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Interface effects during consolidation in titanium alloy components locally reinforced with matrix-coated fibre composite
Acta Materialia 50 (2002) 4981–4993 www.actamat-journals.com
Interface effects during consolidation in titanium alloy components locally reinforced with matrix-coated fibre composite J. Carmai a 1, K.H. Baik b, F.P.E. Dunne a,∗, P.S. Grant b, B. Cantor b a
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK b Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Received 20 February 2002; received in revised form 11 July 2002; accepted 15 July 2002
Abstract An investigation has been made of diffusion bonding at the interface between a local reinforcing metal matrix composite and a monolithic engineering material. Diffusion bonding occurs during the consolidation of the composite during component manufacture. In this study, the composite is made up from Ti–6Al–4V titanium alloy coated SiC fibres, and the monolithic engineering material is also Ti–6Al–4V, but with a different microstructure. An interface model is presented which takes account of diffusion bonding and which is able to describe the deformation behaviour at the interface between composite and monolithic material during composite consolidation. The model is developed from an existing diffusion bonding theory, and is implemented into finite element software. The finite element simulations, and results of experiments, show that diffusion bonding can lead to localised deformation, the inhibition of consolidation, and a resulting inhomogeneous distribution of consolidated and unconsolidated regions during component manufacture. A further effect of the diffusion bonding is to increase the level of component distortion which results from the constraint imposed on the consolidating composite. The interface model presented enables the simulation of practical forming processes so that process variables such as temperature and pressure can be chosen to ensure appropriate finished component properties. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Consolidation; Metal matrix composites; Diffusion bonding
1. Introduction One of the likely applications of matrix-coated fibre composites is in the local reinforcement of Corresponding author. Tel.: +44-1865-273-140; fax: +441865-273-905. E-mail address: [email protected] (F.P.E. Dunne). 1 Current address: King Mingkut’s Institute of Technology North Bangkok, 1518 Piboolsongkram Rd, Bangkok 10800, Thailand ∗
engineering components. During the consolidation of the local reinforcement within components, the coated fibres can be in intimate contact with the walls of the recess, as shown schematically in Fig. 1, and can therefore diffusion bond to the component. This phenomenon can then inhibit subsequent consolidation resulting in a non-uniform consolidation as shown, for example, in Fig. 2. The top coated fibre located next to the die wall at the interface deforms more than the bottom one, and has diffusion bonded to the die, inhibiting deformation in the coated fibres below it.
1359-6454/02/$22.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 2 ) 0 0 3 0 7 - 5
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Fig. 1.
Schematic diagram showing the interface region between recess and coated fibres.
Fig. 2. Micrograph showing a matrix-coated composite specimen consolidated under constant pressure 20 MPa at 900 °C after 1 h using Ti–6Al–4V die and punch.
The surfaces of the component material and the matrix coating are not entirely flat as shown schematically in Fig. 1. When they are brought into contact, micro-cavities can result. As time proceeds, the asperities on the contacting surfaces are flattened and the micro-cavities reduce in size. When the surfaces are fully bonded (that is, all the micro-cavities are removed), the matrix coating on the bond plane will no longer be able to slide relative to the contacting surface. The interface is, therefore, in a state of ‘sticking friction’. The effects of diffusion bonding at the interface between the bulk and composite materials are,
therefore, significant and must be understood. It is important, from a processing point of view, to have a consolidation process model that is capable of capturing surface interaction effects. In practical forming processes, the composite material is consolidated within cavities machined into the unreinforced matrix material. The unreinforced matrix material also deforms at elevated temperature. The effects of deformable die and punch on consolidation behaviour of the composite are also important for process design. They will also be investigated here. In this paper, diffusion bonding behaviour at the interface of the bulk material and composite is investigated. An interface model has been developed which takes account of diffusion bonding and which is able to describe the surface interaction behaviour at the interface. The interface model is then implemented into a finite element solver by mean of a user subroutine. The investigations of the diffusion bonding effects at the interface are carried out using a micromechanical finite element model. Simulation results are compared with experimental results. The effects of deformable die and punch on the consolidation are also studied using a micromechanical finite element model. In the next section, an overview of diffusion bonding processes and models is given.
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2. Diffusion bonding processes and models Generally, diffusion bonding is used advantageously for joining materials which are difficult to join by conventional fusion welding, e.g. titanium and its alloys [1]. In addition, it is often combined with superplastic forming to produce complex components [2]. The process involves holding the cleaned surfaces to be joined together under an applied pressure at elevated temperature. Application of the applied pressure brings the surfaces into intimate contact. A number of theoretical models for diffusion bonding have been developed [1–7]. They are used to predict the time required to form a bond between surfaces of given surface roughness. The first diffusion bonding model was developed by Hamilton [3]. Garmong et al. [4] developed Hamilton’s approach further to describe a more realistic surface geometry. Their model included the final part of the closure and removal of microvoids as a combined creep and volume-diffusion problem. The closure of microvoids was accomplished by a combination of creep, which was modelled as the shrinkage of a thick-walled plastic sphere under hydrostatic pressure, and interfacial diffusion in which the theory of Coble sintering [8] was adopted. The total bonding time was taken to be the sum of the two sequential stages. Derby and Wallach [1,5] introduced a further refinement of the plane strain model of diffusion bonding. The bonding process was still divided into two stages; the collapse of a triangular ridge, followed by the sintering of cylindrically symmetric voids. Unlike the previous models, all possible bonding mechanisms were included and considered to operate simultaneously. Pilling et al. [2] employed a different geometry from the previous models. They assumed that the surfaces to be bonded consisted of parallel, semicircular grooves, which result from a typical surface finish created by machining or grinding processes. They also assumed that surface diffusion at the bonding temperature was sufficiently rapid to maintain circular cross-section. At the starting point for bonding, the points of contact are assumed to deform instantaneously due to very
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high contact stress until their combined area is sufs ficient to support the applied load, i.e. Af ⫽ . A sy parallel array of cylindrical cavities was then generated. The bonding time was determined from
冕 t
dt ⫽ ⫺2pr20
0
冕 1
1⫺Af dA (dV / dt)total f
(1)
s sy
dV dt is the total rate of closure of the interfacial cavities. The closure rate of the interfacial cavities was determined using the coupled diffusion-power law creep theory of cavity growth developed by Chen and Argon [9] but with the law modified to a cylindV drical geometry. was expressed as a function of dt the material constants, the applied pressure, temperature and the initial surface roughness. It was also developed to take into account the effect of material grain size which is of significance in superplastic materials. This is due to the fact that the grain size can be smaller than the interfacial void size. Hence, several grain boundaries can be involved in mass transfer to sinter the cavity. The cavity closure rate would be correspondingly increased. The grain boundaries were assumed to lie parallel to the bond interface where the spacing between each path is the same as the grain size ‘d’. The predicted bonding time for Ti–6Al–4V was found to be slightly greater than that determined experimentally over a range of temperatures and pressures. Guo and Ridley [6] employed an alternative geometric assumption for the shape of the interfacial cavities; they were assumed to be of lenticular shape. All of the possible mechanisms for diffusion bonding were taken into account apart from the mass transfer in the vapour phase. The effects of grain size on diffusion bonding were also accounted for. Improved agreement with experiments was obtained compared with the earlier models. However, Guo and Ridley’s model requires a number of numerical integrations. All the models reviewed above assume that the where Af is the total area fraction bonded and
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length of the cavity is large and that plane strain conditions therefore apply. In commercial diffusion bonding processes, the sheets to be bonded are welded around their peripheries and placed in a sealed box which can be pressurised resulting in a state of isostatic compression. Plane strain conditions therefore no longer hold. Pilling [7] developed a diffusion bonding model for an isostatic state of stress. The same physical mechanisms of void closure were operative as in Pilling et al. [2] but the kinetics of each process were different. New rate equations describing the kinetics of diffusion bonding under an isostatic state of stress were developed. It was found that on average the rate of void closure during isostatic bonding was a factor of two to three times greater than that for plane strain bonding [7]. A number of existing diffusion bonding models have been reviewed. The earlier model predictions are not entirely satisfactory either because they ignore mechanisms which are important for diffusion bonding processes or they require a change in geometry from the first stage to the second stage which leads to discontinuity in predicted bonding rate at the transition. The subsequent models, in particular those of Pilling et al. [2] and Guo and Ridley [6], give better predictions. Furthermore, their model predictions were verified with the experimental data on the diffusion bonding of Ti– 6Al–-4V which is the material of interest in the present work. The predictions obtained from the model of Guo and Ridley for Ti–6Al–4V are slightly better than those of Pilling et al [2] but the model is more complicated. In order to develop an interface model which takes account of diffusion bonding behaviour, a model will be employed to described the diffusion bonding behaviour at the composite-matrix material interface. The plane strain diffusion bonding model is of interest for the present work since it is assume that the micro-cavities resulting from the roughness of the two contacting surfaces are very long compared with their width. Thus, in this work, we consider the diffusion bonding model of Pilling et al [2] which gives reasonably good predictions for the diffusion bonding of Ti–6Al–4V. The development of the interface model and its finite element implementation are described next.
