# Impact of porosity on the magnitude of trapped magnetic field and mechanical stress in HTS bulks during PFM

Santiago Guijosa, Kévin Berger, Frédéric Trillaud, Víctor Rolando

Jara-González, Melika Hinaje

► To cite this version:

Santiago Guijosa, Kévin Berger, Frédéric Trillaud, Víctor Rolando Jara-González, Melika Hinaje. Impact of porosity on the magnitude of trapped magnetic field and mechanical stress in HTS bulks during PFM. Superconductor Science and Technology, 2025, 38 (7), pp.075020. 10.1088/1361-6668/adedc0 . hal-05155442v2

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Submitted on 14 Aug 2025

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# Impact of porosity on the magnitude of trapped magnetic field and mechanical stress in HTS bulks during PFM

Santiago Guijosa<fnref n="1" /> ID, Kévin Berger<fnref n="1" /> ID, Frederic Trillaud<fnref n="1" />,<fnref n="2" />,* ID,
Víctor Rolando Jara-González<fnref n="3" /> ID and Melika Hinaje<fnref n="4" /> ID

E-mail: ftrillaudp@ii.unam.mx

Received 26 February 2025, revised 1 July 2025

Accepted for publication 9 July 2025

Published 18 July 2025

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CrossMark

## Abstract

During pulsed-field magnetization (PFM) processes, high-temperature superconducting (HTS) bulks are subjected to significant mechanical stress due to rapid changes in magnetic flux (Lorentz force) and temperature (thermal stress), potentially leading to mechanical failures. This is particularly emphasized by the inherent fabrication-induced defects such as cracks, grain boundaries, and pores. These defects often result in non-uniform trapped magnetic fields across the bulk, which can strongly impact its trapping performance. The present study focuses on the impact of porosity on the trapped magnetic field, temperature, and mechanical stress of REBCO disk-shaped bulks through a 2D transient multiphysics finite element model using COMSOL Multiphysics®. The results show that porosity causes significant losses by locally densifying the current, leading to high local temperatures, which in turn reduces the critical current density, infusing more local losses. As a result, lightning-like thermal pathways passing through the pores radiate from the edge of the bulk to its center as the applied magnetic field increases, therefore limiting the magnitude of the trapped field. Finally, porosity causes two distinct peaks in maximum tensile stress during PFM: one attributed to thermal effects and the other to electromagnetic forces. Higher porosity often results in the increase of both peaks of tensile stresses, potentially compromising the bulk's physical integrity. It highlights the importance of controlling porosity to improve the trapping performance of HTS bulks. One mitigating action is to fill the pores with silver, which is shown to remarkably reduce both the temperature rise and stress.

Keywords: bulk, multiphysics finite element analysis, porosity, pulsed-field magnetization, REBCO

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<footnote n="1">¹ Université de Lorraine, GREEN, F-54000 Nancy, France</footnote>

<footnote n="2">² Instituto de Ingeniería, Universidad Nacional Autónoma de México, CDMX 04510, México</footnote>

<footnote n="3">³ Prosgrado en Ingeniería, Universidad Nacional Autónoma de México, CDMX 04510, México</footnote>

<footnote n="4">⁴ Université de Lorraine, CNRS, LRGP, F-54000 Nancy, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
# 1. Introduction

Magnetized high-temperature superconducting (HTS) bulks can be used in electrical and transport applications acting as pseudo-permanent magnets when operated at low temperatures [1, 2]. Typically, the magnitude of their trapped magnetic flux density ranges from a few Tesla to more than ten times the remanent magnetic flux densities of conventional potent permanent magnets [3]. The trapped magnetic field is generated by induced superconducting currents that persist in the absence of external magnetic field. Its intensity is proportional to the average critical current density $J_c$ over the bulk and to its radius for a disk-shaped configuration. Advances in manufacturing processes, such as the top-seeded melt growth (TSMG) and the infiltration growth (TSIG), have enhanced flux pinning and homogeneity boosting the $J_c$. Despite these achievements, defects that may encompass non-oxygenated regions, pores, grain boundaries and cracks are still present [4, 5]. These non-superconducting regions can affect the distribution and the magnitude of the current density induced during magnetization, consequently affecting the homogeneity and intensity of the resulting trapped magnetic field [6, 7]. Those two parameters, homogeneity and intensity, are keystones in promoting the reliable use of HTS bulks in applications.

In the present study, particular interest is given to pores and their distribution in disk-shaped REBCO bulks and their impact on the trapped magnetic field during pulsed-field magnetization (PFM) [8–13]. Porosity $P$ is defined as the ratio of the area of pores $A_p$ to the total surface area of the bulk $A_t$, i.e. $P = 100 \times (A_p/A_t)$ in percent. Indeed, irrespective of the shape of the bulk, pores are formed during fabrication, either when high pressure is applied to compact the precursor powders or during the fusion process, as gas diffuses through the material. The size of pores resulting from compaction typically ranges between 1 and 10 µm, while those formed from gaseous diffusion span between 50 and 250 µm [6, 14]. The size strongly depends on the heating rate [15]. Gaseous-originating pores are effectively regarded as void spaces within the material since they are significantly larger than the coherence length and unable to act as pinning centers. Pores and cracks also act as pathways for oxygen to penetrate the sample in order to achieve the desired orthorhombic superconducting phase later in the fabrication process. In the TSMG method, the volume fraction of pores typically ranges from 10% to 30% of the bulk volume. In contrast, the TSIG method, due to infiltration that produces a dense material, exhibits a notably lower volume fraction of pores, usually around 5% [14].

