Simulation of residual stresses in cemented carbides

Simulation of residual stresses in cemented carbides

    Simulation of residual stresses in cemented carbides Wolfgang Kayser, Alexander Bezold, Christoph Broeckmann PII: DOI: Reference: S0...

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    Simulation of residual stresses in cemented carbides Wolfgang Kayser, Alexander Bezold, Christoph Broeckmann PII: DOI: Reference:

S0263-4368(15)30313-9 doi: 10.1016/j.ijrmhm.2016.04.001 RMHM 4219

To appear in:

International Journal of Refractory Metals and Hard Materials

Received date: Revised date: Accepted date:

7 December 2015 29 March 2016 3 April 2016

Please cite this article as: Kayser Wolfgang, Bezold Alexander, Broeckmann Christoph, Simulation of residual stresses in cemented carbides, International Journal of Refractory Metals and Hard Materials (2016), doi: 10.1016/j.ijrmhm.2016.04.001

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ACCEPTED MANUSCRIPT Simulation of Residual Stresses in Cemented Carbides Wolfgang Kayser1, a, Alexander Bezold1, b and Christoph Broeckmann1, c 1

Institute for Materials Applications in Mechanical Engineering (IWM) RWTH Aachen University; Augustinerbach 4, 52062 Aachen, Germany

[email protected], [email protected], [email protected]

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Abstract. This paper outlines how the residual stresses in WC20Co hardmetal after cooling down from sintering temperature to ambient temperature are predicted by a numerical model. A mesoscopic, viscoplastic FEM approach is pursued with an artificial 2.5D representative volume element (RVE). The model has a hardmetal microstructure with a cobalt content of 20 wt-% and an average carbide grain size of 2 µm. It is assumed that the Co binder is subjected to viscoplastic deformation at elevated temperatures, whereas the WC-phase is assumed to exhibit linear elastic isotropic material behaviour over the whole temperature regime. The temperature-dependent material data for this binder phase is obtained experimentally from a coarse-grained model cobalt binder alloy, manufactured in a vacuum melting process followed by pressure assisted cooling to avoid porosity. The mismatch in the coefficients of thermal expansion is assumed to be the sole mechanism which induces a residual stress state. In order to obtain an eigenstrain state on which no artificial strains are superimposed, free expansion boundary conditions were applied to the unit cell in order to ensure periodic or linear displacement boundary conditions. Despite an incomplete description of the constitutive behaviour of the phases, the model provides acceptable results in comparison with experimental literature data obtained via neutron diffraction experiments that allow for a correlation of the stress evolution over temperature. A partially viscoplastic model enables the dependence of residual and internal stresses on cooling rate to be determined. The results are made more representative by averaging stress values over more than one geometrical model. Keywords: Finite Element Method (FEM), Representative Volume Element (RVE), residual stress, hardmetals

1.

Mathematical symbols and abbreviations Mathematical symbols:

Temperature-dependent Norton multiplier Temperature-dependent multiplier for the hyperbolic cosine creep term Temperature-dependent coefficient for the decay of the exponential strain hardening Temperature-dependent coefficient for the linear part of the strain hardening equation Temperature-dependent coefficient for the exponential part of the strain hardening equation Coefficients for the piecewise hermite polynomial Temperature-dependent elastic modulus of phase X Yield function Temperature-dependent Norton exponent Temperature-dependent time exponent for Norton Bailey law Temperature-dependent uniaxial yield limit Temperature-dependent coefficient for the hyperbolic cosine creep term

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Plastic strain component of phase X Plastic strain rate of phase X

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Creep strain component of phase X

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Thermal strain component of phase X

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Temperature Reference temperature for piecewise hermite polynomial Time Volume of the phase X Superscript for one of the constituent phases of the hard metal (here WC or Co) Temperature-dependent isotropic coefficient of thermal expansion for phase X Kronecker delta Change in temperature Total strain component of phase X Elastic strain component of phase X

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Equivalent plastic strain Time derivative of the plastic multiplier Temperature-dependent Poisson’s ratio of phase X Stress components of phase X Von Mises equivalent stress Deviatoric stress components Volume averaged phase specific stress tensor components Volume averaged phase specific hydrostatic stress Chemical elements and molecules:

2D 2.5D 3D CPU CTE EBSD FEM FIB RVE

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Cobalt Tungsten mono carbide