3. Development of the interface model and its finite element implementation Diffusion bonding between the composite and bulk matrix materials can occur rapidly at elevated temperature. As time proceeds, the bonded area enlarges resulting in higher overall shear strength of the interface. When full intimate contact at every point on the bond plane is developed, the shear strength of the interface will be approximately the same as that of the parent material. The matrix coating at the interface will, therefore, no longer be able to slide relative to the die. It is in a state of sticking friction which can lead to the constraint of the deformation of the matrix coating during consolidation which in turn can inhibit subsequent consolidation. The interaction behaviour between the composite and bulk matrix material is important as it also affects the consolidation behaviour of the composite. An interface model has, therefore, been developed to describe the interaction between the consolidating composite and the bulk material. The diffusion bonding process largely governs the frictional behaviour at the diematrix coating interfaces. The interface model, therefore, takes account of diffusion bonding. By employing a diffusion bonding model, it becomes possible to determine when the surfaces are fully bonded and hence when a state of sticking friction takes place. The bonding time can be calculated using Pilling’s model [2]. The total rate of change of the cavity volume with time is given by
冘 | |
(Ni⫺1) / 2 dV dV dV ⫽ ⫹2 dttotal dt dt 0
||
r
(2)
r⬘
This consists of mass transfer from the bond line and the sum of the mass transfers from each grain boundary intersecting the cavity. The first term is given by
|
dV 16DBd⍀sc ⫽ × dt BAfkT
||
r
1 B⫹r r ⫺ 3⫺ r r⫹B
|
冉 冊 冋 冉 冊 册冋 冉 冊 册
4ln
2
1⫺
r r⫹B
2
(3)
where sc is the contact stress, DB is the grain boundary diffusivity, ⍀ is the atomic volume, δ is
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the grain boundary width, r is the instantaneous radius and B is the diffusion distance defined as 1
DBd⍀|sc| 3 B⫽ AfkTe˙
|
|
(4)
冘| |
(Ni⫺1) / 2
The second term,
0
dV , is the contribution dt r⬘
to cavity closure rate from the intersection between grain boundaries and the cavity surface. If the grain size is smaller than the interfacial cavity size, a number of paths exist by which diffusion can occur to sinter the cavity. The number of paths contributing to the closure of the cavity depends on the intantaneous radius (r) and the grain size (d) as the grain boundaries are assumed to lie parallel to the bond interface where the spacing between each layer is the same as the grain size. Thus, the number Nj of paths contributing to the closure of the cavity at any time is
冉冊
Nj ⫽ 2int
r ⫹1 d
(5)
The rate of change of cavity volume due to mass transfer from each grain boundary intersecting with the cavity can be determined from Eqs. (3) and (4) by replacing instantaneous radius r and the area fraction bonded Af by the apparent cavity radius r⬘ and the apparent area fraction bonded A⬘f in which r⬘ ⫽ 冑r2⫺(jd)2
(6)
r⬘ A⬘f ⫽ 1⫺ r0
(7)
where r0 is the initial cavity radius and j is an Nj⫺1 integer between 0 and . 2 Eq. (1) has been derived for calculating a bonding time for a particular applied stress. In the present problem, the contact pressure between the coated fibres and the die walls varies during the consolidation. It generally decreases with time because the contact area between the coated fibres and the die walls increases with time. Additionally, not all of the surface of the matrix coating at the bond plane is fully bonded at the same time since
冉 冊
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the contact area progressively increases with deformation. Some parts on the bond plane may have already developed full interfacial contact but others may have just come into contact. In order to implement the diffusion bonding model, the area fraction bonded, Af, is employed to identify the state of full bonding, i.e. Af ⫽ 1. By rearranging Eq. (1), the increment in area fraction bonded (dAf) for each time increment (dt) and for a given stress (σc) at that increment may be obtained dAf ⫽
冉
dt (dV / dt)total ⫺2pr20 1⫺Af
冊
(8)
dV is given in Eq. (2). Af is the area fraction dttotal bonded at the beginning of the time increment. Af at the end of the time increment can then be obtained by adding dAf to the current Af. The finite element software, ABAQUS, provides a user subroutine FRIC to enable users to define alternative friction behaviour between contacting surfaces. Thus, the diffusion bonding model discussed here can be implemented into the finite element solver by means of this user subroutine. The increment in area fraction bonded, dAf, is determined using information passed into the subroutine by ABAQUS, i.e. time increment (dt) and contact pressure (σc). The area fraction bonded, Af, is assigned as a solution state variable and is updated every time increment. It is difficult to quantify at what value of Af, relative sliding between the contacting surfaces ceases. However, it is certain that at the fully bonded state, (i.e. Af ⫽ 1), the friction condition between the contacting surfaces becomes ‘sticking’. In the initial development of the interface model, it is assumed that the friction condition is sticking when full interfacial contact is achieved. Otherwise, frictionless sliding is assumed. The value of Af is used as a variable to specify when sticking friction occurs. If Af is less than 0.99999 (Af cannot be exactly equal to 1 for numerical reasons) then the friction condition is specified to be frictionless sliding. If Af is greater than 0.99999 then the friction condition is specified to be sticking. Whilst this appears to be rather crude, it is shown later that it can be reasonably effective. In addition, different
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values of Af at which sticking friction is assumed to occur have also been investigated. 4. Micromechanical model for the interface problem A finite element model has been developed to study the diffusion bonding effects at interfaces between die walls and coated fibres. Modelling a complete specimen is very much computationally prohibitive. Therefore, only a single column of coated fibres, which is in contact with a die wall, is considered. The interface area consists of a rigid die wall and five coated half fibres as shown in Fig. 3. The die is modelled using two-dimensional
rigid elements (left boundary) for which the reference nodes are not allowed to move or rotate. Fibres are assumed to be rigid and are defined as analytical surfaces in contact with the matrix. Perfect bonding is assumed for the fibre–matrix interfaces. Finite element meshes for the matrix material consist of four-noded, two-dimensional plane strain elements. Modelling contact problems in ABAQUS always causes some difficulties especially for contacts between two deformable bodies. Fine meshes are preferable for contact simulations but can give rise to computational cost. However, coarse meshes can lead to unstable analyses. A few trial simulations have been carried out to obtain reasonable numbers of elements for the matrix-coated fibres, each of which contains 150 elements. The matrix is assumed to obey power-law creep behaviour. It has been found from the surface roughness measurement of a matrixcoated fibre that surface roughness of the matrix coating is very small, i.e. in the submicron range [10]. It is believed that the matrix coating surfaces will quickly diffusion bond to one another. Therefore, it is reasonable to assume sticking friction between the matrix–matrix interface. The diffusion bonding behaviour at the die–composite interface is described by the model detailed in the previous section. An upper movable boundary is used for application of the distributed load, simulating the pressure state during the consolidation process. It is modelled using a two-dimensional rigid element. Its movement is restricted to be in the y direction. Frictionless conditions are applied to the interface between the upper movable boundary and the composite. All nodes on the right boundary including reference nodes of each fibre are only allowed to move in the y direction.
5. Theoretical study of parameters which influence friction conditions
Fig. 3. Schematic diagram showing the boundary and loading conditions imposed in the finite element micromechanical model.
