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Brazing residual stresses in components of different metallic materials
Materials Science and Engineering, A 174 ( 1994 ) 95-101
95
Brazing residual stresses in components of different metallic materials Klaus Bing a, Bernd Eigenmann a, Berthold Scholtes b and Eckard Macherauch a "~Institutfiir Werkstoffkunde L Universitiit Karlsruhe (TH), W-7500 Karlsruhe (Germany) blnstitut fiir Werkstofftechnik, Metallische Werkstoffe, Gh Kassel-Universitiit, W-3500 Kassel (Germany) (Received March 8, 1993;in revised form June 15, 1993)
Abstract Brazing residual stresses are a consequence of differences between the thermal expansion coefficients of the brazed partners and the solder. In this study, components of copperr, austenitic steel and a nickel-base alloy brazed on ferritic steel are investigated. Using simplifying assumptions, the resulting stress states are estimated and compared with experimental X-ray stress analyses and curvature measurements. This gives clear evidence of complicated inhomogeneous plastic deformations in the compounds, which occur during cooling down from the brazing temperature. In addition, X-ray residual stress determinations allow a quantitative description to be given of edge effects in the compounds examined.
1. Introduction
Cu).
trated schematically in Fig. 1 for a two-layer compound of the components A and B, with aA th > aB th. At the brazing temperature, the compound is stress flee. Cooling to room temperature causes different thermal shrinkages of the individual layers; however, the shrinkage is hindered by the bonding condition at the interface, as shown in the right-hand part of Fig. 1. As a consequence, an overall bending moment of the compound results, as do local bending effects, which are most pronounced on the side faces of the components near the interface. To achieve strain compatibility at the interface, as well as equilibria of the forces and momenta, a relatively complex residual stress state develops. In the case of a compound with components of approximately equal thickness and similar elastic behaviour, tensile (compressive) residual stresses near the interface of layer A (B) and compressive (tensile) residual stresses at the surfaces of the components A (B) occur. In addition, compressive (tensile) residual stresses develop on the side face of layer A (B) near the interface perpendicular to the plane of the plates.
2. General aspects of brazing residual stresses
3. Experimental details
In brazed multilayered compounds with different thermal expansion coefficients ~ith of the individual layers i, residual stresses of the I (macroscopic residual stresses) kind develop when cooling from the brazing temperature to room temperature [1, 2]. This is illus-
Square plates of Ck45 with dimensions 20 mm × 20 m m x 5 mm were brazed to plates of X10CrNiMoTil810, NiCr22Co12Mo9 or copper of dimensions 20 mm x 20 mm but with varying thicknesses. The brazings were carried out with foils of an AgCu
Brazing is a well-established joining method for materials with different properties, regardless of their shapes. After brazing and cooling down to room temperature, distortions and complex residual stress states always occur in the compounds, and these depend on the mechanical properties and the thicknesses of the two components, as well as on the difference between the thermal expansion coefficients and the type of brazing material. Since residual stresses affect the strength of the joints and distortions change the shape of the compounds, a quantitative knowledge of both factors is of great importance for practical purposes. This paper deals with analytical and experimental determinations of the residual stress states and distortions in brazed compounds consisting of differently sized plates of a plain carbon steel (German grade Ck45) and plates of a stainless steel (German grade X10CrNiMoTi1810), a nickel-base alloy (German grade NiCr22Co12Mo9), or pure copper (99.9 vol.%
0921-5093/94/$7.00
© 1994 - Elsevier Sequoia. All rights reserved
K. Bing et al. / Brazing residual stresses
96 brazing temperature ~
room temperature
TABLE 1. Parameters used for X-ray residual stress determinations
Material
0t~h > [l~h
Lattice planes
Ck45 X10CrNiMoTi1810 NiCr22Col 2Mo9 Cu (99.9 vol.%)
E Ihk~l v Ihktl
{hkl}
20o (deg)
(GPa)
{211} {220} {220} {220}
156.4 129.5 129.5 127.5
210 207 224 137
0.28 0.28 0.30 0.34
Fig. 1. Thermally induced distortions and corresponding signs of residual stresses in an idealized two-layer compound shown schematically.
braze alloy (AgCu28) 0.1 mm thick at 1050 °C in the case of X10CrNiMoTi 18 10 and at 850 °C in the other cases. The residual stresses were determined by X-ray diffraction using a computerized ~p diffractometer of the Karlsruhe type. The {hkl} planes of the individual materials shown in Table 1 were investigated with Cr K a radiation. The appertaining 2 00 values are listed in the third column. The residual stresses were calculated according to the sin 2 ~0 method for X-ray residual stress determinations [3], using the {hkl}-dependent X-ray values of Young's modulus Ethk/I and Poisson's ratio v/hkllsummarized in the last two columns of Table 1. The elastic anisotropy was taken into account by averaging the elastic constants calculated from singlecrystal elastic constants [4] according to the Voigt and Reuss models. In addition to the surface residual stresses of the compounds, the residual stress distributions over the thickness of the components were also determined in the centre of the plates by successive electrolytical removal of layers of material and subsequent residual stress measurements. The measured values were corrected for changes in the residual stress equilibrium resulting from materials removed [5]. To obtain information about the brazing-induced distortions, curvature measurements were carried out on the surfaces of the compounds, using a perthometer (Perthen GmbH, G6ttingen).