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Co WC

Abbreviations:

Two dimensional Within this context used for geometrical models which are extruded 2D models or which are lacking representativeness in one dimension Three dimensional Central processing unit Coefficient of thermal expansion Electron back scatter diffraction Finite element method Focused ion beam Representative volume element

Introduction

Hardmetals are wear-resistant two phase composite materials consisting of carbides, such as tungsten carbide (WC) and a metallic binder, such as cobalt (Co). Tungsten carbide materials are widely used as cutting tools and inserts, as wear resistant parts for mining and earth works and are becoming increasingly popular for mechanically highly loaded components [1, 2]. Consolidated by liquid phase sintering [3], hardmetals are exposed to the build-up of high residual stresses between

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the different components, namely the carbide and the binder, while they cool from sintering temperature to ambient temperature. Accordingly, efforts have been made in recent years to estimate the residual stress within the phases of these materials using either analytical [4–7] or finite element models [8–19]. To begin with the analytical models, Sayers [4] describes the residual stresses in dependence on the lattice orientations a and c using a WC single crystal embedded in an isotropic and infinite WCCo-matrix. Taking into account the different elastic behaviour and the mismatch in the coefficient of thermal expansion (CTE) between WC and the WC-Co-matrix, the model shows that the WC inclusion is under compression in all directions with a lower stress component along the c-direction under the influence of a temperature change of 700 K. Seol and Krawitz [5] used a multiphase model for the interpretation of neutron diffraction experiments to investigate the anisotropy and relaxation of residual stresses following uniaxial, compressive loading. Their investigations demonstrated that the stresses in the carbides partially relax but that a higher amount of stresses remain in the loading direction, starting from a nearly hydrostatic residual stress state in tungsten carbide. Golovchan [6] developed two sets of equations which were used to calculate the thermal residual stresses in dependence on the binder content and the contiguity of the carbide phase. The two separate sets of equations are used to predict either the surface stresses or the volumetric stresses in both phases. For an increasing binder content, his model indicates higher compressive stress in the WC phase and decreasing tensile stresses in the binder. With increasing contiguity the compressive internal loading of the carbides increases. Further work on the dependence of thermal residual stresses on the binder content of hardmetals has been conducted by Litoshenko [7]. He focusses on the stresses formed in a tungsten carbide polycrystalline which is embedded in an infinite hard metal matrix. In agreement with [6] he finds that the compressive residual stresses intensify with increasing binder content. Furthermore, taking into account the single crystal values of the carbide grains in the polycrystalline inclusion, he concludes that the single crystal stresses exceed volume-averaged values which might lead to failure of the WC-WC boundaries. The first models based on the finite element method of predicting thermal residual stresses originate from research groups around Schmauder [8–10] and Krawitz [11–13, 18, 19]. The group around Schmauder used two dimensional (2D) models derived from real microstructures of hardmetals [8–10] in their analysis. In [8] the calculation of residual stresses supports the numerical investigation of pore formation in the binder phase. The geometrical models presented in [9, 10] were previously used to predict crack propagation under cyclic loading in hardmetals. All three models use an elastoplastic description, based on a Voce type hardening law and the hard phase in each one is described as isotropic linear elastic material. Embedded cell boundary conditions were used for each model, which means that the microstructural model lies within a framework of elements incorporating the same material properties as the hard metal grade under investigation [10]. The thermal loading is a temperature difference of 800 K [8–10] because it is assumed that the residual stresses will relax at higher temperatures [8]. The increasingly viscous behaviour of the binder suggests that this assumption is valid. Plane strain assumptions were used for the calculation of residual stresses. The results of the simulation show that there are localised regions in the carbides where tensile residual stresses form. Moreover, the results of the simulation provide information about the regions where highest compressive loading in the carbides and the maximal tensile stresses in the binder occur, namely at the locations where the smallest binder mean free path between the carbides exists, and at the edges of the carbide grains. At the same time as the Schmauder group, Podoroga et. al. [20] set up a model using a temperature-dependent elastoplastic description of the binder phase using a linear hardening formulation for the binder and describing intergranular carbide failure. A detailed investigation was also conducted into the distribution of the stresses over the different phases. On the basis of a single 2D model he found that intergranular failure is always caused by the internal stresses formed during cooling. In addition, he stated that using material data for a model binder alloy instead of pure cobalt leads to an increase in the overall stress level.

