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УДК 539.5
Поведение чугуна при термической нагрузке: влияние морфологии графита
E.N. Palkanoglou, K.P. Baxevanakis, V.V. Silberschmidt
Университет Лафборо, LE11 3TU, Великобритания
Чугун является важным конструкционным материалом, широко используемым в промышленности. Изделия из чугуна часто эксплуатируются в условиях сложного термомеханического нагружения, что затрудняет оценку эксплуатационных характеристик с учетом их сложной микроструктуры. В данной работе для изучения поведения чугуна при тепловой нагрузке использован микромеханический подход. На основе известных характеристик чугуна создаются двумерные представительные элементы объема, которые анализируются с использованием метода конечных элементов. Представительные элементы объема состоят из ферритной матрицы с эллиптическими включениями графита. Разработанный численный метод позволяет проводить параметрический анализ влияния морфологии графита и граничных условий. Для оценки точности результатов расчета теплового воздействия проведены расчеты с использованием модели, в которой частицы графита рассматриваются как пустоты. Полученные результаты могут быть полезны для дальнейших исследований чугуна.
Ключевые слова: микромеханика, чугун, частица графита, тепловая нагрузка, термомеханическое поведение, разрушение
DOI 10.24412/1683-805X-2021-5-109-121
Performance of cast iron under thermal loading: Effect of graphite morphology
E.N. Palkanoglou, K.P. Baxevanakis, and V.V. Silberschmidt
Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, LE11 3TU, United Kingdom
Cast iron is an important engineering material, used extensively in industrial applications. Despite being often exposed to complex thermomechanical loading, its performance under such conditions is not fully identified because of its complex microstructure. In this work, a response of cast iron to pure thermal loading is studied employing a micromechanical approach. Specifically, 2D representative volume elements (RVEs) are generated employing material characterisation data and analysed using a finite-element approach. These RVEs comprise a ferritic matrix enveloping a single graphite particle, represented as an ellipse. The developed numerical strategy allows a parametric analysis of the effects of graphite morphology and boundary conditions applied. In addition, a model with a graphite particle modelled as a void is investigated to study the accuracy of the obtained results for the thermal load. The obtained results are expected to be useful in the future design of this engineering alloy.
Keywords: micromechanics, cast iron, graphite particle, thermal loading, thermomechanical behaviour, failure
1. Introduction
Cast iron is one of the most widespread industrial materials. It is used extensively in a diversity of applications ranging from machinery to automotive components, such as engine parts and brake discs. It is estimated that about 70% of the total globally pro© Palkanoglou E.N., Baxevanakis K.P., Silberschmidt V.V., 2021
duced castings are made of various types of cast irons [1]. Its extensive use is thanks to its excellent casting qualities, good mechanical and thermal properties, as well as its competitive price. Cast iron is a wide group of ferrous alloys, which main alloying elements are iron, carbon and silicon. The contents of
carbon and silicon range from 2 to 4 wt % (in the form of either graphite or carbide) and 1 to 3 wt %, respectively [2]. Different types of cast iron are defined based on the varying morphology of graphite inclusions. Specifically, flake graphite are present in grey (flake) iron, nodular graphite particles are met in ductile iron (spheroidal graphite iron) and a combination of both particle types—in compacted graphite iron (CGI).
Although cast irons were used in a number of industrial applications for many decades, the first numerical modelling approaches were developed much later [3]. In recent years, several methodologies based on different assumptions were proposed to model the mechanical response of cast irons. Most modelling schemes follow either a macroscopic approach or a micromechanical one. Macroscopic models employ empirical relationships to describe the global material behaviour while micromechanical approaches incorporate directly material constituents at the level of microstructure to assess the macroscopic mechanical response, often resulting in more accurate descriptions [4].
Following a macroscopic modelling approach, a yield surface and a hardening parameter were often modified to take into consideration cast iron complex microstructural features. McLaughlin and Frishmuth [3] developed a plasticity model for grey iron, modifying the classical von Mises yield surface. Josefson and Hjelm incorporated the tension-compression asymmetry of grey iron in existing modelling scheme of the material [5]. Adopting a Drucker-Prager yield criterion, modified accordingly in tension and compression, a pressure-dependent yield surface was obtained, and a linear kinematic hardening was proposed compared to an isotropic one used by McLaughlin and Frishmuth. Altenbach [6] suggested a nonassociated flow rule to model the elastoplastic deformation of grey iron, after observation of unrealistic transverse strains under uniaxial tension using an associated plasticity flow rule.
