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Stability analysis of the adhesive strength between substrate and a thin film
COMPUTATIONAL MATERIALS SCIENCE ELSEVIER
Computational Materials
Science
9 (1997)
1 16- 120
Stability analysis of the adhesive strength between substrate and a thin film Th. Richter ’ Werksfo~h~sikalische
Gruduiertenkolleg
Modellierung’, D-09596
H. Glkser
*,
Freiberg
Unicersify
Freiberg,
of Mining
und Technology,
Gustuc-Zeuner-Srr.
5.
Germany
Abstract In the present work we show a simple way to describe the mechanical behaviour of an interface. In the recent history there was a lot of discussion about the modelling of multiphase materials. Generally, the interface is assumed as an ideal rigid boundary between matrix and inclusion or layer and substrate. In the following paper we will concentrate on the case of a layer-substrate interface. We will show a simple way to obtain a parameter specifying the interface from experimental results. This parameter is an essential part for the FE-calculation. 0 1997 Elsevier Science B.V.
1. Background The modelling of the behaviour of a thin film on a substrate is necessary to develop new tools. At the Freiberg University of Mining and Technology many experiments were performed with the material combination TIN on steel. Therefore, we have a lot of experimental data available [I]. With respect to the geometrical values, the following research can also be used for multilayers or matrix/inclusion combinations. For these combinations mostly unit cells were used in FE-calculations. However, these elements do not describe the contact zone between the different phases. Contrary to most other researchers we perform the stability and not the crack analysis (e.g. [2]). Hence, we only get the critical point for our system without any failure history. However, we assume that if this point matches exactly with the experimental data, our simulation is correct.
’ Corresponding author. E-mail: [email protected]. 0927-0256/97/$17.00
PII
SO927-0256(97)00065-7
0
1997
Elsevier Science
B.V.
All
During the testing of the pressure behaviour of TiN with a four point bending test, two completely different crack patterns were observed (see Fig. 1). During these tests the samples were constantly observed with X-rays. Therefore, we are able to determine the failure stress exactly. In both cases the value for the failure stress has the same dimension. The crack pattern in the case of delamination is similar to Euler’s buckling. Hence, the delamination is the starting point of our calculation. We have to introduce in the wellknown buckling theory a parameter which represents the interface behaviour. This parameter is realized with a spring. The reason to follow this way are the specifications of the available FE-programm. In commercial codes one or more spring elements are included in the element library. For our calculation we use the following data. As the substrate the steel Ck15 is used. Its Young’s modulus is 210 GPa and Poisson’s ratio is 0.3. In a tensile test we measured the stress-strain data. For TIN the Young’s modulus is 420 GPa and the Poisson’s ratio 0.2. TiN is pure elastic between - 7.6 and 1 GPa.
rights reserved.
T.h. Richter, H. Gliiser/Computational
Materials Science 9 (1997) 116-120 Table 1 Geometrical
data [3]
F Distance between delamination Length of delamination Height of the layer Height of the substrate
‘Regions of delamination
F
117
Parameter
Value
I, 1 h H
1.9 mm 40 pm 2.8 pm 2.3 mm
axial movement in x-direction the applied boundary conditions
is possible. are
Hence,
u(0) = u’(0) = U( 1) = u’(f) = 0.
Fig. 1. Sketch of the different failure pattern after the four point bending test: (a) thick film; (b) thin film.
All material properties were assumed to be isotropic. The geometrical data are shown in Table 1.
2. Stability
analysis
The model was simplified to the configuration shown in Fig. 2. This means, that we have reduced the configuration from three to two dimensions and normalised the parameter b (width of the delamination) to I pm. The required force Fp can be calculated from the measured stress. The boundary conditions for the layer are as follows. For the left end all three degrees of freedom are restrained. For the right end, only an
(1) A finite distance between layer and substrate was introduced to be able to apply the springs in the FE-calculation. However, in reality, for an undamaged layer this distance will be equal to zero. The goal of the analytical solution is the determination of the spring parameter c. This parameter represents a continuous load along the whole length 1. At a first step we determine the bending line. Then we introduce the boundary conditions and perform the stability analysis. The substrate material so far has only an influence on the applied load required in the bending test, but not on the analytic solution. We can get the differential equation for the bending line similar to the way described in [4]
“(X)“” + ;“(x)” +iu(
(2)
In Eq. (2) E represents the Young’s modulus and I is the area moment of inertia. The general solution of Eq. (2) is U(X) = C, sin( Ax) + C, sin( Bx) + C, cos( Ax)
linear spring with parameter Fig. 2. Simplification
x) = 0
+ c, cos( Bx)
c
of the problem for the analytical
solution.