4. Analytical estimation of brazing residual stresses If a multilayered plate of infinite size and different thermal expansion coefficients a~th, of the components is cooled from an initially stress-free state at an elevated temperature T* to room temperature, the distribution of thermal residual stresses can be calculated analytically. This is achieved by assuming a linear elastic materials behaviour of the individual layers i and using Bernoulli's hypothesis of bending; the strains are taken as proportional to the distance from the
neutral axis and inversely proportional to the radius of curvature [6]. Neglecting the temperature dependence of ai th, Young's moduli Ei and Poisson's ratios vi, the surface parallel residual stresses of any component i can then be determined over their thickness as a function of zg, referring to the central plane of each individual layer as
(Yx(Zi):m l-
(
EiO'l-zi'l- (~it h A T
(1)
R
with e~° the strains in the centre planes of the components, R the bending radius of the compound and A T = T * - T2oo being the temperature difference between the solidification temperature T* of the braze alloy and room temperature. The materials data for the calculations were averaged over the temperature range 20 °< T< 500 °C (see ref. 6) and the values are listed in Table 2. Since data for the real braze alloy AgCu28 were not available, the values for pure silver have been used. The residual stresses, estimated according to eqn. (1) and normalized to the temperature difference A T, are shown in Fig. 2 as a function of the ratio ath/ack45 tn for a compound of Ck45-partner (5 mm). In the figure, for simplification, the same Young's modulus E i = Efk45 was assumed for all three layers. The normalized residual stresses at the surface and near the interface of the components, as well as those in the braze alloy, vary linearly with ath/acg45 th. The values of this ratio smaller than unity result in tensile residual stresses at the surface of the partner material and near the interface of Ck45. In contrast, compressive residual stresses are calculated at the surface of Ck45 and near the interface of the partner. For aith/ O~Ck45th> 1, the signs of these residual stresses change. Over the entire range 0.5 < -- O~ith/0~Ck45th< 1.5, however, tensile residual stresses are calculated for the braze alloy. The residual stress distribution over the thickness in the centre of a compound of Ck45-X10CrNiM o T i l 8 1 0 (5 mm) is shown in Fig. 3. Within the individual components, linear variations of the residual stresses are obtained. Since ctith/CtCk45th=l.31, in
K. Bing et al.
/
Brazing residual stresses
TABLE 2. Materials data of the components used for the calculation of thermal residual stresses Material
•i
(~ith
12, th/ol~Ck45
th
( x I O - ~ K -I )
Ck45 X10CrNiMoTil810 NiCr22Co12Mo9 Cu (99.9 vol. %) AgCu28
188 200 212 120 73
0.3 0.3 0.3 0.34 0.3
14.5 19.0 16.5 17.5 20.0
1 1.31 1.14 1.21 1.38
/
/
CkL5
Ei
97
braze alloy
sfoinl, steel (5ram)
i
/
/
I
-10
I
-05
0
~IAT .__
surf(ace
"''~.... A
-
.......
partner braze alloy
--ekes
.i
01s
" 1-b
'
110
'
I0
[MPalK]
Fig. 3. Normalized thermal residual stresses over the thickness of a compound of Ck45 and stainless steel (5 mm).
interface
-2
05
Ckt~5 -- . . . . braze alloy . . . . shainl, sfeel
interface
llS
Fig. 2. Normalized thermal residual stresses vs. ratio of thermal expansion coefficients.
-1 -
surface
o agreement with Fig. 2, the residual stresses are tensile at the surface of Ck45 and compressive at the surface of stainless steel. Figure 4 shows how variations of the thickness of the stainless steel plate change the normalized residual stresses at the surfaces and near the interfaces of compounds C k 4 5 - X 1 0 C r N i M o T i 18 10. For small thicknesses, tensile residual stresses are calculated for the entire stainless steel plate and for the surface of Ck45, whereas Ck45 reveals compressive residual stresses near the interface. Greater thicknesses of the stainless steel component lead to compressive residual stresses over the entire thickness of Ck45. Compressive residual stresses at the surface and tensile residual stresses near the interface are then predicted for the stainless steel. Independent of the thickness of stainless steel, tensile residual stresses are always calculated for the braze alloy.
1~
2'o
~o
3'0
so
thickness of stainless steel [mm]
Fig. 4. Normalized thermal residual stresses in compounds of Ck45 brazed to stainless steel of increasing thickness.
braze alloy 200-
[k t~5
~
El a_
IO01
'
Imrfner (5mm) ~*%
0-
-100mc~lSul~men~ on
compounds with ----Cu (AT-- 20OK) o [u -300- ~ e e l ( A T ; / d ] O K ) ,, stoinl.steel 0 2 /, t~ 2 0 distance from the surfaces [mm]
~-200-
Fig. 5. Measured and calculated residual stress depth distributions of compounds of Ck45-X10CrNiMoTi1810 (5 mm) and Ck45-Cu (5 mm).
