# Microstructural stability of cemented carbides at high temperatures: Modeling the effect on the hot hardness

Victor Lamelas, Manon Bonvalet Rolland, Martin Walbrühl, Annika Borgenstam

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Victor Lamelas, Manon Bonvalet Rolland, Martin Walbrühl, Annika Borgenstam. Microstructural stability of cemented carbides at high temperatures: Modeling the effect on the hot hardness. International Journal of Refractory Metals and Hard Materials, 2024, International Journal of Refractory Metals and Hard Materials, 124, pp.106805. 10.1016/j.ijrmhm.2024.106805. hal-04991564

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# Microstructural stability of cemented carbides at high temperatures:
Modeling the effect on the hot hardness

V. Lamelas<sup>a, *</sup>, M. Bonvalet Rolland<sup>a, b</sup>, M. Walbrühl<sup>c</sup>, A. Borgenstam<sup>a</sup>

## ARTICLE INFO

### Keywords:

ICME
Cemented carbides
Hot hardness
Alternative binders
γ-Carbides
Modeling

## ABSTRACT

There are several semi-empirical models available in literature that correlate the intrinsic hardness of cemented carbides' constitutive phases and certain microstructural parameters, such as mean WC grain size and Co volume fraction, with the hardness of the cemented carbide. Nonetheless, such empirical relations fall short on predicting the behavior of materials other than WC-Co which they were fitted to, limiting their applicability on materials with diverse particle size distributions, alternative binder systems or with additional carbides (γ-carbides). Additionally, current models are limited to the prediction of room temperature hardness. Framed in an Integrated Computational Materials Engineering (ICME) approach, this work proposes several models to be integrated into an already validated semi-empirical approach to describe the hardness of cemented carbides as a function of temperature. First, new microstructural descriptors on the particle and binder size distributions are proposed to enable a better understanding of the influence of polydispersity and of the addition of γ-carbides on the hard-to-soft phase reinforcement. Second, a validated Peierls-Nabarro-based model is used to describe the intrinsic softening of the hard phases with temperature. And finally, the importance of the microstructural changes happening under stress at high temperatures is highlighted and its effect on hot hardness is introduced into the model. These upgrades increase the theoretical and physical base of the modeling tool providing a physical meaning to all the modeling parameters, lowering the need for numerical fitting, making the model more generic and bringing additional information into the micromechanics involved in the softening of cemented carbides.

## 1. Introduction

Cemented carbides are composite materials composed of a metallic matrix embedding a high volume fraction of hard ceramic particles. Their main application are wear-bearing applications, where high hardness and toughness are needed. Both properties are negatively correlated, and materials designed to have higher harnesses tend to have lower toughness and vice versa. In order to optimize the design of new cemented carbides that maximize both properties simultaneously, a good understanding of the correlation between structure and properties is critical. In the current work a better understanding of cemented carbides hardness is targeted.

Numerous studies have previously modeled the effect of different microstructural parameters on the hardness of cemented carbides [1–7]. These models use simple scalar descriptors of microstructures such as

the mean grain size and the volume fraction of the different phases present to help understanding and predicting cemented carbides behavior. Information relating to the spatial distribution [2–5] or the variation of the mean microstructural quantity [1–4,6,7] are commonly not considered. To compensate for not considering any higher-order microstructural descriptors that would help to describe the mechanisms ruling the macroscopic hardness, empirical fitting parameters with limited physical meaning are commonly used [2–7]. The empirical nature of such models limits their universality and applicability to systems other than those they have been calibrated to. The two main limitations that need to be addressed when building a new cemented carbides hardness model are:

a) Currently existing models are restricted in composition. These models have been developed and optimized to describe state of the

<footnote><sup>a</sup> KTH Royal Institute of Technology, Department of Materials Science and Engineering, Brinellvägen 23, SE-100 44 Stockholm, Sweden</footnote>

<footnote><sup>b</sup> Univ. Lille, CNRS, INRAE, Centrale Lille, UMR 8207 - UMET - Unité Matériaux et Transformations, F-59000 Lille, France</footnote>

<footnote><sup>c</sup> AB Sandvik Coromant R&D, Lerkrogsvägen 19, SE-126 80 Stockholm, Sweden</footnote>

<footnote>* Corresponding author.
E-mail address: vlc@kth.se (V. Lamelas).</footnote>
art cemented carbides where only WC and Co are present in the microstructure. This fact makes their applicability to cemented carbides containing alternative binder alloys or additional carbides limited.

b) Current models are restricted in temperature. Existing models describe the hardness of cemented carbides at room temperature (RT). However, cemented carbide tools are subject to high contact pressures and may reach flash temperatures of up to 1200 °C during operation conditions [8]. Under these conditions the hardness decreases, and the extent of this softening can't be directly predicted from the material's hardness at room temperature. For instance, two cemented carbide grades with the same hardness at room temperature but different binder content will experience different softening as temperature increases [9]. Therefore, a model that expands the knowledge on the relationship between microstructural details and hot hardness is needed.

The main objective of the present work is to narrow the mentioned gaps by proposing a model which has a clear physical interpretation and allows a direct connection between micro-mechanisms and mechanical properties. Further, the model aims to serve as a base, generic enough, to host further extensions that consider details that are out of the current scope such as WC interfaces nature (Σ2) or WC grain shapes.

To develop the aimed model, the chosen starting point has been the RT hardness model for Co-based cemented carbides presented by Engqvist [2]. The main reasons behind this choice are, firstly, that the Engqvist model operates only with parameters that are known prior to the design process, fulfilling the requirements of the materials design concept [10]. Secondly, the Engqvist model has a straightforward and general physical interpretation that relies on the fact that the hardness measured for a thin film (independently of the film material) tends towards the hardness of the substrate as its thickness tends to zero. Or conversely, if the film thickness tends to infinite, the measured hardness will approach the one for the bulk film material. The nature of this model makes it a good candidate to be used in the design of new cemented carbides unrestricted in composition if some features that are currently hindering its generality are improved.

The estimated hardness of the cemented carbide in the Engqvist model follows the equation:

$$H_{CC} = (H_{WC} - H_{Co}) \cdot e^{\frac{\lambda}{k}} + H_{Co} = \left( \left( 693 + \frac{2680}{\sqrt{2.1 + d_{WC}}} \right) - 825 \right) \cdot e^{\frac{d_{WC} \cdot V_{Co}}{0.35 \cdot V_{WC}}} + 825 \quad (1)$$

where $H_{WC}$ is the hardness of WC, modeled through a Hall-Petch (H—P) relation; $H_{Co}$ is the intrinsic hardness of the Co-base binder-like alloy; $d_{WC}$ is the WC mean grain size, $V_{Co}$ and $V_{WC}$ are the Co and WC volume fractions respectively; $\lambda$ is the mean free path defined for a microstructure of null contiguity as $\frac{d_{WC} \cdot V_{Co}}{V_{WC}}$ and $k = 0.35$ is a fitting parameter that defines the hardening range.

To increase the applicability of the Engqvist model to a broader range of compositions, two main aspects have to be addressed:

- The treatment given to the intrinsic hardness of the constitutive phases must adopt a more generic approach. Thus, $H_{WC}$ and $H_{Co}$ are now addressed as $H_{Hard-Phase}$ ($H_H$) and $H_{Binder}$ ($H_B$) respectively. Specifically, for the binder hardness treatment to be consistent with the physical basis of the model, any effect related to the microstructural reinforcement needs to be detached from the definition of the intrinsic (bulk) hardness. Thus, for the limit where $V_{Hard-Phase} = 0$ (binder thickness layer $\rightarrow \infty$), the predicted hardness has to correspond to the bulk hardness of the binder-like alloy. Engqvist predicts an intrinsic hardness of Co-based binder like alloys of 825 HV. This value is too high when compared to the intrinsic hardness reported for Co-based alloys in literature, 304 HV [1], 161–325 HV [11], 175 HV [12] or 388–460 HV [13]. The overestimation of the $H_{Co}$ value in the Engqvist model suggests that part of the hardening related to the binder constraint is already accounted for in the definition of the binder intrinsic hardness. This constraint depends on the hard phase size and the binder volume fraction and chemistry, thus including its effect as an invariant $\Delta H$ in the definition of the binder intrinsic hardness leads to modeling inaccuracies [14]. The description of $H_B$ as a bulk hardness, helps to separately study the effect of the different softening mechanisms happening simultaneously, as well as increases generality of the model regarding composition.

- The fitting parameter $k$ has to be defined in such a way that the hardening effect of the hard phase for any type of binder is described. Additionally, the definition of the hardening range factor should not depend on the dataset that was used for its calibration.

To increase the applicability of the Engqvist model to broader temperature ranges; it is essential, first, to separately quantify the contribution of the different mechanisms acting at different length-scales and temperatures to the softening. Up until reaching approximately 70% of the melting temperature of the binder (defined in K), the softening of cemented carbides is caused by the intrinsic softening of its constitutive phases [15]. The intrinsic softening of the binder and the hard phases are characterized by distinctive points at approximately 50% and 30% of their respective melting temperatures [16]. At temperatures higher than ≈70% of the melting temperature of the binder, the mechanical response of the metallic binder is strongly dominated by power-law creep [17,18]. From these temperatures onwards, the main decrease in hardness is governed by the rearrangement of the hard-phase skeleton through grain boundary sliding and the ability of the binder to deform and to accommodate these new microstructural arrangements without forming voids. This rearrangement has a characteristic length-scale related to the microstructure, in contrast to the intrinsic softening which acts in the atomic-scale. By including all the microstructural information in the exponential term of eq. 1 and the intrinsic hardness in the other terms, the intention is to provide a framework that allows to account for high temperature mechanisms such as the grain boundary sliding.

