezoelectric coupling term approaches zero as quartz (being the only piezoelectric phase) phase undergoes damage while, by (19), the isotropic permittivity of all three minerals approaches to that of vacuum (Eqs. (18) and (19) are mineral phase specific). It is emphasized that all mineral phases in the numerical rock description have permittivity, i.e. nonzero matrix \({\varvec{\upepsilon}}\) while only quartz is piezoelectric, having nonzero matrix d. The physical meaning of these matrices can be readily understood from the constitutive equation of linear elastic piezoelectric material: \({D}_{i}={e}_{ikl}{\varepsilon }_{kl}-{\epsilon }_{ij}{E}_{j}\), where \({D}_{i}\) is the electric displacement (C/m²), \({\varepsilon }_{kl}\) is the mechanical strain tensor, \({e}_{ikl}\) is the piezoelectric coupling tensor (C/m²), \({E}_{k}\) is the electric field (V/m), and \({\epsilon }_{ij}\) is the dielectric constants tensor (F/m = C/Vm).

The solution of this problem with an electric potential boundary condition (the HF-HV-AC actuation) is here performed explicitly in time using a (globally) non-iterative staggered (partitioned) solution method (Ramegowda et al. 2019; Saksala 2021):where the Euler scheme is used here for time stepping. An implicit, globally iterative method would not be practical, as the high frequency of the AC voltage to be used in the simulations necessitates an extremely short time step.

The material properties, representing a generic granite like rock, and the model parameters are given in Table 1. The elasticity constants (E, ν) are generated with the MTEX software (Mainprice et al. 2014), and the rest of the material properties are by Hokka et al. 2016, Newnham 2005 and Mahabadi 2012. The dielectric constants in Table 1 are given relative to the permittivity of vacuum \({\epsilon }_{0}\). Moreover, the MH criterion parameter values, B and n, match the quasi-static experiments (Hokka et al. 2016) for Kuru granite. The viscosity moduli values are chosen so as, on one hand, not to cause significant strain rate hardening and, on the other hand, to stabilize the local stress integration. For more details, see Saksala (2021) and Saksala et al. (2017).

The piezoelectric constants and the form of matrix d are given in Fig. 1. These are values in the local crystal coordinate system shown in Fig. 1, which in the present finite element context is the local element coordinate system. Therefore, when other orientations are tested, matrix d is rotated to the desired orientation using e.g. Euler angles (see Saksala (2021) for more details). In this case, the rotation matrix \({\varvec{\Omega}}\) = \({\varvec{\Omega}}\)₃(ψ)\({\varvec{\Omega}}\)₂(θ)\({\varvec{\Omega}}\)₁(φ) where the three rotation matrices \({\varvec{\Omega}}\)_i performs the Euler rotation indicated by their arguments as follows: The first rotation is about the z-axis; the second rotation is about the new x′-axis; and the third rotation is about the new z″-axis.

This test was originally developed for understanding the dynamic crushing of a rock ball (sphere) using the Hopkinson bar apparatus (Huang et al. 2014; Huang 2016). As the application in mind in the present paper is comminution by crushing, a lower impact velocity is applied here. The setup modelled here consists of two rigid platens between which the rock sample is compressed by moving the upper platen downwards with a constant velocity. The modelling principle is illustrated in Fig. 10.

The platen geometry is a virtual and its contact with the rock finite element mesh is modeled by imposing kinematic contact constraints between the rock nodes and the nodes representing the platens. These constraints are of form: u_i,z − u_RBn,z = b_n, where u_i,z and u_RBn,z are, respectively, the platen node (i = 1,2) and rock ball contact node (n = 1 to number of contacts) displacements in z-direction, and b_n is the initial distance between them. Moreover, contact forces P₁ and P₂ are solved along with the global solution with a method described in Saksala (2010). Dashpots, i.e. viscous dampers, are required by the method to absorb possible stress waves generated when the velocity v₀ is significant. The equations of motion of the platen nodes arewhere m₁ and m₂ are computational, not real, masses of the platens and \({c}_{\mathrm{b}}{\rho }_{\mathrm{b}}{A}_{\mathrm{b}}\) is the impedance of the platen defined by the bar velocity of a stress wave in slender rod, density, and the computational area of the platen of the Hopkinson bar device. In the following simulations, values of steel are used for density (7800 kg/m³) and the bar velocity c_b = 5150 m/s, while m₁ = m₂ = 0.1 kg. Other material parameters and model parameters are as above. The interesting outcome of the simulations here are the contact (reaction) forces since they are the loading carried out by the consumable tools in comminution. Figure 11 presents simulation results for crushing the numerical rock balls in Fig. 10c when v₀ = 0.5 m/s.

The failure modes for both numerical rock balls exhibit the experimentally observed (Huang 2016) split of the sample in three or four slices. Moreover, some secondary (meaning mild value of the damage variable) crack-like tensile damage patterns can be inferred from both samples. Substantial compressive damaging can be observed (Fig. 11a) at the poles of the sphere (i.e. in the contact areas). The contact reaction forces, reaching a magnitude of 18 kN, show some fluctuations (Fig. 11d) due to the transient component of the loading (v₀ = 0.5 m/s at t = 0). Finally, Fig. 11e shows the average of the contact forces for both rock samples, and as can be observed, they are almost identical.