It is important to understand the sensitivity of the change in the area fraction bonded to surface roughness geometry, material property and material grain size values. In this section analyses have been carried out to study the effects of
J. Carmai et al. / Acta Materialia 50 (2002) 4981–4993
material diffusivity on the evolution of the area fraction bonded, Af, and hence diffusion bonding time. All simulations were carried out at a constant temperature of 900 °C. A constant pressure of 20 MPa was imposed onto the upper movable boundary generating a macroscopically constrained uniaxial stress state in the deforming material. The yield stress (σy) is assumed to be 54 MPa [11]. Material properties for α Ti and β Ti at 900 °C are given in Table 1. The properties of the alloy were obtained using the law of mixtures. 5.1. Influence of grain boundary diffusivity (DB) on the rate of diffusion bonding Simulations have been carried out for various values of the grain boundary diffusivity (DB). The initial cavity radius (r0) and the grain size were set to be 1 and 10 µm respectively to be physically representative. Since grain size was set to be greater than the initial cavity radius, the only path contributing to the diffusion is from the bond line. Comparisons of the evolution of area fraction bonded (Af) over time at point P, shown in Fig. 3, for grain boundary diffusivity, DB, of 6 × 10⫺8, 3 × 10⫺10, 1.4 × 10⫺9 and 7 × 10⫺9, ⫺11 2 ⫺1 m s are shown in Fig. 4. The first 2.4 × 10 three values were arbitrarily chosen while the last two were taken from Pilling et al. [2] and Hamilton et al. [12] respectively for Ti–6Al–4V. The graphs show that increasing the grain boundary diffusivity, DB, accelerates the rate of change in the area fraction bonded. The simulation with relatively high grain boundary diffusivity (i.e. D B ⫽
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6 × 10⫺8 m2s⫺1) reaches the state of full bonding first. Fig. 5(a)–(c) show profiles of the matrixcoated fibres at the interface after they have been consolidated for 300 s. Fig. 5(a) is obtained from the simulation with the grain boundary diffusivity, DB, of 2.4 × 10⫺11 m2s⫺1. The die and the matrixcoated fibre surfaces have not yet fully bonded to each other. The friction condition at the interface is still frictionless sliding as Af is still less than 1, in particular, Af ⫽ 0.5508. The array of coated fibres uniformly consolidates. The void sizes are approximately the same. Fig. 5(b) is obtained from the simulation with the grain boundary diffusivity, DB, of 1.4 × 10⫺9 m2s⫺1. Some parts of the interface areas have become fully bonded after consolidation for 109 s. At these locations, therefore, a state of sticking friction occurs. The matrix coating of the top coated fibre which has fully diffusion bonded to the die has, as a result, been prevented from moving downward. The upper voids are smaller in area than the lower ones, i.e. V5b ⬍ V4b ⬍ V3b, due to the fact that the movement of the matrix at the interface is constrained by the sticking friction state. This results in localised consolidation. The results for the grain boundary diffusivity, DB, of 6 × 10⫺8 m2s⫺1 are shown in Fig. 5(c). In this case the interface area diffusion bonds more quickly, i.e. within 20 s. Thus, the sticking friction state has been imposed more quickly. This leads to earlier localised consolidation. The coated fibres at the bottom of the fibre array in Fig. 5(c) consolidate less than those shown in Fig. 5(b) whereas the region of the top coated fibre in Fig. 5(c) consolidates more. In addition, the height of
Table1 Material properties for α Ti and β Ti (after Pilling et al. [2]) Properties
α Ti
β Ti
Atomic volume (⍀), m3 Grain boundary width (δ), m Boundary diffusion coefficient (pre exponent), (DB0), m2s⫺1 Boundary diffusion activation energy, (QB), kJ mol⫺1 Lattice diffusion coefficient (pre exponent), (Dv0), m2s⫺1 Lattice diffusion activation energy, (Qv), kJ mol⫺1
1.76 × 10⫺29 5.9 × 10⫺10 6 × 10⫺5
1.81 × 10⫺29 5.72 × 10⫺29 9 × 10⫺6
97
153
8.6 × 10⫺10
1.9 × 10⫺7
150
153
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Fig. 4. Graph showing the variation of area fraction bonded with time at point ‘P’ with average consolidation pressure 20 MPa and at temperature 900 °C for a range of grain boundary diffusivities.
the fibre array in Fig. 5(b) is less than that of Fig. 5(c) since its downward movement has been inhibited earlier. An early state of sticking friction is not desired in the processing of a reinforced composite component, as it leads to non-uniform consolidation and, as shown later, component distortion. An early state of sticking friction can be avoided by reducing the rate of diffusion bonding. The grain boundary diffusivity, DB, is a temperature dependent material property normally given by ⫺Q
DB ⫽ DB0eRTK
(9)
which increases with temperature. As shown previously in Fig. 4, the lower the grain boundary diffusivity, the smaller the rate of change in fractional area bonded, and hence the lower the rate of diffusion bonding. A lower consolidation process temperature therefore reduces grain boundary diffusion and hence inhibits the onset of sticking friction. 5.2. Influence of initial surface roughness on rate of diffusion bonding Comparisons of the bonded area fraction (Af) evolution over time for initial cavity radii of 1, and 5 µm have been obtained from the simulations with of a grain boundary diffusivity DB,
1.4 × 10⫺9 m2s⫺1 and grain size of 10 µm. Fig. 6(a) and (b) show profiles of the matrix-coated fibres at the interface for initial radii of 1 and 5 µm respectively after consolidation for 1400 s. The simulation with a larger initial cavity radius shows a uniform consolidation. On the other hand, the simulation with a smaller initial cavity radius shows highly non-uniform consolidation. At 1400 s, the composite in Fig. 6(b) is nearly fully consolidated whereas only the top void is filled for that in Fig. 6(a). The lower coated fibres are less consolidated. The surface preparation of the parts to be bonded is therefore particularly important. Large initial cavity radius (r0) leads to a slow diffusion bonding at the interface. Small initial surface roughness causes the die wall and the coated fibres to attain full interfacial contact more quickly which can result in unconsolidated dead zones.
6. Comparison of the model predictions with experimental data Consolidation experiments have been conducted by Baik [10] to establish the validity of the model. In these experiments, a Ti–6Al–4V channel die without lubricant was used, into which was inserted unconsolidated composite material. In this way, the diffusion bonding behaviour between the
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Fig. 5. Computed finite element profiles after consolidation for 300 s under a consolidation pressure of 20 MPa at 900 °C for (a) D B ⫽ 2.4 × 10⫺11 m2s⫺1 (b) D B ⫽ 1.4 × 10⫺9 m2s⫺1 (c) D B ⫽ 6 × 10⫺8 m2s⫺1.
Ti–6Al–4V die and the coated fibres could be investigated. The matrix-coated fibres were aligned and inserted into a Ti–6Al–4V channel die to form a composite specimen with square array packing. The radius of the coated fibre is 0.1225 mm while the radius of the fibre only is 0.0705 mm, i.e. the volume fraction of fibres is about 33%. The punch material used in this experiment was a nickel-based superalloy. The punch surface was lubricated. The specimens were then consolidated by uniaxial, vacuum hot pressing under a constant pressure of 20 MPa and at a constant temperature of 900 °C. The test was interrupted after 300 s. The specimen was sectioned after testing and the resulting microstructure is shown in Fig. 7(a). The micrograph shows
that sticking friction has already occurred since the void sizes are different. The top layer of the coated fibres has been deformed more than the lower ones. The same finite element model described earlier was used to simulate the experiment. The surface roughness of the die is about 1 µm. The matrix grains are plate-like with 1 µm width and 10 µm length. The grain size is, therefore, just larger than the cavity radius. Thus, it is assumed that it has no effect on the diffusion bonding. Two values of grain boundary diffusivity (DB), 1.4 × 10⫺9 m2s⫺1 (taken from Pilling et al. [2] for αTi and βTi at 900 °C) and 6 × 10⫺8 m2s⫺1, were used. Fig. 7(a) and (b) show comparisons of the predicted profiles with the experimental micro-
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Fig. 6. Comparison of computed profiles after consolidation for 1400 s under a consolidation pressure of 20 MPa at a temperature of 900 °C obtained from finite element simulations with (a) r 0 ⫽ 1 µm and (b) r 0 ⫽ 5 µm.
