## Abstract

In this work, we show the development of a numerical model to investigate the 3D interactions between microwave radiation and basalt, granite, and sandstone rock samples. In particular, we assign sample heterogeneity based on the Weibull statistical distribution, and invoke a damage model for elemental tensile and compressive stresses based on the maximum tensile stress and the Mohr–Coulomb theories, respectively. Model implementation is facilitated by the use of comsol for use in coupling the electromagnetic, thermal, and solid displacement relations. Various parametric studies are conducted related to variable input power and waveguide port alignment, with model validation conducted with respect to damage resulting from a uniaxial compression test. The results indicate that relatively high induced temperatures will promote damage potential, but its impact must be placed within the context of the sample strength to quantify the true potential damage evolution of a given rock mass. As observed herein, a mechanically weaker rock may be prone to mechanical damage; however, it may also possess a relatively large relative permittivity, enabling it to absorb the least amount of microwave radiation thus yielding comparatively low overall damage profiles compared to a more mechanically competent rock mass.

## 1 Introduction

The mechanical breakdown or comminution of bulk mineral deposits using traditional grinding methods for purposes of energy extraction or other forms of geotechnical mining processes is known to be highly inefficient. In fact, various sources cite that in some countries (including the United States, Australia, and South Africa), these traditional mining processes may require up to 2% of the total national energy asset [1] and return on average only 1% of this energy for useful purposes [2,3]. Energy losses due to excessive noise and heat constitute the primary liabilities, but other factors, including machine wear, pollution, and rising project costs must also be considered [4–6]. The reduction of wear, maintenance, and associated energy costs on tunnel boring machines, subterranean exploration equipment, and the construction of hard rock underground facilities aligns with core functions within the US Army, i.e., development of novel means to improve energy efficiency of subterranean tunneling and construction activities.

One resource that has been shown to potentially reduce these high energy costs by improving efficiencies in mining or drilling processes, is microwave energy. In particular, the preliminary weakening and degradation of the bulk mineral by means of direct exposure to microwave radiation. Microwaves are a form of electromagnetic energy currently used in a variety of geo-mechanical/geothermal applications, including mineral processing [7–9], heating [10,11], and drying [12]. Of particular importance, is the radiation of rock samples, and the associated micro cracking that results from microwave induced thermal stresses. These induced thermal stresses contribute to both tensile and shear components, and occur as a result of the non-uniform heating and consequent temperature gradients that are prevalent in heterogeneous materials. A large number of experiments, conducted at various scales, have supplied ample evidence of these findings, including the works of Refs. [13–16].

Complementary multiscale numerical modeling efforts have typically lagged behind the physical experiments. Complications arise due to the variations in the spatiotemporal electromagnetic field, the non-uniformity of the rock samples, and other factors, including numerical instabilities associated with the equations governing the combined effects of electromagnetics, heat transfer, and mechanical displacements. Potential insights that numerical models may offer, include quantitative, parametric studies related to the effects of sample heterogeneity, composition, input power and frequency, and various other assessments subject to scale, physics, and underlying model assumptions. Although limited, previous research in this area includes various two-dimensional thermo-electric models [17–20]. The development of three-dimensional models, including the contributions from microwave induced thermal stresses, have benefited greatly from the utility of multiphysics, finite element (FEM) based software, including comsol [21], abaqus [22], and others [23–25]. These programs conveniently allow the practitioner a straightforward coupling approach to the electromagnetics, heat transfer, and solid mechanics. Some of this latter three-dimensional computational research includes that of Li et al. [26], Toifle et al. [27], Huang et al. [28], and Xu et al. [29]. Of these, Xu et al. [29] also investigated the effects of heterogeneous samples, using statistical distributions.

In this work, we further investigate the three-dimensional interactions between microwave radiation and various rock samples. In particular, similar to Xu et al. [29], we assign sample heterogeneity based on the Weibull statistical distribution, and invoke a damage model for elemental tensile and compressive stresses based on the maximum tensile stress and the Mohr–Coulomb theories, respectively. We further use the FEM based, multiphysics facility of comsol [21] for use in coupling the electromagnetic, thermal, and solid displacement relations. Unique to this work, is the characterization of the damage model, and the subsequent application and investigation of several different mineral samples, including sandstone, granite, and basalt (each with variable strength and varying capacities for microwave absorption). Additionally, we invoke various parametric studies related to variable input power and waveguide port alignment. Model validation is conducted with respect to damage resulting from a uniaxial compression test, comparing the present model with results from physical experiments of Yang et al. [30].