On the other hand, following a micromechanical modelling scheme, the focus was shifted to modelling either graphite or metallic matrix. Recent micromechanical models [7] focus primarily on graphite particles, in an effort to identify their unique internal structure and propose accurate methodologies to incorporate them in microstructure-based modelling. Using transmission electron microscopy (TEM) and nanoindentation testing of graphite particles, the authors concluded that a single graphite nodule consisted of three parts: (i) a small nucleus; (ii) a thick
external coat surrounding the nucleus and forming the largest part of the particle by volume and (iii) an outer superficial graphite layer. This structure of graphite particles indicates a high anisotropic behaviour, which was incorporated into finite-element simulations, by assuming hexagonal unit cell for graphite [7, 8]. However, the obtained results proved that the incorporation of anisotropy did not play a significant role in prediction of cast-iron performance, notwithstanding significant computational efforts.
Some micromechanical models considered graphite as a void. Kuna and Sun first adopted this approach, soon followed by other of researchers [9-13]. According to this method, cast iron was treated as a porous material; therefore, a Gurson-type model was used for the metallic matrix [14]. Gurson was first to propose a constitutive law to describe a plastic matrix comprising spherical inclusions. This model was later modified by Tvergaard [15], leading to the well-established Gurson-Tvergaard-Needleman (GTN) model. The GTN model takes into consideration void interaction as well as void coalescence. This approach considers the effect of hydrostatic pressure on the void growth; however, effect of shear stresses was ignored. The material was also assumed as stress-free at room temperature; therefore, manufacturing-induced residual stresses were also neglected. However, this approach cannot capture some effects that occur during the deformation of ductile iron. First, fatigue cannot be predicted accurately using a linear-elastic fracture mechanics framework [16-22] due to different damage mechanisms in the matrix and graphite [23, 24]. In addition, large differences were observed for shapes of the deformed nodules in both tension and compression tests at different temperatures [25]. This contradicts to the following conclusion since graphite stiffness and strength are significantly lower than those of the matrix in the entire range of temperatures considered, nodules should always deform in the same way [26]. Besides, the variation in the Young's modulus of ductile iron with strain even at early stages of deformation cannot be explained by plasticity caused only spherical voids
[27].
Although there is a variety of micromechanical models proposed for cast irons, none of them focused solely on thermal phenomena. Generally, cast irons are vulnerable to high temperatures associated with long-term loading, which may lead to failure by initiation and growth of a network of surface cracks
[28]. Application of pure thermal loading causes expansion of the matrix, exceeding that of graphite in-
elusions with a lower coefficient of thermal expansion and leading to their deeohesion from the metal-lie matrix [17, 23, 24]. However, in eases of thermo-meehanieal loading, debonding might not be the primary failure meehanism. The development of high meehanieal stresses ean eause fraeture of graphite even prior to partiele debonding from the metallie matrix [29, 30]. Even though modern east irons are exposed in eomplex thermomeehanieal loading eon-ditions, their performanee is not fully understood or eonneeted to their mierostruetural features. The effeet of graphite morphology on the thermomeehanieal behaviour of east irons was not studied mueh and when investigated it was mostly related to tensile properties at room temperature [31].
This laek of knowledge may be partially attributed to the eomplex mierostrueture of east irons, whieh is very diffieult eharaeterise and model adequately. This study foeuses on the performanee of a seleeted east iron (CGI) under pure thermal loading and the effeet of graphite morphology.
2. Methodology
2.1. Microstructural characterisation
The eomplex mierostrueture of CGI is the key faetor underpinning the high uneertainty of predieti-ons of its thermomeehanieal behaviour. It eonsists of two materials with a signifieantly different meehani-eal and thermal behaviour, speeifieally, in the form of graphite partieles of different shapes and sizes embedded in a metallie matrix (Fig. 1). Three different shapes are reeognised for graphite partieles: nodular, vermieular and flake. Flake graphite partieles tend to be very long and thin whereas vermieular ones are identified as an intermediate shape between flake and nodular. On the other hand, two different metallie
^ jU * I'V '
...C - '
A v
-J '¿r-CS
Vermicular graphite
■ Flake v
graphite, * > tT V^X; /V
f " A yS fX Nodular
• V «* graphite
Fig. 1. Mierostrueture of eompaeted graphite iron with different shapes of graphite partieles (eolor online)
phases ean eoexist in the matrix: ferrite and pearlite. Ferrite is a duetile eonstituent with a low yield point while pearlite has high strength but not as duetile. Their fraetions are primarily determined by the ma-nufaeturing proeess and the alloying elements added [31-33].