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The first finite element models developed by the group around Krawitz are likewise based on actual microstructural images of hardmetals [11, 12]. The model presented in [11] was used to develop a better understanding of the peak position for lattice planes using neutron diffraction. As a spin-off, the residual stresses were predicted using plane strain conditions and an isotropic elastic material description for both phases. Results for the residual stress state were underestimated in comparison to measured values by 22 % to 45 % [11]. It was also stated that the models are sensitive to the morphology of the microstructure and that it is, therefore, essential to investigate the validity of a greater number of models [11]. In a subsequent analysis [12] the constitutive model was enriched by an elastoplastic description of the binder. Results show that the carbides tend to be under internal tensile load in the regions of the carbide/binder boundaries and the carbide edges [11]. The results obtained by Spiegler [10] concerning the location of the magnitude of the maximum stresses were confirmed by Weisbrook [11]. Later models developed by the Krawitz group were designed to investigate the binder deformation [18] and to study the effects of external loading on the residual stress state [19]. The geometry of the models [18, 19] considers idealised three dimensional (3D) representations of the hard metal microstructure. The carbides are modelled as hexahedral plates, bordered by a rectangular binder skeleton [18], as WC cubes in a binder cube or as cross shaped binder regions separating the carbide hexahedrons and as multi cell models based on the idealised microstructures previously listed [19]. The volumetric phase distribution of the phases as well as the contiguity (multi cell model only) can readily be described [19]. However, the complex geometry of the carbides cannot be reproduced. The thermal load of ΔT = 800 K was used for both models. The binder was described using an elastoplastic constitutive formulation and the carbides were described as an isotropic elastic material. The idealization of the morphology resulted in underestimation of the thermal residual stresses by up to a factor of two [18]. The models in [19] were suitable for predicting the plastic deformation of the binder phase under the superposition of external loads. In contrast to neutron diffraction results, the deformation mechanisms during loading and unloading of the material were clarified. However, these models were likewise unable to quantify the residual stresses. Kim et. al. [15, 16] characterised the mechanical behaviour of WC-Co composites, taking account of the anisotropic behaviour of the carbides. Statistically-idealised 2D models and 2D models derived from electron back scatter diffraction (EBSD) were used to perform the simulations. Results show that the maximum loading capacity is obtained in hardmetals with minimum contiguity, wide grain boundary angles, equal orientation of the carbide grains and low variation in the grain size of the carbides [15]. The minimisation of residual stresses caused by the anisotropic thermal expansion of the carbides is cited as the explanation for the positive effects of a carbide grain texture [15]. In the special case of equal orientations of the carbide grains, a high contiguity is preferred in order to lower the residual stresses [16] because the number of WC/Co boundaries is reduced. The studies conducted by Kim provided valuable information regarding the morphological optimization of the hardmetals in order to minimise residual stresses, but the prediction still had a qualitative character. The first 3D estimates of residual stresses using complex geometrical description of the carbides and the binder phase were established by Magin and Gérard [21]. The WC30vol-%Co (WC20wt-%Co) model they developed was created using the Digimat program, e-Xstream engineering SA, Hautcharge, Luxembourg. The results were obtained by applying a temperature change of 1000 K [21] to the model then evaluated by giving the statistical distribution of hydrostatic and maximum principle stresses of the carbides. The outcome was an average hydrostatic stress level in the carbides of approximately -326 MPa [21]. As presented in this brief overview of the residual stress prediction of hardmetals, it has become clear that previous work focused mainly on microstructural influences on the thermal residual stress state to aid explanation of the effects observed after neutron diffraction measurements or for the evaluation of the stress propagation within the different phases. So the main influences on the internal or residual stresses of random heterogeneous composite materials, namely the

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3.

Constitutive modelling and boundary conditions

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3.1.

Theory and modelling aspects

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microstructure and the difference in CTE [22], have already been addressed. The focus of this study represents a move towards examining the temperature and time dependency of the residual stresses in response to the identification of viscoplastic effects as a potentially important influence on the residual stress state in composite materials [22, 23]. In the following sections, the approach presented includes viscoplastic effects in a 2.5 D model of an artificial and statistically-based hard metal microstructure.

