UDK 621.791.3:62-426:539.55 ISSN 1580-2949 Izvirni znanstveni članek MTAEC9, 39(5)155(2005) GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES WHEN COMPRESSING A REINFORCED BRAZED JOINT GEOMETRIJSKI MODELI DEFORMACIJE OKROGLIH ŽIC ARMATURE PRI STISKANJU ARMIRANEGA SPAJKANEGA SPOJA Borut Zorc1, Ladislav Kosec2 1Welding Institute, Ptujska 19, 1000 Ljubljana, Slovenia 2Faculty of Natural Science and Technology, Aškerčeva c. 12, 1000 Ljubljana, Slovenia agnes.brezovnikŽguest.arnes.si Prejem rokopisa – received: 2004-09-25; sprejem za objavo - accepted for publication: 2005-07-05 A reinforcement that coalesces with the parent metal will considerably improve the toughness, strength, and crack resistance of a brazed joint. Coalescence in a large area is obtained by the application of compression to test pieces during brazing, which results in deformation of round reinforcement wires into flat profiles. Various theoretical geometrical models of wire flattening that might be applicable for an advance estimate of the width of the coalescence between the reinforcement wires and the parent metal are shown and analysed for comparison. Keywords: brazed joint, reinforcement, wire deformation Armatura, ki se je zrasla z osnovnim materialom, občutno poveča žilavost, trdnost in odpornost proti razpokanju spajkanega spoja. Zraščanje na veliki površini dosežemo s stiskanjem vzorcev med spajkanjem. To ima za posledico deformacijo žic armature v ploske profile. Prikazani in primerjalno analizirani so različni teoretični geometrijski modeli sploščenja žice, kar se lahko uporabi za vnaprejšnjo oceno širine zraščenja armaturnih žic z osnovnim materialom. Ključne besede: spajkan spoj, armatura, deformacija žice 1 INTRODUCTION Reinforcements in the form of a plate, a mesh, fibres, and particles are added to a brazed joint in order to compensate for the thermal stresses in ceramics/metal joints, which are a result of different thermal extensions of the materials brazed 1-3. Similar composite joints can also be produced by diffusion welding or solid-state bonding 4,5. As a rule none of the hitherto known methods of reinforcement of a brazing joint eliminates its characteristic imperfections, i.e., low toughness and low resistance to crack initiation and propagation. This is a result of the absence of coalescence of the reinforcement and the parent metal. External loads are transmitted to the reinforcement through the brazing metal; therefore, the reinforcement plays no active role in the joint. The joint properties are determined by the brazing metal and its coalescence with the parent metal. The joint thus reinforced will mainly improve the shear strength, but not the other types of strength. In this case the brazing metal is distributed in the same way as in conventional brazed joints, i. e., across the total joint plane. Thus, it can be concluded that the mechanical properties of the brazed joint depend on the brazing-metal distribution. If the brazing metal, however, is prevented from distributing continuously across the joint plane, and a strong and tough connection with the parent metal is made to form at barrier locations, the imperfections of the brazed joint can be eliminated. On the basis of the above findings a reinforced brazed joint with a reinforcement consisting of parallel wires with a round cross-section (comb system) has been developed 6. Such a reinforcement eliminates all the imperfections of the brazed joint in metal materials simultaneously, provided that it has coalesced with the parent metal. Because of a different distribution, the brazing metal cannot affect the joint properties decisively. They are now dependent on the width of coalescence cn of an individual reinforcement wire with the parent metal (Figure 1): the greater the width, the better the mechanical properties of the joint. Thus, the toughness properties of common brazed joints on low-carbon and austenitic steels brazed with silver brazing filler metal, L-Ag40Cd (DIN 8513), will improve from 10 J/cm2 to as high as 150 J/cm2 if the inserted steel-wire reinforcement coalesces with the parent metal. The carrier of the mechanical properties of the joint will now be the reinforcement and not the brazing metal, which makes the reinforcement play an active role in the transmission of loads across the joint. This, however, was not a characteristic of the brazed joints known till now. The coalescence between the reinforcement and the parent metal occurs in the molten pool during diffusion brazing, in which case the brazing-metal eutectic MATERIALI IN TEHNOLOGIJE 39 (2005) 5 155 B. ZORC, L. KOSEC: GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES Figure 1: Scheme of a reinforced, brazed joint with a coalesced reinforcement of parallel flattened wires with a round cross-section Slika 1: Shema armiranega spajkanega spoja z zraščeno armaturo iz vzporednih sploščenih