5. X-ray residual stress determinations X-ray residual stress determinations have been carried out to describe the residual stress state of real compounds. Figure 5 shows measured residual stress depth distributions of compounds of C k 4 5 - X 10CrNiM o T i l 8 1 0 (5 mm) and C k 4 5 - C u (5 mm). In the case of the c o m p o u n d of C k 4 5 - C u , the residual stresses
vary almost linearly over the thickness of the components and are generally relatively small. This can be explained by the similar thermal expansion coefficients of copper and Ck45, as well as by the low Young's modulus and yield strength of copper. In contrast, the residual stress distributions in the c o m p o u n d of Ck45X 1 0 C r N i M o T i l 8 1 0 show higher residual stress
K. Bing et aL
98
/
Brazing residual stresses
values, also varying mainly linearly over the thickness of the components, with the exception of the interface region. The residual stress distributions calculated analytically according to eqn. (1) for the corresponding compounds are also plotted in Fig. 5. By choosing the temperature difference A T as an adjustable parameter of the calculations, plastic deformations were taken into account only in a general way. Calculated and measured residual stress distributions fit best for A T= 200 K in the case of the compound of Ck45-Cu and for A T = 400 K in the case of Ck45-X 10CrNiMoT i 1810. However, considerable discrepancies between calculations and measurements remain in both compounds, mainly in the interface regions. Also, A T is much smaller than the solidification temperature of the braze alloy. These facts indicate that plastic deformation occurs if, during cooling to room temperature, the thermally induced multiaxial brazing stresses reach the temperature-dependent yield strength of the respective component. Apparently, this is most pronounced in the interface regions of the components [7] and probably in the braze alloy. However, this effect in the braze alloy could not be confirmed experimentally up to now. To examine the effect of the finite geometry of the compounds, residual stress distributions over the square surfaces of Ck45 were determined for various compounds of Ck45 and stainless steel. The left-hand part of Fig. 6 shows the distribution of the residual stress component oxRs in a direction parallel to Ck45 edge of a compound of Ck45-X10CrNiMoTi1810 (5 mm). The point (x = 0, y = 0) represents the centre of
2oo, /
the surface. The diagram demonstrates impressively the inhomogeneous residual stress distribution resulting from complex elastic and plastic deformations in real brazed compounds with finite geometries. The maximum measured residual stress (about 110 MPa) is not found in the centre of the plate but at a distance of about 3-4 mm. A slight minimum of the residual stress values occurs in the centre of the plate. Near the edges perpendicular to the x direction, the residual stresses decrease to about - 3 0 MPa. Along the edges in the x direction (y = _ 10 mm), the residual stresses decrease from 70 MPa at x = 0 to - 3 0 MPa near the corner. For comparison, the results for the residual stress component oyRs are shown in the righthand part of Fig. 6. The inhomogeneous residual stress distributions imply that the plastic deformations are also not distributed homogeneously in the plane of finite plates. This is qualitatively in agreement with the fact that the shear stresses parallel to the interface, which occur in the interface region as a result of the bonding condition, must be zero in the centre of the plate for reasons of symmetry. Moreover, they must vanish for reasons of stress equilibrium on a side face perpendicular to their direction of acting. Consequently, a maximum shear stress must occur between the centre and the edge. In that area, plastic deformations will be most pronounced. It is obvious that plastic deformations which are inhomogeneously distributed in directions parallel to the interface will cause inhomogeneous bending of the plates. It must be assumed that a triaxial residual stress state with residual stress components ozRs perpendicu-
2o0
~t [l~t~ ]
-10
Sm°tFig. 6. Measured residual stresses ax Rs and %~s at the surface of Ck45 of a compound of C k 4 5 - X 10CrNiMoTi 1810 (5 mm).
K. Bing et al.
/
Brazing residual stresses
Ck45 than does the compound in Fig. 6. Nearly constant values of crxrS= - 4 0 MPa were found. In agreement with these findings, Fig. 4 predicts a relatively low stress level in Ck45 of compounds of Ck45-X10CrNiMoTi1810 (10 mm). The low residual stress level in Ck45 and the relatively high yield strength at high temperatures of stainless steel which, according to Fig. 4, will show relatively high residual stress values near the interface, might be the reasons for smaller plastic deformations in this compound and, therefore, more homogeneous residual stress distributions in the plane of the plates. In addition to the surface residual stress states, the residual stresses on the side faces perpendicular to the interface have been investigated on compounds of Ck45-X10CrNiMoTi1810 (10, 5 and 2 mm), as sketched in the upper right-hand part of Fig. 8. The results are shown in Figs. 8(a)-8(c). Near the interface, in general, tensile residual stresses were found in Ck45 and compressive residual stresses in the stainless steel.
iar to the interface will occur. However, this can only be proved by extensive neutron diffraction studies and/or elastic-plastic finite element calculations. As shown in Fig. 7, a compound of Ck45X10CrNiMoTi1810 (10 mm) reveals a more homogeneous residual stress distribution at the surface of
L
10
-100~,
x
[mini
99
lC
Fig. 7. Measured residual stresses OxRs at the surface of Ck45 of a compound of Ck45-X10CrNiMoTi1810 (10 mm).
100 braze alloy 0
-100
~
-200 Ck ~S (Smm)
@ -300
I
0
I
I
I
I
I
I
I
I
I
I
I
8 6 6 distance from surface [mm]
2
2
-broze alloy
.....-brazealloy 0 n %-
~A
J
-IOQ
A
.~ -200
Ck6S
Ck6S
(~) (5mm) -300
I
I
2
I
C) I
6
I
I
I
I
(Smm} I
6 2 0 0 dislunce from surfoce [mm]
I
2
i
I
6
Fig. 8. Measured residual stresses on the side face perpendicular to the interface of compounds of Ck45-X10CrNiMoTi1810 with stainless steel of thickness (a) 10 mm, (b) 5 mm and (c) 2 mm.