graphs for the grain boundary diffusivities of 1.4 × 10⫺9 m2s⫺1 and 6 × 10⫺8 m2s⫺1 respectively. Fig. 7(a) shows that the void areas predicted by the model are comparable with the experimental results but the predicted height of the specimen is smaller than that in the experiment. A better comparison can be seen when using the grain boundary diffusivity of 6 × 10⫺8 m2s⫺1 as shown in Fig. 7(b). However, the predicted height of the specimen is still slightly smaller than that of the experiment. The difference between the measured and the predicted height results from the sticking friction
state which may actually take place before full diffusion bonding occurs at every point on the bond plane (Af ⫽ 1). When the level of area fraction bonded, Af, at which the friction condition becomes sticking, was changed from 0.99999 to a smaller value, better agreement could be obtained. However, it has not yet been possible to quantify the level of the area fraction bonded, Af, at which the friction condition becomes ‘sticking’. It should also be noted that friction effects before diffusion bonding occurs are ignored in the model. Further experimental investigations are required to quantify the frictional behaviour of the materials at the interface prior to diffusion bonding. It is also useful to note that the grain boundary diffusivity is dependent on the proportion of β and α phase [2] at a particular temperature. The composition of each phase changes with temperature. The diffusivity data for Ti–6Al–4V taken from Pilling et al. [2] were calculated from those of the constituent phases by the law of mixtures. The volume fraction of β was assumed to be that given by experimentally determined temperature–volume fraction data for the β-phase. As a result, the grain boundary diffusivity for PVD Ti–6Al–4V may be quite different from that of conventional Ti–6Al–4V. Indeed, microstructural examination shows that PVD Ti–6Al–4V has a much finer grain size and exhibits much higher creep rates than the conventional Ti–6Al–4V [13]. A larger diffusivity ( 6 × 10⫺8 m2s⫺1) for this material may well be justified.
7. The effect of deformable die and punch on consolidation In the manufacturing of partially reinforced composite components, dies and punches can be made of Ti–6Al–4V. They are deformable, at elevated temperature. This can lead to component distortion if pressure and temperature are not appropriate. A finite element model was developed to simulate the experiment, taking account of the deformation of punch and die, which were previously assumed to be rigid. Three columns of five layers of the matrix-coated fibres are modelled as shown in Fig. 8. The finite element meshes for the
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Fig. 7. Comparison of predicted and experimentally observed consolidation of Ti–6Al–4V coated fibres which are in contact with a Ti–6Al–4V die (on the left). Consolidation was carried out for 300 s with an average pressure 20 MPa at 900 °C. In the calculations, diffusivities of (a) 1.4 × 10⫺9 m2s⫺1 and (b) 6 × 10⫺8 m2s⫺1 were used, and sticking friction is specified to occur when the area fraction bonded is equal to 0.99999.
matrix coatings are the same as described earlier. The bottom of the die is in contact with an analytical surface and the friction between them is defined to be sticking. The die–punch interface is specified to be frictionless. The composite is allowed to diffusion bond to the die and punch. Their interaction is described by the interface model detailed in Section 3. The movement of the right boundary is constrained to zero in the x direction. The die and the punch are assumed to deform by power-law creep. The creep parameter and the creep exponent for the die and the punch are taken to be 1.5 × 10⫺6 MPans⫺1 and 1.43 [2] respectively to represent a conventional Ti–6Al–4V material which has a lower creep rate than the PVD Ti– 6Al–4V material. The initial cavity radius is assumed to be 1 µm and the grain boundary diffusivity is taken to be 6 × 10⫺8 m2s⫺1. The other material parameters remain unchanged. Fig. 9 shows a qualitative comparison of a computed profile and a micrograph of the specimen after consolidation for 300 s under a constant
pressure of 20 MPa and at 900 °C. The computed profile shows that the punch as well as the die deform during consolidation. The bottom of the punch and the channel, which are in contact with the coated fibres, become distorted (that is they change in geometry relative to the original geometry). The coated fibres on the far left (close to the die wall) consolidate least. This is because the coated fibres have diffusion bonded to the die wall, thus inhibiting downward movement. The coated fibres which are not in contact with the die wall are still able to move downward. The height of the far right coated fibre column is lower than that on the far left which is located next to the die wall. The predicted profiles of the bottom part of the channel die and the punch are qualitatively comparable with the experimental micrograph. Both the experimental micrograph and the computed profile show that the bottoms of the punch and the die are not flat, and hence that some permanent die and punch distortion has occurred.
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8. Conclusions
Fig. 8. Schematic diagrams showing the region of finite element modelling together with the boundary conditions.
A simple interface model was developed by adopting an existing diffusion bonding model to describe the friction behaviour at the interface between bulk matrix material and matrix coated fibre composites. By consideration of the area fraction bonded between the two contacting surfaces, friction conditions were specified. A micro-mechanical finite element model of a single column of coated fibres, which was located at the interface, was developed. The interaction behaviour between the contacting surfaces was described using the interface model. It was found that the grain boundary diffusivity, initial surface roughness and the material grain size all affect the diffusion bonding, and hence the friction condition at the interface. The higher the grain boundary diffusivity, the higher the rate of change in area fraction bonded, hence the earlier sticking friction between the contacting surfaces was achieved. Large initial surface roughness leads to a longer time to achieve full bonding. Thus, the state of sticking friction is delayed. Diffusion bonding between local reinforcing composite and monolithic matrix has been shown to lead to inhomogeneous consolidation.
Fig. 9. Comparison of level of distortion of die and punch obtained from finite element simulation and micrograph obtained from experiments conducted at a temperature of 900 °C, pressure of 20 MPa, after consolidation for 300 s.
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The finite element predictions were also compared with experimental results. With the grain boundary diffusivity taken from Pilling et al. [2], the void areas predicted by the model are in reasonable agreement with the experimental results but the predicted height of the specimen is shorter than that found in the experiments. The difference is possibly due to the fact that the state of sticking friction occurs before the contacting surfaces attain 100% bonding. However, it has not yet been quantified at which level of area fraction bonded is required for relative sliding between the contacting surfaces to cease. Better comparisons can be obtained using higher values of the grain boundary diffusivity. The effect of deformable die and punch on consolidation behaviour was also investigated. The micromechanical finite element analyses show that consolidation of the matrix coated fibres using a deformable die and punch without a diffusion bonding inhibitor can lead to component distortion.
[3]
[4]
[5] [6] [7]
[8]
[9] [10] [11]
[12]
References [13] [1] Derby B, Wallach ER. Theoretical model of diffusion bonding. Metal Science 1982;16:49–56. [2] Pilling J, Livesey DW, Hawkyard JB, Ridley N. Solid
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state bonding in superplastic Ti–6Al–4V. Metal Science 1984;18:117–22. Hamilton CH. Pressure requirements for diffusion bonding titanium. Titanium science and technology. New York: Plenum, 1973. Garmong G, Paton NE, Argon AS. Attainment of full interfacial contact during diffusion bonding. Metallurgical Transactions A 1975;6A:1269–79. Derby B, Wallach ER. Diffusion bonding: development of theoretical model. Metal Science 1984;18:427–31. Guo ZX, Ridley N. Modelling diffusion bonding of metals. Materials Science and Technology 1987;3:945–52. Pilling J. The kinetics of isostatic diffusion bonding in superplastic materials. Materials Science and Engineering 1988;100:137–44. Coble RL. Diffusion models for hot pressing with surface energy and pressure effects as driving forces. Journal of applied physics 1970;41(12):4798. Chen IW, Argon AS. Diffusive growth of grain-boundary cavities. Acta Metallurgica 1981;29:1759–68. Baik, K.H., 2000. Personal communication, Department of Materials Science, University of Oxford. Schuler S, Derby B, Wood M, Ward-Close CM. Matrix flow and densification during the consolidation of matrix coated fibres. Acta Materialia 2000;48:1247–58. Hamilton CH, Ghosh AK, Mahoney MW. In: Hasson DF, Hamilton CH, editors. Advanced processing methods for Titanium Alloys. Warrendale, PA: TMS-AIME; 1982. p. 129–44. Warren J, Hsiung LM, Wadley HNG. High temperature deformation behaviour of physical vapour deposited Ti– 6Al–4V. Acta Metallurgica et Materialia 1995;43(7):2773–87.