In the methodology section, the changes introduced into the Engqvist model to build the new Hot Hardness model are presented. These changes have the goal of achieving a new model with a broader compositional and thermal application range. The results obtained from applying the new Hot Hardness model for a wide range of cemented carbides within the operational temperature range are presented and discussed in relation to the Engqvist model. Both, methodology and results sections, are structured so that the different mechanisms relevant in the cemented carbides softening at each temperature range are described in the same order as they appear as temperature increases. Fig. 1 shows a schematic summarizing the current approach, where the incremental improvements applied to the model are introduced as temperature increases.

## 2. Methodology

### 2.1. Room temperature hardness – Microstructural relations

#### 2.1.1. WC room temperature hardness

WC is the constituent with highest volume fraction in cemented carbides, thus understanding its contribution to the effective hardness of the cemented carbides is key. The dependency of WC hardness with the carbide grain size is undoubtedly established, and several Hall-Petch formulations have been proposed [2,6,1]. The main difference between these formulations is if the WC hardness is considered to have a finite value when the grain size approaches zero or not. From a physical perspective, the hardness of WC cannot be infinite, and H—P relations such as the one proposed by Lee and Gurland [1] are deemed to overestimate the hardness of very fine grades. The introduction of a finite
<figure><img src="image_4.png" /><figcaption>Fig. 1. Mechanisms accounted in the model as a function of temperature.</figcaption></figure>

value at sizes approaching zero seems reasonable but it may be complex to determine unequivocally.

The H-P relation proposed by Engqvist [2] converges towards a hardness value of 2542 HV when the grain size is zero, and 693 HV for infinite grain size. Engqvist builds this relationship on the fitting of four binderless samples with different mean grain sizes, and approximately 10% volume of other carbides such as Mo₂C or (Ti, Ta, Nb)C. The high-volume fractions of transition metal carbides in addition to WC in these samples make their description of the intrinsic properties of WC ambiguous. For instance, multiple studies on binderless WC in the nanometric regime have consistently reported hardness values above 2542 HV [19–28] [29–35], even at grain sizes around 400 nm [31]. These experimental observations are considered relevant in the definition of a more precise H-P relation.

In this work, a dataset consisting of 68 binderless WC samples, with a maximum porosity of 2% and a maximum content of 1 wt% of other alloying compounds is considered to define a new H-P relation describing the hardness of WC as a function of its grain size.

## 2.1.2. Room temperature hardening parameter k - f(PSD)

### 2.1.2.1. Literature review and trends.
Prior to introducing new microstructural descriptors into the Engqvist model, the back-calculation of the k value leading to a perfect prediction of the hardness and the study of the calculated trends can help in providing this parameter with additional physical meaning. To analyze these trends, a curated dataset consisting of 581 samples with known grain sizes and Co-based binders derived from [36] is chosen (Dataset A from now on). To do so, in

kBack-calc is fitted as:

$$k_{\text{Back-calc}} = \frac{-\lambda}{\ln\left(\frac{H_{\text{CC}}^{\text{Exp}} - H_B}{H_{\text{WC}} - H_B}\right)} \quad (2)$$

where all parameters have the same meaning as in the original Engqvist equation (Eq. 1) but here H_B has been calculated using the model proposed by Walbrühl et al. in [37]. In this model, the hardness of bulk alloys free of any constrainment is estimated by accounting for the intrinsic hardness of each alloy constituent together with the strengthening resulting from having several elements in solid solution.

The resulting data obtained from the back-calculation of k from Dataset A is presented in Fig. 2.

There is a clear correlation between k and the WC grain size when considering Dataset A. Additionally, a less prominent (but still clear) dependency of kBack-calc with the binder volume fraction is observed in Fig. 2.b). The fluctuations observed in both figures are a result of the intrinsic variability of experimental data taken from several sources and a result of k being a function of both the grain size and the binder volume fraction, showing that for the same grain size kBack-calc will be higher the higher the binder content. The original model’s fitting value of 0.35 µm is represented by a red dashed line in both Fig. 2 a) and b), showing that this value consistently underestimates the required value for a correct fitting as grain size and volume fraction increases.

### 2.1.2.2. Physical interpretation and definition of k.
The proposed Hot hardness model aims to enhance the physical interpretation given to the

<figure><img src="image_5.png" /><figcaption>Fig. 2. Effect of a) WC grain size and b) % volume binder on the back-calculated k parameter.</figcaption></figure>
exponential factor governing the hard-to-soft load share, defined as “hardening effect”, in the Engqvist model. Upon reformulation, it becomes clear that $e^{-\frac{\lambda}{k}}$ acts as a rule of mixture coefficient encapsulating microstructural details where $\lambda$ contains information on the grain size and on the binder volume fraction, and $k$ determines how steeply the hardening effect diminishes with increasing binder thickness [2]. For instance, with the original fitting, $k = 0.35 \mu m$, the relative hardening decreases by 0.1% per each additional atomic layer added to the mean free path. To provide a physical interpretation to the $k$ parameter, different hardening mechanisms related to the binder constraint or to the WC skeleton force-transmission can be considered:

a. The trend in Fig. 2 indicates that $k$ is influenced by the WC grain size and that it could be considered to describe the binder constraint. A useful equation that links the constraintment factor of a ductile layer in between rigid plates can be found in reference [38]:

$$ \frac{\sigma_{eff}}{\sigma_0} = \left[ 1 + \frac{2K}{3} \cdot \left( \frac{d}{2h} \right) \right] \quad (3) $$

where $\sigma_{eff}$ and $\sigma_0$ are the effective yield stress and intrinsic yield stress of the constrained ductile material, respectively, $K$ is a constant defined as the maximum shear factor, $h$ is the thickness of the ductile layer and $d$ is the width of the rigid plates.

The constraintment calculated from eq. 3 can be considered to describe approximately the behavior of a binder layer in between two WC grains. In this case, $d$ is equivalent to $d_{WC}$ and $h$ to $\lambda$. Under these assumptions, the constraintment of the binder (or $k$) is proportional to $(1 - V_B)/V_B$ when considering the Engqvist expression for $\lambda$. If one considered the expression for $\lambda$ proposed by other authors [1] then the constraintment would also depend on the contiguity (C) of the hard phase skeleton, being proportional to $[(1 - C)(1 - V_B)/V_B]$. In both cases the constraintment depends on the binder content, but in the second case it also depends on the contiguity between the WC grains, which leads to the following consideration.

b. The hard phase constitutes a fully percolated skeleton where the formed force chains play a critical role in the load-sharing between binder and reinforcement, and hence in the mechanical properties. In this sense, $k$ can be considered to be related to the WC skeleton. When deformed, various structural rearrangements of the load paths occur at different length scales. Locally, the number of neighboring particles varies, and in the meso-scale the force chains arrangement, known as the material's fabric in the granular materials literature [39,40], undergoes significant changes that lead to varying mechanical response [41]. The dynamic behavior of the force chains during a compression or hardness test is a complex study beyond this publication's scope. However, insights on the fabric evolution from granular materials literature [42–45] can be used to qualitatively understand the mechanisms acting in cemented carbides during deformation.

The force transmission along heterogeneous materials is directly influenced by factors related to its fabric like branch length, inter-particle distance and branch interconnectivity. These factors are notably affected by microstructural parameters including the particle size distribution and the binder content. For instance, in materials with polydisperse reinforcing particles, such as cemented carbides, larger grains exhibit higher coordination numbers [46]. So, despite that coarse grains being intrinsically softer, they act as nodes in the force chains that are more stable against rearrangement and that are subjected to higher loads [43,44,47]. In this context, $k$ can be considered as a parameter that is a function of the microstructure that rules the resistance to force chain rearrangement, or that determines the resistance of force chains to buckle [48].

In summary, $e^{-\frac{\lambda}{k}}$ is the factor accounting for the cumulative effect of both discussed hardening mechanisms. Binder constraintment and force transmission in the hard-phase force chains occur simultaneously, with synergetic effects on the load share between phases. Thus, on one side, the effect of constraintment is associated to a characteristic lengths-scale acting on the binder, the mean free path, where the constraintment increases with decreasing mean free path. On the other side, the force transmission in the percolated hard-phase is associated to a characteristic length-scale acting on the hard-phase skeleton. This length-scale characterizes the interaction of particles participating in a force chain. In this sense, the mean branch vector length between hard particles emerges as a significant distance ruling the force transmission characteristics in the skeleton [49,50]. To model this distance effectively, it has been considered relevant to approximate $k$ as the mean nearest-neighbor distance between particles.

To partially address the complex micro-scale mechanisms while keeping the applicability of the model simple, several simplifications are needed. The exponential term in the Engqvist model ($e^{-\lambda/k}$) is reformulated to account for the particle size distributions (PSD) of hard phases in cemented carbides as:

$$ e^{-\frac{\lambda}{k}} \rightarrow e^{-\frac{l_c(PSD)}{l_p(PSD)}} = e^{-\frac{\frac{4}{3} \cdot V_{Bind} \cdot \sqrt[3]{-\ln(V_{Bind})} \cdot \exp(2\sigma_L^2)}{r\left(\frac{4}{3}\right)}} \quad (4) $$

where $l_c(PSD)$ is the mean chord-length defined as the mean length of the segments laying in the binder bounded by the intersections of an infinitely long line with a binder-hard phase interface [51], $l_p(PSD)$ is mean nearest-neighbor distance between particles and $\sigma_L$ is the standard deviation characterizing the scatter of a lognormal distribution.