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The constitutive material description is based on a small strain formulation which allows for additive strain decomposition. For both phases the strain is described as follows. ...................................................................................................... (1)

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In Equations 1 and 2, , , and denote the tensor components of the elastic strain, thermal strain, the plastic component of the strain and the creep strain for the corresponding phases WC and Co. The temperature-dependent description of the elastic and thermal strain is isotropic and is formulated as: ............................................................................................................ (3)

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........................................................................................................................ (4) The superscript X is a placeholder for one of the phases, is the Young’s modulus, the Poisson ratio, the isotropic thermal expansion coefficient, ΔT the temperature change between two time steps of the incremental loading and the stress tensor component. The Einstein summation convention has been used, which means that terms have to be added when indices appear twice in the formulation. is the Kronecker-Delta. The plastic strain is given in the form stated below: ................................................................................................................................... (5) -

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In Eq. 5 the plastic strain rate component is given in dependence on the time derivative of the plastic multiplier and the yield function f which uses a Voce type formulation to describe the strain hardenening of the cobalt. In Eq. 6,

is the von Mises stress,

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deviatoric part of the stress tensor, the equivalent plastic strain, is a temperature-dependent constant describing the linear part of the strain hardening, and are constants for the exponential part and the saturation stress of the hardening. The parameters were obtained for a model binder alloy (87.8 wt-% Co, 11.68 wt-% W, 0.14 wt-% C and 0.38 of minor elements), similar to the binder specifications mentioned in [24], at 1200°C, 800°C, 400°C and 25°C. The temperature-dependent plastic description for each parameter is achieved by connecting the

ACCEPTED MANUSCRIPT experimentally determined discrete data points with a piecewise cubic hermite polynomial in the form: ............................................................ (7)

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In Eq. 7 is the reference temperature of the temperature interval. The commercially available MATLAB R2014a software developed by The MathWorks Inc., Natick, Massachusetts, was used to fit the parameters of the Voce Law and the piecewise cubic hermit polynomials. Values for all parameters are given in Table 1.

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Table 1: Parameters for piecewise cubic hermite polynomials for three temperature intervals

C2(T)

bk [-]:

25 400 800 25 400 800 25 400 800 25 400 800

3.44E+2 311E+2 1.36E+2 1.58E+3 1.37E+3 1.01E+3 1.14E+2 1.36E+2 1.01E-01 2.03E+2 7.07E+1 5.18E+1

0 -1.46E-1 -3.18E-1 -4.16E-1 -6.94E-1 -1.28E+0 1.27E-1 0 -1.17E-1 -5.00E-1 -8.37E-2 0

-3.19E-4 -1.75E-3 1.07E-4 -4.96E-6 4.73E-6 -3,34E-3 -2.22E-4 -3.67E-4 -1.84E-4 7.15E-5 6.56E-5 6.71E-4

2.20E-7 2.56E-6 1.58E-7 2.23E-7 -1.24E-6 1.96E-6 9.33E-8 3.66E-7 9.22E-8 8.59E-7 6.51E-8 -4.61E-7

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Parameter of Eq 6:

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The viscous part of the strain is given in the form of a Norton-Bailey type equation expanded by a term [25] which describes the creep behaviour under low stress conditions: .................................................. (8)

3.2.

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Isotropic temperature-dependent elastic material data and thermal expansion material data of the tungsten carbide phase is taken from Reeber and Wang [26]. Due to the lack of a sufficient number of specimens, isotropic elastic data and thermal expansion data for the binder are taken from the Co-based alloy Hanes 25, accessible in the MPDB materials database software, JAHM Software Inc., North Reading USA. Creep parameters were taken from [27] for the temperature interval of 950 °C > T >800 °C. The previously described creep and hardening behaviour of the binder were implemented in the ABAQUS 6.13-1 finite element program, Dassault systems, VélizyVillacoublay, France, using CREEP and UHARD subroutines. Geometrical modelling