To eharaeterise the different morphologieal features of CGI, a set of images of mierostruetures was produeed using seanning eleetron mieroseopy (SEM) and analysed with the image proeessing software Image J. This tool ean effeetively identify various mierostruetural eonstituents based on their grayseale intensity and ealeulate several geometrieal eharaeter-isties sueh as the fraetion of graphite partieles, as well as their orientations and distanees between them. Statistieal data based on the obtained results ean be then used to eonstruet a statistieally equivalent unit eell [34-36].
In this study, ten images of CGI mierostrueture were analysed. Critieal to the analysis proeess was the applieation of a threshold for the transformation of an image from a grayseale speetrum to a binary one and the adjustment of the outline shape of parti-eles. Herein, an ellipse was seleeted as it allows better eonsisteney together with estimation of nodular-ity—a dimensionless parameter defined as the frae-tion of inelusion area to perimeter. This parameter ranges from zero to one, with values greater than 0.80 eorresponding to nodular partieles and lower to elliptieal. Besides, values lower than 0.20 indieate very thin elliptieal partieles of needle shape, used here to represent the flake partieles (Fig. 2). Following a proeess based on pixel reeognition, various geometrieal features were identified, with an uneer-tainty of 0.5 pixel/mm; the results of the image proe-essing software are presented in Table 1 and Fig. 2.
M
mm
0.00-0.19 0.40-0.59 0.80-1.00 0.20-0.39 0.60-0.79 Nodularity
Fig. 2. Fraetions of graphite partieles ranging with their average nodularity (eolor online)
Table 1. Minimum and maximum values of geometrical parameters of graphite particles in CGI [2]
Volume fraction of graphite, % Perimeter, ^m Area, ^m2 Major axis, ^m Minor axis, p,m
5.2-11.37 3.54-315.88 0.99-6086.96 0.602-67.96 0.52-28.51
Apparently, most inclusions exhibited an average nodularity varying from 0.20 to 0.59 (Fig. 2), supporting the evidence that the graphite particles in CGI microstructure are mostly vermicular [37]. Finally, the volume fraction of nodular graphite was slightly higher compared to that of flakes graphite, indicating that CGI tends to have a microstructure closer to that of ductile iron than the flake one [2]. The microstructural features were incorporated into micromechanical finite element models.
2.2. Microstructure-based modelling 2.2.1. Geometry
In general, a studied volume element encloses a large number of inclusions [38]. However, the complex microstructure of cast irons makes the study of the effect of graphite morphology on the mechanical performance of the material very difficult due to nontrivial shapes of particles and the interaction between them. Hence, the generation of a unit cell with a single inclusion embedded with effective properties assumed for the surrounding metallic matrix to consider the contribution of the remaining particles, is a better option. Considering the volume fraction of graphite in CGI, which is much lower than that of ferrite, the assumption is well supported. Based on the results obtained from microstructural characterisation and the discussed assumptions, a two-dimensional unit cell was generated, comprising a square domain that represents the metallic matrix and an elliptical inclusion that can represent graphite particles of different morphology [2]. As already dis-
cussed, CGI contains different volume fractions of nodular, vermicular and flake graphite particles. Therefore, ellipses with different geometrical characteristics were selected to describe each of these cases of graphite particles. Figure 3 depicts the geometrical features of the model studied. Assuming an average fraction of graphite to be 8.2% (see Table 1), the dimensions of the inclusions were calculated in order the percentage of graphite to be constant throughout.