The given reformulation starts from the assumption that the PSDs of the hard phases can be described through a lognormal distribution following the expression:

$$ f(d_H) = \frac{1}{d_H \sqrt{2\pi}\sigma_L^2} \exp\left(-\frac{\ln\left(\frac{d_H}{\langle d_H \rangle}\right)^2}{2\sigma_L^2}\right) \quad (5) $$

where $d_H$ is the hard-phase diameter, $\langle d_H \rangle$ is a parameter denoting central tendency measure (chosen here to be the mean grain size of the hard-phase) and $\sigma_L^2$ is the variance of $\ln(d_H)$.

For the redefinition of the characteristic length-scale associated to the binder, the mean free path in the Engqvist model is instead defined through an analogous microstructural parameter that accounts for the PSD of the hard phase. This parameter is $l_c$, and it is a function of the specific surface for an arrangement of particles in 3D, calculated as:

$$ l_c = \frac{4V_{Bind}}{s} = \frac{4^* \langle v_H \rangle^* V_{Bind}}{\langle s_H \rangle} \quad (6) $$

where $s$ corresponds to the ratio between surface area of hard-phase particles ($\langle s_H \rangle$) and their volume ($\langle v_H \rangle$). Therefore, assuming a 3D arrangement of spherical particles with a lognormal size distribution, these quantities can be calculated as:

$$ \langle s_H(R) \rangle = 4\pi\langle R^2 \rangle \quad (7) $$

and

$$ \langle v_H(R) \rangle = \frac{4}{3} \pi \langle R^3 \rangle \quad (8) $$

where $\langle R^2 \rangle$ and $\langle R^3 \rangle$ are the n$^{th}$ moments (second and third respectively) of the lognormal distribution given by:

$$ \langle R^n \rangle = \exp\left(\frac{n^2 \sigma^2}{2}\right) \langle R \rangle^n \quad (9) $$
For the redefinition of the characteristic length-scale associated to the hard-phase skeleton, the fitting parameter k is instead defined as $l_p$. This definition then contains information on the microstructural inter-connectivity and its value is proportional to the stability of the force chains. The calculated value of $l_p$ is related to the probability of finding the nearest-neighbor to a reference particle at a distance $r$ ($H(r)$), which can be defined for spatially uncorrelated spheres as:

$$H(r) = \rho s_H(r) \exp(-\rho v_H(r)) \quad (10)$$

where $\rho$ is the number density of particles.

$l_p$ is the first spatial moment of $H(r)$, which in the case of particles that can overlap arbitrarily and following a Poisson distribution arrangement in 3D, can be obtained by:

$$l_p = \int_0^\infty r \cdot H(r) \cdot dr = - \frac{\langle d_H \rangle \Gamma(\frac{4}{3})}{2 \cdot \ln(V_{Bind})^{\frac{1}{3}}} \quad (11)$$

where $\Gamma$ is the gamma function.

While the assumption of considering spatially uncorrelated spheres (fully penetrable particles) to determine $l_p$ may appear imprecise, this choice is consistent with the assumptions considered for the calculation of the binder characteristic length scale, $l_c$. Under the current assumptions, the obtained mean-chord length considers an even distribution of the binder around all the particles, assuming null contiguity and thus represents the lower bound for this characteristic length. This is easily seen when considering stereological variations for the calculation of the mean free path such as the one used by Lee & Gurland [1]. These authors reach higher accuracies on modeling the mean free path of experimental samples when dividing Engqvist expression for $\lambda$ by a factor of $(1 - C)$ [52]. Therefore, for real microstructures with $C > 0$ the actual mean free path will always be greater than $\lambda$. Thus, in the current modeling approach, where the binder-related scale is described by its lower bound, it is convenient to describe the hard phase-related scale using also its lower bound. This is achieved by determining $l_p$ considering fully penetrable spheres where the effect of contiguity is approximated by the overlapping of the particles [53] and the minimum interparticle distance is not limited to two particle radii. In addition to the need of congruency between microstructural descriptors, the current description of $l_p$ “dodges” the complexity of defining the non-trivial percolation properties of the microstructure, inherent to materials with high fractions of spatially correlated particles.

In conclusion, the exponential term in the Engqvist model ($e^{-\lambda/k}$) has been reformulated to incorporate the effect of the particle size distribution, and characteristic length-scale of the hard-phase skeleton on the load distribution between hard phase and binder. The new expression for the exponential term (Eq. 4), analogously to the expression for binder constraint presented in eq. 3, does not depend on the mean grain size.

## 2.2. Mid temperature range hardness – Intrinsic softening of constituents

Part of the softening experienced by cemented carbides at high temperatures is attributed to the intrinsic softening of its constitutive phases. In the current model, the softening of metallic phases (binder) and of ceramic phases (hard- phase) are modeled simultaneously using two different strategies.

For the modeling of the binder hardness as a function of temperature, the model presented in [37] is selected due to three main advantages. Firstly, the model determines the intrinsic hardness of multicomponent alloys independently of composition, making it especially suitable when modeling cemented carbides with alternative binders. Secondly, the model accounts for the strengthening effect resulting from having alloying elements in solid solution, making it especially suitable for cemented carbides binders where dissolution of W is not negligible. And

finally, this model includes the temperature related softening of multi-component metallic alloys. However, it is important to highlight that other hardening mechanisms such as phase transformations (e.g.: martensitic binders), work hardening or precipitation hardening (e.g.: intermetallic precipitates such as Ni₃Al, FeAl) are not considered in the current framework.

On the other hand, the intrinsic softening of the hard-phases is described through a Peierls-Nabarro (P—N) based model presented in [54]. In the published work only the intrinsic softening of FCC γ-carbides is discussed, thus an adaptation of the P—N based model to WC will be summarized and discussed.

The primary difference in applying the P—N based model to WC, in comparison to γ-carbides, lies in its HCP crystal structure. Unlike FCC, HCP lacks slip planes fulfilling Von Mises criteria, which requires a minimum of 5 independent slip systems for compatible deformation. Therefore, the modeling of WC deformation requires the consideration of additional slip planes. Experimental observations indicate that multiple slip systems (up to 15) are involved in the defamation of WC [55], and that their contributions vary with temperature. In the present work, only the four slip systems contributing the most to plastic deformation in the prismatic planes and two additional slip systems in the basal planes have been considered due to computational constraints. The considered slip systems are summarized in Table 1.

To calculate the critically resolved shear stress (CRSS), $\tau_c^T$, the adapted P—N model uses the generalized stacking fault energy (GSFE) of every given slip system as input data. Further, the dislocation core properties are determined using the PNADIS [56] open software. Estimate of additional parameters such as elastic quantities, Debye frequency and dislocation density are obtained from literature and summarized in Table 2. These sources set the parametrization needed to calculate $\tau_c^T$ on each slip system as:

$$\frac{\tau_c^T}{G} = C_1 \left[ C_2 \cdot \ln \left( C_3 \frac{G}{\tau_c^T} \right) - \frac{k_B T}{a^3 G} \ln \left( \frac{\rho_m b \lambda_b \nu_D}{\dot{\epsilon}} \right) \right]^2 \quad (12)$$

where $\tau_c^T$ is the critical shear stress for dislocation slip at a given temperature, G is the elastic shear modulus, $k_B$ corresponds to the Boltzmann constant, T is the temperature, b is the burgers vector, $\nu_D$ is the Debye frequency, $\rho_m$ is the dislocation density, $\lambda_b$ is the mean free path of dislocations slipping over obstacles and $\dot{\epsilon}$ is the strain rate. $C_1$, $C_2$ and $C_3$ are constants dependent on the slip system and dislocation character.

Once $\tau_c^T$ of each slip system is calculated, the hardness is approximated assuming that slip happens in those slip systems with the most favorable geometrical orientation (maximum Schmid factor) and using a Tabor relation modified for ceramic materials. The reader is referred to [54] for more specific details on the followed methodology.

All the input needed for WC GSFE was generated using the framework of Density-Functional Perturbation Theory (DFPT) as implemented in the Vienna ab initio simulation package (VASP). The exchange-correlation functional was described using Perdew, Burke and Ernzerhof (PBE) parametrization. The P6-m2 unit cell of WC was first relaxed and equilibrium lattice constants of 2.914 Å and 2.866 Å were obtained for a and c respectively. For the calculation of the GSFE at 0 K, supercells oriented adequately for each of the slip systems (slipped plane facing the z axis) were slipped along the fault plane that maximized the distance between the C and W atoms (leading to a more probable slip [60]). At least 10 displacements were considered for each slip, and the atomic positions were relaxed along the Z-axis, minimizing the forces to <0.03 eV/Å on each atom. The criterion for total internal energy convergence was set to $10^{-5}$ eV, and a cutoff energy of 500 eV was used. A vacuum width of 20 Å was added to periodical slabs for all configurations. The GSFE supercell K-points resolution and number of atoms per super cell are summarized in Table 3.
<table><caption>**Table 1**
Crystallographic and dislocation character defining parameters.</caption><thead><tr><th>Family</th><th>Plane</th><th>Slip Direction</th><th>S/E (Beta)</th><th>b</th><th>h</th><th>r<sub>Cut-off</sub> (a<sub>0</sub>0K)</th><th>MAX Schmid</th></tr></thead><tbody><tr><td rowspan="6">Basal</td><td rowspan="6">{0001}</td><td>a:<br>1/3 &lang;2110&rang;<br>=</td><td>S (0°)</td><td rowspan="3">a</td><td>√3a/2</td><td>1.420</td><td rowspan="3">0.5</td></tr><tr><td>a + b:<br>1/3 &lang;1120&rang;</td><td>E (90°)</td><td>a/2</td><td>1.576</td></tr><tr><td>a-b:<br>1/3 &lang;1100&rang;</td><td>S (0°)<br>E (90°)</td><td>√3a</td><td>0.713<br>0.803</td><td>0.36</td></tr><tr><td>a-b:<br>1/3 &lang;1100&rang;</td><td>Mix (30°)<br>Mix (60°)</td><td>√3a/3</td><td>0.937<br>0.985</td><td>0.49</td></tr><tr><td>a + c:<br>1/3 &lang;1213&rang;</td><td>S (0°)<br>E (90°)</td><td>√a<sup>2</sup> + c<sup>2</sup></td><td>0.701<br>1.047</td><td>0.49</td></tr><tr><td>a + c:<br>1/6 &lang;1213&rang;</td><td>S (0°)<br>E (90°)</td><td>√a<sup>2</sup> + c<sup>2</sup>/2</td><td>0.394<br>0.594</td><td>0.49</td></tr><tr><td rowspan="4">Prismatic</td><td rowspan="4">{1010}</td><td>c:<br>1/3 &lang;2110&rang;</td><td>S (0°)<br>E (90°)</td><td>c</td><td>a/2√2</td><td>1.104<br>1.648</td><td rowspan="2">0.5</td></tr><tr><td>a:<br>1/3 &lang;2110&rang;</td><td>S (0°)<br>E (90°)</td><td>c</td><td>a/2√2</td><td>0.701<br>1.045</td><td rowspan="2">0.43</td></tr><tr><td>c:<br>1/3 &lang;2110&rang;</td><td>S (0°)<br>E (90°)</td><td>a</td><td>a/√3a</td><td>1.028<br>1.538</td><td>0.5</td></tr><tr><td>c:<br>1/3 &lang;2110&rang;</td><td>S (0°)<br>E (90°)</td><td>c</td><td>a/√3a</td><td>1.028<br>1.538</td><td>0.5</td></tr></tbody></table>