Interface effects during consolidation in titanium alloy components locally reinforced with matrix-coated fibre composite J. Carmai a 1, K.H. Baik b, F.P.E. Dunne a,∗, P.S. Grant b, B. Cantor b a
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK b Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Received 20 February 2002; received in revised form 11 July 2002; accepted 15 July 2002
Abstract An investigation has been made of diffusion bonding at the interface between a local reinforcing metal matrix composite and a monolithic engineering material. Diffusion bonding occurs during the consolidation of the composite during component manufacture. In this study, the composite is made up from Ti–6Al–4V titanium alloy coated SiC fibres, and the monolithic engineering material is also Ti–6Al–4V, but with a different microstructure. An interface model is presented which takes account of diffusion bonding and which is able to describe the deformation behaviour at the interface between composite and monolithic material during composite consolidation. The model is developed from an existing diffusion bonding theory, and is implemented into finite element software. The finite element simulations, and results of experiments, show that diffusion bonding can lead to localised deformation, the inhibition of consolidation, and a resulting inhomogeneous distribution of consolidated and unconsolidated regions during component manufacture. A further effect of the diffusion bonding is to increase the level of component distortion which results from the constraint imposed on the consolidating composite. The interface model presented enables the simulation of practical forming processes so that process variables such as temperature and pressure can be chosen to ensure appropriate finished component properties. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Consolidation; Metal matrix composites; Diffusion bonding
1. Introduction One of the likely applications of matrix-coated fibre composites is in the local reinforcement of Corresponding author. Tel.: +44-1865-273-140; fax: +441865-273-905. E-mail address: [email protected] (F.P.E. Dunne). 1 Current address: King Mingkut’s Institute of Technology North Bangkok, 1518 Piboolsongkram Rd, Bangkok 10800, Thailand ∗
engineering components. During the consolidation of the local reinforcement within components, the coated fibres can be in intimate contact with the walls of the recess, as shown schematically in Fig. 1, and can therefore diffusion bond to the component. This phenomenon can then inhibit subsequent consolidation resulting in a non-uniform consolidation as shown, for example, in Fig. 2. The top coated fibre located next to the die wall at the interface deforms more than the bottom one, and has diffusion bonded to the die, inhibiting deformation in the coated fibres below it.
1359-6454/02/$22.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 2 ) 0 0 3 0 7 - 5
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Fig. 1.
Schematic diagram showing the interface region between recess and coated fibres.
Fig. 2. Micrograph showing a matrix-coated composite specimen consolidated under constant pressure 20 MPa at 900 °C after 1 h using Ti–6Al–4V die and punch.
The surfaces of the component material and the matrix coating are not entirely flat as shown schematically in Fig. 1. When they are brought into contact, micro-cavities can result. As time proceeds, the asperities on the contacting surfaces are flattened and the micro-cavities reduce in size. When the surfaces are fully bonded (that is, all the micro-cavities are removed), the matrix coating on the bond plane will no longer be able to slide relative to the contacting surface. The interface is, therefore, in a state of ‘sticking friction’. The effects of diffusion bonding at the interface between the bulk and composite materials are,
therefore, significant and must be understood. It is important, from a processing point of view, to have a consolidation process model that is capable of capturing surface interaction effects. In practical forming processes, the composite material is consolidated within cavities machined into the unreinforced matrix material. The unreinforced matrix material also deforms at elevated temperature. The effects of deformable die and punch on consolidation behaviour of the composite are also important for process design. They will also be investigated here. In this paper, diffusion bonding behaviour at the interface of the bulk material and composite is investigated. An interface model has been developed which takes account of diffusion bonding and which is able to describe the surface interaction behaviour at the interface. The interface model is then implemented into a finite element solver by mean of a user subroutine. The investigations of the diffusion bonding effects at the interface are carried out using a micromechanical finite element model. Simulation results are compared with experimental results. The effects of deformable die and punch on the consolidation are also studied using a micromechanical finite element model. In the next section, an overview of diffusion bonding processes and models is given.
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2. Diffusion bonding processes and models Generally, diffusion bonding is used advantageously for joining materials which are difficult to join by conventional fusion welding, e.g. titanium and its alloys [1]. In addition, it is often combined with superplastic forming to produce complex components [2]. The process involves holding the cleaned surfaces to be joined together under an applied pressure at elevated temperature. Application of the applied pressure brings the surfaces into intimate contact. A number of theoretical models for diffusion bonding have been developed [1–7]. They are used to predict the time required to form a bond between surfaces of given surface roughness. The first diffusion bonding model was developed by Hamilton [3]. Garmong et al. [4] developed Hamilton’s approach further to describe a more realistic surface geometry. Their model included the final part of the closure and removal of microvoids as a combined creep and volume-diffusion problem. The closure of microvoids was accomplished by a combination of creep, which was modelled as the shrinkage of a thick-walled plastic sphere under hydrostatic pressure, and interfacial diffusion in which the theory of Coble sintering [8] was adopted. The total bonding time was taken to be the sum of the two sequential stages. Derby and Wallach [1,5] introduced a further refinement of the plane strain model of diffusion bonding. The bonding process was still divided into two stages; the collapse of a triangular ridge, followed by the sintering of cylindrically symmetric voids. Unlike the previous models, all possible bonding mechanisms were included and considered to operate simultaneously. Pilling et al. [2] employed a different geometry from the previous models. They assumed that the surfaces to be bonded consisted of parallel, semicircular grooves, which result from a typical surface finish created by machining or grinding processes. They also assumed that surface diffusion at the bonding temperature was sufficiently rapid to maintain circular cross-section. At the starting point for bonding, the points of contact are assumed to deform instantaneously due to very
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high contact stress until their combined area is sufs ficient to support the applied load, i.e. Af ⫽ . A sy parallel array of cylindrical cavities was then generated. The bonding time was determined from
冕 t
dt ⫽ ⫺2pr20
0
冕 1
1⫺Af dA (dV / dt)total f
(1)
s sy
dV dt is the total rate of closure of the interfacial cavities. The closure rate of the interfacial cavities was determined using the coupled diffusion-power law creep theory of cavity growth developed by Chen and Argon [9] but with the law modified to a cylindV drical geometry. was expressed as a function of dt the material constants, the applied pressure, temperature and the initial surface roughness. It was also developed to take into account the effect of material grain size which is of significance in superplastic materials. This is due to the fact that the grain size can be smaller than the interfacial void size. Hence, several grain boundaries can be involved in mass transfer to sinter the cavity. The cavity closure rate would be correspondingly increased. The grain boundaries were assumed to lie parallel to the bond interface where the spacing between each path is the same as the grain size ‘d’. The predicted bonding time for Ti–6Al–4V was found to be slightly greater than that determined experimentally over a range of temperatures and pressures. Guo and Ridley [6] employed an alternative geometric assumption for the shape of the interfacial cavities; they were assumed to be of lenticular shape. All of the possible mechanisms for diffusion bonding were taken into account apart from the mass transfer in the vapour phase. The effects of grain size on diffusion bonding were also accounted for. Improved agreement with experiments was obtained compared with the earlier models. However, Guo and Ridley’s model requires a number of numerical integrations. All the models reviewed above assume that the where Af is the total area fraction bonded and
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length of the cavity is large and that plane strain conditions therefore apply. In commercial diffusion bonding processes, the sheets to be bonded are welded around their peripheries and placed in a sealed box which can be pressurised resulting in a state of isostatic compression. Plane strain conditions therefore no longer hold. Pilling [7] developed a diffusion bonding model for an isostatic state of stress. The same physical mechanisms of void closure were operative as in Pilling et al. [2] but the kinetics of each process were different. New rate equations describing the kinetics of diffusion bonding under an isostatic state of stress were developed. It was found that on average the rate of void closure during isostatic bonding was a factor of two to three times greater than that for plane strain bonding [7]. A number of existing diffusion bonding models have been reviewed. The earlier model predictions are not entirely satisfactory either because they ignore mechanisms which are important for diffusion bonding processes or they require a change in geometry from the first stage to the second stage which leads to discontinuity in predicted bonding rate at the transition. The subsequent models, in particular those of Pilling et al. [2] and Guo and Ridley [6], give better predictions. Furthermore, their model predictions were verified with the experimental data on the diffusion bonding of Ti– 6Al–-4V which is the material of interest in the present work. The predictions obtained from the model of Guo and Ridley for Ti–6Al–4V are slightly better than those of Pilling et al [2] but the model is more complicated. In order to develop an interface model which takes account of diffusion bonding behaviour, a model will be employed to described the diffusion bonding behaviour at the composite-matrix material interface. The plane strain diffusion bonding model is of interest for the present work since it is assume that the micro-cavities resulting from the roughness of the two contacting surfaces are very long compared with their width. Thus, in this work, we consider the diffusion bonding model of Pilling et al [2] which gives reasonably good predictions for the diffusion bonding of Ti–6Al–4V. The development of the interface model and its finite element implementation are described next.