2.2.2. Constitutive law
In simulations, the metallic matrix is considered as isotropic and ductile; hence, assuming an elasto-plastic behaviour, a classical J2 flow theory of plasticity was used for its constitutive description [39]. On the other hand, graphite is a soft, brittle material; however, there is evidence that it exhibits a limited plastic part; so, the classical plasticity theory was for it as well [26, 39, 40]. The basic constitutive equations used for both materials are presented next with their parameters depicted in Table 2. The ferritic matrix was assumed to have the effective material properties, derived from in house mechanical testing on macroscopic specimens of CGI. It is a common practice in analysis of heterogenous materials to represent their nonhomogeneous constituents with a 'homogenised' (effective) medium, with properties derived from those of the individual phases and microstructure. Additionally, employing the effective properties for the matrix material, the contribution of the remaining graphite particles in the microstructure can be considered.
Fig. 3. Geometry of selected unit cells [2]
Table 2. Constitutive parameters for ferritic matrix and graphite inclusions
Yield point, MPa Yield strain, % Temperature, °C Young's modulus, GPa Poisson's ratio Coefficient of thermal expansion
Ferritic matrix 324.0 0.209 25 150 0.25 [42] 1.2 x 10-5 [43]
316.9 0.195 150
301.2 0.225 300
265.9 0.178 400
257.7 0.179 500
Graphite [41] 27.565 0.184 50 15.85 0.2 2.9 x 10-6
The thermoelastic constitutive law is described using tensorial notation as follows
a. = Cmea +P,. (9-00), (1)
where a. and ekl are the stress and strain tensors respectively, Cijkl is the elasticity tensor, P. are coefficients of thermal stress, 9 and 90 are the current and reference temperatures.
The strain decomposition in an elastic s® and plastic sp. parts is given as
(2)
„tot e
8j =SV
V
Assuming an isotropic hardening response, the yield function is given by
(3)
f (a) ^/-ajaj - (av - Ka),
where ay is the yield stress, K is the tangent modulus, and a. are the components of the deviatoric stress
tensor.
Especially for the matrix material, the yield function is assumed to be temperature-dependent, therefore Eq. (3) was modified to the following form:
(4)
f (a) = J-ajaj -(ay(0)-Ka).
A flow rule was also adjusted, given by d8p = dsp-3 a
(5)
2^3/2 a: a '
where dsp and e"p are the increments of equivalent plastic strain and its rate, respectively.
Due to their brittle nature, graphite particles in CGI tend to debond from the metallic matrix [34]; this decohesion, combined with stress concentration at their sharp edges can lead to crack initiation [44]. The appearance of cracks is soon followed by their coalescence and propagation along the interface between graphite and metallic matrix, eventually leading to total failure [45]. Considering this behaviour of CGI as experimentally observed under mechanical
loading [46], a damage criterion was applied to graphite that causes deletion of elements, for which it was fulfilled. Damage estimation in the numerical code was performed by measuring the value of damage variable at all Gauss points of a finite element. Therefore, an element was considered damaged after ©d became nonzero in any Gauss point and the element was deleted when the set critical value was exceeded at all integration points. The basic equations of the selected damage criterion are [47]:
«d = f d ^ = 1, (6)
A^d =;
Ä Sd) As!
-> 0,
(7)
e~dd(n, sj)
where Sjp is the equivalent plastic strain at the onset of damage, defined as a function of stress triaxiality ^ (ratio of hydrostatic pressure to the von Mises equivalent stress) and equivalent plastic strain rate, sdp. The value of parameter sj was selected to be the strain of graphite at its yield, 0.184%, following the concept that graphite is very brittle and cannot bear any plastic deformations [40]. Hence, was calculated at every increment and the difference between successive increments (A©d) must be positive for the damage criterion to be met.
2.2.3. Model details
The abovementioned unit cell was modelled with four-node plane-strain elements (CPE4). Following a mesh-convergence study, 12218 elements were used, with a total of 24 456 degrees of freedom. In simulations, different boundary conditions (BCs) were applied to the model to investigate their effect on the results. The cases of BCs examined were: (i) fully fixed; (ii) pinned; (iii) periodic. Both fixed and pinned boundary conditions were applied to the four edges of the square domain, constraining both displacements and rotations across both axes in the for-
mer case whereas for the latter one, only displacements were constrained. Apparently, for these two cases, the boundary conditions were straightforward to define; however, this was not the case for the periodic ones. This type of BCs ensures continuity of both displacements and tractions across neighbouring unit cells. Generally, a unit cell can be described as the smallest part of the microstructure that can be used to represent it, in terms of its characteristics, assuming it has a regular pattern. Therefore, the periodic boundary conditions (PBCs) at the cell edges are used in order to simulate accurately the deformation field [12]. Considering two points at any two opposing facing edges of a unit cell of length d, the periodic boundary conditions on them are stated as [48] u(x, d) = u(x) + sd, (8)
t( x, d) = -t( x), (9)
where u and t are the displacement and traction, respectively, and s is the average infinitesimal strain over the volume.