## 2 Model Development and Numerical Considerations

### 2.1 Model Assumptions.

As with all numerical/theoretical assessments attempting to replicate a physical process, there will inevitably exist some limiting assumptions. In this work, we assume the following:

The total strain consists only of elastic and thermal components (i.e., $\epsilon T=\epsilon el+\epsilon th$). Plastic strain ($\epsilon p$) and creep strain ($\epsilon cr$), for example, are excluded since the material is assumed brittle, and for purposes of simplicity.

Material heterogeneity is modeled with respect to Young’s modulus and the material compressive and tensile strength, with element values sampled from a statistical Weibull distribution. For each sample, the tensile strength is proportionate to the compressive strength.

Damage is assumed elastic and isotropic for each element as described by the elastic constitutive law with damage thresholds in tension and compression prescribed by the maximum tensile stress and Mohr–Coulomb criterion, respectively.

### 2.2 Model Equations.

Subject to the above assumptions, the primary model framework relies upon the traditional conservation equations applicable to the determination of electromagnetic wave energy, heat transfer, and mechanical displacements. The following outlines the primary equations and derivations associated with each of the aforementioned contributions.

#### 2.2.1 Electromagnetic Wave Energy.

*e*

^{iωt}time dependence. Since the time derivative of

*e*

^{jωt}is

*jωe*

^{jωt}, we replace ∂/∂

*t*with

*jω*and ∂

^{2}/∂

*t*

^{2}with −

*ω*

^{2}to get

**is the electric field vector,**

*E**ω*is the angular velocity (i.e.,

*ω*= 2

*πf*

_{0}),

*λ*is the wave number, $\epsilon 0$ is the permittivity of free space ($\epsilon 0=8.85\xd710\u221212F/m$),

*μ*

_{r}and $\epsilon r$ are the respective relative permeability and permittivity,

*σ*

_{e}is the electrical conductivity, and $j=\u22121$.

#### 2.2.2 Heat Transfer.

*T*is the temperature,

*ρ*is the density,

*c*

_{p}is isobaric heat capacity,

*k*is the thermal conductivity, and

*Q*is the volumetric heat source. In this case,

*Q*represents an energy source term coupling Eq. (6) with the aforementioned electromagnetic microwave energy

**is the current density,**

*J***is the magnetic field intensity, and**

*H***is the magnetic flux density. Aside, we note that convective heat transfer may be omitted with respect to microwave radiation.**

*B*#### 2.2.3 Mechanical Constitutive Relations and Damage.

*E*) is dependent on the damage variable (

*D*), such that

#### 2.2.4 Damage.

As stated previously, the damage (for a given element) is quantified for both tension and compression by means of the maximum tensile strength criterion and the Mohr–Coulomb criterion, respectively. As indicated in Fig. 1, for the purposes of this work, an element is considered in tension for negative values of stress and strain, while an element is compressed for positive values.

*σ*

_{t}= 0, where

*σ*

_{c}= 0, where

*σ*

_{1}and

*σ*

_{3}are the respective maximum and minimum principles stresses, $\epsilon \xaf$ is the equivalent strain (i.e., $\epsilon \xaf=\epsilon 12+\epsilon 22+\epsilon 32$),

*ϕ*is the internal friction angle of the material,

*σ*

_{c0}and

*σ*

_{t0}are the uniaxial compressive and tensile strengths, $\epsilon c0$ and $\epsilon t0$ are the respective elastic compressive and tensile strain limits,

*κ*is the maximum value of $\epsilon \xaf$ in the load history, and the parameter

*e*

_{f}is defined as

*G*

_{f}is the fracture energy per unit volume, and

*h*

_{cb}is the characteristic element size.

### 2.3 Model Geometry, Mesh, and Boundary Conditions.

Apart from the initial validation test, as shown in Fig. 2, the simulations of this work were composed of two primary geometries, differentiated only by the location of the waveguide input port. As indicated, these differences correspond to a horizontal port alignment (see Fig. 2(a)) and vertical alignment (Fig. 2(b)). The sample height and radius used in this work were fixed at 158 mm and 25 mm, respectively, while the inlet port waveguide was modeled with dimensions (*l*_{w} × *d*_{w} × *h*_{w}) of 78 mm × 50 mm × 18 mm, respectively.