Increasing pore density weakens the mechanical strength of the ceramic superconductor [16] and affects the trapped field when magnetized [6]. REBCO bulks, as ceramic materials, show high compressive strength (of the order of ≈100 MPa) but low tensile strength (of the order of ≈10 MPa), depending on the quality and processing of the sample. Hence, a significant challenge in using HTS as pseudo-permanent magnets is their proneness to fracture during magnetization. Indeed, high electromagnetic- or/and thermal-induced tensile stresses

are generated during such processes. It has been shown that the tensile strength can be improved by adding silver in the precursor powder thereby filling the pores. Thus, the porosity decreases and crack propagation lessens as reported in [17–20]. In addition, drilled holes in the precursor powder can provide escape routes for the gas leading to a reduction of porosity [21, 22]. In practice, mechanical reinforcements in the shape of metal rings are usually used to accommodate tensile stresses by acting as an external compressive support [23–25].

During PFM, rapid magnetic flux motions within the bulk body lead to intense and abrupt increases in temperature. Thermal expansion combined with a strong Lorentz force gives rise to large mechanical stresses [26]. Such stresses can initiate and propagate fractures. Although there is substantial research on the mechanical behavior and fracture of bulks [20, 23, 24, 26–29], there is yet no comprehensible study on the impact of pores and their distribution on the combination of mechanical behavior of the bulk, temperature response during magnetization, intensity of the trapped field and its uniformity during PFM. In [29], the authors focused on pure mechanical analysis implementing voids to simulate pores in a 2D phase-field model with mechanical reinforcement, showing that such voids provide enough energy for crack nucleation at very high magnetic fields. In [30, 31], they used a peri-dynamic model to study the impact of cracks and millimeter-size voids in 2D for PFM and in 3D for field cooling magnetization [32]. In particular, they looked at the dynamic evolution of fractures during magnetization. Similarly to pores in 2D, millimeter-size holes and their impact on magnetization were studied in [33], but from an electromagnetic point of view, only. Smaller sizes of pores, more representative of actual pore’s average dimension, remain to be investigated relying on a multiphysics approach. Hence, the present study proposes to fill the gaps by including the presence of micrometer-size pores in a 2D finite element analysis that integrates an electromagnetic model using the H-formulation coupled to a thermomechanical model, both implemented in COMSOL Multiphysics® version 6.2 [34]. The present work clearly unveils the necessity to master the pore’s distribution as their impact is undeniable on the mechanical and electromagnetic behavior of REBCO bulks.

After introducing the 2D finite element model (FEM) of a disk-shaped REBCO bulk and its underlying assumptions in section 2, the four porosity case studies are presented next in section 3. Section 4 deals with the results and their discussion. Section 5 concludes the present study.

# 2. 2D multiphysics FEM

In the proposed 2D model, illustrated in figure 1, the domain corresponding to the HTS bulk $\Omega_{sc}$ is considered as an infinitely long cylinder (along the z direction) encapsulated by an ‘Air’ domain $\Omega_a$. The domain $\Omega_{sc}$ includes the presence of pores $\Omega_p$ in the shape of small ellipses and circles. These shapes are representative of the general build of actual pores while removing the extra complexity associated with the
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Schematic drawing of the 2D model showing the different domains and their boundaries. The geometry lies in the x-y plane. Domains and boundaries: HTS bulk domain $\Omega_{sc}$, ‘Air’ domain $\Omega_a$ and pores’ domain $\Omega_p$ (blue inclusions). The drawing is not to scale, the micrometer-size pores appear here larger than they are actually in the model.</figcaption></figure>

meshing of non-regular and non-smooth geometries. Those geometries are difficult to tackle with the proposed finite element method without requiring a very dense and extremely fine mesh resulting in a drastic increase in computation time.

## 2.1. H formulation

The electromagnetic behavior is modeled using the ‘Magnetic Field’ module of COMSOL Multiphysics®, solving for the magnetic field **H** according to,

$$ \mu_0 \frac{\partial \mathbf{H}}{\partial t} + \nabla \times (\rho \nabla \times \mathbf{H}) = 0 \quad (1) $$

where $\mu_0$ is the permeability of the vacuum and $\rho$ is the electrical resistivity of the media. The latter is equal to a fictitious ‘Air’ resistivity $\rho_a$ for both $\Omega_a$ and $\Omega_p$, and equal to the resistivity $\rho_{sc}$ for the superconducting domain $\Omega_{sc}$. For the superconductor, the electrical resistivity is inferred from the power law. It is recalled in (2),

$$ \rho_{sc} = \frac{E_c}{J_c} \left( \frac{\|\mathbf{J}\|}{J_c} \right)^{n-1}. \quad (2) $$

$E_c$ is the critical electric field equal to $1 \mu V \text{ cm}^{-1}$; $n$ is the index of transition; **J** is the current density; and $J_c$ is the critical current density. $J_c$ depends on the local magnetic flux density **B** and temperature $T$ as follows,

$$ J_c(\mathbf{B}, T) = \frac{J_{c0}(T)}{\left(1 + \frac{\|\mathbf{B}\|}{B_0}\right)} \quad (3) $$

where $B_0$ is a flux density constant and $J_{c0}(T)$ is the temperature-dependent critical current density at zero applied field (self field) defined as,

$$ J_{c0}(T) = J_{c00} \left( 1 - \left( \frac{T}{T_c} \right)^2 \right)^{\frac{3}{2}}. \quad (4) $$

<table><caption>**Table 1.** Parameters for the simulation [9, 14, 39].</caption><thead><tr><th>Parameter</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td colspan="3">Geometric and electrical properties (section 2.1)</td></tr><tr><td><i>D</i></td><td>Bulk's diameter</td><td>29.1 mm</td></tr><tr><td><i>n</i></td><td>Power law exponent</td><td>21</td></tr><tr><td><i>E</i><sub>c</sub></td><td>Electric field criterion</td><td>1 &mu;V cm<sup>-1</sup></td></tr><tr><td><i>J</i><sub>c0</sub></td><td>Current density constant</td><td>500 A mm<sup>-2</sup></td></tr><tr><td><i>J</i><sub>c00</sub></td><td>Critical current density at <i>T</i><sub>0</sub></td><td>177 A mm<sup>-2</sup></td></tr><tr><td><i>B</i><sub>0</sub></td><td>Flux density constant</td><td>1.3 T</td></tr><tr><td><i>B</i><sub>m</sub></td><td>Max. applied field magnitude</td><td>4 T</td></tr><tr><td>&tau;</td><td>Pulse time constant</td><td>13 ms</td></tr></tbody></table>