As well known, cast iron is vulnerable to thermal loading due to its heterogeneous microstructure. According to Table 2, the coefficient of thermal expansion for the matrix material is nearly an order of magnitude larger than that of graphite. This mismatch may lead to fracture at the microscale under temperature changes, as graphite cannot deform as much as the matrix and debonds from it. Hence, the application of pure thermal loading can trigger this phenomenon and help to identify the parameters that affecting it. A pure thermal load was applied as a field to the entire unit cell, represented as a linear increase of temperature in the material from 50 to 500°C.
3. Results and discussion
3.1. Evolution of plastic zone in matrix and graphite degradation
The finite-element simulations demonstrated that the morphology of graphite particles played a significant role in the evolution of plastic zone as depicted in Fig. 4 for the three shapes of graphite studied: nodular, vermicular and flake (all graphs are plotted in the undeformed configuration with the graphite domain removed). The equivalent plastic strain at each temperature level was normalised with the plastic strain at failure—0.01107. A full model for the spherical inclusion was analysed instead of a quadrant for direct comparison with the other two shapes.
The mismatch in the coefficients of thermal expansion of the two constituents affected the graphite expansion. This triggered stress concentration in the
matrix area adjacent to the circumference of the inclusion, causing the onset of the plastic deformations in the matrix material (Fig. 4, a). The strains developed were around 10% of the plastic strain at failure. As temperature increased beyond 221°C, stresses above the yield point propagated from the interface to the edges of the unit cell, causing the plasticisation of the entire matrix. Despite full plasticisation, the strain profile in the unit cell was not uniform, as higher plastic stains were still observed in the matrix area around the inclusion (Fig. 4, b). This was to expect, as the plastic strains in matrix declined with the radial distance to retaining this to pattern under further increase of temperature. A similar character had distributions of von Mises stresses (Fig. 5). At 500°C, the strain field started becoming nearly uniform. This could be attributed to the temperature-dependent mechanical behaviour of matrix with increases in temperature affecting its hardening character.
For a vermicular inclusion, restricted graphite expansion due to the mismatch of the coefficients of thermal expansion, induced stresses, mainly in the areas around the covertices of the inclusion, where the interface between matrix and graphite was lengthy, accelerating local plasticisation (Fig. 4, e). At temperature above 220°C, stress concentration, due to limited deformation of graphite, intensified, leading to a fully plasticised unit cell (Fig. 4, f). Still, a large area, with higher plastic strains (compared to the rest of the domain), is noticeable. This area was also observed for a nodular inclusion; however, both its shape and size were different. In this case, it propagated from the covertices of the inclusion along the secondary diagonal up to the unit-cell boundaries (Fig. 4, g). The strains developed in this area were 25% of the strain at failure, whereas in the rest of the domain reached up to 20% of that magnitude. Finally, at 500°C, the phenomenon of plastic strains equalisation took place: as a result, highest plastic strains developed in an elliptical area around the co-vertices of the inclusion (Fig. 4, h).
Similarly, high stresses concentrating around the covertices of the flake particle were also detected, easing the appearance of plastic deformations much earlier compared to the other two graphite shapes, at about 216°C (Fig. 4, i). The entire volume element was fully plasticised after the temperature exceeded 320°C (Fig. 4, j); however, the phenomenon of a nonuniform plastic strain pattern was also observed for a flake particle. The area with higher plastic strains initiated from the middle point of the coverti-ces of the ellipse propagating slightly along the se-
Fig. 4. Evolution of plastic zone in matrix for three characteristic graphite morphologies: nodular (a-d), vermicular (e-h), flake (i-l) (color online)
condary diagonal (Fig. 4, k). At 500°C, a totally uniform plastic strain field was obtained (Fig. 4, l), mainly attributed to the elongated shape of this graphite inclusion. This morphology triggers the development of high stresses in the entire volume element, contrary to the previous two cases, where only a part of the volume was affected.