(3)
T.h. Richter.
118
H. Gliiser/
Computationul
Muterds
Science
9 (1997)
3. Effect of the substrate
with the substitutions
B= /m
(5)
Now, we have to perform the stability analysis under the restriction of Eq. (6). Eq. (6) represents the determinant of the coefficients Ci, which result from the boundary conditions.
A first FE-calculation with the above determined value C.analytic shows that the behaviour of the model is too weak. The construction of the model will be discussed later. For now we are only interested in the result. The weak behaviour can be explained by the fact, that a second spring was automatically introduced by the program when the substrate was applied to our model. The overall spring parameter in this case can be calculated from the following equation (8)
det(C,)
=
116-120
0
0
1
I
A
B
0
0
sin( AI)
sin( BI)
A sin( Al)
Bsin( BI)
cos( Al) -
cos( BI)
A cos( A/)
- Bcos(
BI) (6)
From Eq. (6) we can evaluate equation for the system. det( C;) = ( A2 + B’)sin( + 2AB(cos(
Al)sin( Al)cos(
the characteristic
B1) Bl) - 1) =0
(7)
This equation can be solved analytically only in some cases. A numerical solution can always lead to a result. Eq. (7) represents the function det(Ci) =f(c) and can be plotted (Fig. 3). The first zero-point will be the unknown critical value c. From Fig. 3 we can determine c,,,,~,~~ = 19.2 MPa for the combination TIN and steel. dd (C,,
r-1
c NW 1000
2000
30000
4000 1 Fig. 3. Graph of the numerical solution.
In a first step we assume that the analytical result will be equal to the value for the interface spring in the ANSYS calculation plus the additional term which is unknown so far. The result of the (‘substrate’ calculated stress has to be lower compared to the experimental result. Therefore, we have to change the parameter in the FE-simulation until we get the failure ~~~~~~~btained from the experiments. With Eq. (8) we determine csubstrats= 1 1.5 MPa.
4. FE-calculation The calculations were done by the commercial FE-code ANSYS on a HP9000 workstation [S]. For the 2D-calculations we use simple elements. In addition to the structural elements and spring elements we also use contact elements, which are necessary to avoid penetration of the layer into the substrate. Problems are hidden in the geometrical data. The real thickness of the substrate exceeds the thickness of the layer by a factor of 1000. These fact would require a meshing for the substrate that leads to computing times beyond reasonable limits. Further, it is necessary to apply as many springs as possible on the interface since we have to simulate a uniform line load. Hence, we decided to model substrate and layer with the same height. This is possible with respect to the applied boundary conditions. With these conditions we do not reduce the substrate effect. At the
T.h. Richter, H. Gkiser/Computational Materials Science 9 (1997) 116-120
” P z 1 m
7800 7600 ‘. 7400.. 7200.. moo,. ,/” ,’ 6000 .. / ,..’ / omo~~ ,d’ 6400 .*__...----:
6200. 1
:
:
: 5
:
:
:
A--.----/” .T’
:
: : 10
:
:
:
: 16
:
I
Timesteps
Fig. 4. Working of the automatic time step procedure.
delamination point the substrate will be plastified in reality and in our model. Boundary conditions were applied, such that the left side of layer and substrate is fixed in x-direction, the bottom of the substrate is fixed in y-direction and the substrate is coupled to the layer at the left (X = 0) and right (X = I) hand side of the model. A displacement is applied at the right end of layer and substrate in negative x-direction. To obtain the line load we have to calculate the the constant for each value cAfNSYS,representing single spring. The line load will be activated by a movement of the layer normal to it’s interface. The load of the combined springs is the equivalent to the line load. If we compare the resulting forces in the analytical model and in the FE- calculation, we get the equation C analytic -=C6kUS
119
the exact point of the delamination we use the automatic time step algorithm. The effect of this procedure, the approaching of the solution to the critical value, is shown in Fig. 4. One can see the shape of the deformed layer in Fig. 5. In Fig. 5 one also can see which region have changed from a homogeneous pressure to tensile stress. Since the critical tensile stress for TiN is only 1 GPa, it is likely that mode I cracks will be generated in this stress regions.