žic s prvotno okroglim presekom transforms into the solid solution. The process is controlled by diffusion, e. g., of phosphorus from the nickel-base brazing metal BNi-7 (Figure 2A) or copper-base brazing metal BCuP-2 (Figure 2B) into the parent metal and the reinforcement. In diffusion brazing, the brazing metal plays an active role in the solid and tough coalescence between the reinforcement and the parent metal. Another way of coalescing the two is by means of solid diffusion welding. In this case the brazing metal does not participate in the coalescence between the reinforcement and the parent metal. It is vital to remove the brazing metal from the coalescence location, which may be achieved by compressing the test pieces. This process is controlled by diffusion between the reinforcement and the parent metal and the processes of recrystallisation and growth of the crystal grains beyond the contact surface between the reinforcement and the parent metal. An example of this is the brazing of steel with a silver brazing metal and a steel reinforcement (Figure 2C). In order to obtain the required width of the coalescence between the reinforcement and the parent metal, the test pieces should be compressed during brazing so that the wires concerned will flatten or get impressed into the parent metal. If the reinforcement is more ductile, i. e., deformable, than the parent metal, the wires will flatten with a negligible impression into the parent metal (Figure 2). The state of the flattened wires in the joint is similar to that in the solid bonding of ceramics or glass with a wire or a wire ring 7,8, where the ceramic or glass part is absolutely nondeformable. In practice it is vital to control the width of the coalescence between the reinforcement and the parent metal cn with the known joint compression to dimension b. This is possible if all the mutually related geometrical relations of the deformed wire are known. If the width of the coalescence of the individual wire with the parent metal is known, the mechanical properties of the reinforced brazed joint can be estimated in advance. 2 MODELS In the field of welding, there are numerous mathematical, statistical, and physical-chemical models 156 describing welding processes. A number of different approaches and methods are known for preparing models, but in practice two of them are preferred, i. e., the statistical one and the physical one. In engineering metallurgy, models for the calculation of forces and stresses in deformation are known. Figure 2: Flattened wires of the reinforcement in a reinforced brazed joint (A – brazing material: BNi-7, reinforcement and parent metal: austenitic steel AISI 304, h/b ˜0.36; B – brazing material: BCuP-2, reinforcement and parent metal: copper, h/b ˜ 0.33; C – brazing material: BAg-5, reinforcement: steel with 0.7 % carbon, parent metal: steel with 0.16 % carbon, h/b ˜ 0.28) Slika 2: Sploščene žice armature v armiranem spajkanem spoju (A – spajka: BNi-7, armatura in osnovni material: avstenitno jeklo AISI 304, h/b ˜0,36; B – spajka: BCuP-2, armatura in osnovni material: baker, h/b ˜ 0,33; C – spajka: BAg-5, armatura: jeklo z 0,7 % ogljika, osnovni material: jeklo z 0,16 % ogljika, h/b ˜ 0,28) MATERIALI IN TEHNOLOGIJE 39 (2005) 5 B. ZORC, L. KOSEC: GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES In our case, the geometrical relationship between the deformed wire and the known compression is of primary importance since the mechanical properties of the reinforced brazed joint can be controlled. In the literature, such an approach has not yet been found. Geometrical models were prepared on the basis of the following assumptions: • the cross-section S of a circle is the S of a flat profile, • during compression the wires only flatten and do not elongate; this is the actual state due to a much greater wire length l in comparison to its diameter d (l/d >> 1), • only the wires, and not the parent metal, will deform. Figure 3 shows the deformation of a round wire into a flat profile. The active bearing cross-section of each wire in the joint, S1, is the rectangular part determined by flattening cn and the height b of the profile, which is equal to the joint width. The inactive parts of the flat profile are the rounded ends, S2. During wire compression the bearing cross-section S 1 transforms from an upright rectangle (cn < b) through a square (cn= b) into a lying rectangle (cn > b, in the case of strong compression: cn >> b). With the increase in cn the bearing cross-section S1 increases too at the expense of the inactive part S2. In other words, the inactive cross-section of the round wire S may transform, by means of a strong compression, into the active cross-section S 1 ˜ S. The experimental results obtained with the reinforced joints (Figure 2) in copper and different steels show that the area S2 has a shape similar to a segment but very rarely to a semicircle. This is to say that the rounding-off radius is greater than b/2, and the area S2 is smaller than a semicircle with the radius of b/2. The area S2 is presumably semicircular only at the beginning of compression when cn << b. A similar shape is also obtained in the case when a square wire is compressed 7,8. Several models of the deformation of the round-wire cross-section into a flat profile were prepared, i. e., • a model with a semicircular area S2 (model index: n = 1), • a model of a rectangle (model index: n = 2), • a model with a segment area S2 (model index: n = 3), • a model with a parabolic ending of the area S2 (model index: n = 4), • an empirical mathematical model (model index: n = 5). 