**is the inward normal unit vector. The surface of the cylindrical sample was assigned a surface to ambient radiation condition $(\u2212n\u22c5q=\epsilon \sigma sb(Tamb4\u2212T4))$. With $\epsilon $ and**

*n**σ*

_{sb}, representing the emissivity (see Table 2) and Stefan–Boltzmann constant, respectively. The port inlet condition was maintained at a cutoff frequency of 2.45 GHz and with a transverse electric (TE) fundamental waveguide mode of 1.0 (i.e., TE

_{10}mode type), as is typical for most rectangular waveguides. The input power in certain cases was varied from 1 kW to 4 kW.

Finally, the physics controlled mesh was composed of 24,230 tetrahedral elements, ranging in size from 0.73 mm to 18 mm.

Figure 3 provides a flowchart of the simulation process for a given finite element. As shown, subsequent to the assignment of material heterogeneity by means of the Weibull statistical distribution (see Sec. 2.5), the electromagnetic wave energy (** E**) is first solved within the frequency domain (see Eq. (5)). Solutions for

**are then coupled to the thermal energy equation (Eq. (6)) allowing for the computation of elemental temperature and thermal stress-strain (i.e., Eq. (9)). Finally, the damage evolution proceeds according to the maximum tensile strength and Mohr–Coulomb criterion, which conditionally affects the elastic contributions to the stress-strain constitutive equation (see Eq. (10)) by means of the modulus of elasticity and its dependency on the damage variable**

*E**D*. This pattern is repeated for each element within the domain prior to moving to the next time interval at which time the entire process is again repeated.

### 2.4 Thermophysical Properties.

In this work, three different rock samples are numerically investigated, sandstone, basalt, and dry granite, with specific mineral concentrations (vol%) as shown in Table 1. These particular samples were chosen because of their variable microwave absorption properties. Basalt, for example, is known to absorb relatively large amounts of microwave radiation, particularly due to its high concentration of pyroxene, while granite and sandstone are known to absorb much less, due their relative abundance of quartz [31]. Table 2 lists the various thermophysical properties used within the model. As indicated, many properties, including those corresponding to thermal conductivity (*k*), density (*ρ*), heat capacity (*c*_{p}), and relative permittivity ($\epsilon r$) assume a temperature dependence. Of primary importance is the dielectric behavior of the samples, as quantified by the complex value for relative permittivity. Figure 4 shows the temperature dependent relative permittivity for both the real and the imaginary components used in this work. In particular, the imaginary component ($\epsilon \u2033$) is known to strongly influence absorption as a function of temperature [31].

Mineral | Sandstone (vol%) | Basalt (vol%) | Granite (vol%) |
---|---|---|---|

Quartz | 55 | 5 | 27 |

Feldspar | 19 | 61 | 53 |

Micas and chlorite | 28 | — | 20 |

Pyroxene | — | 23 | — |

Carbonate | 1 | 8 | — |

Ilmenite | — | 2 | — |

Mineral | Sandstone (vol%) | Basalt (vol%) | Granite (vol%) |
---|---|---|---|

Quartz | 55 | 5 | 27 |

Feldspar | 19 | 61 | 53 |

Micas and chlorite | 28 | — | 20 |

Pyroxene | — | 23 | — |

Carbonate | 1 | 8 | — |

Ilmenite | — | 2 | — |

Material properties | Sandstone | Basalt | Granite | |
---|---|---|---|---|

Thermal conductivity (W/(m·K)) Reference | k | −9 × 10^{−15}T^{5} + 3 × 10^{−11}T^{4}–3 × 10^{−8}T^{3} + 2 × 10^{−5}T^{2}–0.0118T + 5.7144[31] | −1 × 10^{−15}T^{5} + 6 × 10^{−12}T^{4}–8 × 10^{−9}T^{3} + 4 × 10^{−6}T^{2}–0.0002T + 1.4458[31] | −2 × 10^{−14}T^{5} + 9 × 10^{−11}T^{4}–1 × 10^{−07}T^{3} + 8 × 10^{−5}T^{2}–0.0286T + 5.8411[31] |

Density (kg/m^{3})Reference | ρ | −4 × 10^{−10}T^{4} + 2 × 10^{−6}T^{3}–0.0019T^{2} + 0.8093T + 2292.8[31] | −2 × 10^{−10}T^{4} + 5 × 10^{−7}T^{3}–0.0006T^{2} + 0.1777T + 2885.2[31] | −1 × 10^{−9}T^{4} + 3 × 10^{−6}T^{3}–0.0035T^{2} + 1.3525T + 2578.9[31] |