5. Conclusion In the present the delamination To do this with mine the spring analysis. During
work we show a way to calculate of an TiN layer on a steel substrate. an FE-code, we first need to deterparameter cmalyric from a stability the FE-calculation we determine the parameter Csubstrate for the substrate effect (weaker performance of the FE-model as the analytical model) in an iterative way. This way is generally useful, if the crack pattern allows a simplification into a two dimensional model.
Acknowledgements
n
1,
The parameter n is the number of springs at the critical length 1. The calculation was done in a framework of a nonlinear buckling analysis. To get
The presented work was performed within the Graduiertenkolleg ‘ Werkstoffphysikalische Modellierung’ at Freiberg University of Mining and Technology and supported by the ‘Deutsche Forschungsgemeinschaft’.
~
Load direction
pplied displacement
Fig. 5. FEM-result after delamination (deformed geometry).
120
T.h. Richter,
H. Gkiser/
Compututionul
References [I] H. Oettel, R. Wiedemann, Residual Stresses in PVD Hard Coatings, Proc. of ICMCTP, San Diego, 1995. [2] A.G. Evans, J.W. Hutchinson, The thermomechanical integrity of thin films and multilayers, Acta Metall. Mater. 43 (1995)
Muterids
Science
9 (1997)
116-120
2507-2530. [3] Th. Bertram,
H. Oettel. Stresses and mechanisms for failure of TIN coatings, J. Vat. Sci., in preparation. 141 G. Biirgermeister, H. Steup, Stabilitatstheorie mit Erlluterungen zur DIN41 14-Teil I, Akademie-Verlag. Berlin, 1957. [5] ANSYSO, Version 5.1 Documentation, ANSYSO, 1994.
Computational Materials
Science
9 (1997)
1 16- 120
Stability analysis of the adhesive strength between substrate and a thin film Th. Richter ’ Werksfo~h~sikalische
Gruduiertenkolleg
Modellierung’, D-09596
H. Glkser
*,
Freiberg
Unicersify
Freiberg,
of Mining
und Technology,
Gustuc-Zeuner-Srr.
5.
Germany
Abstract In the present work we show a simple way to describe the mechanical behaviour of an interface. In the recent history there was a lot of discussion about the modelling of multiphase materials. Generally, the interface is assumed as an ideal rigid boundary between matrix and inclusion or layer and substrate. In the following paper we will concentrate on the case of a layer-substrate interface. We will show a simple way to obtain a parameter specifying the interface from experimental results. This parameter is an essential part for the FE-calculation. 0 1997 Elsevier Science B.V.
1. Background The modelling of the behaviour of a thin film on a substrate is necessary to develop new tools. At the Freiberg University of Mining and Technology many experiments were performed with the material combination TIN on steel. Therefore, we have a lot of experimental data available [I]. With respect to the geometrical values, the following research can also be used for multilayers or matrix/inclusion combinations. For these combinations mostly unit cells were used in FE-calculations. However, these elements do not describe the contact zone between the different phases. Contrary to most other researchers we perform the stability and not the crack analysis (e.g. [2]). Hence, we only get the critical point for our system without any failure history. However, we assume that if this point matches exactly with the experimental data, our simulation is correct.
’ Corresponding author. E-mail: [email protected]. 0927-0256/97/$17.00
PII
SO927-0256(97)00065-7
0
1997
Elsevier Science
B.V.