The distribution of von Mises stresses along the semidiagonal AB, at the same temperature levels listed earlier are depicted in Fig. 5 for the three types of studied particles. Apparently, the stresses were maximum in the matrix area around the inclusion and decreased away from it. It is also evident that there is no high variation at the developed stresses for different temperature levels, mainly because there are no other defects in the domain that would trigger stress concentrations in the matrix.
Next, the onset and evolution of graphite degradation were studied for the three types of graphite particles. As discussed earlier, graphite is a soft and brittle material. It cannot expand as much as the matrix material and, due to the mismatch in coefficients of thermal expansion, debonds from the metallic matrix. This damage mechanism appears only when pure
thermal loading is applied. In this case, stresses developed in the inclusion are in the elastic region, far below the yield point; therefore, no particle fracture is expected and debonding appears first. On the other hand, under combined thermomechanical loads, high stresses can develop inside the inclusion, triggering its fracture. However, thermal cycles might cause an in-particle cracking, as during unloading local volumetric tension could appear even under triaxial compression. Although in-particle cracking can be found after a number of cycles, it never appears before debonding under monotonic thermal loading.
The results of this investigation are presented in Fig. 6 with the temperature damage starts normalised with that of the plasticisation start Tp. The onset and evolution of damage was affected by the morphology of graphite inclusions. Stiffness degradation initiated after plasticisation (T/Tp > 1) for the spherical-shaped graphite particle and caused a reduction in stiffness by almost 19% at the end of thermal loading (500°C). As the shape of the inclusion diverged from the nodular, the degradation of graphites occurred before the matrix plasticisation. For the vermicular particle, the onset of damage was observed earlier compared
Fig. 5. Distribution of von Mises stress along AB for nodular (a), vermicular (b) and flake (c) inclusions (color online)
Fig. 6. Evolution of graphite degradation for three inclusion shapes (color online)
to the spherical one (T/Tp < 1) leading to a final degradation of material stiffness by 22%. In case of the flake particle, its very thin elliptical shape and sharps edges, triggered the onset of degradation significantly earlier compared to the other two shapes: T/Tp = 0.95. The final reduction of stiffness for this graphite morphology reached up to 25%, higher than for the other cases.
In order to further explore the effect of inclusion shape factor on the matrix plasticisation and the onset of damage in graphite, a parametric investigation was implemented. This variable, defined as the ratio of the major axis to the minor one of the particle, ranged from 1 up to 20, as indicated by the statistical analysis of microstructure characterisation. A shape factor of 1 describes a nodular particle, while the
Fig. 7. Effect of inclusion's shape factor on onsets of damage, plasticisation, and maximum stiffness degradation (color online)
values above 6 correspond to flake particles; intermediate values ranging from 3 to 6 are of vermicular ones.
The onset of matrix plasticisation and damage in graphite as well as the total stiffness degradation at the end of thermal excursion to 500°C are presented with respect to shape factor in Fig. 7. The morphology of graphite particles influenced both matrix plasticisation and damage in graphite. Spherical particles (a/b = 1) thanks to their round shape, created a circumferentially uniform stress evolution in the adjacent matrix material, causing a delay in the onset of matrix plasticisation compared to elliptically shaped inclusions. Besides, the first plastic strains in the matrix were found before the initiation of stiffness degradation in graphite for the round-shaped particle. This phenomenon implies a ductile behaviour, as the material exhibited some plastic deformation before graphite particles start degrading, finishing with 22% reduction of their initial stiffness.
On the other hand, elliptical graphite particles (1.5 < a/b < 20), exhibited a totally different behaviour. The matrix material around these particles was plasticised at lower temperature. The edges that became much sharper as the shape factor increased, triggered stress concentrations in the matrix material, causing earlier plasticisation. Additionally, graphite degradation preceded matrix plasticisation in unit cells with elliptical particles. This observation corresponds to a brittle response, since graphite lost gradually its load capacity before any inelastic deformation appeared in the matrix, causing a reduction in cast iron overall strength at the macroscale.