Heat capacity (J/(kg·K)) Reference | c_{p} | −1 × 10^{−6}T^{3} + 0.0013T^{2} + 0.3653T + 599.33[31] | 2 × 10^{−7}T^{3}–0.0009T^{2} + 1.2683T + 511.28[31] | −8 × 10^{−7}T^{3} + 0.0011T^{2} + 0.4087T + 610.22[31] |

Electrical conductivity (S/m) Reference | σ_{e} | 1.0 × 10^{−3}[32] | 1.0 × 10^{−4}[32] | 1.0 × 10^{−5}[32] |

Relative permeability | μ_{r} | 1 | 1 | 1 |

Relative permittivity (real part) Reference | $\epsilon \u2032$ | −7 × 10^{−15}T^{5} + 3 × 10^{−11}T^{4}–6 × 10^{−8}T^{3} + 4 × 10^{−5}T^{2}–0.0159T + 6.9977[31] | 1 × 10^{−13}T^{5}–5 × 10^{−10}T^{4} + 7 × 10^{−7}T^{3}–0.0005T^{2} + 0.1531T−10.299[31] | 1 × 10^{−16}T^{6}–5 × 10^{−13}T^{5} + 9 × 10^{−10}T^{4}–8 × 10^{−7}T^{3} + 0.0004T^{2}–0.0917T + 14.267[31] |

Relative permittivity (imaginary part) Reference | $\epsilon \u2033$ | 2 × 10^{−15}T^{5}–8 × 10^{−12}T^{4} + 9 × 10^{−9}T^{3}–4 × 10^{−6}T^{2} + 0.0006T + 0.0971[31] | 5 × 10^{−14}T^{5}–2 × 10^{−10}T^{4} + 3 × 10^{−7}T^{3}–0.0002T^{2} + 0.0601T−6.6219[31] | 6 × 10^{−17}T^{6}–3 × 10^{−13}T^{5} + 5 × 10^{−10}T^{4}–4 × 10^{−7}T^{3} + 0.0002T^{2}–0.0513T + 4.9273[31] |

Mean Young’s modulus (GPa) | E_{y0} | 20.0 | 50.0 | 60.0 |

Poisson’s ratio Reference | ν | 0.2 [33] | 0.23 [33] | 0.20 [33] |

Coefficient of thermal expansion (K^{−1})Reference | α | 10.0 × 10^{−6}[31] | 5.4 × 10^{−6}[31] | 8.0 × 10^{−6}[31] |

Mean uniaxial compressive strength (MPa) | σ_{c0} | 10.0 | 40.0 | 50.0 |

Ratio compressive to tensile strength | σ_{c0}/σ_{t0} | 10 | 10 | 10 |

Emissivity Reference | $\epsilon $ | 0.8 | 0.99 [34] | 0.8 |

Input parameters | ||||

Port input power (kW) | P_{in} | 1–4 | ||

Input frequency (GHz) | f | 2.45 | ||

Mode number | n | 10 | ||

Mode phase | θ_{in} | 0 | ||

Shape (heterogeneity) index | m | 3 |

Material properties | Sandstone | Basalt | Granite | |
---|---|---|---|---|

Thermal conductivity (W/(m·K)) Reference | k | −9 × 10^{−15}T^{5} + 3 × 10^{−11}T^{4}–3 × 10^{−8}T^{3} + 2 × 10^{−5}T^{2}–0.0118T + 5.7144[31] | −1 × 10^{−15}T^{5} + 6 × 10^{−12}T^{4}–8 × 10^{−9}T^{3} + 4 × 10^{−6}T^{2}–0.0002T + 1.4458[31] | −2 × 10^{−14}T^{5} + 9 × 10^{−11}T^{4}–1 × 10^{−07}T^{3} + 8 × 10^{−5}T^{2}–0.0286T + 5.8411[31] |

Density (kg/m^{3})Reference | ρ | −4 × 10^{−10}T^{4} + 2 × 10^{−6}T^{3}–0.0019T^{2} + 0.8093T + 2292.8[31] | −2 × 10^{−10}T^{4} + 5 × 10^{−7}T^{3}–0.0006T^{2} + 0.1777T + 2885.2[31] | −1 × 10^{−9}T^{4} + 3 × 10^{−6}T^{3}–0.0035T^{2} + 1.3525T + 2578.9[31] |