3. Development of the interface model and its finite element implementation Diffusion bonding between the composite and bulk matrix materials can occur rapidly at elevated temperature. As time proceeds, the bonded area enlarges resulting in higher overall shear strength of the interface. When full intimate contact at every point on the bond plane is developed, the shear strength of the interface will be approximately the same as that of the parent material. The matrix coating at the interface will, therefore, no longer be able to slide relative to the die. It is in a state of sticking friction which can lead to the constraint of the deformation of the matrix coating during consolidation which in turn can inhibit subsequent consolidation. The interaction behaviour between the composite and bulk matrix material is important as it also affects the consolidation behaviour of the composite. An interface model has, therefore, been developed to describe the interaction between the consolidating composite and the bulk material. The diffusion bonding process largely governs the frictional behaviour at the diematrix coating interfaces. The interface model, therefore, takes account of diffusion bonding. By employing a diffusion bonding model, it becomes possible to determine when the surfaces are fully bonded and hence when a state of sticking friction takes place. The bonding time can be calculated using Pilling’s model [2]. The total rate of change of the cavity volume with time is given by
冘 | |
(Ni⫺1) / 2 dV dV dV ⫽ ⫹2 dttotal dt dt 0
||
r
(2)
r⬘
This consists of mass transfer from the bond line and the sum of the mass transfers from each grain boundary intersecting the cavity. The first term is given by
|
dV 16DBd⍀sc ⫽ × dt BAfkT
||
r
1 B⫹r r ⫺ 3⫺ r r⫹B
|
冉 冊 冋 冉 冊 册冋 冉 冊 册
4ln
2
1⫺
r r⫹B
2
(3)
where sc is the contact stress, DB is the grain boundary diffusivity, ⍀ is the atomic volume, δ is
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the grain boundary width, r is the instantaneous radius and B is the diffusion distance defined as 1
DBd⍀|sc| 3 B⫽ AfkTe˙
|
|
(4)
冘| |
(Ni⫺1) / 2
The second term,
0
dV , is the contribution dt r⬘
to cavity closure rate from the intersection between grain boundaries and the cavity surface. If the grain size is smaller than the interfacial cavity size, a number of paths exist by which diffusion can occur to sinter the cavity. The number of paths contributing to the closure of the cavity depends on the intantaneous radius (r) and the grain size (d) as the grain boundaries are assumed to lie parallel to the bond interface where the spacing between each layer is the same as the grain size. Thus, the number Nj of paths contributing to the closure of the cavity at any time is
冉冊
Nj ⫽ 2int
r ⫹1 d
(5)
The rate of change of cavity volume due to mass transfer from each grain boundary intersecting with the cavity can be determined from Eqs. (3) and (4) by replacing instantaneous radius r and the area fraction bonded Af by the apparent cavity radius r⬘ and the apparent area fraction bonded A⬘f in which r⬘ ⫽ 冑r2⫺(jd)2
(6)
r⬘ A⬘f ⫽ 1⫺ r0
(7)
where r0 is the initial cavity radius and j is an Nj⫺1 integer between 0 and . 2 Eq. (1) has been derived for calculating a bonding time for a particular applied stress. In the present problem, the contact pressure between the coated fibres and the die walls varies during the consolidation. It generally decreases with time because the contact area between the coated fibres and the die walls increases with time. Additionally, not all of the surface of the matrix coating at the bond plane is fully bonded at the same time since
冉 冊
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the contact area progressively increases with deformation. Some parts on the bond plane may have already developed full interfacial contact but others may have just come into contact. In order to implement the diffusion bonding model, the area fraction bonded, Af, is employed to identify the state of full bonding, i.e. Af ⫽ 1. By rearranging Eq. (1), the increment in area fraction bonded (dAf) for each time increment (dt) and for a given stress (σc) at that increment may be obtained dAf ⫽
冉
dt (dV / dt)total ⫺2pr20 1⫺Af
冊
(8)
dV is given in Eq. (2). Af is the area fraction dttotal bonded at the beginning of the time increment. Af at the end of the time increment can then be obtained by adding dAf to the current Af. The finite element software, ABAQUS, provides a user subroutine FRIC to enable users to define alternative friction behaviour between contacting surfaces. Thus, the diffusion bonding model discussed here can be implemented into the finite element solver by means of this user subroutine. The increment in area fraction bonded, dAf, is determined using information passed into the subroutine by ABAQUS, i.e. time increment (dt) and contact pressure (σc). The area fraction bonded, Af, is assigned as a solution state variable and is updated every time increment. It is difficult to quantify at what value of Af, relative sliding between the contacting surfaces ceases. However, it is certain that at the fully bonded state, (i.e. Af ⫽ 1), the friction condition between the contacting surfaces becomes ‘sticking’. In the initial development of the interface model, it is assumed that the friction condition is sticking when full interfacial contact is achieved. Otherwise, frictionless sliding is assumed. The value of Af is used as a variable to specify when sticking friction occurs. If Af is less than 0.99999 (Af cannot be exactly equal to 1 for numerical reasons) then the friction condition is specified to be frictionless sliding. If Af is greater than 0.99999 then the friction condition is specified to be sticking. Whilst this appears to be rather crude, it is shown later that it can be reasonably effective. In addition, different
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values of Af at which sticking friction is assumed to occur have also been investigated. 4. Micromechanical model for the interface problem A finite element model has been developed to study the diffusion bonding effects at interfaces between die walls and coated fibres. Modelling a complete specimen is very much computationally prohibitive. Therefore, only a single column of coated fibres, which is in contact with a die wall, is considered. The interface area consists of a rigid die wall and five coated half fibres as shown in Fig. 3. The die is modelled using two-dimensional
rigid elements (left boundary) for which the reference nodes are not allowed to move or rotate. Fibres are assumed to be rigid and are defined as analytical surfaces in contact with the matrix. Perfect bonding is assumed for the fibre–matrix interfaces. Finite element meshes for the matrix material consist of four-noded, two-dimensional plane strain elements. Modelling contact problems in ABAQUS always causes some difficulties especially for contacts between two deformable bodies. Fine meshes are preferable for contact simulations but can give rise to computational cost. However, coarse meshes can lead to unstable analyses. A few trial simulations have been carried out to obtain reasonable numbers of elements for the matrix-coated fibres, each of which contains 150 elements. The matrix is assumed to obey power-law creep behaviour. It has been found from the surface roughness measurement of a matrixcoated fibre that surface roughness of the matrix coating is very small, i.e. in the submicron range [10]. It is believed that the matrix coating surfaces will quickly diffusion bond to one another. Therefore, it is reasonable to assume sticking friction between the matrix–matrix interface. The diffusion bonding behaviour at the die–composite interface is described by the model detailed in the previous section. An upper movable boundary is used for application of the distributed load, simulating the pressure state during the consolidation process. It is modelled using a two-dimensional rigid element. Its movement is restricted to be in the y direction. Frictionless conditions are applied to the interface between the upper movable boundary and the composite. All nodes on the right boundary including reference nodes of each fibre are only allowed to move in the y direction.
5. Theoretical study of parameters which influence friction conditions
Fig. 3. Schematic diagram showing the boundary and loading conditions imposed in the finite element micromechanical model.