Further, more elongated graphite particles (6 < a/b < 20) followed similar trends in but with some clear differences. It was noted that as the shape factor increased both phenomena took place at lower tem-
peratures. For a/b > 12, the temperatures that both phenomena saturated (neither matrix plasticisation nor graphite degradation occur below 212 and 203°C, respectively). Moreover, the stiffness degradation increased for higher values of the shape factor: the reduction in stiffness did not exceed 23% for particles 3 < a/b < 6; however, for a/b > 6 the stiffness degradation of 24% was reached, with values up to 29% reported for shape factors higher than 12. This trend was induced by the morphology of these particles: as their shape became more elongated and their edges sharper, higher stresses were recorded in the matrix, causing graphite damage at lower temperatures, with a higher total degradation at the end of applied loading.
3.2. Effect of boundary conditions
The effect of boundary conditions on the obtained results is discussed in this section. Generally, for pure thermal spatially uniform loading, two extreme cases are possible: (i) fully free thermal deformations, causing only thermal strains in homogeneous materials; (ii) fully constrained cases, causing only thermal stresses. Many applications have intermediate cases since other parts can also have some thermally-induced deformations. The implementation of periodic boundary conditions is often time-consuming and computationally expensive, as denser meshes could be required to obtain accurate results. Hence, a comparison with simpler BCs was attempted to identify the respective sensitivity of the model. Three-unit cells, containing the three characteristic shapes of graphite particles as shown in Fig. 3 were additionally analysed using two different types of boundary conditions: fixed and pinned. The constitutive behaviour of both the matrix and graphite followed Eqs. (1)-(6).
The levels of temperatures at which either graphite degradation (Td) or matrix plasticisation (Tp) occurred are reported in Table 3 for the three different types of boundary conditions studied—periodic,
Table 3. Effect of different boundary conditions for three different graphite shapes
Graphite shape Periodic Fixed Pinned
Td, °C T 1p' °C Td, °C Tp, °C Td, °C Tp, °C
Nodular 230 221 150 123 149 122
Vermicular 215 218 93 102 95 95
Flake 211 216 69 91 68 90
fixed and pinned. For a spherical particle matrix, plasticisation took place when 21% of the total thermal loading was applied, regardless of using fixed or pinned BCs; graphite degradation for the same graphite shape followed after this, taking place at 26% of the total thermal excursion. For the vermicular inclusion, the corresponding temperature levels for matrix plasticisation were 16 or 15% of the total thermal load for the fixed and pinned BCs, respectively. Graphite degradation preceded matrix plasticisation, as already discussed when elliptical particles are examined, and damage initiated at 14 or 15% of the thermal load applied depending on whether the fixed or pinned BCs were used. Finally, for the flake particle, the temperatures that the first plastic strains appeared in the matrix and damage in graphite was reported corresponded to 14 and 9% of the total thermal loading, respectively, for the fixed and pinned boundary conditions.
Apparently, the results for either fixed or pinned BCs effectively coincide. Both BCs were applied to all edges of the unit cell, hence, not allowing it to deform freely. This restriction induces additional stresses in both constituents, as they were unable to expand, causing both phenomena to initiate earlier. These two BCs produced conservative results, compared to the BCs. In case of the periodic BCs, the edges for the unit cell can deform according to the requirement of the continuity of displacements, leading to the higher temperatures for the appearance of both phenomena in microstructure.
3.3. Modelling graphite as void
Previous findings in the literature on the early debonding of graphite particles from the matrix supported the approach of modelling graphite as a void. As already discussed, this was a dominant approach during past decades; however, it was proven inadequate for some types of loading conditions. During thermal exposure, there is a deformation mismatch between graphite and the metallic matrix leads to decohesion. After full debonding, graphite does not
Table 4. Comparison of three different void shapes for different boundary conditions
Periodic Tcp, Fixed Tcp, Pinned Tcp,
OC OC OC
Nodular 22б 113 113
Vermicular 22б 8б 8б
Flake 22б 77 77
carry any load and interact with ferrite, which affects the material performance.