**Thermal properties (section 2.2)**

<table><tr><td><i>T</i><sub>c</sub></td><td>Critical temperature</td><td>92 K</td></tr><tr><td><i>T</i><sub>0</sub></td><td>Operating temperature</td><td>65 K</td></tr><tr><td>&kappa;</td><td>Thermal conductivity</td><td>20 W (m&middot;K<sup>-1</sup>)</td></tr><tr><td>&alpha;</td><td>Thermal expansion coefficient</td><td>10<sup>-5</sup> K<sup>-1</sup></td></tr><tr><td><i>c</i><sub>p</sub></td><td>Specific heat capacity</td><td>See [9]</td></tr></table>

**Mechanical properties (section 2.3)**

<table><tr><td><i>v</i></td><td>Poisson's ratio</td><td>0.33</td></tr><tr><td><i>Y</i></td><td>Young's modulus</td><td>103 GPa</td></tr><tr><td><i>&sigma;</i><sub>F</sub></td><td>Fracture strength</td><td>30 MPa</td></tr><tr><td><i>&gamma;</i><sub>m</sub></td><td>Mass density</td><td>5900 kg m<sup>-3</sup></td></tr></table>

$T_c$ is the critical temperature equal to 92 K and $J_{c00}$ is the critical current density extrapolated at $T=0$ K in self field [35]. The different parameters of the above equations are given in table 1, section 3, when the case studies are introduced.

## 2.2. Heat balance equation

To evaluate the temperature in the superconductor (domain $\Omega_{sc}$) in transient, the following equation is solved using the module ‘Heat Transfer in solids’,

$$ \gamma_m c_p \frac{\partial T}{\partial t} - \nabla \cdot (\kappa \nabla T) = Q \quad (5) $$

where $\gamma_m$ is the mass density, $c_p$ is the specific heat capacity at constant pressure, and $\kappa$ is the thermal conductivity of the bulk ($\Omega_{sc}$). $Q$ is the volumetric Joule dissipation expressed as $Q = \mathbf{E} \cdot \mathbf{J}$ arising from the electrical field $\mathbf{E}$ appearing in the superconductor during its magnetization. Thermal insulation boundary conditions are imposed on the pores (considered as voids) and heat convection with sub-cooled liquid nitrogen at 65 K is considered at the surface of the bulk. The heat transfer can be found in [36]. The thermal properties for the case studies are given in table 1, section 3.

## 2.3. Mechanical model

Temperature gradients and Lorentz forces ($\mathbf{F}_L = \mathbf{J} \times \mathbf{B}$) emerge during magnetization, inducing stresses in the superconductor. The resulting displacement field **u**
can be solved according to,

$$ \nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{L}} = \gamma_{\mathrm{m}} \frac{\partial^2 \mathbf{u}}{\partial t^2} \qquad (6) $$

where $\boldsymbol{\sigma}$ symbolizes the 2D stress tensor related to the total strain $\varepsilon$ (mechanical and thermal) via the Hook's law $\boldsymbol{\sigma} = \mathbf{C}(Y, \nu) : \varepsilon$; $\mathbf{C}$ is the fourth-order stiffness tensor that contains the material properties [27], such as the Young's modulus $Y$ and Poisson's ratio $\nu$. The linearized thermal induced strain is expressed as $\alpha\Delta T$, with $\alpha$ the thermal expansion coefficient. The thermomechanical properties for the case studies are provided in table 1, section 3. The mechanical behavior was implemented using the 'Solid Mechanics' and 'Thermal Expansion' modules of COMSOL. Free boundary conditions are imposed on the bulk and pores. Since ceramics typically experience brittle fractures when one of the normal stress components exceeds the material's tensile or fracture strength $\sigma_F$, the primary mechanical parameter used in this work is the maximum stress, defined as $\sigma_{\max} = \max(\sigma_r, \sigma_{\theta})$ in a 2D cylindrical coordinate system. This metric, often referred to as the Rankine criterion, is widely used in mechanical analysis, especially for ceramics, as it simplifies complex stress states into a single value that can be directly compared to the fracture strength of the material [37]. Thus, if the maximum stress exceeds the fracture strength of the material, failure occurs. In the case of REBCO bulks under magnetization, research has shown that the maximum tensile stress responsible for fracture is the hoop stress $\sigma_{\theta}$ [24, 26, 38].

## 2.4. Numerical implementation

A two-step time-dependent solver was implemented. From $t=0$ s to the point where the flux creep is minimal ($\approx 300\tau$), a segregated solver was employed. For the rest of the simulation (for $t > 300\tau$ until the final time $t_f = 10000\tau$), a fully coupled solver was utilized. This approach was chosen because the segregated solver demonstrated better convergence and stability during the pulse phase, where variables change rapidly, allowing for more efficient handling of individual variables. Subsequently, as the flux creep subsides and a steady state is reached, the fully coupled solver achieved faster convergence while handling larger time steps. To ensure an accurate representation of the magnetization, the time steps are chosen according to the pulse shape. It is designed to capture both rapid and gradual changes keeping a fast convergence. The first interval used millisecond time steps to effectively resolve dynamic changes during the pulse. The subsequent interval used a logarithmic progression, to capture the slow flux creep behavior over longer timescales. After conducting a mesh refinement study, since computational times can easily reach several hours or days, the best strategy was to set the maximum mesh size in $\Omega_{sc}$ to be comparable to the size of an individual pore (of the order of one hundred micrometer) and the minimum mesh around a quarter of this value. With this restriction on the mesh size, the relative error $e_r$ in the trapped

field is estimated to be less than 0.06%, less than 0.36% for the maximum temperature, and less than 3.58% for the maximum stress in comparison to the finest mesh. This approach ensures optimal computational efficiency without sacrificing accuracy and was applied to the subsequent case study on a porous bulk.