<table><caption>**Table 2**
Elastic constants and Debye frequency as a function of temperature and average dislocation density as obtained in Refs. [55], [57]–[58].</caption><thead><tr><th></th><th>Ref.</th></tr></thead><tbody><tr><td>Shear Modulus - G(T)</td><td>293.44 – 4.075 • 10<sup>-6</sup> • T<sup>2</sup></td></tr><tr><td>Bulk Modulus - B(T)</td><td>400.52 – 9.545 • 10<sup>-6</sup> • T<sup>2</sup></td></tr><tr><td>Debye frequency - ν<sub>D</sub>(T)</td><td>(651.5 – 4.059 • 10<sup>-6</sup> • T<sup>2</sup>) • k<sub>b</sub></td></tr><tr><td>Dislocation Density - ρ</td><td>2.5 • 10<sup>14</sup> – 6.7 • 10<sup>16</sup></td></tr></tbody></table>

<table><caption>**Table 3**
Computation details used in the calculation of GSFE of WC using VASP.</caption><thead><tr><th>Family</th><th>Slip Plane</th><th># atoms</th><th>K-mesh</th></tr></thead><tbody><tr><td>Basal</td><td>{0001}</td><td>12</td><td>M-P 12x12x2</td></tr><tr><td rowspan="2">Prismatic</td><td>{1010}</td><td>24</td><td>M-P 12x12x1</td></tr><tr><td>{1120}</td><td>24</td><td>M-P 6x12x2</td></tr></tbody></table>

## 2.3. High temperature range – Microstructural stability

### 2.3.1. High temperature hardening parameter k - f(T, PSD, x<sub>Bind</sub>)

#### 2.3.1.1. Literature review and trends.
Similar to the treatment followed for room temperature, and assuming that the descriptions of the model for the softening of the binder and the hard phases are predicting the proper trends and order of magnitudes, it is possible to back-calculate the value of k that would lead to a perfect prediction of the hardness at high temperature. In this case, the expression for k can be written as:

$$k_{\text{Back-calc}}(T) = \frac{-\lambda}{\ln\left(\frac{H_{\text{CC}}^{\text{Exp}}(T)-H_{\text{B}}(T)}{H_{\text{H}}^{\text{N}}(T)-H_{\text{B}}(T)}\right)} \quad (13)$$

where $H_{\text{CC}}^{\text{Exp}}(T)$ is the experimentally measured hardness, $H_{\text{B}}(T)$ is the hardness of the binder modeled through the model in reference [37] and $H_{\text{H}}^{\text{N}}(T)$ is the hardness of the hard phase modeled through the P–N

based model.

To analyze trends in $k_{\text{Back-calc}}(T)$, a dataset consisting of 44 Co-based and 3 (60Co/20Ni/20Fe)-based samples with varying grain sizes and binder fractions has been considered from references [13,61–66] (Dataset C from now on). The considered temperature range is between 25 and 900 °C. To isolate the impact of temperature, a relative value is defined as:

$$k_{\text{Relative}} = \frac{k_{\text{Back-calc}}(T)}{k_{\text{Back-calc}}(25 C)} \quad (14)$$

The trends resulting from the analysis of the selected dataset are shown in Fig. 3.

It is first important to acknowledge the significant data scatter in Fig. 3, typical of the combination of data from various experimental studies. Nevertheless, some key trends can still be identified and summarized as:

- Finer grain sizes maintain higher hardness levels at intermediate temperatures but exhibit a sharper drop in hardness when a certain critical temperature is reached.

- Grades with lower binder contents effectively maintain higher hardness, both relative and absolute, as temperature increases.

Thus, microstructure plays a significant role in the softening properties of cemented carbides.

#### 2.3.1.2. Microstructural changes high temperatures – effect on k(T, PSD, x<sub>Bind</sub>).
When cemented carbides are deformed most of the strain takes place in the binder phase, and the macroscopic deformation must be accommodated by the compatible movement of the hard grains [67]. The extent of this rearrangement depends on the capacity of the binder to accommodate strain, which increases significantly with increasing temperature. This leads to phenomena such as dilatancy [68], that results in the formation of inner cavities due to incompatibilities in the WC grains translations [69,70]; or the appearance of binder lamellae [71,72,73] caused by the infiltration of binder in WC-WC contacts, reducing contiguity [74,75,59]. According to the high temperature
<figure><img src="image_6.png" /><figcaption>Fig. 3. Relative change of back-calculated k with temperature for a) different grain sizes b) different binder contents.</figcaption></figure>

mechanisms, and with the current interpretation of k being the characteristic length scale acting on the hard-phase force chains, which is approximated as the nearest neighbor distance; its value is expected to decrease with temperature. This decrease is physically interpreted as the force chains being cut by binder infiltration and void formation caused by high degrees of rearrangement.

Given the microstructural changes during deformation, it is clear that the resistance of the binder and the WC-Binder and WC-WC interfaces are critical in the macroscopic response of the material. The explicit introduction of such factors into the current modeling scheme is complex, but their impact on the hardness can anyway be considered with parameters related to the microstructure.

An adequate set of mathematical expressions that serve to describe interface mediated processes are sigmoidal functions. These expressions have successfully been used to model phenomena where percolation is key [76]. For instance, this type of functions can be found in the modeling of powders compression [77] or in the prediction of the thermal conductivity of cemented carbides (where the role of interfaces is key) [61]. Consequently, the following expression is proposed to characterize the change in cemented carbides microstructures under compression (hardness tests) as a function of temperature:

$$l_p(T, \text{PSD}_H, x_{\text{Bind}}) = l_p^T + \frac{l_p^{RT}(\text{PSD}_H, V_B) - l_p^T}{1 + e^{q*(T - T_{\text{III}}(d_H, V_B, x_B))}} \quad (15)$$

where $l_p^T$ and $l_p^{RT}$ (PSD$_H$, V$_B$) (defined in section 2.1.2) are the characteristic length scale acting on the hard-phase force chains at melting and room temperature respectively, $q(d_H)$ is the parameter governing the temperature dependence of $l_p$ and $T_{III}(d_H, V_{Bind}, x_{Bind})$ is the characteristic temperature above which severe creep of the binder starts.

At $T_m$, the binder melts, and it is reasonable to assume that the liquid covers all the hard-phase grains, causing the microstructure to lose its texture, and no effective force-chains are present ($l_p^T = 0$). The definition of $q$ and $T_{III}$ shares some analogy with the theoretical framework developed for the modeling of glass transition in glass physics. Indeed, despite the deformation of cemented carbides and the rheology of glasses may seem unrelated, several authors [78,79] have been able to find some sort of universality in jammed solid systems, that has allowed the findings developed on a certain system to be used across a broad length-scale span. Thus, using this relationship to our advantage, concepts used in glass physics such as fragility [80] or glass transition temperature can be used with reasonable analogy in the current treatment of $l_p(T)$. Under this approach, $q$ may be regarded as the fragility, which is a material property that sets how steeply the viscosity ($\eta$) of a material changes as it approaches the glass transition temperature, and where $T_{III}$ may be regarded as the glass transition temperature formally

defined as the temperature at which the viscosity of the system is $10^{13} Pa \cdot s$. During deformation of cemented carbides, above this viscosity threshold the material behaves as a nearly incompressible continuum material (moving from a Tresca to Von Mises behavior as temperature increases). On the other side, below this viscosity threshold, the material behaves as a dilatant material, similarly to a highly cohesive granular material [81].

Therefore, in this framework, the determination of an effective viscosity of the cemented carbide as a function of temperature becomes critical. This viscosity is approximated as the viscosity of a metal influenced by the presence of discrete particles. Raj and Ashby [82] studied the rheology of materials experiencing grain boundary sliding with presence of discrete particles and proposed the grain boundary viscosity to be:

$$\eta_{GB} = \frac{k_B T d^4}{8 b^3 L^2 D_{GB}} \quad (16)$$

where d is the particle size, b is the burgers vector of the slip system ruling the plasticity of the binder, L is the basic periodicity of the boundaries and $D_{GB}$ is the grain boundaries diffusion.