All
During the testing of the pressure behaviour of TiN with a four point bending test, two completely different crack patterns were observed (see Fig. 1). During these tests the samples were constantly observed with X-rays. Therefore, we are able to determine the failure stress exactly. In both cases the value for the failure stress has the same dimension. The crack pattern in the case of delamination is similar to Euler’s buckling. Hence, the delamination is the starting point of our calculation. We have to introduce in the wellknown buckling theory a parameter which represents the interface behaviour. This parameter is realized with a spring. The reason to follow this way are the specifications of the available FE-programm. In commercial codes one or more spring elements are included in the element library. For our calculation we use the following data. As the substrate the steel Ck15 is used. Its Young’s modulus is 210 GPa and Poisson’s ratio is 0.3. In a tensile test we measured the stress-strain data. For TIN the Young’s modulus is 420 GPa and the Poisson’s ratio 0.2. TiN is pure elastic between - 7.6 and 1 GPa.
rights reserved.
T.h. Richter, H. Gliiser/Computational
Materials Science 9 (1997) 116-120 Table 1 Geometrical
data [3]
F Distance between delamination Length of delamination Height of the layer Height of the substrate
‘Regions of delamination
F
117
Parameter
Value
I, 1 h H
1.9 mm 40 pm 2.8 pm 2.3 mm
axial movement in x-direction the applied boundary conditions
is possible. are
Hence,
u(0) = u’(0) = U( 1) = u’(f) = 0.
Fig. 1. Sketch of the different failure pattern after the four point bending test: (a) thick film; (b) thin film.
All material properties were assumed to be isotropic. The geometrical data are shown in Table 1.
2. Stability
analysis
The model was simplified to the configuration shown in Fig. 2. This means, that we have reduced the configuration from three to two dimensions and normalised the parameter b (width of the delamination) to I pm. The required force Fp can be calculated from the measured stress. The boundary conditions for the layer are as follows. For the left end all three degrees of freedom are restrained. For the right end, only an
(1) A finite distance between layer and substrate was introduced to be able to apply the springs in the FE-calculation. However, in reality, for an undamaged layer this distance will be equal to zero. The goal of the analytical solution is the determination of the spring parameter c. This parameter represents a continuous load along the whole length 1. At a first step we determine the bending line. Then we introduce the boundary conditions and perform the stability analysis. The substrate material so far has only an influence on the applied load required in the bending test, but not on the analytic solution. We can get the differential equation for the bending line similar to the way described in [4]
“(X)“” + ;“(x)” +iu(
(2)
In Eq. (2) E represents the Young’s modulus and I is the area moment of inertia. The general solution of Eq. (2) is U(X) = C, sin( Ax) + C, sin( Bx) + C, cos( Ax)
linear spring with parameter Fig. 2. Simplification
x) = 0
+ c, cos( Bx)
c
of the problem for the analytical
solution.
(3)
T.h. Richter.
118
H. Gliiser/
Computationul
Muterds
Science
9 (1997)
3. Effect of the substrate
with the substitutions
B= /m
(5)
Now, we have to perform the stability analysis under the restriction of Eq. (6). Eq. (6) represents the determinant of the coefficients Ci, which result from the boundary conditions.
A first FE-calculation with the above determined value C.analytic shows that the behaviour of the model is too weak. The construction of the model will be discussed later. For now we are only interested in the result. The weak behaviour can be explained by the fact, that a second spring was automatically introduced by the program when the substrate was applied to our model. The overall spring parameter in this case can be calculated from the following equation (8)
det(C,)
=
116-120
0
0
1
I
A
B
0
0
sin( AI)
sin( BI)
A sin( Al)
Bsin( BI)
cos( Al) -
cos( BI)
A cos( A/)
- Bcos(
BI) (6)
From Eq. (6) we can evaluate equation for the system. det( C;) = ( A2 + B’)sin( + 2AB(cos(
Al)sin( Al)cos(
the characteristic
B1) Bl) - 1) =0
(7)
This equation can be solved analytically only in some cases. A numerical solution can always lead to a result. Eq. (7) represents the function det(Ci) =f(c) and can be plotted (Fig. 3). The first zero-point will be the unknown critical value c. From Fig. 3 we can determine c,,,,~,~~ = 19.2 MPa for the combination TIN and steel. dd (C,,
r-1
c NW 1000
2000
30000
4000 1 Fig. 3. Graph of the numerical solution.