100
K. Bing et al.
/
According to Fig. 1, this is a consequence of local bending effects on the side faces near the interface [2]. Smaller thicknesses of the stainless steel plate led to lower residual stress maxima in both components; however, these were more pronounced on the side face of the stainless steel. They also decrease the width of the maximum tensile residual stresses on the side face of Ck45.
Brazing residual stresses
1
"l T convex / 1.01/,~.'~
o Ni-bose alloy
_
0. -0.5. surfoce of Ck/+5
-1.0"Ida-re "-
-1.51~ concave
----
lb
0
6. Measurements of the radius of curvature of the compounds
tx stoinl,steel.
surface
of partner
i'5
z0
fhickness of brazing partner [mm]
Fig. 9. Measured reciprocal curvature radius of compounds
with various brazing partners vs. thickness of the partner. The overall bending effects of compounds of Ck45 brazed to stainless steel, the nickel-base alloy or copper were analysed by curvature measurements on both surfaces. The thickness of the brazed partner was varied in the range 0.5-20 mm. In Fig. 9, the reciprocal radius 1/R for both surfaces is plotted vs. the thickness of the brazed partner for the materials combinations mentioned above. On the surface of Ck45, the measured values of 1 / R increase with decreasing thickness of the joining partner down to about 2 mm, where maximum values are observed. They decrease again by smaller thicknesses. Furthermore, as one would expect according to the ratios of thermal expansion coefficients, bending of compounds of Ck4 5 - X 10CrNiMoTi 1810 is generally more pronounced than that of compounds of C k 4 5 NiCr22Co12Mo9 or Ck45-Cu. The 1 / R values measured on the surfaces of the joining partners show mainly similar dependences but opposite signs. However, 1 / R reveals maximum values for the smallest thicknesses investigated. The results show that, for the compounds in Fig. 9, overall bending deformation is most pronounced in the case of thin brazing partners, and decreases rapidly with increasing thickness of the partner. Together with the results presented in Fig. 8, this shows that increasing overall bending of the plates, resulting from the decreasing thickness of the brazing partner, reduces the local bending effects and residual stresses on the side faces of the compounds. Moreover, the curvature measurements confirm that the bending deformation in the compounds is, in most cases, not homogeneous, since the magnitude of 1 / R is different on both surfaces of the individual compounds.
7. Concluding remarks The combination of X-ray measurements and calculations allows a complete description to be made of the thermally induced residual stresses of brazed multilayered compounds. Moreover, a comparison of the
measured and calculated residual stress states and curvature measurements give clear evidence of complicated, inhomogeneous plastic deformations in the compounds during cooling from the brazing temperature. The residual stress development is characterized by the superposition and interaction of overall bending of the plates as well as local bending effects on the side faces of the interface, both depending on the thickness of the components. X-ray residual stress determinations allow a quantitative description of the edge effects of the compounds examined. Since considerable discrepancies occur between analytically calculated brazing residual stresses using a simple elastic model and the results of X-ray residual stress measurements, plastic deformations during the brazing process cannot be neglected for the calculations. Consequently, an exact calculation of the brazing residual stress states is only possible using three-dimensional finite element calculations for finite plates, taking into account elastic-plastic materials deformation behaviour and temperature-dependent properties.
Acknowledgment The financial support by the Deutsche Forschungsgemeinschaft for the research work presented is gratefully acknowledged.
References 1 E. Macherauch, H. Wohlfahrt and U. Wolfstieg, Zur zweckm~il3igen Definition van Eigenspannungen, HartereiTechnische Mitteilungen, 28 ( 1973 ) 201-211. 2 B. Eigenmaim, B. Scholtes and E. Macherauch, X-ray residual stress determination in soldered ceramic/metal compounds, in W. Kraft (ed.), Joining Ceramics, Glass and Metal DGM Informationsgesellschaft mbH, Oberursel, 1989, pp. 249-255.
K. Bing et al.
/
Brazing residual stresses
3 E. Macherauch and P. Miiller, Das sin 2 ~p-Verfahren der r6ntgenographischen Spannungsmessung, Z. Angew. Phys., 13 (1961)305-312. 4 Landolt-B6rnstein, in K.-H. Hellwege and A. M. Heilwege (eds.), Physikalisch-Chemische Tabellen, Vols. 111 11 and 111 18, 1979, 1984. 5 M. G. Moore and W. P. Evans, Mathematical correction for stress in removed layers in X-ray diffraction residual stress analysis, Trans. SAE, 66 ( 1958) 340-345. 6 0 . Iancu, D. Munz, B. Eigenmann, B. Scholtes and E. Macher-
101
auch, Residual stress state of brazed ceramic/metal compounds, determined by analytical methods and X-ray residual stress measurements, J. Am. Ceram. Soc., 73 (1990) 1144-1149.
7 L. Pintschovius, N. Pyka, R. Kussmaul, D. Munz, B. Eigenmann and B. Scholtes, Residual stresses in brazed ceramic-metal compounds, in M. T. Hutchings and A. D. Krawitz (eds.), Measurement of Residual and Applied Stress Using Neutron Diffraction, Kluwer, Dordrecht, 1992, pp. 473-477.