## 3. Case studies

Accounting for the presence of numerous pores in a mesh-based model is challenging due to their small dimensions. To ease the computational burden without degrading the accuracy of the results, the pores are treated as regular and smooth geometries. It is the cornerstone for using FEM that avoids spurious local solutions. In addition to this inherent numerical problem, adding pores requires a multiphysics approach. Indeed, the number and dispersion of pores can affect local temperature rise and stress. Hence, there is an interest to quantify factors such as pore shape, size, number, and distribution that may result in impacting individually and collectively the trapped field. Unfortunately, exploring and isolating these parameters would not always show practical insights due to difficulties in experimental verification caused by the stochastic nature of the defect formation during bulk fabrication. In that regard, the present work aims to simplify the study while maintaining relevance by studying how pores fundamentally affect key quantities in PFM. These key quantities are the trapped field, maximum temperature rise and maximum stress. The goal is to unveil trends related to pore size and number, and analyzing strategies that may enhance the mechanical integrity of the bulk during PFM.

Four case studies, detailed below, are proposed to understand the impact of pores in HTS bulks during PFM, with the ideal case (non-porous) as reference. These studies are: single pore, sensitivity against pore size, increasing porosity and silver addition. For the cases 2–4, a random distribution is generated within a cylindrical region representing 90% of the overall cross-section of the bulk (in 2D), therefore excluding its rim. Indeed, the pores are expected to be denser around the middle of the bulks, between the rim and the center as the gas can escape more easily from the former than from the latter during manufacturing. It should be noted that the porosity treated in the study is in the lower range of actual void fraction in bulks and the pore sizes are within the upper range observed in practice. The purpose is to ease the computation, in particular on the mesh side of the FEM, while keeping realistic bulks' features to be able to assess in a reliable manner the impact of pores on the trapped magnetic field and the mechanical behavior of REBCO bulks. Due to the main assumptions, especially the infinite long cylinder, the work focuses on trends providing still a more realistic model than previously proposed in the literature as discussed in the introduction (section 1).

The disk-shaped bulks are made of REBCO and are assumed to have been prepared according to a TSMG derived method. For both case studies, the PFM is implemented in the model as an external pulsed magnetic field $B_a(t)$, given in (7),
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Schematic drawing of the different locations for the single pore case. Red dots show the center, middle and edge position of the 1 mm diameter pores *D*.</figcaption></figure>

applied at the air boundary $Γ_a$ along the axial direction of the infinite cylinder (z-axis),

$$B_a(t) = B_m \frac{t}{\tau} \exp \left( 1 - \frac{t}{\tau} \right). \qquad (7)$$

$B_m$ is the peak magnetic flux density and $τ$ is the duration required to reach the peak of the pulse [8]. It is important to note that the simulation values in table 1 are in the range of expected magnitudes for generic REBCO bulks and PFM setups [9, 14].

### 3.1. Case 1: single pore

The first case study considers a single pore represented by a circle in 2D having a diameter of 1 mm. The size of the pore is chosen purposely to emphasize any effects of its presence. The idea is to assess the impact of its position within the bulk: at its edge, at its middle, or at its center (see figure 2).

### 3.2. Case 2: sensitivity against pore size

The porosity *P* is fixed at 0.5%, with circular pores of identical diameter randomly distributed. The pore diameters vary from 75 to 300 µm, with five simulations carried out per diameter (five simulations for 75 µm), totaling 25 simulations. This study aims to analyze how quantities of interest evolve with increasing pore size while maintaining a constant porosity.

### 3.3. Case 3: increasing porosity

Since the impact of increasing porosity is the central focus of this article, this case study investigates aleatory distributions of elliptical pores as porosity *P* increases from 0.5% to 2% in steps of 0.5%. The major and minor axes of the pores are randomly selected within a size range of 100 - 250 µm. Ten simulations are performed per porosity to capture trends, resulting in a total of 40 simulations. The objective is to analyze how increasing porosity and the randomness in pore sizes impact the quantities of interest.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Pore size distribution based on experimental data [6], used for case 4 (1% porosity). A 50 µm offset was added to all diameters to avoid computational issues. It follows a skewed profile. The equivalent parameters are a mean value equal to 90.3 µm, a standard deviation of 33.62 µm, and a skewness of 8.6.</figcaption></figure>

### 3.4. Case 4: silver addition

The fourth case study computes a more realistic manufacturing condition by using a pore size distribution sampled from experimental data [6]. The dataset provides individual pore areas in mm², which are converted in diameter assuming a circular shape. Thus, a list of diameters is built and from that list, a random diameter is assigned to the major or the minor axis of each elliptical pore in the model. To alleviate the computational burden, 50 µm was added to all pore diameters. A skewed Gaussian distribution was used to represent the histogram of the pore diameter distribution as illustrated in figure 3. Here, pores are randomly filled with silver, a common fabrication procedure aimed at enhancing mechanical properties. Silver is present in two amounts, corresponding to 65% and 100% of the pores. The 65% fill represents a more realistic scenario, as not all pores are typically filled with silver [40], while the 100% fill serves as an ideal case. For this study, a single distribution of pores with porosity equal to 1% was considered. The physical properties of silver were given by the COMSOL material properties library.