It is important to understand the sensitivity of the change in the area fraction bonded to surface roughness geometry, material property and material grain size values. In this section analyses have been carried out to study the effects of
J. Carmai et al. / Acta Materialia 50 (2002) 4981–4993
material diffusivity on the evolution of the area fraction bonded, Af, and hence diffusion bonding time. All simulations were carried out at a constant temperature of 900 °C. A constant pressure of 20 MPa was imposed onto the upper movable boundary generating a macroscopically constrained uniaxial stress state in the deforming material. The yield stress (σy) is assumed to be 54 MPa [11]. Material properties for α Ti and β Ti at 900 °C are given in Table 1. The properties of the alloy were obtained using the law of mixtures. 5.1. Influence of grain boundary diffusivity (DB) on the rate of diffusion bonding Simulations have been carried out for various values of the grain boundary diffusivity (DB). The initial cavity radius (r0) and the grain size were set to be 1 and 10 µm respectively to be physically representative. Since grain size was set to be greater than the initial cavity radius, the only path contributing to the diffusion is from the bond line. Comparisons of the evolution of area fraction bonded (Af) over time at point P, shown in Fig. 3, for grain boundary diffusivity, DB, of 6 × 10⫺8, 3 × 10⫺10, 1.4 × 10⫺9 and 7 × 10⫺9, ⫺11 2 ⫺1 m s are shown in Fig. 4. The first 2.4 × 10 three values were arbitrarily chosen while the last two were taken from Pilling et al. [2] and Hamilton et al. [12] respectively for Ti–6Al–4V. The graphs show that increasing the grain boundary diffusivity, DB, accelerates the rate of change in the area fraction bonded. The simulation with relatively high grain boundary diffusivity (i.e. D B ⫽
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6 × 10⫺8 m2s⫺1) reaches the state of full bonding first. Fig. 5(a)–(c) show profiles of the matrixcoated fibres at the interface after they have been consolidated for 300 s. Fig. 5(a) is obtained from the simulation with the grain boundary diffusivity, DB, of 2.4 × 10⫺11 m2s⫺1. The die and the matrixcoated fibre surfaces have not yet fully bonded to each other. The friction condition at the interface is still frictionless sliding as Af is still less than 1, in particular, Af ⫽ 0.5508. The array of coated fibres uniformly consolidates. The void sizes are approximately the same. Fig. 5(b) is obtained from the simulation with the grain boundary diffusivity, DB, of 1.4 × 10⫺9 m2s⫺1. Some parts of the interface areas have become fully bonded after consolidation for 109 s. At these locations, therefore, a state of sticking friction occurs. The matrix coating of the top coated fibre which has fully diffusion bonded to the die has, as a result, been prevented from moving downward. The upper voids are smaller in area than the lower ones, i.e. V5b ⬍ V4b ⬍ V3b, due to the fact that the movement of the matrix at the interface is constrained by the sticking friction state. This results in localised consolidation. The results for the grain boundary diffusivity, DB, of 6 × 10⫺8 m2s⫺1 are shown in Fig. 5(c). In this case the interface area diffusion bonds more quickly, i.e. within 20 s. Thus, the sticking friction state has been imposed more quickly. This leads to earlier localised consolidation. The coated fibres at the bottom of the fibre array in Fig. 5(c) consolidate less than those shown in Fig. 5(b) whereas the region of the top coated fibre in Fig. 5(c) consolidates more. In addition, the height of
Table1 Material properties for α Ti and β Ti (after Pilling et al. [2]) Properties
α Ti
β Ti
Atomic volume (⍀), m3 Grain boundary width (δ), m Boundary diffusion coefficient (pre exponent), (DB0), m2s⫺1 Boundary diffusion activation energy, (QB), kJ mol⫺1 Lattice diffusion coefficient (pre exponent), (Dv0), m2s⫺1 Lattice diffusion activation energy, (Qv), kJ mol⫺1
1.76 × 10⫺29 5.9 × 10⫺10 6 × 10⫺5
1.81 × 10⫺29 5.72 × 10⫺29 9 × 10⫺6
97
153
8.6 × 10⫺10
1.9 × 10⫺7
150
153
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Fig. 4. Graph showing the variation of area fraction bonded with time at point ‘P’ with average consolidation pressure 20 MPa and at temperature 900 °C for a range of grain boundary diffusivities.
the fibre array in Fig. 5(b) is less than that of Fig. 5(c) since its downward movement has been inhibited earlier. An early state of sticking friction is not desired in the processing of a reinforced composite component, as it leads to non-uniform consolidation and, as shown later, component distortion. An early state of sticking friction can be avoided by reducing the rate of diffusion bonding. The grain boundary diffusivity, DB, is a temperature dependent material property normally given by ⫺Q
DB ⫽ DB0eRTK
(9)
which increases with temperature. As shown previously in Fig. 4, the lower the grain boundary diffusivity, the smaller the rate of change in fractional area bonded, and hence the lower the rate of diffusion bonding. A lower consolidation process temperature therefore reduces grain boundary diffusion and hence inhibits the onset of sticking friction. 5.2. Influence of initial surface roughness on rate of diffusion bonding Comparisons of the bonded area fraction (Af) evolution over time for initial cavity radii of 1, and 5 µm have been obtained from the simulations with of a grain boundary diffusivity DB,
1.4 × 10⫺9 m2s⫺1 and grain size of 10 µm. Fig. 6(a) and (b) show profiles of the matrix-coated fibres at the interface for initial radii of 1 and 5 µm respectively after consolidation for 1400 s. The simulation with a larger initial cavity radius shows a uniform consolidation. On the other hand, the simulation with a smaller initial cavity radius shows highly non-uniform consolidation. At 1400 s, the composite in Fig. 6(b) is nearly fully consolidated whereas only the top void is filled for that in Fig. 6(a). The lower coated fibres are less consolidated. The surface preparation of the parts to be bonded is therefore particularly important. Large initial cavity radius (r0) leads to a slow diffusion bonding at the interface. Small initial surface roughness causes the die wall and the coated fibres to attain full interfacial contact more quickly which can result in unconsolidated dead zones.
6. Comparison of the model predictions with experimental data Consolidation experiments have been conducted by Baik [10] to establish the validity of the model. In these experiments, a Ti–6Al–4V channel die without lubricant was used, into which was inserted unconsolidated composite material. In this way, the diffusion bonding behaviour between the
J. Carmai et al. / Acta Materialia 50 (2002) 4981–4993
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Fig. 5. Computed finite element profiles after consolidation for 300 s under a consolidation pressure of 20 MPa at 900 °C for (a) D B ⫽ 2.4 × 10⫺11 m2s⫺1 (b) D B ⫽ 1.4 × 10⫺9 m2s⫺1 (c) D B ⫽ 6 × 10⫺8 m2s⫺1.
Ti–6Al–4V die and the coated fibres could be investigated. The matrix-coated fibres were aligned and inserted into a Ti–6Al–4V channel die to form a composite specimen with square array packing. The radius of the coated fibre is 0.1225 mm while the radius of the fibre only is 0.0705 mm, i.e. the volume fraction of fibres is about 33%. The punch material used in this experiment was a nickel-based superalloy. The punch surface was lubricated. The specimens were then consolidated by uniaxial, vacuum hot pressing under a constant pressure of 20 MPa and at a constant temperature of 900 °C. The test was interrupted after 300 s. The specimen was sectioned after testing and the resulting microstructure is shown in Fig. 7(a). The micrograph shows
that sticking friction has already occurred since the void sizes are different. The top layer of the coated fibres has been deformed more than the lower ones. The same finite element model described earlier was used to simulate the experiment. The surface roughness of the die is about 1 µm. The matrix grains are plate-like with 1 µm width and 10 µm length. The grain size is, therefore, just larger than the cavity radius. Thus, it is assumed that it has no effect on the diffusion bonding. Two values of grain boundary diffusivity (DB), 1.4 × 10⫺9 m2s⫺1 (taken from Pilling et al. [2] for αTi and βTi at 900 °C) and 6 × 10⫺8 m2s⫺1, were used. Fig. 7(a) and (b) show comparisons of the predicted profiles with the experimental micro-
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Fig. 6. Comparison of computed profiles after consolidation for 1400 s under a consolidation pressure of 20 MPa at a temperature of 900 °C obtained from finite element simulations with (a) r 0 ⫽ 1 µm and (b) r 0 ⫽ 5 µm.