Heat capacity (J/(kg·K)) Reference | c_{p} | −1 × 10^{−6}T^{3} + 0.0013T^{2} + 0.3653T + 599.33[31] | 2 × 10^{−7}T^{3}–0.0009T^{2} + 1.2683T + 511.28[31] | −8 × 10^{−7}T^{3} + 0.0011T^{2} + 0.4087T + 610.22[31] |

Electrical conductivity (S/m) Reference | σ_{e} | 1.0 × 10^{−3}[32] | 1.0 × 10^{−4}[32] | 1.0 × 10^{−5}[32] |

Relative permeability | μ_{r} | 1 | 1 | 1 |

Relative permittivity (real part) Reference | $\epsilon \u2032$ | −7 × 10^{−15}T^{5} + 3 × 10^{−11}T^{4}–6 × 10^{−8}T^{3} + 4 × 10^{−5}T^{2}–0.0159T + 6.9977[31] | 1 × 10^{−13}T^{5}–5 × 10^{−10}T^{4} + 7 × 10^{−7}T^{3}–0.0005T^{2} + 0.1531T−10.299[31] | 1 × 10^{−16}T^{6}–5 × 10^{−13}T^{5} + 9 × 10^{−10}T^{4}–8 × 10^{−7}T^{3} + 0.0004T^{2}–0.0917T + 14.267[31] |

Relative permittivity (imaginary part) Reference | $\epsilon \u2033$ | 2 × 10^{−15}T^{5}–8 × 10^{−12}T^{4} + 9 × 10^{−9}T^{3}–4 × 10^{−6}T^{2} + 0.0006T + 0.0971[31] | 5 × 10^{−14}T^{5}–2 × 10^{−10}T^{4} + 3 × 10^{−7}T^{3}–0.0002T^{2} + 0.0601T−6.6219[31] | 6 × 10^{−17}T^{6}–3 × 10^{−13}T^{5} + 5 × 10^{−10}T^{4}–4 × 10^{−7}T^{3} + 0.0002T^{2}–0.0513T + 4.9273[31] |

Mean Young’s modulus (GPa) | E_{y0} | 20.0 | 50.0 | 60.0 |

Poisson’s ratio Reference | ν | 0.2 [33] | 0.23 [33] | 0.20 [33] |

Coefficient of thermal expansion (K^{−1})Reference | α | 10.0 × 10^{−6}[31] | 5.4 × 10^{−6}[31] | 8.0 × 10^{−6}[31] |

Mean uniaxial compressive strength (MPa) | σ_{c0} | 10.0 | 40.0 | 50.0 |

Ratio compressive to tensile strength | σ_{c0}/σ_{t0} | 10 | 10 | 10 |

Emissivity Reference | $\epsilon $ | 0.8 | 0.99 [34] | 0.8 |

Input parameters | ||||

Port input power (kW) | P_{in} | 1–4 | ||

Input frequency (GHz) | f | 2.45 | ||

Mode number | n | 10 | ||

Mode phase | θ_{in} | 0 | ||

Shape (heterogeneity) index | m | 3 |

With respect to the strength parameters, due to the wide range of possible experimental values corresponding to Young’s modulus and compressive strength, and the fact these particular values will be distributed in accordance with the statistical Weibull distribution, the mean quantities shown in Table 2 assume comparatively typical magnitudes specific to this study. Finally, the input operational parameters are also shown in Table 2. As shown, while the mode and frequency remained fixed in all cases, the input power is allowed to vary from 1 kW to 4 kW.

### 2.5 Heterogeneity Assignment.

*m*> 0 is the shape parameter and

*λ*> 0 is the scale parameter. Integrating Eq. (18), we obtain the cumulative distribution function (i.e., $F(x)=\u222b\u2212\u221exf(t)dt$), such that

*U*is a random number between zero and one. Utilizing Eq. (20), parameters such as Young’s modulus and compressive strength may be sampled as follows:

*E*

_{y0}, and mean compressive strength

*σ*

_{c0}are used as input parameters. As indicated in Fig. 5(a), larger values of

*m*correspond to higher profile symmetries, with more data points centered near the mean. Conversely, lower values of

*m*tend to offer skewed profiles with greater degree of spread. For practical purposes, this suggests that the value of

*m*can be tailored to act as a type of heterogeneity index, offering a means for assigning heterogeneity (i.e., in terms of average Young’s modulus or strength) depending on the characteristics of the physical sample. Figure 5(b), for example, illustrates the distribution of Young’s modulus within two circular right cylinders for

*m*= 1 and

*m*= 2, showing a comparative decrease in skewness for the latter case. In this work, unless otherwise specified, the heterogeneity value was fixed at a value of three.