## 4. Results

To assist in the analysis of the results, figure 4(a) presents the evolution of the magnetic flux density at the center of the bulk for the ideal case (no pores with homogeneous and isotropic superconducting properties). In the PFM process, there are three main stages. The first stage (1) is related to shielding and flux penetration with an initial set of currents that originated from the edge of the bulk and are moving toward its center. The second stage (2) is flux trapping. Once the pulse reaches its maximum and starts to decrease, another set of currents is induced in the opposite direction to the initial shielding
<figure><img src="image_7.png" /><figcaption>**Figure 4.** (a) PFM in the ideal case, blue dots mark key magnetization moments with corresponding times. Impact of the pore location on: (b) the evolution of the maximum stress, (c) the Lorentz force contribution to tensile stress, (d) the thermal expansion contribution to tensile stress, (e) the temperature evolution, and (f) the normalized critical current density obtained here for the minimum critical current computed in the bulk.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 5.** (a) Magnetic flux density distribution and **J** circulation (arrows) during pulse ascent. (b) Normalized critical current density *J*<sub>c</sub>/*J*<sub>c00</sub> near the pore located at the edge of the bulk. One can observed a 'discontinuity' in the distributions caused by the presence of the pore (modeled as void). This discontinuity follows a parabolic pattern illustrated as a black dashed line on the figures [41–43].</figcaption></figure>

currents thereby conserving the magnetic flux. There is then a redistribution of the magnetic flux in the direction of the center of the bulk. In the third stage (3), flux creep sets in, following the relocation of the flux arising from the persistent currents; demagnetizing effects reduce the trapped field until a steady-state regime is reached. These different regimes would correspond closely to times $t_1$ (1), $t_2 = \tau$, $t_3$ (2) and $t_4$ (3) shown in figure 4(a). The same feature appears during PFM, independently of the case studies.

## 4.1. Case 1: single pore

Figure 4(b) presents the maximum stress during magnetization for each single pore location alongside the ideal case (no pores, taken as reference). Figures 4(c) and (d) present the electromagnetic and thermal contributions to the tensile stress. As discussed in section 2.3, the analysis focuses on the tensile (positive) maximum stress responsible for brittle fractures. Therefore, compressive (negative) stresses are not shown in the figures to avoid confusion. In the ideal case, during the pulse ascent, electromagnetic stresses do not contribute to the tensile stress, whereas thermal stresses play a significant role in its tensile component. During the descending phase, as flux redistributes and superconducting currents evolve, both thermal and electromagnetic stresses contribute to the maximum tensile stress.

As observed in figure 4(b), the presence of a single pore significantly affects the mechanical behavior during magnetization, regardless of its location. A pore located at the edge induces high tensile thermal stresses during the rise of the pulse, reaching a maximum around $t_1$ when the magnetic flux reaches the center. The middle pore exhibits a similar, though less pronounced effect with slightly higher electromagnetic stresses during the descent than the edge pore. Finally, the central pore generates lower tensile thermal stresses and the electromagnetic stresses are higher than other pore locations,

having a maximum at $t_3$ when the magnetic flux moves to the center. To better understand the location relationship with each stress contribution, the temperature rise is analyzed.

Figure 4(e) presents the evolution of the maximum temperature in the HTS bulk for each pore location. The stress behavior for the edge and middle pores mirrors the temperature increase, indicating that the pores near the edge of the superconductor induce higher tensile thermal stresses during the ascending stage of a PFM, reaching a maximum value around $t_1$. Furthermore, figure 4(e) displays the average temperature throughout the pulse, showing that despite the sharp temperature spikes near the pores located at the edge, the average temperature in all cases remains nearly the same. This suggests that the temperature increase is localized.

Finally, the magnetic flux and induced current behaviors are studied to understand the origin of the temperature and electromagnetic-induced stresses, taking as example the edge pore for which the impact is the most pronounced in comparison to the other cases. Figure 5(a) shows the distribution of the norm of the magnetic flux density $|\mathbf{B}|$ and the arrows indicate the direction of the current density **J**, during the pulse rise corresponding to $B_a = 0.35B_m$ and just before $t_1$. A parabola-shaped discontinuity line (d-line [41–43]) drawn from the pore toward the center of the bulk can be identified. Inside the parabola, the current density is distorted, while outside it follows the conventional flow found in the ideal case, as previously reported in [33]. The presence of the pore creates local concentrations of current density around it. This increases regionally the resistivity, generating local losses. Consequently, the temperature rises locally, and $J_c(\mathbf{B},T)$ drops accordingly, as shown in figure 5(b). In turn, it escalates the resistivity further leading to a localized thermal avalanche. Inside this parabola, the heat accompanied with increasing temperature will diffuse towards the center of the bulk due to the reduction in $J_c$. This thermal runaway is more significant for pores found near the edge of the bulk, as it disturbs the magnetic flux in
<table><caption>**Table 2.** Statistical analysis for 25 simulations with increasing pore size (5 per level) at fixed porosity, *P* = 0.5%. The average values ‘⟨⟩’ and the standard deviations ‘±’ for the trapped field *B*<sub>t</sub>, the maximum temperatures, and the two stress maxima obtained about *t*<sub>1</sub> and *t*<sub>3</sub> are compiled.</caption><thead><tr><th>Pore size (μm)</th><th>⟨B<sub>t</sub>⟩ (T)</th><th>⟨T<sub>max</sub>⟩ (K)</th><th>⟨σ<sub>1,max</sub>⟩ (MPa)</th><th>⟨σ<sub>2,max</sub>⟩ (MPa)</th></tr></thead><tbody><tr><td><strong>Ideal</strong></td><td><strong>1.552</strong></td><td>69.7</td><td>0.005</td><td>3.3</td></tr><tr><td>75</td><td>1.531 ± 0.001</td><td>74.1 ± 3.3</td><td>7.5 ± 7.2</td><td>14.3 ± 7.0</td></tr><tr><td>125</td><td>1.532 ± 0.003</td><td>79.2 ± 1.7</td><td>9.3 ± 0.4</td><td>22.9 ± 24.6</td></tr><tr><td>175</td><td>1.531 ± 0.004</td><td>78.6 ± 2.1</td><td>13.2 ± 9.1</td><td>8.6 ± 1.8</td></tr><tr><td>250</td><td>1.522 ± 0.008</td><td>80.2 ± 2.6</td><td>11.1 ± 3.1</td><td>7.6 ± 0.6</td></tr><tr><td>300</td><td>1.525 ± 0.005</td><td>80.6 ± 2.0</td><td>12.4 ± 2.5</td><td>7.7 ± 1.3</td></tr></tbody></table>