Eight artificial microstructures were created using the Digimat FE 4.5.1 program, e-Xstream engineering SA, Hautcharge, Luxembourg. The WC particles were modelled as inclusions using an average size of 2 µm and allowing a size reduction of 25 % from the initial value. The data for the geometrical shape (truncated prisms) of the carbides were taken from [28]. Topology-related restrictions and the carbide grain orientations were not given in order to speed up the convergence of the random placement of the carbides within the predefined volume of 20 µm x 20 µm x 1 to 1.5 µm. A 3D microstructure with a small extension in thickness direction was generated. In contrast to a full 3D model with an extension of a similar order of magnitude in all three dimensions, the model created has an altered homogenised stiffness response along the thickness direction. Normally the contiguity of grains causes the phases to constrain their movements. In the present case, the contiguity cannot be fully captured in one dimension because of the reduced thickness of the models which results in a significant change in the homogenised stiffness response of the model. The models presented are, therefore, referred to as 2.5D models. If the 2.5D model were constrained by periodic or linear boundary conditions, the homogenised reaction of the model would show increased artificial stiffness along the thickness direction, hence leading to an increased stress response. Taking into account the fact that in the present simulation the boundary of the

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model is unconstrained, the homogenised stiffness of the model in the thickness direction is lower than that of a full 3D model. Accordingly, the stress response of the 2.5D model along the thickness direction is expected to be lower in this study than it would be if full 3D models were used. The 2.5D approach is far more realistic in terms of the homogenised material response [29] than the extremal 2D assumptions of plane strain or plane stress conditions and is thus more capable of capturing the residual stress response. It can, therefore, be concluded that the potential loss of accuracy when the 2.5D approach is pursued rather than a full 3D one is lower than it would be if plane stress or plane strain assumptions were made. These drawbacks in terms of the accuracy of the stress results in comparison with that achieved when full 3D models are used, are accepted as the price to be paid for faster geometrical modelling and a higher probability of achieving good mesh quality. In addition to this, the use of 2.5D models can save CPU time. A 2.5D modelling strategy is a good compromise in terms of accurate results, computational efficiency and pre-processing effort. Seven models with a 20 wt-% cobalt content (WC20Co) and one containing a WC30 wt % Co grade (WC30Co) were created. The intention behind the use of more than one model (applies only to the WC20Co grades) is to increase the representativeness of the averaged residual stress response, by using statistical methods in order to average the residual stress values calculated, over more than one model. The statistical method used was introduced by Chen et al. [30] and has been shown to increase the representatives of results even when smaller-scale models are used. This approach can help to regain some of the accuracy lost due to the application of the 2.5D modelling approach. To mesh the models, Digimat exports the geometrical data to ABAQUS and uses its meshing capabilities. Adaptive meshing refines the mesh automatically in regions of sharp phase edges thus leading to a good but not perfect mesh in terms of the geometrical properties of the elements. Linear tetrahedrons for coupled thermal displacement (C3D4T) were chosen as element type. The number of elements varies depending on the model in the range of 75,599 to 145,779 elements per model. The resulting models are shown in Figure 1. Further specifications regarding the models such as thickness, binder content and number of elements are given in Table 2.

Figure 1: Models M1 to M8 created using the Digimat FE software.

ACCEPTED MANUSCRIPT Table 2: Specifications of the artificial microstructural models

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Width [µm]:

Depth [µm]:

20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20

1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0

Boundary conditions and thermal loading

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Height [µm]:

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Co-content [wt-%]: 20 20 20 20 20 20 20 30

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Model:

Element count: 83,757 84,623 82,714 102,854 75,316 145,779 144,595 75,599

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The models are allowed to expand freely. Neither linear displacement nor periodic boundaries are applied to the surfaces of the models because artificial strain effects would alter the results of the estimation of internal stresses. The magnitude of the additional strain caused by the boundary conditions selected, is frequently higher than the strain induced by the mismatch of the thermal expansion coefficients. The only option that allows for homogenisation of the stresses over both composite phases is the embedded cell approach [10, 31–34]. When this type of boundary condition is set, the microstructural model is surrounded by an additional frame, which has the homogeneous material properties of the composite, thus leading to a more natural strain response at the boundaries of the RVE. Because the type II residual stresses of the carbides, which are the stresses equilibrating over grain scale [35], are of particular interest, boundary conditions enabling a homogenisation approach are not necessarily required. Furthermore, homogenisation over both phases, which would yield information comparable to the type I residual stress response, is not considered in this study. Type I stresses are the stresses, which equilibrate over the whole component [35]. Displacement boundary conditions are applied only to single nodes at three of the edges of the model to prevent translation or rotation. The model is subjected to homogeneous, time-dependent thermal loading. Accordingly, no temperature gradients form between the phases or over the model. The assumption that there are no thermal gradients over distances of a few microns is justified because the thermal conductivity and specific heat capacity of the compound are suitably high [36], resulting in rapid equilibration of thermal gradients and slow exchange of heat to the atmosphere. The temperature profile used to obtain the residual stress results is given in Figure 2.