Their proposed theory can be adapted to cemented carbides by considering that the binder covers all the hard grains and that the units sliding against each other correspond to volume elements with diameter being $d_{WC} + l_c$, establishing the boundary periodicity. Considering this arrangement, the boundaries involved in the sliding are not the binder grain boundaries, but the bulk binder enclosed between two grains. Thus, the diffusion quantity ruling the rheology/creep of such configuration is the chemical diffusion coefficient of the major element in the binder like alloy ($D(x_{Binder})$). Here it is important to note that one binder grain can cover hundreds of hard grains, and that the impact that these boundaries may have on the creep properties in comparison to the high number of binder interlayers present in the microstructure is expected to be small. With these adaptations, the viscosity of the cemented carbide can be approximated as:

$$\eta_{H-B-H} = \frac{k_B T d^2 (1 - V_B)^2}{8 b^3 D(x_{\text{Binder}})} = \frac{k_B T d^2 (1 - V_B)^2}{8 b^3 D_0 \cdot e^{-Q/RT}} \quad (17)$$

This, viscosity can be used to find the temperature ($T_{III}$) at which the cemented carbide will experience a change in its rheological behavior, set at a value of $10^{13} Pa \cdot s$. Note that the transition temperature will be greatly determined by diffusion in the binder, and additionally by the microstructure of the cemented carbide, being higher for coarser and higher fraction of hard phase.

After the determination of $T_{III}$, the microstructure fragility ($q$) can be
fitted to the available high temperature experimental data (Dataset C in this work).

## 2.4. Integration of all modeling elements into a single framework

The previous sections have detailed various modifications and extensions to the Engqvist model. It is important to understand how these adjustments collectively form a comprehensive phenomenological equation. The integration of all different modeling parts leads to the following equation:

$$H_{CC}(T) = \left( \left( \frac{V_{WC}}{V_H} H_{WC}(d_{WC}) \cdot f_{WC}(T) + \frac{V_{\gamma}}{V_H} H_{\gamma}(d_{\gamma}) \cdot f_{\gamma}(T) \right) - H_B(T) \right) \cdot e^{-\frac{l_c(PSD_H, V_B)}{l_p(T, PSD_H, V_B, x_B)}} + H_B(T) \quad (18)$$

where $V_{WC}$ and $V_{\gamma}$ are the volume fractions of WC and γ-carbides respectively, and $V_H$ is the sum of both. $H_{WC}(d_{WC})$ and $H_{\gamma}(d_{\gamma})$, correspond to the H—P relations determined at room temperature for WC and γ-carbides respectively. $f_{WC}(T)$ and $f_{\gamma}(T)$ are the softening functions determined through Peierls-Nabarro methodology for each hard phase, set to be 1 at room temperature and decreasing with temperature. $H_B(T)$ is the intrinsic hardness of the binder as a function of temperature determined following methodology described in [37]. The exponential term is defined following the methodology described in sections 2.1.2.2 and 2.3.1.2.

In eq. 18, the effect of the addition of γ-carbides on the hardness of cemented carbides is considered in two ways:

1. The addition of γ-carbides affects the intrinsic hardness of the constituents in the composite material proportionally to their volume fraction. Thus, their effect on the composite hardness is accounted for through a rule of mixtures. This approximation has been used

previously in cemented carbides with third phases with good results [83–85].

2. Additional γ-carbides usually have a very different PSD from the major WC. The presence of a population of hard particles with distinct size and shape, clearly modifies the force chains formed in the hard-phase skeleton. This is considered in the calculation of $l_c$ and $l_p$, which rule the load distribution between the binder and the hard-phase skeleton.

# 3. Results and discussion

## 3.1. Room temperature hardness

### 3.1.1. WC hardness

Fig. 4 shows the experimental data used to establish a new H—P relationship for the intrinsic hardness of WC (Eq. 19). The H—P relations proposed by Lee & Gurland [1] and by Engqvist [2] are also shown as reference. The obtained H—P fitting used in the current model follows the expression:

$$H_{WC} = 545 + \frac{2391}{\sqrt{1.03 + d_{WC}}} \quad (19)$$

Eq. 19 complies with fine grades having a finite hardness and predicts a maximum hardness of 2901 HV as the grain size approaches zero. Conversely, on the lower bound the proposed H—P relation assigns a value of 545 HV for infinitely large WC grains. This lower limit serves as a theoretical cap, as there will be a practical limit in terms of minimum HV and maximum WC grain size that can be achieved in real materials. In this sense, the maximum grain size used in the current fitting is 5 µm, which is big enough to cover the relevant application range. Applications involving bigger grain sizes may require additional data to fine tune the HV values obtained from the current H—P fitting.

The current fitting uses a larger dataset of binderless WC materials and is more restrictive than Engqvist’s in accepting the presence of

<figure><img src="image_7.png" /><figcaption>Fig. 4. Hardness of binderless WC materials as a function of grain size from references [34–50] and prediction of WC hardness for different H—P relations proposed in literature [2,1].</figcaption></figure>
carbides other than WC in the tested materials. Thus, the current fitting is expected to be a better representative of the dependency of WC hardness as a function of its grain size. Nonetheless, the H—P relation used in the current work still depends strongly on the available experimental data, and further experiments would enhance its reliability. In case of getting additional experimental data for binderless WC, especially on the coarse end, the refined fitting could directly substitute Eq. 19 in the bigger framework (Equation18). In line with this, the current modeling approach allows for the direct substitution or of H—P relations describing the hardness of carbides other than WC into Eq. 18. The data in literature reporting a methodical study on the dependency of hardness as a function of the grain size for binderless γ-carbides materials is scarce. Nevertheless, a rough estimation for some of the H—P relations that were found in literature are reported in Table 4.

### 3.1.2. Effect of PSD and skeleton length-scale on hardness

Before evaluating how the new modeling features perform in predicting the hardness, it is relevant to check if the considered assumptions are reasonable. In section 2.1.2, *k* has been defined as a characteristic length relative to the hard-phase skeleton, and has been approximated to be the nearest-neighbor distance between grains, calculated through Eq. 11. The validation of this approximation can be performed through the measurement of the nearest neighbor distance in real microstructures, and its comparison with the calculated *l*<sub>p</sub> and with the value of *k* needed for a perfect fitting. The nearest neighbor distance is measured through the image analysis of skeletonized microstructures in ImageJ software [96] and their further analysis by means of AnalyzeSkeleton plugin [97]. The nearest neighbor distance is measured as the mean branch length multiplied by the ratio of junctions and branches present in the skeleton [51]. This treatment takes into consideration side-side as well as vertex-side contacts between the hard particles, describing in a suitable way the locations where transmission of force in the force chains takes place.

Fig. 5.a shows the microstructures for a set of samples with varying grain sizes and constant binder volume fractions and the overlapped hard-phase skeletons obtained through the described methodology. The centrality and dispersion values describing the PSD of the hard phase are also included for reference. In Fig. 5.b the measured branch length and calculated *l*<sub>p</sub> are compared to the *k* value providing a perfect fitting of the hardness. The rather good agreement between the measured and calculated *l*<sub>p</sub> confirms that Eq. 11 is a good approximation for the nearest neighbor distance of particles in cemented carbides and the close agreement to *k*<sub>Back-calc</sub> indicates that the calculated *l*<sub>p</sub> is a good parameter to be used in the modeling of the hardness of cemented carbides.

To assess the effectiveness of the proposed changes in the modeling of cemented carbides room temperature hardness, the model predictions have been evaluated against two different datasets, each serving different purposes. Dataset B (N<sub>B</sub> = 230), which includes information on the samples PSD, was used for calibration. This dataset not only validated the model's capability to incorporate PSD information in the prediction of hardness, but also helped to identify the common trends in WC polydispersity to later be used in bigger datasets. The use of the trends obtained from dataset B for WC polydispersity has enabled the validation of dataset A.2 (N<sub>A.2</sub> = 823) where information on the PSD is not available. This dataset is the same as Dataset A, presented in section

<caption>**Table 4**</caption>

Hall-Petch relations proposed for carbides commonly present in cemented carbides.

<table><thead><tr><th>Phase</th><th>Experimental Hall – Petch</th><th>References used</th></tr></thead><tbody><tr><td>TiC</td><td>2207 + <sup>678</sup>&frasl;<sub>&radic;0.35 + d<sub>Nc</sub></sub></td><td>[86–88]</td></tr><tr><td>NbC</td><td>915 + <sup>1742</sup>&frasl;<sub>&radic;0.15 + d<sub>NbC</sub></sub></td><td>[89,90–92]</td></tr><tr><td>TaC</td><td>1365 + <sup>366</sup>&frasl;<sub>&radic;0.15 + d<sub>TaC</sub></sub></td><td>[93–95]</td></tr></tbody></table>

#### 2.1.1, but it also includes samples alternative binders.

The impact of WC polydispersity on the hardness of cemented carbides is included in the model in eq. 4, using the standard deviation of a lognormal PSD σ<sub>L</sub>. All samples in dataset B report either the WC PSD or the centrality and dispersion values characterizing the WC PSD. This information can be used directly in eq. 18 to predict hardness. Fig. 6 shows the room temperature hardness calculated for samples in dataset B using the Engqvist and the current model, Fig. 6.a and Fig. 6.b respectively. The results for the current approach replicate the experimental trends and show that the use of PSD information, i.e. σ<sub>L</sub>, in the modeling of the room temperature hardness of cemented carbides with Co-based and alternative binders clearly improves the accuracy of the predictions. The reporting of PSD information is more common in fine grained materials with high hardnesses. Therefore, the results shown in Fig. 6 highlight a significant improvement of the current model with respect to the Engqvist model, which can be seen for samples with experimental hardness approaching 2200 HV. The new H—P relationship proposed for the intrinsic WC hardness (eq. 19) extends the upper limit of the hardness modeling range from 2542 HV to 2901 HV. Considering that values up to 2720 HV [5] and 3189 HV [6] have been reported in literature for conventional and spark plasma sintered samples respectively, this extension is considered beneficial for the modeling of a wider range of cemented carbides, especially for small-grained materials with low binder fractions.