<table><caption>**Table 3.** Statistical analysis for 40 simulations with increasing porosity (10 per *P*) and the ideal case. Each value shows the average ‘⟨⟩’ and the standard deviation ‘±’ for the trapped field *B*<sub>t</sub>, maximum temperature *T*<sub>max</sub>, and two stress maxima, *σ*<sub>1,max</sub> and *σ*<sub>2,max</sub>.</caption><thead><tr><th>Porosity (%)</th><th>⟨B<sub>t</sub>⟩ (T)</th><th>⟨T<sub>max</sub>⟩ (K)</th><th>⟨σ<sub>1,max</sub>⟩ (MPa)</th><th>⟨σ<sub>2,max</sub>⟩ (MPa)</th></tr></thead><tbody><tr><td><strong>Ideal</strong></td><td><strong>1.552</strong></td><td>69.7</td><td>0.005</td><td>3.3</td></tr><tr><td>0.5</td><td>1.528 ± 0.002</td><td>80.7 ± 2.0</td><td>14.7 ± 3.6</td><td>8.2 ± 0.6</td></tr><tr><td>1.0</td><td>1.504 ± 0.002</td><td>81.6 ± 1.1</td><td>13.9 ± 1.7</td><td>13.5 ± 8.1</td></tr><tr><td>1.5</td><td>1.477 ± 0.009</td><td>82.9 ± 1.4</td><td>17.3 ± 3.6</td><td>16.1 ± 12.7</td></tr><tr><td>2.0</td><td>1.454 ± 0.009</td><td>81.4 ± 1.3</td><td>17.2 ± 3.3</td><td>27.3 ± 13.6</td></tr></tbody></table>

the early stages of magnetization with additional power dissipation. Thus, the closer the pore is to the edge of the HTS bulk, the greater the reduction in the critical current density $J_c(\mathbf{B}, T)$ to be expected, as shown in figure 4(f).

## 4.2. Case 2: sensitivity against pore size

Table 2 presents the statistical analysis compared to the ideal case, showing the following tendencies with increasing pore size for a given porosity equal to 0.5%: smaller trapped field, higher maximum temperature rise, higher maximum stress during pulse ascend (about $t_1$), and lower maximum stress during pulse descend (about $t_3$). As highlighted in the previous study, see section 4.1, a first stress peak associated with thermal expansion arises around $t_1$, followed by a second peak at $t_3$ primarily induced by electromagnetic forces. These peak stresses are subsequently referred to as $\sigma_{1,max}$ and $\sigma_{2,max}$, respectively. While the exact timing of these events may vary slightly across simulations due to the randomness in pore distributions, all cases exhibit peak responses near these times, making them suitable reference points for comparison.

The reduced trapped field with increasing pore size may result from a broader current redistribution around larger pores, causing higher local losses lowering $J_c$. This is consistent with the increased temperature rise. As noted earlier, higher temperature during pulse ascend induces greater thermal stresses, which are observed with larger pores. Finally, the lower maximum stress during flux trapping for large pores may be due to the dominance of electromagnetic stresses. This suggests that more compact clusters of pores generate larger electromagnetic stresses because the pores are close to each other. This will be further discussed at the end of the next section 4.3.

## 4.3. Case 3: increasing porosity

For the following analysis, ten simulations per porosity were conducted. Each simulation represents a given spatial distribution of pores. Table 3 presents the statistical results comparing the ideal to the porosity cases. It shows average center trapped field, maximum temperature, and maximum stress. The average center trapped field decreases linearly with increasing porosity, as shown in figure 6(a). In the 2D approximation and according to the simulation parameters, given a porosity *P*, the trapped field drops by a factor of 3.236*P* in percent. Similarly, a linear increase of 10.65*P* MPa is observed in $\sigma_{2,max}$ at $t_3$ arising from electromagnetic stresses as shown in figure 6(b). Higher porosity leads to greater maximum temperature and stress. However, this trend should be interpreted with caution, as the pore distributions are random in nature. The increasing standard deviations with larger porosity reflect this variability. Thus, the spatial arrangement of pores plays a significant role, and more extensive statistics would be required to draw a clear conclusion.

For the subsequent group of analysis, we study the cases within each *P* level corresponding to those with the higher maximum stress. Figure 7 shows the magnetic flux distribution at the end of the magnetization for the chosen set of porosity and a given spatial distribution of pores. It is observed that with increasing porosity the magnetic field distribution becomes more and more distorted compared to the regular concentric rings shown by the ideal case. At the same time, the trapped field at the center decreases, due to the local redistribution of the induced current and the impact of thermal runaway in $J_c$, as discussed in the previous section 4.1.