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T1 T2

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Figure 2: Temperature loading over time

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Note that these profiles were picked at random only to verify if the model shows a distinct dependence on the processing time. These curves are not relevant to industrial processing. Volume averaging and evaluation process

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To facilitate comparison with residual stress values obtained by neutron diffraction, the stress components of a single phase are volume-averaged to obtain an averaged stress response of a single phase. ....................................................................................................................... (9)

In Eq. 9 is the stress tensor at every integration point of the phase X, denotes the volume of X the WC or Co Phase and i represents the volume-averaged stress tensor of phase X. Averaging over separate partial volumes within a closed model leads not to a stress value in a strict physical sense, but to an equivalent stress tensor component for the corresponding phase under examination. In order to facilitate comparison of the simulated residual or internal stresses with the experimental values, the hydrostatic part of the volume-averaged stress was calculated as presented in Eq. 10. ............................................................................................................................. (10) In the case of the WC20Co models, arithmetic mean values and standard deviations of the averaged hydrostatic part of the stresses were calculated over all models and are shown in the results section. 4.

Results and Discussion

Because the definition of residual stresses implies a constant temperature over time and region [35, 37], only stresses calculated for 25°C are referred to as residual stresses. Stresses resulting from higher temperatures are referred to as internal stresses. In Figure 3, the hydrostatic part of the

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volume- averaged internal stresses is given over temperature. As previously mentioned in the evaluation section, data for the WC20Co grade is given as mean values of the hydrostatic part of the volume- averaged stress tensors of all models. The data for the single WC30Co model are included in the diagram.

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-500 -700

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Temperature [ C]

WC-T1-WC20Co2 µm Co-T1-WC20Co-2 µm WC-T2-WC20Co2 µm Co-T2-WC20Co-2 µm WC-T2-WC30Co2 µm Co-T2-WC30Co-2 µm

WC-WC20Ni-1,5 µm WC-WC30Ni-2,4 µm

Figure 3: Volume-averaged hydrostatic stresses of all models.

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In Figure 3, black data points denote the hydrostatic part of the stress tensor for the carbides, grey indicates the hydrostatic part of the equilibrating stress tensor of the cobalt phase. T1 and T2 within the legend denote the temperature profile given in Fig. 2. Symbols with white filling represent neutron diffraction data from literature. Values and error ranges for WC20Ni were taken from [38] and those for WC30Ni were obtained via linear interpolation of data given in [39]. Error bars over the simulation results specify the standard deviation of the stresses from the mean value given for the WC20Co models. The average carbide grain size is also included in the legend. As shown, the model is not sensitive to cooling time. Both of the WC20Co curves match perfectly for temperature profiles T1 and T2. The results are not surprising in relation to the temperature region below 825°C because viscoplastic behaviour is deactivated at these low temperatures. Only in the temperature interval 950 ≥ T ≥ 825°C can differences in the model possibly be achieved by varying the cooling rate, but even in this interval the results do not deviate from each other. On the basis of the current implementation of the model, the results give rise to the conclusion that as long as the order of magnitude of the cooling rate lies within the same range, namely hours, days or minutes, the internal stresses of hardmetals can be considered to be rateindependent. However, while viscoplasticity and, therefore, rate-dependent material behaviour both occur within a narrow range, the conclusion referenced above should be verified by enlarging the range of the viscoplastic description and adding rate-dependent plastic behaviour in the temperature range below 400°C. Alternatively, the rate dependency could be verified experimentally. For the following discussion of the simulation results in direct comparison with experimental data [38, 39] it is important to note that the information obtained can be only regarded as being indicative of, not as a fully statistically valid measure for the model quality. This is because the comparison is based on a very low volume of data which do not reflect every aspect (e. g. grain size or base component of the binder alloy) of the modelled hard metal grades under investigation). For the subsequent comparison of experimental with simulation data, it is assumed that the residual