The successful introduction of σ<sub>L</sub> as a parameter to calculate cemented carbides hardness allows to explore the effect of polydispersity on the contribution of the hard phase to the effective composite hardness. Fig. 7 shows the value of eq. 4 calculated for different constant PSD standard deviations together with the back-calculated values of the exponential term for samples dataset A. The results show that the contribution of the hard phase to the composite hardness decreases as WC grain size distribution broadens, which is consistent with the trends reported in [98]. In addition, it can also be seen how the influence of size polydispersity in hardness is higher the higher the binder volume fraction. Further, Fig. 7 also shows that most of the points in Dataset A can be described using a σ<sub>L</sub> in the range [0.1–0.7].

An initial examination of size polydispersity trends in dataset B allows one to estimate the range of σ<sub>L</sub> values commonly found in cemented carbides. Fig. 8 shows the standard deviation of particle sizes as a function of the mean size of samples in dataset B. Most of the reported σ<sub>L</sub> values fall within the range of 0.2–0.67, which is consistent with ranges documented in literature for polycrystalline materials [99] and granular materials [100]. The matching of the range of commonly observed σ<sub>L</sub> values with the σ<sub>L</sub> range required to model dataset A (Fig. 7) shows the potential of the model to be applied to such big a dataset. However, the hardness prediction of materials with missing PSD information still requires the use of fittings. These fittings may provide a value of σ<sub>L</sub> as a function of another known microstructural parameter, i.e. mean size, or at least may narrow down the possible combinations of mean size and dispersion values to those close to real materials.

Fig. 8.a reports the dispersion of the grain sizes (σ<sub>N</sub>), while Fig. 8.b reports the dispersion of the logarithm of the grain size (σ<sub>L</sub>), measured in dataset B. When considering a normal scale, the standard deviation of samples increases with the mean grain size approximately linearly. Consequently, the log-normal standard deviations follow a logarithmic trend. The parameter used in eq. 4 is σ<sub>L</sub>, in dataset B this parameter has a mean value of 0.43, and its relationship with mean grain size can be described by the logarithmic fitting:

$$ \sigma_L = 0.0393 \cdot \ln(d_{WC}) + 0.4367 \quad (20) $$

The substantial data scatter in Fig. 8.b somewhat diminishes the significance of the logarithmic trend fitted in eq. 20, but still enables the validation of the model by estimating the hardness of larger sample groups with unknown PSD, such as Dataset A.2. Fig. 9 compares the estimated hardness of Dataset A.2 samples using the Engqvist model (Fig. 9.a) and the Hot Hardness model (Fig. 9.b) with experimentally
<figure><img src="image_8.png" /><figcaption>Fig. 5. a) Skeletonized microstructures of cemented carbides with constant binder volume and different grain sizes. b) Relation between $k_{Back-Calc}$, $l_p$ and measured mean branch length.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Calculated hardness for calibration dataset B samples compared to experimental measurements. a) Engqvist model b) Current model.</figcaption></figure>

measured hardness. The differences are significant, and they reveal that the usage in the Engqvist of a constant fitting parameter *k* independent of the grain size clearly shows a negative effect on the predicted hardness trends for broad ranges of microstructures. The Engqvist model overestimates the hardness of grades with hardness in range [1550–2000] HV and underestimates low hardness grades. In the considered dataset, multiple samples have hardness values below the lower limit set by Engqvist (825 HV), thus leading to clear inaccuracies. The proposed change in the definition of the intrinsic binder hardness, from a constant value of 825 HV to the intrinsic hardness of a binder like alloy free of any constrainment improves and extends the applicability of room temperature hardness modeling, particularly for coarse-grained

materials with high binder volume fractions. Additionally, this change also enables the modeling of the hardness of cemented carbides with alternative binder systems. In Fig. 9.a, the datapoints corresponding to alternative binder systems mostly underestimate the experimental hardness. This is expected, as the Engqvist model is not originally meant to work with the binder's intrinsic properties. However, it is challenging to establish a hardness that accounts for both intrinsic hardness and microstructural constraints for alternative binders, analogous to the approach used for Co-based systems with a value of 825 HV. When using the new $H_B$ definition, the calculated room temperature hardness of alternative binder cemented carbides match the experimental trends. But their accuracy is worse compared to Co-binders. Particularly the
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Exponential term ruling the contribution of WC and binder hardness to the effective composite hardness determined for experimental samples from dataset A, and calculated exponential term as a function of size polydispersity.</figcaption></figure>

hardness of Fe-based binders is slightly underestimated. This could be related to the importance of considering additional hardening mechanisms other than solid solution strengthening in the binder hardness

description i.e. martensitic transformation.

The use of the fitted Eq. 20 not only allows one to study bigger datasets but also to explore the trends in room temperature hardness with the variation of WC grain size and the binder volume fraction. Fig. 10 presents room temperature hardness for Co-based cemented carbides as a function of these two structure design parameters. Calculations from the current model using the fitted $\sigma_L$ and the Engqvist model are compared. The main differences observed are:

- The aforementioned extension of the hardness upper limit from 2542 to 2901 HV. Considering that values up to 2720 HV [25] and 3189 HV [101] have been reported in literature for conventional and spark plasma sintered samples respectively, this extension is beneficial for the modeling of the whole range of materials sintered through conventional routes. However, it may be insufficient in the modeling a small set of grades processed under very specific conditions.

- The effect of increasing binder volume fraction on the hardness of fine grades is significantly different in the two models. In ultrafine grades the plastic deformation is mainly attributed to the metal binder [58], and its deformation occurs earlier than in coarser

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Standard deviation of the particle size distribution as a function of the mean grain size for samples in Dataset B. b) Log-normalized standard deviation as a function of mean grain size for samples in Dataset B.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Calculated hardness data for validation dataset A.2 samples compared to experimental measurements a) Engqvist model. b) Current model.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** a) Hardness vs. carbide grain size for different binder volume fractions. b) Hardness vs. binder volume fraction for different carbide grain sizes. Comparison between trends obtained according to Engqvist [2] and the present model.</figcaption></figure>

grades. This is a direct result of the skeleton configuration of both grades, in which for coarse grades, the contact network between hard grains promotes more contribution of the hard phase to the global deformation. An intuitive example showing the qualitative improvement of the current model compared to the Engqvist model would be the cemented carbide grade with a grain size of 0.1 µm and 40% volume binder. According to the Engqvist model such grade, in which the hard phase is even below the percolation limit, should mostly behave as a binder with distributed precipitates, and would have a hardness around 2200 HV. This value is clearly overestimated. Contrarily, the current approach predicts a hardness

around 1400 HV, which is closer to experimental values reported [102].

## 3.2. Mid temperature hardness

### 3.2.1. Hard phase intrinsic softening

Fig. 11 presents the GSFE calculated by slipping a WC crystal along the considered slip planes and directions. To the authors knowledge, there are very few publications that report GSFE of WC being [103] and very recently [104]. The results shown in Fig. 11 are in good qualitative agreement with both references, but quantitatively they lie between the SFE values reported in [103] (lower bound) and [104] (higher bound).

The higher value of SFE the more difficult to slip, thus Fig. 12 shows that slip along $\langle a+c \rangle$ and $\langle c \rangle$ in the prismatic planes $\{10\overline{1}0\}$ is relatively easy. While within the same plane, slip along $\langle a \rangle$ is less probable due to its higher SFE. The secondary prismatic plane $\{1\overline{1}20\}$ shows an intermediate SFE. Regarding the slip systems studied in the basal plane, these present higher SFE and therefore a lower probability of slip. These observations are in good agreement with multiple experimental works that identify the slip in the prismatic plane $\{10\overline{1}0\}$ as the primary slip system in WC [105,60,55,106,58,107].

Two of the studied slip systems ($\{10\overline{1}0\}\langle a+c \rangle$ and $\{0001\}\langle a \rangle$) present an intrinsic stacking fault (ISF) characterized by a local minimum in the GSFE curve. This ISF indicates a trend towards partial splitting in these slip systems. In the case of partial splitting in the prismatic plane, the partial dissociation of $\langle a+c \rangle$ dislocations has been extensively reported experimentally, and is generally accepted to be the principal slip mechanism in WC [105,60,55,106,58,107]. On the other hand, despite being energetically possible, the slip of partial dislocation in the basal plane is rarely reported [55] in experimental slip characterization studies.