Figure 8(a) shows the evolution of the maximum temperature during magnetization for each selected porosity level, reaching almost the same maximum temperature for all cases. As the single pore study pointed out, this temperature rise will
<figure><img src="image_9.png" /><figcaption>**Figure 6.** (a) Trapped field decreases linearly with increasing porosity. (b) The maximum electromagnetic stress at $t_3$ grows linearly with porosity. In both cases, each data point represents the average of ten simulations with different random distributions of pore.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Comparison of trapped field distributions after magnetization ($t = t_4$) for increasing porosity. It exposes a reduction in the center trapped field magnitude and a distorted magnetic flux density distribution.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** (a) Maximum temperature evolution during PFM for selected porosity. The porosity increases the temperature rise drastically, even though values are similar between porosity. (b) Maximum normalized stress showing different profiles for each percentage but a clear increase in tensile stress in comparison with the ideal case, with two peaks $\sigma_{1,max}$ and $\sigma_{2,max}$, around times $t_1$ and $t_3$ respectively. Tensile stresses reach the fracture strength of the material in some cases.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Comparison of maximum stress ranges between 0.5% and 2% porosity levels, shown as shaded areas representing envelopes obtained from ten simulations per pore distribution.</figcaption></figure>

occur locally around each pore and propagate towards the center of the bulk. Additionally, figure 8(b) provides the maximum stress during the PFM. High tensile stresses are reached around $t_1$ and a second peak of tensile stress appears during the pulse descent, between $\tau$ and $t_3$. The induced tensile stresses can reach the prescribed magnitude of fracture strength, here equal to 30 MPa, a typical value for ceramic bulks [4, 16, 17, 20, 21]. It strongly suggests that porosity facilitates the genesis and/or propagation of fractures and cracks for non-reinforced and low mechanical quality bulks. Further work would be necessary on the nucleation and propagation of cracks with pores relying on a more dedicated mechanical study. In general, a high porosity leads to high tensile stresses. Nevertheless, there is no direct or obvious relation between the peaks of tensile stress and the porosity from the present study since two physical phenomena are involved to explain the mechanical behavior of the bulks during PFM. This aspect is discussed next.

As previously mentioned, the impact on the stress will highly depend on the location of the pores in the bulk and their relative distances. Since each distribution is random, different behaviors can be observed even within the same porosity. Figure 9 shows the dispersion of the maximum stress for the ten simulations aforementioned and two porosity equal to 0.5% and 2%, respectively (see figure 6). It is recalled that the ten simulations correspond to ten different spatial distributions of pores. All the simulations are encompassed by two envelops for each porosity thereby delimiting a blue area for 2% and a rose area for 0.5%. Overall, the tensile stresses increase from 0.5% to 2% with a slight overlap between $t_1$ and $t_3$. The same pattern is found for all cases, a stress peak is reached at $t_1$ ($\sigma_{1,max}$) and a second peak appears at $t_3$ ($\sigma_{2,max}$). The first peak, as discussed in section 4.1, is attributed to an

abrupt temperature rise in the bulks. The second peak grows in amplitude with increasing porosity indicating that higher pore densities boost the electromagnetic tensile stresses at the flux trapping and flux creep stages. It is worth noting that these time stamps, namely $t_1$ and $t_3$, are used throughout the study to identify typical stages of the magnetization process. Figure 10 shows a comparison of the magnetic flux density, normalized critical current density, temperature, and normalized maximum stress distributions for increasing porosity at $t_1 = 4$ ms. Interestingly, the heat diffuses along thermal pathways having a lightning appearance, remarkably similar to the thermal instabilities in superconducting thin films reported in [44]. These dendritic thermal structures evolve from the edge of the bulk through the closest pores, leading a path towards the center where $J_c$ is minimum due to the local temperature rise, enabling magnetic flux to penetrate farther into the sample. In particular, compressive stress follows these lightning-like thermal pathways, while tensile stress emerges at their boundaries, corresponding to the initial temperature and stress peaks in figure 8. The thermal pathways are characterized by a higher temperature than the surroundings. The local rise in temperature is accompanied by a local thermal expansion of the material at the expense of the ‘cooler’ surroundings. As a result, the material inside the pathways experiences compressive stress as it is constrained by the ‘cool’ regions. At the boundaries between the hot and cool regions, tensile stress develops from differential expansion. Thus, it is highly expected that fractures and cracks, seeding at the edge of the bulk, would easily propagate along these thermal pathways.

Figure 11 shows the same comparison but at the end of the flux trapping phase, $t_3 = 40$ ms. The temperature diffuses along the pathways drawn by the pores across the bulk, still
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Comparison of the distributions of magnetic flux density, normalized critical current density, temperature and normalized maximum stress for increasing porosity at $t_1 = 4$ ms. Lightning-like patterns of temperature diffusion can be seen between the rim and the center of the bulk passing through the pores. With more porosity, the number of pathways or lightning-like structures surges, triggered by local thermal runaways discussed in section 4.1. In turn, it produces tensile stresses and field discontinuities following the same pattern.</figcaption></figure>

impairing $J_c$ and thus the capability to trap the magnetic flux. The tensile stress is concentrated where the magnetic flux is the highest, due to Lorentz forces. Regions with tensile and compressive stress are still visible, arising from localized remanent heat, corresponding to the second temperature and tensile stress peak in figure 8.

To gain a deeper understanding of the localized maximum stress peaks and their relationship to pores, figure 12 presents a magnification of regions of interest (edge and center) at times $t_1$ and $t_3$ for the 2% porosity. At $t_1$, the maximum tensile stress is observed at the boundaries of the thermal pathways, with the stress significantly higher at the edge of the bulk.