2.1. Model with a semicircular area S2 The model is simple and represents a boundary condition with regard to the segment. With the compression of the wire and the joint, respectively, from d to b a semicircle always has a larger area S2 than a segment, which gives the smallest width c1 and the largest width a1. On the basis of Figure 3 it can be stated: Figure 3: Geometrical relations between the circle and the flat profile (n = model index) Slika 3: Geometrijsko razmerje med krogom in ploščatim profilom (n = indeks modela) S = S +2S;hn From this it is obtained: c =c =— (d 2 -b 2 ) = —-0.785b n 1 4b 4b an = a1 = c1 +b pd2 4b + 0.215b (1) (2) 2.2 Model of a rectangle The model is unreal with regard to the actual state, but it represents another boundary condition with regard to the segment. In the compression of the wire and the joint, respectively, from d to b we have only the rectangular bearing cross-section S1, which gives the largest width c2. The model gets more real with a very strong compression when c2 >> b. On the basis of Figure 3 it can be stated: S = S1;hn =h2 =0 From this it is obtained: pd2 4b (3) (4) 2.3 Model with a segment area S2 According to reference 9 and Figure 4 the approximate calculation of the segment area S2 equals S h 3 (6b + 8t) 15 (5) MATERIALI IN TEHNOLOGIJE 39 (2005) 5 157 B. ZORC, L. KOSEC: GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES and + h 2 (6) The approximate formula of the segment area S2 does not depend on the radius of the relevant circle R and the angle q>, which both determine the length of the segment arc (see Figure 5). Both parameters are the unknown and are difficult to choose optionally. The unknown width of protuberance h3 is easier to determine since it varies between 0 < h3 < b/2 (half-height of the profile and half-width of the joint respectively). On the basis of Figure 3, the area of the flat profile in Figure 4 is: S = 2--S 1- + 2S2 By taking into account Eq. (6) and Eq. (5) it is found that the total width of flattening is (7) (8) With the selected dimensions h3 in the above given range, a number of curves for c3 are obtained. They vary between c1 (item 2.1) and c2 (item 2.2). Eq. (7) indicates that in selecting h3 = 0 flattening c3 = c2 is obtained, which is the model of a rectangle. In selecting h3 = b/2, however, a model with a semicircular area is obtained, which is hard to observe directly due to the approximate segment area (Eq. (5)). But it is valid if h3 = b/2 is inserted into the area S2 (from Eqs. (5), (6), and Figure 4) and if a real number is inserted for b. From this it is found that the double area S2 equals the area of the circle with a diameter b, which makes a model with a semi-circular area: nd2 2h3 4b 15b 6b + 8- ifbY c3 +2h3 2S 2) 15 6b + 8 nb2 2.4 Model with a parabolic ending of the area S2 The model was prepared on the basis of Figure 5 with a sector. The flat-profile cross-section S consists of three partial areas, i.e.: • S1 - area of the rectangle between the two centres of the circles O 1 and O2 • S2 - area of four right-angled triangles between the sector and the rectangle • S3- area of the two sectors. The cross-section of the flat profile is equal to S = S1 + S2 + S3. S = 4b(a-R) + 2b Jr2 -b2 + 2R2 arcsin-4- (9) 4 4 -V 4 R The cross-section of the flat profile can also be expressed as S = 4ab4-b2f(x) (10) With the theorem of altitude, which indicates the relationship between the altitude and the longest side of a right-angled triangle, in accordance with Figure 5 (below), R and, consequently, x are expressed as functions of a, c, and b4. If x is treated by the half-angle tangent formula (universal substitution), the expression tan a = a-c/b4 is obtained, where a is the angle between the half-altitude of the compressed wire and the half of its arc (Figure 5). If instead of function f a new function g taking into account this dependence is used, we get: Figure 4: Model of the flat profile with a segment Slika 4: Model ploščatega profila s krožnim odsekom Figure 5: Model of the flat profile with a sector being a basis for a parabolic rounding Slika 5: Model ploščatega profila z izsekom, ki jeosnova za