## 3 Discussion and Results

### 3.1 Validation Case: Uniaxial Compression Test.

Preliminarily, the model was validated for the case of uniaxial compression, with comparisons conducted from the physical experiments of Yang et al. [30]. Specifically, a 2D axisymmetric cylinder of basalt was examined, having a height and radius of 100 mm and 25 mm, respectively, and with the stated thermophysical properties corresponding to basalt shown in Table 2. Sample heterogeneity, derived from the specification of the average Young’s modulus *E*_{y0}, and average compressive strength *σ*_{c0} were sampled from the Weibull distribution with heterogeneity index *m* = 5, and a compressive to tensile strength ratio (*σ*_{c0}/*σ*_{t0}) of ten was imposed. A displacement of 1 mm was applied on the upper surface of the cylinder with the lower end fixed.

Figures 6(a)–6(c) show the damage evolution over several displacement intervals (0.34–0.48 mm) applied to the circular cylinder. As indicated, the damage (0 ≤ *D* ≤ 1) shows maximum values (∼1) along curvilinear intervals located within the lower quadrant. At a displacement of 0.48 mm, the damage intervals become fully continuous and the sample fails along a clearly distinguishable line of fracture. Figure 6(d) depicts this damage as a function of axial strain for a characteristic point located within this particular damage region. As shown, the logarithmic damage reaches a maximum value of unity at approximately 0.9% axial strain. Finally, Fig. 6(e) shows axial stress versus axial strain comparisons between this work and the physical experiments of Yang et al. [30]. For this work, the results reflect the average value along the vertical (*z*-direction) of stress and strain sampled over all elements. As indicated, the results compare favorably, showing only some disparity between the current work and Yang et al. [30] for values of the maximum stress, with approximate magnitudes of 119 MPa and 130 MPa, respectively. Both results show sample failure occurring at approximately 0.9% axial strain.

### 3.2 Microwave Induced Damage Simulations.

As mentioned previously, the electromagnetic wave equation (Eq. (5)) was used to solve for the electric field vector (** E**) within the frequency domain. Figure 7 shows contours of the E-field norm (|

**|) along the**

*E**XZ*central plane for the two geometries considered, namely the horizontal inlet port alignment (Fig. 7(a)), and the vertical port alignment (Fig. 7(b)). The simulations were conducted at a power level

*P*= 3 kW. As shown, the E-field is non-uniform with horizontal band structure and maximum intensity of 9.5 × 10

^{4}V/m at the inlet port. Average intensities at the surface of the cylindrical sample for both port alignments were approximately equal (2.5 × 10

^{4}V/m). Differences between the two cases were observed primarily within cavity regions near the sample, showing distinctly higher magnitudes of |

**| in the case of vertical port alignment.**

*E*Figure 8 shows contours of temperature and damage for the basalt, granite, and sandstone cases at time *t* = 30 s (*P* = 2 kW, *m* = 3) at the sample surface as well as an interior *xy* plane. As shown, the results reveal a significant increase in the maximum interior temperature compared with the exterior surface temperature (i.e., for basalt this difference is approximately 40 K). This comparatively high interior heating (as opposed to surface heating) is a characteristic signature of microwave induced radiation, as explained in various other works [13,25], and can result in sample core melting after sufficient duration and intensity (i.e., the melting temperature of basalt; *T _{m}* = 1448–1623 K).

The complementary damage contours (0 ≤ *D* ≤ 1; where 1.0 represents complete cell damage) are shown in Fig. 9, and reveal the opposite trend to those just observed with respect to temperature. Here the average interior, core damage is in general much smaller than the exterior surface damage. As shown, the surface damage contours vary considerably according to sample, but tend to run along circumferential, horizontal bands at various positions along the sample length. The interior damage, in contrast, is shown to be greatest along locations near the sample perimeter.