The assumption of neglecting graphite contribution is discussed in this section with a view to examine its suitability for pure thermal loading. The geometrical characteristics of the unit cells are depicted in Fig. 3 and the three types of boundary conditions presented in the previous section were employed to enable a direct comparison of the obtained results. The temperatures of matrix plasticisation were found for three void shapes and three different boundary conditions (Table 4). The void morphology had no effect on matrix plasticisation when periodic boundary conditions are applied. However, its effect was significant for the fixed and pinned BCs; in these cases, the plasticisation temperature was reduced as the void geometry became more elongated, as a result of concentration of thermal stresses
Moreover, a comparison of the results obtained with the two approaches (Tables 3 and 4) shows that the lowest spread of the results PBCs were employed. Apparently, neglecting the graphite contribution can result in overestimation of temperatures for the onset of matrix plasticisation regardless of the shape examined. Despite this, the difference between the two approaches can be described as acceptable; hence, neglecting graphite can be an option when high accuracy is not required.
The insufficiency of this approach to produce accurate results for PBCs might be associated with the stress triaxiality developed in graphite during loading (Fig. 8). The vermicular and flake graphite particles exhibited lower stress triaxiality compared with the nodular one. Generally, high stress triaxiality (>2-3) is linked with ductile modes of failure such as dimple fracture, while low triaxiality is associated with shear slip. Hence, this approach can be reasonable for high
2 5
0.5 I-■-'-■-'-■-'-■-'-■-
0 100 200 300 400 T, °C
Fig. 8. Average stress triaxiality for three graphite morphologies under periodic boundary conditions (color online)
values of stress triaxiality; however it should not be used for stress triaxiality values below 2, as the obtained results would be underestimated. Low stress triaxiality values are also observed when either pure shear is applied, or the hydrostatic part of the stress tensor is negative, leading to an overall negative value for the ratio. Therefore, the approach of neglecting graphite in any of these cases should be avoided.
On the other hand, modelling graphite as a void with the fixed or pinned boundary conditions showed sensitivity to the particle morphology and larger differences were observed for the two methods. The temperatures of matrix plasticisation were lower than those obtained when graphite was included in the simulations, regardless of the particle shape. So, neglecting graphite when boundary conditions are not periodic is not recommended, since a significant underestimation of the temperatures Although it is not recommended to use, it is still a conservative approach.
4. Conclusions
The effect of graphite morphology on the performance of cast irons under pure thermal loading was investigated in this study. Adopting a microme-chanical approach, a two-dimensional unit cell comprising a single inclusion was generated based on the statistical results, derived from microstructure characterisation. An elastoplastic behaviour was assigned to both constituents while a damage model accounting for the soft and brittle nature of graphite was used. Three known graphite morphologies were selected in order to identify the influence of particle shape on both matrix plasticisation and stiffness degradation of graphite. The effect of boundary conditions on the performance of the unit cell under pure thermal loading was also investigated for these graphite particles. Finally, the accuracy of the approach of modelling the inclusion void was examined under thermal loading.
A few interesting conclusions are derived from this study. The influence of graphite morphology on the onset of both matrix plasticisation and stiffness degradation of graphite was confirmed. In this study, the idealised shape of graphite inclusion was selected to avoid interaction between different effects. Still, irregularity of particle shapes was shown to affect the mechanical behaviour of metal-matrix composite [49]. The presence of very thin elliptical particles deteriorated the performance of cast iron; this effect was more pronounced for higher values of shape fac-
tor. Their shape triggered stress concentration in matrix areas, causing plasticisation at lower temperatures. On the other hand, for more round particles, the developed stresses in the metallic matrix were not as high and, hence, the matrix plasticisation occurred later for vermicular and nodular graphite inclusions. The effect of boundary conditions on modelling the thermomechanical behaviour of cast iron was also examined. Both fixed and pinned BCs produced more conservative results than PBCs. Finally, modelling graphite as a void is an adequate approach when modelling spherical-shaped graphite particles using PBCs. However, neglecting the contribution of elliptical graphite particles should be avoided, as it produced large differences. The model with idealised particle shapes presented here will be used as a reference for the future study with direct incorporation of microstructure.
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Received 05.04.2021, revised 11.05.2021, accepted 12.05.2021
Сведения об авторах
Evangelia Nektaria Palkanoglou, PhD researcher, Loughborough University, UK, E.N.Palkanoglou@lboro.ac.uk Konstantinos P. Baxevanakis, Lecturer, Loughborough University, UK, K.Baxevanakis@lboro.ac.uk Vadim V. Silberschmidt, Professor, Loughborough University, UK, V.Silberschmidt@lboro.ac.uk