The GSFE data in Fig. 11 is used to estimate the CRSS for each of the considered slip systems and dislocation characters, shown in Fig. 12.a. The most noticeable result is the difference between the CRSS calculated for screw and edge dislocations. For all the studied slip systems, edge dislocations showed a lower CRSS compared to their screw counterparts. Hence, in this work the plastic deformation of WC is only described by considering the effect of edge dislocation slipping. Among these edge dislocations, the slip in $\{10\overline{1}0\}$ planes along $\langle a+c \rangle$ and $\langle c \rangle$ directions; slip in the basal plane along $\langle a \rangle$ and slip along $\{1\overline{1}20\}$ planes and $\langle c \rangle$ direction show similar CRSS values, becoming the primary slip systems at low temperature. Experimental values of CRSS as a function of temperature obtained by H. Jones in [106] through micropillar compression validate this observation. The slip of edge dislocations in $\{10\overline{1}0\}$ planes along $\langle a \rangle$ direction exhibits higher CRSS values, implying a lower contribution to bulk deformation. Yet, its significance increases with rising temperatures due to its higher sensitivity to temperature. The glide of perfect edge dislocations in the basal plane along $\langle a-b \rangle$ direction exhibits the highest CRSS among all edge dislocations, and similarly to screw dislocations its contribution to plasticity is not considered. At high temperatures, the gliding of partial dislocations in this slip system becomes favorable, partly explaining WC's softening behavior. In fact, the CRSS of partial dislocations as a function of temperature in WC differs notably from that of perfect dislocations. At low temperatures, very high stresses are needed for slip through dissociated partials, while around 600–700 K partial dislocations slip along the $\{10\overline{1}0\}\langle a+c \rangle$ slip system becomes easier and governs the softening of WC. At even higher temperatures, 900 K, the dissociation of $\langle a-b \rangle$ partials in the basal plane also gains relevance in the overall deformation of WC. The activation of partial dislocation gliding is in good agreement with the mechanism proposed by H. Hu in [55] where the gliding of partial dislocation enables the reaction of dislocations in planes that would supply a fifth degree of freedom for the slip of WC (activating additional prismatic planes).

The trends observed for CRSSs in Fig. 12.a are also observed for the
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** GSFE for all considered slip systems at 0 K. The coordinate is normalized by the length of the burgers vector corresponding to each slip system.</figcaption></figure>

hardness estimated for each slip system shown in Fig. 12.b, however such trends are slightly conditioned by the maximum Schmid factors and Tabor constants acting on each slip system. The contribution to the bulk hardness of each slip system is weighted considering their relative value (with higher probability for lower CRSS) and their occurrence frequency in WCs crystallography [54]. The calculated hardness range matches experimental values reported by G.E. Hollox and J.J. Gilman [108,109], however it's important to note that the nominal value of WC hardness depends strongly on the grain size (following a H—P relation). Hence, the significance of the proposed P-N-based model lies more in understanding the temperature-dependent softening rather than the exact value of predicted hardness. With respect to the softening behavior of WC, the calculated values align qualitatively with experimental findings in [13], though with a slight underestimation in the predicted softening transition temperature. This divergence may be a result of an overrepresentation of the partial dislocations contribution (see Fig. 13) or due to the considered strain rate for hardness indentation being lower than the experimental one. The variation in softening transition temperature resulting from changing $\dot{e}$ from $10^{-3}$ to $10^{-2}$ s$^{-1}$, can be seen by comparing the black and grey lines in Fig. 12.b.

Overall, despite its simplified approach, the obtained description of the softening of WC is considered adequate; especially given the limited high-temperature hardness data available for pure polycrystalline WC. In addition, the adopted methodology relies only on input from DFT calculations, enabling its applicability to other hard phases, and aligning with the compositional universality scope used in the materials design framework.

### 3.3. High temperature hardness

#### 3.3.1. Effect of high temperature microstructural change on hardness

The categorization into mid or high temperature range in cemented carbides is determined by their deformation behavior. In the low and

mid temperature range, the material behaves as a continuum with a brittle behavior at low temperatures and increasing plasticity as temperature increases. The temperature is considered high, when the continuum approximation is no longer valid, and when the plasticity is high enough for the deformation to resemble a highly cohesive granular material. Fig. 14 shows the hardness as a function of temperature modeled only making use of the intrinsic softening of the material constituents. It can be seen how the calculations further overestimate the hardness at higher temperatures. Up to a range of approximately 500 °C, the sole use of the intrinsic softening of the constituents, disregarding any microstructural rearrangement, is enough to model the softening. At temperatures above 500 °C, the intrinsic softening of the binder and the hard phase is not enough to explain the softening of the composite. Thus, it is clear that additional mechanisms are activated as temperature increases.

As already described, the mechanisms acting at high temperatures involve the rearrangement of the microstructure through grain boundary sliding and consequently binder redistribution. In this work, the transition temperature at which this mechanism is fully activated ($T_{III}$) is determined by setting a critical value for the material viscosity ($10^{13} Pa \cdot s$) considered to characterize creep. The calculated viscosities for a Co-based cemented carbide with 30% volume binder and different grain sizes are plotted in Fig. 15.a to exemplify the procedure followed to determine $T_{III}$. The calculated $T_{III}$'s are shown for a range of cemented carbides with varying binder volume fraction and grain size (Fig. 15.a), and with constant binder content and varying composition and grain size (Fig. 15.b). In both cases, the composition of binder-like alloys accounting for the dissolution of W and other elements (if present) has been considered to determine the diffusion rates ruling the deformation in each case. Regardless of the binder composition, the hard phase grain size has a strong impact on $T_{III}$, decreasing with decreasing grain size. The effect of the binder volume fraction on $T_{III}$ is not as strong as the effect of the grain size, but it is clear that severe microstructural
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** a) Calculated CRSS as a function of temperature for all the considered slip systems and dislocation characters. (experimental data from [106]) b) Hardness as function of temperature estimated for all the considered slip systems and dislocation character and calculated bulk hardness of polycrystalline WC (experimental data from [13,108,109]).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Relative contribution of each slip system to the total macroscopic slip as a function of temperature.</figcaption></figure>

rearrangement in cemented carbide grades with lower binder volume fraction will occur at higher temperatures than for their high binder containing counterparts. Both of these trends are in good qualitative agreement with the trends observed for hardness as a function of temperature measured on cemented carbides with varying grain size and binder content [9,62]. Regarding the effect of the binder composition (Fig. 15.b), the model captures the changes in $T_{III}$ between different binder systems. For instance when comparing Co, Ni/Fe and Co/Ni/Fe-based binders it is clear that Co has a positive effect on pushing $T_{III}$ towards higher temperatures. However, the current model is less accurate at capturing the effects of minor compositional differences on the creep properties of specific minor alloying elements. A clear example of this can be found when comparing Co-based binders with or without minor additions of Cr and V. It is well known that the addition of Cr effectively improves the creep performance of cemented carbides through the formation of Cr based complexions in the Hard-binder interfaces [110,111]. This improvement should be observed as an increase in $T_{III}$ but in the current framework cannot capture this effect.

For all the studied cases, the calculated $T_{III}$ lay in a range that is in good correspondence with the measured temperatures at which relaxation mechanisms in the binder are activated [112].

In Fig. 14, where only the intrinsic softening of the constituents is considered, the temperatures from which the calculated hardness starts to deviate significantly from the experimental one are much lower than the calculated $T_{III}$ (Fig. 15). This is related to the *fragility* of the microstructure, described in this work through the parameter *q*. This parameter rules how rapidly the dynamics of the microstructural rearrangement will change as temperature increases. For high *q* values, the change in material deformation mechanism is sharp and happens within a narrow temperature range. Conversely, for low *q* values the
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Experimental VS Modeled hardness as a function of temperature only considering the intrinsic softening of the constitutive phases and disregarding any microstructural rearrangement.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Calculated rheological behavior transition temperature a) for a cemented carbides with Co-based binder showing the effect of the WC grain size and binder volume fraction. Dashed Lines correspond to the calculated microstructural viscosity for a Co-based cemented carbide with 30% Vol. binder b) for cemented carbides with 10% Vol. binder of compositions found in literature [66,63].</figcaption></figure>
transition is more gradual, and the effects of microstructural rearrangements on the hardness starts to be noted mildly at lower temperatures.

q has been fitted to $k_{Relative}$ from samples in Dataset C after calculating $T_{III}$ for each sample. Fig. 16 shows the results obtained from this fitting as a function of grain size (Fig. 16.a) and % volume binder (Fig. 16.b). A clear dependency of q with the grain size is observed, and it is used to model q by means of eq. 21.

$$q(d_H) = 0.00312 \cdot d_H^{-0.3832} \qquad (21)$$

The strong correlation between q and the grain size, makes it challenging to detect any trends on q with respect to the binder volume fraction, still a slight increase in q with increasing binder volume fraction can be observed. Despite the value of the coefficient of determination (R²) not being very high, considering the natural scatter in data coming from several sources and the obtained strong trends, the obtained fitting can be used to discuss the impact of the hard-phase grain size on the resistance of cemented carbides against microstructural rearrangement.

The higher values of q reported for fine grades indicate differences in the stability of the hard-phase skeleton depending on the grain size. Considering q to be a measure of the microstructure fragility, the higher value for finer grades than for coarse grades shows that force chains in fine grades are less stable against buckling because of having more degrees of freedom and lower coordination numbers. These characteristics of the hard-phase skeleton in fine grades make the decrease in $l_p$, and thus in hardness, to be stronger than in coarse grades. This behavior is extensively observed in published data, where fine grades present higher relative hardnesses than coarse grades at intermediate temperature ranges and for which the softening is more pronounced after a certain temperature threshold. Contrarily, the softening in coarse grades is more gradual with temperature [9,62,61].

The definition of both $T_{III}$ and q serves to approximate the effect that the sliding and rotating of hard grains as well as the extensive deformation of the binder have on the material's hardness. This deformation of the binder leads to binder infiltration and lamella formation, up to a point where the deformation is not enough to match the mechanical incompatibilities set by the hard phase skeleton and cracks (low temperatures) or pores (high temperatures) start to form. The increasing loss of modeling accuracy with rising temperature observed in Fig. 14, where only the intrinsic softening of the constitutive phases is accounted for, can be improved by accounting for the effect of microstructural softening.