#### 3.2.1 Effect of Variable Inlet Power.

Contours of surface temperature and surface damage as inlet power are increased from 1 kW to 4 kW as shown in Fig. 10. As indicated, for the *xy*-plane shown, the maximum temperatures appear along the top and central regions of the cylinder are increased from approximately 360 K to 500 K for *P* = 1 kW and *P* = 4 kW, respectively. Likewise, the maximum amount of damage, corresponding to *D* = 1.0, increases from localized bands near the center of the cylinder to generalized regions covering nearly the entire surface for *P* = 1 kW and *P* = 4 kW, respectively.

Figure 11 shows the evolution of the surface and interior temperature by increasing the inlet power intensity from 1 kW to 3 kW for each of the three samples, with all other input parameters remaining constant in accordance with Table 2. Temperature was computed as the average over the entire sample or interior planar surface and was computed over a period of 30 s. As expected, both the surface and interior average temperature were observed to increase with increasing power intensity and increase approximately linearly with time. Considering the results for each individual sample, basalt maintained the highest average surface and interior temperature (for each power intensity) with a maximum (at 30 s and 5 kW) of approximately 355 K, and 413 K, respectively. Granite maintained the lowest overall surface and interior temperatures (at 30 s and 1 kW) with values of approximately 300 K and 313 K, respectively.

The evolution of the average surface and interior damage for each of the samples are shown in Fig. 12. As indicated, the results tend to increase with time and follow logarithmic profiles. From Fig. 12(a), following the previous temperature profiles, basalt is shown to exhibit the greatest average surface damage (*D* = 0.7 at *P* = 3 kW and *t* = 30 s) with granite showing the least damage (*D* = 0.07 at *P* = 1 kW and *t* = 30 s). Likewise, the profiles of interior damage (see Fig. 12(b)) reveal similar comparative trends between the samples, but show that the maximum value (*D* = 0.41) is nearly equivalent for both basalt and sandstone at *t* = 30 s.

The implications of these results may be of interest, particularly when considered within the context of relative permittivity and mechanical tensile/compressive strength. As mentioned previously, the relative permittivity ($\epsilon r$), particularly the imaginary component ($\epsilon \u2033$), is largely responsible for microwave absorption. Of the three samples considered, basalt possesses the highest relative permittivity (dielectric strength) with respect to both the real ($\epsilon \u2032$) and imaginary components ($\epsilon \u2033$). This explains its comparatively high average temperatures both at the surface and the interior. Typically, these relatively high temperatures will promote damage, primarily surface damage as we have seen, but its impact must be placed within the context of the sample strength. With respect to damage evolution, as indicated in Table 2, sandstone possesses the smallest mean compressive and tensile strength rendering it susceptible to significant damage (from strictly a mechanical standpoint). However, it also possesses the smallest relative permittivity, enabling it to absorb the least amount of microwave radiation (allowing for relatively low temperature). These offsetting forces thus contribute to the comparatively low damage profiles shown for sandstone in Fig. 12.

## 4 Conclusions

In this work, we developed a numerical model to investigate the 3D interactions between microwave radiation and basalt, granite, and sandstone rock samples. In particular, we assigned sample heterogeneity based on the Weibull statistical distribution, and invoked a damage model for elemental tensile and compressive stresses based on the maximum tensile stress and the Mohr–Coulomb theories, respectively. Model implementation was facilitated by the use of the FEM based, multiphysics facility of comsol [21] for use in coupling the electromagnetic, thermal, and solid displacement relations. Various parametric studies were conducted related to variable input power and waveguide port alignment, with model validation conducted with respect to damage resulting from a uniaxial compression test. Some of the principal conclusions of this work include:

Mechanical as well as dielectric strength parameters must both be considered when determining the overall damage potential for microwave-rock interactions.

Quality computational results rely significantly on the accuracy of the thermos-electric parameters, particularly the temperature dependent relative permittivity.

The use of the Weibull distribution to induce statistically variable compressive and tensile strength can greatly facilitate sample heterogeneity, and is necessary for the observance of sample damage.

Future evolutions, using this model as a framework, may include such things as the inclusion of creep for use in time-dependent material behaviors, as well as parametric related examinations related to variations in compressive/tensile strength, sample to waveguide port distance, microwave boundary conditions, etc.

## Acknowledgment

This research was conducted on behalf of the US Army Engineer Research and Development Center’s Information Technology Laboratory, and Geotechnical and Structures Laboratory. Permission was granted by the Director, Information Technology Laboratory to publish this information with unlimited distribution. Funding was provided under US Army program 1010a-SBP213 FY21(CA).

## Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.