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stress response of WC-Ni and WC-Co is similar in terms of binder content. This assumption was based on the fact that there was no experimental literature data for WC20 and WC30Co hardmetals with a grainsize of 2 µm available at the time of writing. A comparison of the averaged WC20Co residual stress response with the WC20Ni literature data at T = 27°C shows that the final residual stresses of WC20Co in the carbide phase are underestimated by almost 50 %. In the higher temperature regions from 900°C to 400°C the internal stresses in the carbide phase of the WC20Co grade are overestimated in magnitude, which is a result of the relatively narrow implementation of viscoplastic material behaviour, so stress relaxation cannot occur at lower temperatures. The transition temperature observed in neutron diffraction measurements, above which the internal stresses relax, is found in temperature regions between 526°C [40, 41] and 625°C [42]. In the model, however, the viscous material behaviour is disregarded at temperatures below 825°C. Furthermore, the omission of the viscous term causes the Co-phase to harden at temperatures T < 800°C, which also increases the magnitude of the equilibrating internal compressive stresses in the carbide phase. Below 400°C the magnitude of the internal stresses of the carbide phase of WC20Co start to fall below the experimental values as already observed for the final residual stress state at room temperature. Here it can be argued that the choice of free expansion boundary conditions has an impact because of the unconstrained outer boundaries of the model. Regions of high tensile stresses in the carbide region (Figure 4) at the boundaries of the model might be avoided by the use of other boundary conditions as briefly discussed in the modelling section. A reduction in tensile stresses in the carbides would definitely lead to a higher magnitude of the averaged internal compressive stress response and results would be closer to the experimental values. Nevertheless, the search for the most suitable boundary conditions for the mesoscopic residual stress analysis is beyond the scope of this paper and will require discussion when the model has been further developed as proposed at the end of this section and when a more suitable set of material parameters has been derived. Although the quantitative values deviate from experimental results, the qualitative development of internal stresses over temperature is captured quite well. The model naturally shows a relaxation of stresses due to the viscous behaviour of the binder where the creep model is active. A new formation of internal stresses at even higher temperatures (900°C to 1000°C) as reported by two references [40, 41], which may find its origin in the solution of WC in the binder [41], cannot be captured by the model in its present constitutive formulation. Looking at the WC30Co model, the residual stress level at 25°C is captured quite well, overestimating the stress in the WC by about 15% in comparison with data linearly interpolated between stress values for WC20Ni and WC40Ni hardmetals taken from [39]. The evolution of stress over temperature cannot be compared because, to the knowledge of the authors, there are no temperature-dependent data available for the corresponding WC30Co or WC30Ni grades to be compared with. Now it has to be determined why a single model yields reasonable results and a statisticallybased approach struggles to predict the residual stress state at room temperature (having overcome the limitations in predicting plane stress or plane strain formulations in terms of a pronounced numerical under or overestimation of stresses [29] by using a 2.5D model). Since the constitutive formulation does not change, the reason is most likely to lie in the limited representativeness of the WC20Co models. As already outlined in the brief review, results can vary with individual microstructural sections of actual hard metal structures [11] and the morphology of the microstructure [15, 16] under investigation. Because the models are statistically generated by Digimat FE using a Voronoi approach, the microstructure does not necessarily resemble the morphology of actual hardmetals with respect to contiguity, binder mean free path and grain distribution. It is clear from Figure 1 that some of the models exhibit pronounced grain clustering at the borders of the model (e. g. M3, M5) or in the centre of the model (e. g. M1). This leads to large binder compounds where stresses are lower than in regions with a more balanced phase distribution. This leads to a lower stress response over the whole model and consequently to a lower magnitude of compressive stress in the carbides. Hence, averaging over a limited number of models, none of which is truly representative, will not lead to further improvement of the results. In view of the fact

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that there is no statistical background for the individual WC30Co model present and that the data for verification are interpolated, the explanation above is not presented as a well-established fact, but neither is it out of the question. Besides the discussion of the quantitative results it is also worth taking a closer look at the qualitative stress distribution in the model. Figure 4 shows the distribution of the hydrostatic stresses for one of the WC20Co models. Overall, the qualitative results presented in Figure 4 show that the carbides are exposed to mainly compressive loading and the binder to tensile loading. In regions of low mean free binder path, tensile stresses are highest in the binder, thus leading to high compression in the carbides. The highest compressive stresses in the carbides are observed at acute dihedral angles. This results in high tensile loading in the binder. Furthermore, compressive stresses can be observed in the binder phase on the boundaries of the model. Consequently, tensile stresses are formed in regions where carbides are close to the surface and not constrained by the binder. In general, the qualitative prediction of the residual stress state corresponds to the findings of previous authors regarding the distribution and location of compressive and tensile stresses [9–11, 21]. It can also be observed that the highest magnitude of stresses is localised at the boundaries of phases and quickly decreases towards the centre of the phase.