graphs for the grain boundary diffusivities of 1.4 × 10⫺9 m2s⫺1 and 6 × 10⫺8 m2s⫺1 respectively. Fig. 7(a) shows that the void areas predicted by the model are comparable with the experimental results but the predicted height of the specimen is smaller than that in the experiment. A better comparison can be seen when using the grain boundary diffusivity of 6 × 10⫺8 m2s⫺1 as shown in Fig. 7(b). However, the predicted height of the specimen is still slightly smaller than that of the experiment. The difference between the measured and the predicted height results from the sticking friction
state which may actually take place before full diffusion bonding occurs at every point on the bond plane (Af ⫽ 1). When the level of area fraction bonded, Af, at which the friction condition becomes sticking, was changed from 0.99999 to a smaller value, better agreement could be obtained. However, it has not yet been possible to quantify the level of the area fraction bonded, Af, at which the friction condition becomes ‘sticking’. It should also be noted that friction effects before diffusion bonding occurs are ignored in the model. Further experimental investigations are required to quantify the frictional behaviour of the materials at the interface prior to diffusion bonding. It is also useful to note that the grain boundary diffusivity is dependent on the proportion of β and α phase [2] at a particular temperature. The composition of each phase changes with temperature. The diffusivity data for Ti–6Al–4V taken from Pilling et al. [2] were calculated from those of the constituent phases by the law of mixtures. The volume fraction of β was assumed to be that given by experimentally determined temperature–volume fraction data for the β-phase. As a result, the grain boundary diffusivity for PVD Ti–6Al–4V may be quite different from that of conventional Ti–6Al–4V. Indeed, microstructural examination shows that PVD Ti–6Al–4V has a much finer grain size and exhibits much higher creep rates than the conventional Ti–6Al–4V [13]. A larger diffusivity ( 6 × 10⫺8 m2s⫺1) for this material may well be justified.
7. The effect of deformable die and punch on consolidation In the manufacturing of partially reinforced composite components, dies and punches can be made of Ti–6Al–4V. They are deformable, at elevated temperature. This can lead to component distortion if pressure and temperature are not appropriate. A finite element model was developed to simulate the experiment, taking account of the deformation of punch and die, which were previously assumed to be rigid. Three columns of five layers of the matrix-coated fibres are modelled as shown in Fig. 8. The finite element meshes for the
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Fig. 7. Comparison of predicted and experimentally observed consolidation of Ti–6Al–4V coated fibres which are in contact with a Ti–6Al–4V die (on the left). Consolidation was carried out for 300 s with an average pressure 20 MPa at 900 °C. In the calculations, diffusivities of (a) 1.4 × 10⫺9 m2s⫺1 and (b) 6 × 10⫺8 m2s⫺1 were used, and sticking friction is specified to occur when the area fraction bonded is equal to 0.99999.
matrix coatings are the same as described earlier. The bottom of the die is in contact with an analytical surface and the friction between them is defined to be sticking. The die–punch interface is specified to be frictionless. The composite is allowed to diffusion bond to the die and punch. Their interaction is described by the interface model detailed in Section 3. The movement of the right boundary is constrained to zero in the x direction. The die and the punch are assumed to deform by power-law creep. The creep parameter and the creep exponent for the die and the punch are taken to be 1.5 × 10⫺6 MPans⫺1 and 1.43 [2] respectively to represent a conventional Ti–6Al–4V material which has a lower creep rate than the PVD Ti– 6Al–4V material. The initial cavity radius is assumed to be 1 µm and the grain boundary diffusivity is taken to be 6 × 10⫺8 m2s⫺1. The other material parameters remain unchanged. Fig. 9 shows a qualitative comparison of a computed profile and a micrograph of the specimen after consolidation for 300 s under a constant
pressure of 20 MPa and at 900 °C. The computed profile shows that the punch as well as the die deform during consolidation. The bottom of the punch and the channel, which are in contact with the coated fibres, become distorted (that is they change in geometry relative to the original geometry). The coated fibres on the far left (close to the die wall) consolidate least. This is because the coated fibres have diffusion bonded to the die wall, thus inhibiting downward movement. The coated fibres which are not in contact with the die wall are still able to move downward. The height of the far right coated fibre column is lower than that on the far left which is located next to the die wall. The predicted profiles of the bottom part of the channel die and the punch are qualitatively comparable with the experimental micrograph. Both the experimental micrograph and the computed profile show that the bottoms of the punch and the die are not flat, and hence that some permanent die and punch distortion has occurred.
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8. Conclusions
Fig. 8. Schematic diagrams showing the region of finite element modelling together with the boundary conditions.
A simple interface model was developed by adopting an existing diffusion bonding model to describe the friction behaviour at the interface between bulk matrix material and matrix coated fibre composites. By consideration of the area fraction bonded between the two contacting surfaces, friction conditions were specified. A micro-mechanical finite element model of a single column of coated fibres, which was located at the interface, was developed. The interaction behaviour between the contacting surfaces was described using the interface model. It was found that the grain boundary diffusivity, initial surface roughness and the material grain size all affect the diffusion bonding, and hence the friction condition at the interface. The higher the grain boundary diffusivity, the higher the rate of change in area fraction bonded, hence the earlier sticking friction between the contacting surfaces was achieved. Large initial surface roughness leads to a longer time to achieve full bonding. Thus, the state of sticking friction is delayed. Diffusion bonding between local reinforcing composite and monolithic matrix has been shown to lead to inhomogeneous consolidation.
Fig. 9. Comparison of level of distortion of die and punch obtained from finite element simulation and micrograph obtained from experiments conducted at a temperature of 900 °C, pressure of 20 MPa, after consolidation for 300 s.
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The finite element predictions were also compared with experimental results. With the grain boundary diffusivity taken from Pilling et al. [2], the void areas predicted by the model are in reasonable agreement with the experimental results but the predicted height of the specimen is shorter than that found in the experiments. The difference is possibly due to the fact that the state of sticking friction occurs before the contacting surfaces attain 100% bonding. However, it has not yet been quantified at which level of area fraction bonded is required for relative sliding between the contacting surfaces to cease. Better comparisons can be obtained using higher values of the grain boundary diffusivity. The effect of deformable die and punch on consolidation behaviour was also investigated. The micromechanical finite element analyses show that consolidation of the matrix coated fibres using a deformable die and punch without a diffusion bonding inhibitor can lead to component distortion.
[3]
[4]
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[12]
References [13] [1] Derby B, Wallach ER. Theoretical model of diffusion bonding. Metal Science 1982;16:49–56. [2] Pilling J, Livesey DW, Hawkyard JB, Ridley N. Solid
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state bonding in superplastic Ti–6Al–4V. Metal Science 1984;18:117–22. Hamilton CH. Pressure requirements for diffusion bonding titanium. Titanium science and technology. New York: Plenum, 1973. Garmong G, Paton NE, Argon AS. Attainment of full interfacial contact during diffusion bonding. Metallurgical Transactions A 1975;6A:1269–79. Derby B, Wallach ER. Diffusion bonding: development of theoretical model. Metal Science 1984;18:427–31. Guo ZX, Ridley N. Modelling diffusion bonding of metals. Materials Science and Technology 1987;3:945–52. Pilling J. The kinetics of isostatic diffusion bonding in superplastic materials. Materials Science and Engineering 1988;100:137–44. Coble RL. Diffusion models for hot pressing with surface energy and pressure effects as driving forces. Journal of applied physics 1970;41(12):4798. Chen IW, Argon AS. Diffusive growth of grain-boundary cavities. Acta Metallurgica 1981;29:1759–68. Baik, K.H., 2000. Personal communication, Department of Materials Science, University of Oxford. Schuler S, Derby B, Wood M, Ward-Close CM. Matrix flow and densification during the consolidation of matrix coated fibres. Acta Materialia 2000;48:1247–58. Hamilton CH, Ghosh AK, Mahoney MW. In: Hasson DF, Hamilton CH, editors. Advanced processing methods for Titanium Alloys. Warrendale, PA: TMS-AIME; 1982. p. 129–44. Warren J, Hsiung LM, Wadley HNG. High temperature deformation behaviour of physical vapour deposited Ti– 6Al–4V. Acta Metallurgica et Materialia 1995;43(7):2773–87.