95
Brazing residual stresses in components of different metallic materials Klaus Bing a, Bernd Eigenmann a, Berthold Scholtes b and Eckard Macherauch a "~Institutfiir Werkstoffkunde L Universitiit Karlsruhe (TH), W-7500 Karlsruhe (Germany) blnstitut fiir Werkstofftechnik, Metallische Werkstoffe, Gh Kassel-Universitiit, W-3500 Kassel (Germany) (Received March 8, 1993;in revised form June 15, 1993)
Abstract Brazing residual stresses are a consequence of differences between the thermal expansion coefficients of the brazed partners and the solder. In this study, components of copperr, austenitic steel and a nickel-base alloy brazed on ferritic steel are investigated. Using simplifying assumptions, the resulting stress states are estimated and compared with experimental X-ray stress analyses and curvature measurements. This gives clear evidence of complicated inhomogeneous plastic deformations in the compounds, which occur during cooling down from the brazing temperature. In addition, X-ray residual stress determinations allow a quantitative description to be given of edge effects in the compounds examined.
1. Introduction
Cu).
trated schematically in Fig. 1 for a two-layer compound of the components A and B, with aA th > aB th. At the brazing temperature, the compound is stress flee. Cooling to room temperature causes different thermal shrinkages of the individual layers; however, the shrinkage is hindered by the bonding condition at the interface, as shown in the right-hand part of Fig. 1. As a consequence, an overall bending moment of the compound results, as do local bending effects, which are most pronounced on the side faces of the components near the interface. To achieve strain compatibility at the interface, as well as equilibria of the forces and momenta, a relatively complex residual stress state develops. In the case of a compound with components of approximately equal thickness and similar elastic behaviour, tensile (compressive) residual stresses near the interface of layer A (B) and compressive (tensile) residual stresses at the surfaces of the components A (B) occur. In addition, compressive (tensile) residual stresses develop on the side face of layer A (B) near the interface perpendicular to the plane of the plates.
2. General aspects of brazing residual stresses
3. Experimental details
In brazed multilayered compounds with different thermal expansion coefficients ~ith of the individual layers i, residual stresses of the I (macroscopic residual stresses) kind develop when cooling from the brazing temperature to room temperature [1, 2]. This is illus-
Square plates of Ck45 with dimensions 20 mm × 20 m m x 5 mm were brazed to plates of X10CrNiMoTil810, NiCr22Co12Mo9 or copper of dimensions 20 mm x 20 mm but with varying thicknesses. The brazings were carried out with foils of an AgCu
Brazing is a well-established joining method for materials with different properties, regardless of their shapes. After brazing and cooling down to room temperature, distortions and complex residual stress states always occur in the compounds, and these depend on the mechanical properties and the thicknesses of the two components, as well as on the difference between the thermal expansion coefficients and the type of brazing material. Since residual stresses affect the strength of the joints and distortions change the shape of the compounds, a quantitative knowledge of both factors is of great importance for practical purposes. This paper deals with analytical and experimental determinations of the residual stress states and distortions in brazed compounds consisting of differently sized plates of a plain carbon steel (German grade Ck45) and plates of a stainless steel (German grade X10CrNiMoTi1810), a nickel-base alloy (German grade NiCr22Co12Mo9), or pure copper (99.9 vol.%
0921-5093/94/$7.00
© 1994 - Elsevier Sequoia. All rights reserved
K. Bing et al. / Brazing residual stresses
96 brazing temperature ~
room temperature
TABLE 1. Parameters used for X-ray residual stress determinations
Material
0t~h > [l~h
Lattice planes
Ck45 X10CrNiMoTi1810 NiCr22Col 2Mo9 Cu (99.9 vol.%)
E Ihk~l v Ihktl
{hkl}
20o (deg)
(GPa)
{211} {220} {220} {220}
156.4 129.5 129.5 127.5
210 207 224 137
0.28 0.28 0.30 0.34
Fig. 1. Thermally induced distortions and corresponding signs of residual stresses in an idealized two-layer compound shown schematically.
braze alloy (AgCu28) 0.1 mm thick at 1050 °C in the case of X10CrNiMoTi 18 10 and at 850 °C in the other cases. The residual stresses were determined by X-ray diffraction using a computerized ~p diffractometer of the Karlsruhe type. The {hkl} planes of the individual materials shown in Table 1 were investigated with Cr K a radiation. The appertaining 2 00 values are listed in the third column. The residual stresses were calculated according to the sin 2 ~0 method for X-ray residual stress determinations [3], using the {hkl}-dependent X-ray values of Young's modulus Ethk/I and Poisson's ratio v/hkllsummarized in the last two columns of Table 1. The elastic anisotropy was taken into account by averaging the elastic constants calculated from singlecrystal elastic constants [4] according to the Voigt and Reuss models. In addition to the surface residual stresses of the compounds, the residual stress distributions over the thickness of the components were also determined in the centre of the plates by successive electrolytical removal of layers of material and subsequent residual stress measurements. The measured values were corrected for changes in the residual stress equilibrium resulting from materials removed [5]. To obtain information about the brazing-induced distortions, curvature measurements were carried out on the surfaces of the compounds, using a perthometer (Perthen GmbH, G6ttingen).