Figure 4: Exemplary contour plot of the distribution of the hydrostatic stress of model M7. Distribution in the WC is shown on the left and distribution in the Co-binder on the right. With reference to the previously outlined drawbacks and uncertainties of the current model, the following modifications will be included in future work to improve the predictive quality of the model and to determine whether or not residual stresses in hardmetals are dependent on cooling rate. 

The viscous behaviour of the binder will be investigated over a wider temperature range and rate-dependent plasticity will be taken into account. This requires the investigation of

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the elasto-viscoplastic properties of a model binder alloy to obtain parameters for a suitable, constitutive representation. In order to increase representativeness, the models will have to be derived from actual hard metal microstructures using either sectioning techniques or focused ion beam (FIB) technology combined with electron backscatter diffraction (EBSD) to obtain 2.5D or full 3D models. In addition to the improved model generation techniques, statistical methods will be applied in order to save computational costs and representativeness will be increased by using smaller models [30]. The changes in phase volume caused by precipitation of carbides from the binder during cooling, will be taken into account in order to identify the trend of internal stresses over temperature in the region above 600°C.

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The model presented here is capable of predicting the final residual stresses in the hard metal microstructure, provided that the model is representative of a similar material grade under experimental investigation. The evolution of internal stresses over temperature cannot be predicted quantitatively in satisfactory agreement with experimental data using the model in its current state. This is due to the narrow temperature range in which viscoplastic behaviour tends to occur. However, in a qualitative sense the models can predict the development of residual stresses over temperature. For cooling rates in the same order of magnitude, the model in its current form is unable to resolve a significant dependency of residual stresses. A statistical approach to the generation of models does not seem to be suitable for quantitative prediction of residual stresses, even when combined with statistical methods to average the stress response over a several models.

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The research presented in this paper was funded by Fachverband Pulvermetallurgie Arbeitskreis Hartmetall. The authors would like to thank the Fachverband Pulvermetallurgie and all contributing companies for their financial support and for the provision of sample materials.

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Wolfgang Kayser was born in Cologne (Germany) in 1984. In 2012 he graduated from RWTH Aachen University in Mechanical Engineering in the field of Engineering Design. Since 2012 he has been employed as a scientific researcher at the Institute for Materials Applications in Mechanical Engineering (IWM), RWTH Aachen University and is currently working on the simulation of residual stresses in cemented carbides.

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Alexander Bezold was born in Ebermannstadt (Germany) in 1970. He graduated in Material Sciences and Engineering in 1999 at the University of Erlangen-Nürnberg. Until 2001 he was employed as a scientific researcher at the Chair of Glass and Ceramics at the University of Erlangen-Nürnberg. In 2001 he was engaged as a structural engineer (Hays). In 2003, he moved to the Institute for Ceramic components in Mechanical Engineering (IKKM), RWTH Aachen University and was promoted to the position of Group Leader (Department Design & Simulation) in 2006. He was subsequently appointed Senior Engineer at the Institute for Materials Applications in Mechanical Engineering (IWM), RWTH Aachen University in 2009. Since 2012 he has been the Associate Director at the IWM.

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Christoph Broeckmann was born in Weeze (Germany) in 1963. After graduating in Mechanical Engineering from Ruhr-University Bochum, Germany, in 1990 he was employed at the Institute of Materials Science at Bochum until 2000. He was awarded a PhD in 1994 (Fracture of Carbide-Rich Steels). In 2000 Christoph Broeckmann earned his habilitation (Creep of Particle-Reinforced Materials) and was engaged by the company Köppern GmbH & Co. KG in Hattingen. In 2003, he was appointed Managing Director of “Köppern Entwicklungs GmbH”. In 2008 he took up a professorship at the RWTH Aachen University and has headed the Institute for Materials Applications in Mechanical Engineering (IWM) since 2009.

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Highlights  Temperature dependent simulation of residual stresses in cemented carbides  Implementation of visco-plastic binder behavior  Statistical evaluation of results over several models