In addition, pores within these pathways experience significant tensile stress at the border between hot and cool areas, with stress intensifying if another pore is nearby. At $t_3$, tensile stress shifts toward the center, with much higher stresses at the pore boundaries. This behavior is consistent with the understanding that cracks and fractures typically begin at the tips of defects [4, 5, 19, 20]. These findings suggest that pores play a significant role in fracture during high-field magnetization, either by inducing elevated thermal stresses that propagate from the bulk edge during the ascending phase and follow thermal pathways toward the center or by generating tensile stresses at the center of the bulk during the flux trapping stage.
<figure><img src="image_14.png" /><figcaption>**Figure 11.** Comparison of magnetic flux density, normalized critical current density, temperature and normalized maximum stress distributions for increasing porosity at $t_3 = 40$ ms. As the flux trapping stage ends, the magnetic flux density relocates towards the center of the bulk resulting from a new distribution of current density, producing higher electromagnetic stresses in this region. Increasing porosity impairs the flux trapping capabilities of the HTS bulk.</figcaption></figure>

Both scenarios are particularly critical in regions with a local high density of pores.

substantially reduced, since the silver-filled pores improve thermoelectric homogeneity and reduce the electromagnetic hoop stress concentrations [40]. This highlights the combined thermal and mechanical mitigation effect of silver during magnetization.

### 4.4. Case 4: silver addition

Figures 13 and 14 present the effects of silver-filled pores on the maximum temperature and stresses, respectively. When all pores are filled with silver (Ag), a significant damping effect is observed, leading to a reduction in both temperature rise and stress, with no notable stress peaks from thermal or electromagnetic forces. For the 65% silver-filled case, the maximum temperature remains nearly identical to the non-filled case, resulting in a similar thermal stress peak $\sigma_{1,max}$, as localized thermal avalanches still occur through the unfilled pores. However, the electromagnetic stress peak $\sigma_{2,max}$ is

## 5. Conclusion

A 2D multiphysics FEM model was developed to study the effect of porosity on the thermal, magnetic, and mechanical behavior of REBCO bulks during PFM. Porosity increases local current density, raising temperature and reducing $J_c(B, T)$, which leads to lightning-like thermal pathways through clusters of pores. These structures generate tensile
<figure><img src="image_15.png" /><figcaption>**Figure 12.** Maximum stress distribution magnification for regions of interest (center and edge) at $t_1$ and $t_3$ for 2% porosity. The scale was adapted, now from -0.1 to +0.1, for higher contrast.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 13.** Impact of silver addition on maximum temperature rise. When all pores are filled with silver, the temperature profile matches the ideal case. In contrast, filling only 65% of the pores leads to no significant reduction in the maximum temperature.</figcaption></figure>

stresses that may compromise the bulk's physical integrity during the pulse rise. Higher porosity also induces stronger electromagnetic stresses during the flux trapping stage, especially near the center, emphasizing the importance of pore location and distribution.

Larger pore sizes reduce the trapped field and intensify thermal effects. A 3.236*P* reduction in the center trapped

<figure><img src="image_17.png" /><figcaption>**Figure 14.** Maximum tensile stress evolution for different percentage of silver filling. The presence of silver-filled pores significantly reduces the tensile stress, reducing the likelihood of mechanical failure during PFM.</figcaption></figure>

field was observed with increasing porosity, along with asymmetrical field distributions and current discontinuities around pores. Electromagnetic stress increases linearly with porosity. Filling pores with silver significantly reduces both temperature rise and stress, highlighting the need for mitigation scheme as fillers and mechanical reinforcement during PFM.

The present study is limited to 2D even though some phenomena, such as thermal diffusion, typically occur in 3D for bulks magnetized by PFM and cooled via cryocooler. Therefore, it may lead to an overestimation of the temperature rise. It is a first step towards more complex electromagnetic, thermal and mechanical models. These models with one, two or all the aforementioned physics can then be built on the basis of the present work by extending it to 3D to provide a more realistic representation, particularly when combined with crack propagation models to study fracture. However, the 2D model, as it stands, provides some useful trends on the impact of pores on trapped magnetic field and also points some research direction for additional improvements in the modeling or gathering experimental data. Hence, it already showed good quantitative agreement on the magnitude of the center trapped field when comparing simulation results and experimental PFM data from Namburi *et al* [14]. Using identical parameters, the experiment reports 1.45 T whereas the average value given by the present model is about 1.454 T for 2% *P*, with a relative error of 0.3%. Despite this preliminary success, further modeling and statistical analyses of the effects of general defects beyond just pores are necessary to better understand the limitations they may impose on the practical use of HTS bulks. Undoubtedly, experimental work has yet to be thoroughly performed to generate the relevant statistics on electromagnetic, thermal and mechanical physical properties of HTS bulks. These parameters are necessary to construct the proper models to study their behavior under different operating conditions.
**Data availability statement**

All data that support the findings of this study are included within the article (and any supplementary files).

**Acknowledgments**

F Trillaud would like to thank in part the Dirección General de Asuntos del Personal Académico (DGAPA) of the Universidad Nacional Autónoma de México (UNAM) under Grant DGAPA-PAPIIT 2024 (#IN104124) and in part the Mexican federal fund, Ciencia básica y de frontera 2023-2024 of the Secretaría de Ciencia, Humanidades, Tecnología e Innovación (Secihti) under the Grant CBF2023-2024-2288 for financial support.

**Author contributions**

Santiago Guijosa ID 0000-0002-8064-3684
Data curation (lead), Formal analysis (lead),
Investigation (lead), Methodology (equal), Writing – original
draft (lead)

Kévin Berger ID 0000-0001-8841-917X
Conceptualization (supporting), Formal analysis (equal),
Funding acquisition (equal), Investigation (equal), Project
administration (supporting), Software (lead),
Supervision (equal), Writing – review & editing (equal)

Frederic Trillaud ID 0000-0002-6964-5603
Conceptualization (lead), Formal analysis (equal), Funding
acquisition (equal), Investigation (equal),
Methodology (equal), Supervision (equal), Writing – review
& editing (equal)

Víctor Rolando Jara-González ID 0000-0002-2021-0502
Formal analysis (supporting), Methodology (supporting)

Melika Hinaje ID 0000-0001-5255-2702
Funding acquisition (equal), Project administration (lead),
Supervision (supporting), Writing – review &
editing (supporting)

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