parabolično zaokrožitev 2 t 158 MATERIALI IN TEHNOLOGIJE 39 (2005) 5 B. ZORC, L. KOSEC: GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES S = 4ab 4b42g (a-c | (11) The selected function g is defined by the equation g(tan a) =f(sin 2a) and takes the following form: g(u) 1 + 3u2 1 + u 2 2 arctan u (12) y When approximating this function we take the new function: and require a minimum of the integral 2 J(c+yź+dź2-g(£) ) d By minimizing the integral using the method of least squares, coefficients /3, y, ņ can be calculated: ß = -0.0207023, y = 1.5924766, ō = -0.6956118 A quadratic approximation to the fundamental equation is: S = 4ab4 - b42 ß+7 b4 c\ J a-c \ + ō\ ) \2 b4 ) (13) By solving Eq. (13) and taking into account the total dimensions of the flattened wire, the final equations of the profile widths a4 and c4 are obtained: -(72 4/3Ō) — + ōnd 2 4 4ōb yb izd2 nd2 4b 20 4b 0.92b ¦ +0.223b (14) (15) It is evident that a very complicated method provides a similar result to the model with a semicircular area. If a different function g was selected, the coefficients /3, y, and (5 would change and a different result would be obtained. The method allows a free choice of the form of function g. It turned out that the equation obtained for c4 was the basis of a simple calculation of the active bearing profile width that can replace all the previously shown models. 2.5 Empirical mathematical model The model is based on the model with a parabolic end surface S2 (item 2.4). In Eq. (15) y/20 is a constant with a value of-1.144659. Equation (15) can be written in a general form cn =c5 =a4 -k-b; 0.223 1 (Figure 8). That is to say that the other models approximate the model of the rectangle. For example, when flattening a 1.0 mm wire compressed to a profile height b of 0.2 mm, the ratio c n/an with other models amounts to 0.94-0.98. The important smallest width of wire flattening cn, which determines the coalescence width of the flattened reinforcement wires and the parent metal in a reinforced brazed joint, offers the model of a parabola, and the greatest width the model of a rectangle. The greatest width of wire flattening an offers the model of a parabola, and the smallest width an the model of a rectangle. Consequently, the model of a parabola shows the lowest ratios cn /an (Figure 8) and the highest ratio Figure 6: Width of wire flattening cn (active bearing width of profile) as a function of wire compression to b (height of profile); 1, 2, 3, 4 – model indices Slika 6: Širina sploščenja žice cn (aktivna nosilna širina profila) kot funkcija stiskanja žice na b (višina profila); 1, 2, 3,4 – indeksi modelov u u 0 2 MATERIALI IN TEHNOLOGIJE 39 (2005) 5 159 B. ZORC, L. KOSEC: GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES Figure 7: Width of wire flattening a (max. width of profile) as a function of wire compression to b (height of profile); 1, 2, 3, 4 – model indices Slika 7: Širina sploščenja žice a (maksimalna širina profila) kot funkcija stiskanja žice na b (višina profila); 1, 2, 3,4 – indeksi modelov between the protuberance width and the profile height hn/b. The ratio hn/b is constant with all the models, regardless of the degree of flattening and the initial wire diameter. The ratio hn/b amounts to 0 for the rectangle, to 0.2 for the segment h3 = b/5, to 0.25 for the segment h3 = b/4, to 0.33 for the segment h3 = b/3, to 0.5 for the semicircle h1 = b/2, and to 0.57 for the parabola. The useless area of the flat profile S2 is somewhat more pointed with the model of a parabola (h4/b > 0.5), but at the same time it is the largest since the ratio cn/an is the lowest. The models differ in the profile width an much less than in the profile width cn (Figures 6 and 7). For example, for the 1.0 mm wire the ratios a2/a4 between the model of a rectangle, which gives the smallest an = a2, and the model of a parabola with the highest an = a4 amount to 0.813 (with b = 0.9 mm), 0.934 (with b = 0.5 mm), and 0.989 (with b = 0.2 mm). There is almost no Figure 8: The ratio c/a as a function of wire compression to b (wire d = 1.0 mm); 1, 2, 3, 4 – model indices Slika 8: Razmerje c/a kot funkcija stiskanja žice na b (žica d = 1,0 mm); 1, 2, 3,4 – indeksi modelov 160 Figure 9: Width of wire flattening c5 (width of profile) calculated using the empirical mathematical model Slika 9: Širina sploščenja žice c5 (širina profila), izračunana z uporabo empiričnega matematičnega modela difference between an = a1 (semicircular model) and an = a4 (model of a parabola); therefore, the curves practically coincide. The ratios c4/c2 between the model of a parabola, which gives the smallest cn = c4, and the model of a rectangle with the highest cn = c2 amount to 0.049 (with b = 0.9 mm), 0.706 (with b = 0.5 mm), and 0.953 (with b = 0.2 mm). The model of a parabola is close to the model of the semicircle since the ratios c4/c1 amount to 0.833 (with b = 0.7 