In a first step we assume that the analytical result will be equal to the value for the interface spring in the ANSYS calculation plus the additional term which is unknown so far. The result of the (‘substrate’ calculated stress has to be lower compared to the experimental result. Therefore, we have to change the parameter in the FE-simulation until we get the failure ~~~~~~~btained from the experiments. With Eq. (8) we determine csubstrats= 1 1.5 MPa.
4. FE-calculation The calculations were done by the commercial FE-code ANSYS on a HP9000 workstation [S]. For the 2D-calculations we use simple elements. In addition to the structural elements and spring elements we also use contact elements, which are necessary to avoid penetration of the layer into the substrate. Problems are hidden in the geometrical data. The real thickness of the substrate exceeds the thickness of the layer by a factor of 1000. These fact would require a meshing for the substrate that leads to computing times beyond reasonable limits. Further, it is necessary to apply as many springs as possible on the interface since we have to simulate a uniform line load. Hence, we decided to model substrate and layer with the same height. This is possible with respect to the applied boundary conditions. With these conditions we do not reduce the substrate effect. At the
T.h. Richter, H. Gkiser/Computational Materials Science 9 (1997) 116-120
” P z 1 m
7800 7600 ‘. 7400.. 7200.. moo,. ,/” ,’ 6000 .. / ,..’ / omo~~ ,d’ 6400 .*__...----:
6200. 1
:
:
: 5
:
:
:
A--.----/” .T’
:
: : 10
:
:
:
: 16
:
I
Timesteps
Fig. 4. Working of the automatic time step procedure.
delamination point the substrate will be plastified in reality and in our model. Boundary conditions were applied, such that the left side of layer and substrate is fixed in x-direction, the bottom of the substrate is fixed in y-direction and the substrate is coupled to the layer at the left (X = 0) and right (X = I) hand side of the model. A displacement is applied at the right end of layer and substrate in negative x-direction. To obtain the line load we have to calculate the the constant for each value cAfNSYS,representing single spring. The line load will be activated by a movement of the layer normal to it’s interface. The load of the combined springs is the equivalent to the line load. If we compare the resulting forces in the analytical model and in the FE- calculation, we get the equation C analytic -=C6kUS
119
the exact point of the delamination we use the automatic time step algorithm. The effect of this procedure, the approaching of the solution to the critical value, is shown in Fig. 4. One can see the shape of the deformed layer in Fig. 5. In Fig. 5 one also can see which region have changed from a homogeneous pressure to tensile stress. Since the critical tensile stress for TiN is only 1 GPa, it is likely that mode I cracks will be generated in this stress regions.
5. Conclusion In the present the delamination To do this with mine the spring analysis. During
work we show a way to calculate of an TiN layer on a steel substrate. an FE-code, we first need to deterparameter cmalyric from a stability the FE-calculation we determine the parameter Csubstrate for the substrate effect (weaker performance of the FE-model as the analytical model) in an iterative way. This way is generally useful, if the crack pattern allows a simplification into a two dimensional model.
Acknowledgements
n
1,
The parameter n is the number of springs at the critical length 1. The calculation was done in a framework of a nonlinear buckling analysis. To get
The presented work was performed within the Graduiertenkolleg ‘ Werkstoffphysikalische Modellierung’ at Freiberg University of Mining and Technology and supported by the ‘Deutsche Forschungsgemeinschaft’.
~
Load direction
pplied displacement
Fig. 5. FEM-result after delamination (deformed geometry).
120
T.h. Richter,
H. Gkiser/
Compututionul
References [I] H. Oettel, R. Wiedemann, Residual Stresses in PVD Hard Coatings, Proc. of ICMCTP, San Diego, 1995. [2] A.G. Evans, J.W. Hutchinson, The thermomechanical integrity of thin films and multilayers, Acta Metall. Mater. 43 (1995)
Muterids
Science
9 (1997)
116-120
2507-2530. [3] Th. Bertram,
H. Oettel. Stresses and mechanisms for failure of TIN coatings, J. Vat. Sci., in preparation. 141 G. Biirgermeister, H. Steup, Stabilitatstheorie mit Erlluterungen zur DIN41 14-Teil I, Akademie-Verlag. Berlin, 1957. [5] ANSYSO, Version 5.1 Documentation, ANSYSO, 1994.