4. Analytical estimation of brazing residual stresses If a multilayered plate of infinite size and different thermal expansion coefficients a~th, of the components is cooled from an initially stress-free state at an elevated temperature T* to room temperature, the distribution of thermal residual stresses can be calculated analytically. This is achieved by assuming a linear elastic materials behaviour of the individual layers i and using Bernoulli's hypothesis of bending; the strains are taken as proportional to the distance from the
neutral axis and inversely proportional to the radius of curvature [6]. Neglecting the temperature dependence of ai th, Young's moduli Ei and Poisson's ratios vi, the surface parallel residual stresses of any component i can then be determined over their thickness as a function of zg, referring to the central plane of each individual layer as
(Yx(Zi):m l-
(
EiO'l-zi'l- (~it h A T
(1)
R
with e~° the strains in the centre planes of the components, R the bending radius of the compound and A T = T * - T2oo being the temperature difference between the solidification temperature T* of the braze alloy and room temperature. The materials data for the calculations were averaged over the temperature range 20 °< T< 500 °C (see ref. 6) and the values are listed in Table 2. Since data for the real braze alloy AgCu28 were not available, the values for pure silver have been used. The residual stresses, estimated according to eqn. (1) and normalized to the temperature difference A T, are shown in Fig. 2 as a function of the ratio ath/ack45 tn for a compound of Ck45-partner (5 mm). In the figure, for simplification, the same Young's modulus E i = Efk45 was assumed for all three layers. The normalized residual stresses at the surface and near the interface of the components, as well as those in the braze alloy, vary linearly with ath/acg45 th. The values of this ratio smaller than unity result in tensile residual stresses at the surface of the partner material and near the interface of Ck45. In contrast, compressive residual stresses are calculated at the surface of Ck45 and near the interface of the partner. For aith/ O~Ck45th> 1, the signs of these residual stresses change. Over the entire range 0.5 < -- O~ith/0~Ck45th< 1.5, however, tensile residual stresses are calculated for the braze alloy. The residual stress distribution over the thickness in the centre of a compound of Ck45-X10CrNiM o T i l 8 1 0 (5 mm) is shown in Fig. 3. Within the individual components, linear variations of the residual stresses are obtained. Since ctith/CtCk45th=l.31, in
K. Bing et al.
/
Brazing residual stresses
TABLE 2. Materials data of the components used for the calculation of thermal residual stresses Material
•i
(~ith
12, th/ol~Ck45
th
( x I O - ~ K -I )
Ck45 X10CrNiMoTil810 NiCr22Co12Mo9 Cu (99.9 vol. %) AgCu28
188 200 212 120 73
0.3 0.3 0.3 0.34 0.3
14.5 19.0 16.5 17.5 20.0
1 1.31 1.14 1.21 1.38
/
/
CkL5
Ei
97
braze alloy
sfoinl, steel (5ram)
i
/
/
I
-10
I
-05
0
~IAT .__
surf(ace
"''~.... A
-
.......
partner braze alloy
--ekes
.i
01s
" 1-b
'
110
'
I0
[MPalK]
Fig. 3. Normalized thermal residual stresses over the thickness of a compound of Ck45 and stainless steel (5 mm).
interface
-2
05
Ckt~5 -- . . . . braze alloy . . . . shainl, sfeel
interface
llS
Fig. 2. Normalized thermal residual stresses vs. ratio of thermal expansion coefficients.
-1 -
surface
o agreement with Fig. 2, the residual stresses are tensile at the surface of Ck45 and compressive at the surface of stainless steel. Figure 4 shows how variations of the thickness of the stainless steel plate change the normalized residual stresses at the surfaces and near the interfaces of compounds C k 4 5 - X 1 0 C r N i M o T i 18 10. For small thicknesses, tensile residual stresses are calculated for the entire stainless steel plate and for the surface of Ck45, whereas Ck45 reveals compressive residual stresses near the interface. Greater thicknesses of the stainless steel component lead to compressive residual stresses over the entire thickness of Ck45. Compressive residual stresses at the surface and tensile residual stresses near the interface are then predicted for the stainless steel. Independent of the thickness of stainless steel, tensile residual stresses are always calculated for the braze alloy.
1~
2'o
~o
3'0
so
thickness of stainless steel [mm]
Fig. 4. Normalized thermal residual stresses in compounds of Ck45 brazed to stainless steel of increasing thickness.
braze alloy 200-
[k t~5
~
El a_
IO01
'
Imrfner (5mm) ~*%
0-
-100mc~lSul~men~ on
compounds with ----Cu (AT-- 20OK) o [u -300- ~ e e l ( A T ; / d ] O K ) ,, stoinl.steel 0 2 /, t~ 2 0 distance from the surfaces [mm]
~-200-
Fig. 5. Measured and calculated residual stress depth distributions of compounds of Ck45-X10CrNiMoTi1810 (5 mm) and Ck45-Cu (5 mm).