mm), 0.942 (with b = 0.5 mm), and 0.993 (with b = 0.2 mm). With little compression the model of a parabola differs significantly from the other models and gives very small values of cn = c4 (e.g., the ratio c4/c1 = 0.259 with b = 0.9 mm). Consequently, it is the least real in the first phase of compression. For the calculation and control of the profile width cn, an empirical-mathematical equation can substitute for all the geometrical models (Figure 9). The situation is practically the same as in Figure 6. The width c5 is calculated from a4 (model of a parabola) because, according to the model of a parabola, the dimension a is calculated independently from the dimension c, which is, however, not the case with the other models. With the other models, first c is calculated and from this it is possible to calculate dimension a. The differences between the models obtained with weak compression reduce significantly with strong compression. With the mathematical models the ratio hn/b is constant, regardless of the degree of wire compression. Experimental measurements showed different ratios hn/b, i.e. ranging between 0.20 and 0.48, depending on the degree of flattening. The latter are in the range of the models presented. The main cause should be looked for in different conditions (temperature, rate of deformation, friction) of wire flattening due to the use of different brazing materials having different melting points. Differing materials are also prone to flow differently under the same conditions. It is known that the deformation at high MATERIALI IN TEHNOLOGIJE 39 (2005) 5 B. ZORC, L. KOSEC: GEOMETRICAL MODELS FOR THE DEFORMATION OF ROUND REINFORCEMENT WIRES temperatures is more continuous, which indicates that a metal will deform in a similar way as a very viscous substance. It is known too that the resistance to deformation increases with a higher rate of deformation and a decrease in the temperature. During the formation of the reinforced brazed joint some of these parameters were not controlled because interest was focused on the influence of the coalescence of the reinforcement and the parent metal on the mechanical properties of the brazed joint. The deformability of the wires being compressed became the focus of our attention when a question was raised as to whether it is possible to predict the width of the coalescence cn between the wire and the parent metal. Consequently, the above-mentioned models were prepared. Hopefully, the validity of one of the models in dependence of the parameters of wire deformability or a combination of the models with regard to the degree of wire flattening will be established. 4 CONCLUSIONS The models for a round wire flattening into a flat profile by compression offer the basis for experimental studies to show which model gives the most realistic description of the practical situation. The boundary models and the least real ones are the models of a rectangle and that of a parabola. The models with the profile width an agree very well, while with the profile width cn there are major differences, particularly in the first compression phase. The differences between the different models reduce if stronger compression is applied. Thus, all the models approximate the same point. All the geometrical models shown have a constant ratio between the protuberance width and the profile height hn/b regardless of the degree of wire compression. Experimental studies will show whether this is really the case. Because a round wire is compressed, the beginning of the compression process for the wire is best described by a model with a semicircular area S2, whereas later on the model with a segment area S2 is decisive since the experimentally obtained ratios between the protuberance width and the profile height hn/b are < 0.5. For an easy calculation of the profile width cn an empirical mathematical model with the empirically selected constant k ranging between 0.223 and 1.01 was elaborated. NOTATION a maximum width of the profile b height of the profile c active bearing width of the profile d wire diameter h width of protuberance k constant l wire length R radius of the sector with the model with a parabolic ending S cross-sectional area of a circle and flat profile S1, S2, S3 partial cross-sectional areas of the flat profile a angle between the half-altitude of the compressed wire and the half of its arc ß, y, ō coefficients ACKNOWLEDGEMENT The author wish to thank to mag. Janez Barbič, univ. dipl. mat., for the help in the preparation of the model in Chapter 2.4. 5 REFERENCES 1 Folley, G., Andrews, D. J. Joining ceramics to metals by brazing. In Proceedings of the 3rd Int. Conf. on Brazing, High Temperature Brazing and Diffusion Welding, DVS-Verlag GmbH, Düsseldorf, Aachen, 1992, 258–263 2 Mirski, Z. Composite brazed joints with sintered carbides. 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