Fig. 17 shows the modeling results compared to experimental data obtained when using the new equation derived in this work for the

calculation of cemented carbides hardness (eq. 18). The introduction of the microstructural softening acting at high temperature in the modeling of the hot hardness significantly improves the calculations accuracy with respect to the results shown in Fig. 14. More specifically, a significant improvement is observed at temperatures higher than 500 °C. Despite most of the high temperature hardness data available in literature only reports Co-based cemented carbides without addition of γ-carbides, it is important to note that Dataset C also includes samples with alternative binder compositions [66] and samples containing γ-carbides [63]. The model describes the peculiarities that such features have on the hot hardness of these materials.

Fig. 18 shows the errors of calculated hardness relative to the experimental values for the presented modeling framework. The obtained trends and the centrality of the predictions are within a ± 10% accuracy, however there is a growing scatter with increasing temperature. The current model represents a significant improvement with respect to existing models in the prediction of hot hardness of cemented carbides. Its application could push further the development of new cemented carbide grades with excellent properties at high temperatures. However, achieving higher accuracy in predicting hot hardness would require future work, such as the study of cemented carbides employing Discrete Element Methods (DEM) or micro-mechanical models that explicitly model the particle arrangements and interactions along the whole operation temperature range.

## 4. Hot hardness design guidelines

The results obtained from the present model together with the new perspective that shifts from considering cemented carbides as continuum materials towards treating them as highly cohesive particulate materials as temperature increases, enables one to discuss several design guidelines crucial in the development of cemented carbides resistant to thermal softening. Within the scope of this publication, those material features that enhance the resistance of the cemented carbide against structural softening will be beneficial for keeping high hardness at high temperatures. Specifically, we aim to design cemented carbides that maximize the interconnectivity of the hard-phase skeleton ($l_p$), minimize its temperature sensitivity (q), and increases the rheological transition temperature ($T_{III}$). Additionally, the use of constituents with intrinsically high hardnesses and low thermal softening will enhance the cemented carbide's mechanical properties at high temperatures. These design considerations are detailed in the following points:

- The plastic characteristics of the binder at high temperatures will be ruled by diffusion. Materials with inhibited dislocation motion, low

<figure><img src="image_19.png" /><figcaption>Fig. 16. Fitted q as a function of a) grain size and b) binder volume fraction.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 17. Experimental VS modeled hardness as a function of temperature when considering high temperature microstructural rearrangement.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18. Relative errors for the calculated hardness as a function of temperature. Color legend has the same meaning as in previous figures.</figcaption></figure>

diffusivities, high alloying or presence of precipitates, present high viscosities. This makes the rearrangement of the hard particles difficult, inhibiting microstructure dilation and leading to high values of $T_{III}$.

- Higher binder volume fractions provide more space for hard particles to rearrange, leading to stronger softening. In materials with very high binder fractions (≥30 Vol%), below the percolation threshold, $T_{III}$ will even approach values characteristic of the first relaxation mechanism in the binder around 50% $T_{m,Bind}$. Regarding parameter $q$, this will increase moderately with increasing binder volume fraction.

- Finer materials benefit from a H-P relation and thus present harder intrinsic properties. However, as a rule of thumb, one can consider that the force chains present in the skeleton or fine grades are less resistant against buckling and thus against rearrangement than those in coarser grades. The relation of the force-chain structure with the particle size distribution is complex, but qualitatively one can assume that the lower coordination number of fine particles and the higher number of possible configurations in fine grades respectively to coarse grades, leads to a lower resistance against rearrangement. According to these two extremes, there should exist an optimum PSD that benefits from the higher force-chain stability of coarse particles
and from the intrinsically higher hardness of fine particles. In this microstructure, coarser particles with high coordination numbers serve as anchoring points in the skeleton that are preferential load paths increasing the microstructural stability. This effect is theoretically proven in granular materials with similar particle shape as WC, in which particle arrangements with a PSD dispersion values ($σ_L$) between 0.4 and 0.6 lead to more stable force chains formed without a significative increase of the free fine particles [113]. Experimentally, similar results have been obtained when characterizing the softening characteristics of bimodal cemented carbides [64].

Thus, generically, finer microstructures will lead to higher *q* values and lower $T_{III}$, showing a stronger softening than coarser materials, however this can be tailored through a proper design of the hard phase PSD.

- The effect that adding γ-carbides has on the softening properties of cemented carbides, is in close relation to the one discussed for coarse grains in polydisperse materials. Among the most common γ-carbides present in cemented carbides microstructure some are intrinsically harder than WC (i.e.: TiC, VC), and some are intrinsically softer (HfC, NbC, TaC). Even when adding softer carbides, the addition of these is known to damp the softening of cemented carbides. This is related to γ-carbides being usually bigger than the WC grains and them having a different morphology with a much intricate contact surface than WC-WC straight contacts. These characteristics turn additional carbonitrides into anchoring points in the force transmission chains that are even more effective than coarse grains in stabilizing the skeleton against rearrangement. In addition, the presence of γ-carbides during grain boundary sliding increases the periodicity of the sliding boundaries, changing from individual sliding grains to packs of γ-WC interlocked volumes sliding with respect to each other. Experimentally, it is important that the additional carbides act in the correct length-scale of the hard-phase skeleton, being the coarse grains or γ-carbides in the range of 2 to 5 times the mean size of the WC grains [83].

Thus, the introduction of γ-carbides in the microstructure will increase $l_p$ and $T_{III}$ and will minimize *q*, leading to higher hardnesses at high temperatures.

- The aspect ratio of hard phases is an important factor affecting the resistance of microstructure against rearrangement. Similar to the discussion on γ-carbides morphology, elongated WC particles present higher coordination numbers and increased mechanical interlocking potential, leading to more robust force chains [113,114,7]. In cemented carbides, the shape of WC can be tailored through alloying with elements that anisotropically affect the interfacial energies of the basal and prismatic planes of WC, such as Ti. The formation of WC platelets not only boosts the intrinsic hardness of the hard phase by favoring the presence of the harder basal plane over the prismatic plane but also enhances microstructural geometrical interlocking [7]. Changing the aspect ratio towards elongated particles is expected to decrease the value of *q* and increase $T_{III}$. However, the effect that changing the aspect ratio of the hard phases may have on the dilative behavior of the microstructure and thus in the pore formation with once the material starts rearranging is a complex phenomenon that needs further study if stronger conclusions are required.

- The presence of alloying elements that modify the interfacial chemistry (i.e. Cr, Ti, V) and interfacial energies (i.e. Ru) of cemented carbides will clearly have an impact on the material's hot hardness. Any factor that increases the resistance of the material's interfaces will have a positive impact on keeping the hardness at high temperatures high. The presence of Cr, Ti or V complexions in WC-WC and WC-Binder interfaces has shown improved creep resistance at temperature ranges of 900–1000 °C [110,111].

The presence of interface complexions in cemented carbides is expected to decrease the value of *q* and delay $T_{III}$. Their effect on the

model parameters is not studied in detail in the current work, making it an interesting area for future research.

## 5. Conclusions

In the current paper a model to calculate the hardness as a function of temperature of cemented carbides unrestricted in composition is proposed. The model describes the hardness of Co-based or alternative binder cemented carbides, with potential presence of γ-carbides in the microstructure, for a temperature range between RT to 900 °C. This is achieved by using the Engqvist model as a starting point and applying several modifications to it.

First, any hardening mechanism acting on the binder resulting from the microstructure of cemented carbides is detached from the definition of the intrinsic hardness of the binder. Secondly, all effects of microstructural hardening on the material properties are isolated in the exponential term ruling the load distribution between binder and hard phase. This allows to give a physical interpretation to all the parameters in the model as well as to account for softening mechanisms acting at high temperatures. In the current model, the load distribution between binder and hard-phase depends on the ratio between a characteristic length-scale pertinent to the binder and a characteristic length-scale pertinent to the hard-phase skeleton. The former characterizing the binder constrainment and the latter one characterizing the resistance against microstructural rearrangement of the material. Both quantities are calculated to be the mean-chord length of the binder phase and the nearest-neighbor distance of the hard-phase respectively, being both a function of the PSD of the hard-phase and of the binder content. Their calculation represents an improvement with respect to previous models that only considered the mean WC size.

The physical reasoning behind such microstructural parameters also allows for the micro-mechanical description of mechanisms, i.e.: grain boundary sliding, acting at higher temperature. At these temperatures the material loses integrity and is modeled as a highly cohesive granular jammed material. The microstructural rearrangement at high temperatures causes a decrease in the hard-phase skeleton characteristic length-scale. This decrease physically means that the force chains in the material are less stable as temperature increases. This decrease in microstructural stability is characterized by a fragility coefficient and a characteristic transition temperature that are function of the hard phase grain size and polydispersity, binder volume fraction and binder composition.

Additionally to the effect of cemented carbides microstructure on the composite hardness, the intrinsic softening of the material constituents also contributes to the global softening along the whole temperature range. On one side, the intrinsic hardness of the binder as a function of temperature has been modeled using an existing framework unrestricted on composition presented in [37]. And on the other side, the intrinsic softening of the hard phases has been modeled using the Peierls-Nabarro softening model presented in [54] for γ-carbides, and its use expanded to WC. In this expansion, additional details on the slip systems ruling the softening of WC grains have been presented with special attention to the role of partial dislocation motion activation in the prismatic and basal planes.

### CRediT authorship contribution statement

**V. Lamelas:** Conceptualization, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. **M. Bonvalet Rolland:** Conceptualization, Supervision, Writing – review & editing. **M. Walbrühl:** Conceptualization, Software, Supervision, Writing – review & editing. **A. Borgenstam:** Conceptualization, Funding acquisition, Supervision, Writing – review & editing.
Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

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