5. X-ray residual stress determinations X-ray residual stress determinations have been carried out to describe the residual stress state of real compounds. Figure 5 shows measured residual stress depth distributions of compounds of C k 4 5 - X 10CrNiM o T i l 8 1 0 (5 mm) and C k 4 5 - C u (5 mm). In the case of the c o m p o u n d of C k 4 5 - C u , the residual stresses
vary almost linearly over the thickness of the components and are generally relatively small. This can be explained by the similar thermal expansion coefficients of copper and Ck45, as well as by the low Young's modulus and yield strength of copper. In contrast, the residual stress distributions in the c o m p o u n d of Ck45X 1 0 C r N i M o T i l 8 1 0 show higher residual stress
K. Bing et aL
98
/
Brazing residual stresses
values, also varying mainly linearly over the thickness of the components, with the exception of the interface region. The residual stress distributions calculated analytically according to eqn. (1) for the corresponding compounds are also plotted in Fig. 5. By choosing the temperature difference A T as an adjustable parameter of the calculations, plastic deformations were taken into account only in a general way. Calculated and measured residual stress distributions fit best for A T= 200 K in the case of the compound of Ck45-Cu and for A T = 400 K in the case of Ck45-X 10CrNiMoT i 1810. However, considerable discrepancies between calculations and measurements remain in both compounds, mainly in the interface regions. Also, A T is much smaller than the solidification temperature of the braze alloy. These facts indicate that plastic deformation occurs if, during cooling to room temperature, the thermally induced multiaxial brazing stresses reach the temperature-dependent yield strength of the respective component. Apparently, this is most pronounced in the interface regions of the components [7] and probably in the braze alloy. However, this effect in the braze alloy could not be confirmed experimentally up to now. To examine the effect of the finite geometry of the compounds, residual stress distributions over the square surfaces of Ck45 were determined for various compounds of Ck45 and stainless steel. The left-hand part of Fig. 6 shows the distribution of the residual stress component oxRs in a direction parallel to Ck45 edge of a compound of Ck45-X10CrNiMoTi1810 (5 mm). The point (x = 0, y = 0) represents the centre of
2oo, /
the surface. The diagram demonstrates impressively the inhomogeneous residual stress distribution resulting from complex elastic and plastic deformations in real brazed compounds with finite geometries. The maximum measured residual stress (about 110 MPa) is not found in the centre of the plate but at a distance of about 3-4 mm. A slight minimum of the residual stress values occurs in the centre of the plate. Near the edges perpendicular to the x direction, the residual stresses decrease to about - 3 0 MPa. Along the edges in the x direction (y = _ 10 mm), the residual stresses decrease from 70 MPa at x = 0 to - 3 0 MPa near the corner. For comparison, the results for the residual stress component oyRs are shown in the righthand part of Fig. 6. The inhomogeneous residual stress distributions imply that the plastic deformations are also not distributed homogeneously in the plane of finite plates. This is qualitatively in agreement with the fact that the shear stresses parallel to the interface, which occur in the interface region as a result of the bonding condition, must be zero in the centre of the plate for reasons of symmetry. Moreover, they must vanish for reasons of stress equilibrium on a side face perpendicular to their direction of acting. Consequently, a maximum shear stress must occur between the centre and the edge. In that area, plastic deformations will be most pronounced. It is obvious that plastic deformations which are inhomogeneously distributed in directions parallel to the interface will cause inhomogeneous bending of the plates. It must be assumed that a triaxial residual stress state with residual stress components ozRs perpendicu-
2o0
~t [l~t~ ]
-10
Sm°tFig. 6. Measured residual stresses ax Rs and %~s at the surface of Ck45 of a compound of C k 4 5 - X 10CrNiMoTi 1810 (5 mm).
K. Bing et al.
/
Brazing residual stresses
Ck45 than does the compound in Fig. 6. Nearly constant values of crxrS= - 4 0 MPa were found. In agreement with these findings, Fig. 4 predicts a relatively low stress level in Ck45 of compounds of Ck45-X10CrNiMoTi1810 (10 mm). The low residual stress level in Ck45 and the relatively high yield strength at high temperatures of stainless steel which, according to Fig. 4, will show relatively high residual stress values near the interface, might be the reasons for smaller plastic deformations in this compound and, therefore, more homogeneous residual stress distributions in the plane of the plates. In addition to the surface residual stress states, the residual stresses on the side faces perpendicular to the interface have been investigated on compounds of Ck45-X10CrNiMoTi1810 (10, 5 and 2 mm), as sketched in the upper right-hand part of Fig. 8. The results are shown in Figs. 8(a)-8(c). Near the interface, in general, tensile residual stresses were found in Ck45 and compressive residual stresses in the stainless steel.
iar to the interface will occur. However, this can only be proved by extensive neutron diffraction studies and/or elastic-plastic finite element calculations. As shown in Fig. 7, a compound of Ck45X10CrNiMoTi1810 (10 mm) reveals a more homogeneous residual stress distribution at the surface of
L
10
-100~,
x
[mini
99
lC
Fig. 7. Measured residual stresses OxRs at the surface of Ck45 of a compound of Ck45-X10CrNiMoTi1810 (10 mm).
100 braze alloy 0
-100
~
-200 Ck ~S (Smm)
@ -300
I
0
I
I
I
I
I
I
I
I
I
I
I
8 6 6 distance from surface [mm]
2
2
-broze alloy
.....-brazealloy 0 n %-
~A
J
-IOQ
A
.~ -200
Ck6S
Ck6S
(~) (5mm) -300
I
I
2
I
C) I
6
I
I
I
I
(Smm} I
6 2 0 0 dislunce from surfoce [mm]
I
2
i
I
6
Fig. 8. Measured residual stresses on the side face perpendicular to the interface of compounds of Ck45-X10CrNiMoTi1810 with stainless steel of thickness (a) 10 mm, (b) 5 mm and (c) 2 mm.
100
K. Bing et al.
/
According to Fig. 1, this is a consequence of local bending effects on the side faces near the interface [2]. Smaller thicknesses of the stainless steel plate led to lower residual stress maxima in both components; however, these were more pronounced on the side face of the stainless steel. They also decrease the width of the maximum tensile residual stresses on the side face of Ck45.
Brazing residual stresses
1
"l T convex / 1.01/,~.'~
o Ni-bose alloy
_
0. -0.5. surfoce of Ck/+5
-1.0"Ida-re "-
-1.51~ concave
----
lb
0
6. Measurements of the radius of curvature of the compounds
tx stoinl,steel.
surface
of partner
i'5
z0
fhickness of brazing partner [mm]
Fig. 9. Measured reciprocal curvature radius of compounds
with various brazing partners vs. thickness of the partner. The overall bending effects of compounds of Ck45 brazed to stainless steel, the nickel-base alloy or copper were analysed by curvature measurements on both surfaces. The thickness of the brazed partner was varied in the range 0.5-20 mm. In Fig. 9, the reciprocal radius 1/R for both surfaces is plotted vs. the thickness of the brazed partner for the materials combinations mentioned above. On the surface of Ck45, the measured values of 1 / R increase with decreasing thickness of the joining partner down to about 2 mm, where maximum values are observed. They decrease again by smaller thicknesses. Furthermore, as one would expect according to the ratios of thermal expansion coefficients, bending of compounds of Ck4 5 - X 10CrNiMoTi 1810 is generally more pronounced than that of compounds of C k 4 5 NiCr22Co12Mo9 or Ck45-Cu. The 1 / R values measured on the surfaces of the joining partners show mainly similar dependences but opposite signs. However, 1 / R reveals maximum values for the smallest thicknesses investigated. The results show that, for the compounds in Fig. 9, overall bending deformation is most pronounced in the case of thin brazing partners, and decreases rapidly with increasing thickness of the partner. Together with the results presented in Fig. 8, this shows that increasing overall bending of the plates, resulting from the decreasing thickness of the brazing partner, reduces the local bending effects and residual stresses on the side faces of the compounds. Moreover, the curvature measurements confirm that the bending deformation in the compounds is, in most cases, not homogeneous, since the magnitude of 1 / R is different on both surfaces of the individual compounds.
7. Concluding remarks The combination of X-ray measurements and calculations allows a complete description to be made of the thermally induced residual stresses of brazed multilayered compounds. Moreover, a comparison of the
measured and calculated residual stress states and curvature measurements give clear evidence of complicated, inhomogeneous plastic deformations in the compounds during cooling from the brazing temperature. The residual stress development is characterized by the superposition and interaction of overall bending of the plates as well as local bending effects on the side faces of the interface, both depending on the thickness of the components. X-ray residual stress determinations allow a quantitative description of the edge effects of the compounds examined. Since considerable discrepancies occur between analytically calculated brazing residual stresses using a simple elastic model and the results of X-ray residual stress measurements, plastic deformations during the brazing process cannot be neglected for the calculations. Consequently, an exact calculation of the brazing residual stress states is only possible using three-dimensional finite element calculations for finite plates, taking into account elastic-plastic materials deformation behaviour and temperature-dependent properties.
Acknowledgment The financial support by the Deutsche Forschungsgemeinschaft for the research work presented is gratefully acknowledged.
References 1 E. Macherauch, H. Wohlfahrt and U. Wolfstieg, Zur zweckm~il3igen Definition van Eigenspannungen, HartereiTechnische Mitteilungen, 28 ( 1973 ) 201-211. 2 B. Eigenmaim, B. Scholtes and E. Macherauch, X-ray residual stress determination in soldered ceramic/metal compounds, in W. Kraft (ed.), Joining Ceramics, Glass and Metal DGM Informationsgesellschaft mbH, Oberursel, 1989, pp. 249-255.
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Brazing residual stresses
3 E. Macherauch and P. Miiller, Das sin 2 ~p-Verfahren der r6ntgenographischen Spannungsmessung, Z. Angew. Phys., 13 (1961)305-312. 4 Landolt-B6rnstein, in K.-H. Hellwege and A. M. Heilwege (eds.), Physikalisch-Chemische Tabellen, Vols. 111 11 and 111 18, 1979, 1984. 5 M. G. Moore and W. P. Evans, Mathematical correction for stress in removed layers in X-ray diffraction residual stress analysis, Trans. SAE, 66 ( 1958) 340-345. 6 0 . Iancu, D. Munz, B. Eigenmann, B. Scholtes and E. Macher-
101
auch, Residual stress state of brazed ceramic/metal compounds, determined by analytical methods and X-ray residual stress measurements, J. Am. Ceram. Soc., 73 (1990) 1144-1149.
7 L. Pintschovius, N. Pyka, R. Kussmaul, D. Munz, B. Eigenmann and B. Scholtes, Residual stresses in brazed ceramic-metal compounds, in M. T. Hutchings and A. D. Krawitz (eds.), Measurement of Residual and Applied Stress Using Neutron Diffraction, Kluwer, Dordrecht, 1992, pp. 473-477.