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Stress intensity factors for mixed-mode fracture in adhesive-bonded joints
Stress intensity factors for mixed-mode fracture in adhesive-bonded joints H.L.J. Pang School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263 (Received 8 February 1993; revised 10 January 1994) Finite-element analysis was carried out for a compact mixed mode (CMM) fracture specimen designed for fracture toughness testing of adhesive bonded joints. Stress intensity factors were computed for the homogeneous substrate material and the corresponding adhesive-bonded joint. Using the finiteelement result, a semi-empirical correction factor is applied to the CMM adhesive-bonded joint specimens. (Keywords: adhesive bonds; fracture toughness; mixed mode; stress intensity factor)
The strength of adhesive-bonded joints is a major concern when they are used in structural applications. Adhesive joints usually fail by initiation and propagation of flaws, and under these circumstances fracture mechanics is a practical approach for determining the toughness of adhesive joints. Kinloch I has documented the application of fracture mechanics concepts to adhesive joints, and provided a range of fracture toughness data for different adhesive joint designs. Of particular interest, the determination of mixed mode (mode I and II) fracture toughness was reported for scarf joints, but there is still no standard testing procedure for mixed-mode fracture toughness determination and hence there is scope for further research. This study reports an alternative adhesive joint design for mixed-mode fracture testing. A compact mixed mode (CMM) fracture specimen is proposed. It is capable of determining the complete range of fracture toughness under pure mode I, pure mode II and mixed mode (I and II) loading conditions. The experimental set-up for the CMM fracture specimen is shown in Figure 1, where different modes of loading are indicated on the loading frame. The single-edge cracked specimen shown in Figure 2 is inserted into the loading frame and secured by six loading pins. The concept for the CMM fracture specimen was modified from the compact mode II fracture specimen reported earlier by Banks-Sills and Arcan 2. The development of the CMM fracture specimen provides a comprehensive procedure for determining the fracture toughness of adhesive bonds under pure and mixed modes I and I I of fracture using this unique specimen. This test procedure will provide the toughness data for characterizing the fracture criterion of adhesive bonds subjected to mixed mode I and II loading. This paper will deal only with the finiteelement work. The experimental work is still in progress and will be reported later. 0142-1123/94/060413-04 (~ 1994 Butterworth-Heinemann Ltd
T
/
\;.,o
mixed mode
m o d e II
Figure I C M M
load frame
FRACTURE MECHANICS ANALYSIS In linear elastic fracture mechanics, the mode I fracture toughness parameters such as the stress intensity factor K~c, strain energy release rate Gic and J-integral Jl~ are well established for metallic materials. For a crack in a homogeneous material, a simple relationship exists between these parameters: (gic) 2
JI~=GI~=
(1)
E'
where E' = E for plane stress and E'a -
Ea ( 1 - v 2) for
plane strain. Fatigue, 1994, Vol 16, August
413
Mixed-mode fracture in adhesive-bondedjoints: H.L.J. Pang For a crack at the centre of an adhesive layer, relatively distant from any interface, it can be assumed that the above expression is still valid, and the appropriate value of elastic properties for the adhesive, E~ and Va, are substituted to give Jic(joint) = Gic(joint) where E'~
= Ea
[Klc(joint)] 2 E'a
for plane stress and E'~
ftt
lt
Es , Os
Es , Os
A
(2) Ea (l_v2)
Lt ta
Ea , 9a
for
plane strain. In practice, the adhesive layer thickness is in the range of 0.1-1.0 mm for adhesive-bonded fracture toughness test specimens. A predetermined crack size is placed in the centre of the adhesive layer. This is often a Teflon tape inserted during the bonding process. Although the crack tip is within the adhesive layer, the close proximity of the interfaces (i.e. substrate surfaces) will result in complications, and the stress intensity factor solution must take into account the interaction effects of the adhesive and substrate material properties. There are no clearly established relationships for this problem, and the author is proposing an empirical solution derived from finite-element calibrations described in the procedure below. Consider an edge crack in a homogeneous material of a substrate specimen given in Figure 3a and the corresponding case with a crack at the centre of an adhesive joint shown in Figure 3b. Employing finiteelement analysis for both these models and taking account of the appropriate material properties, it is possible to compute the corresponding stress intensity factors and J-integrals. The relationship of the stress intensity factor to J-integral for the substrate (Ks and Js) and adhesive joint (Ka and J,) under plane strain conditions is given by
K ~ = ,~/ (\J~, ]Es t
(3)
Ka = ~ / \ 1 - ~ /
(4)
Using the finite-element result for these two cases,
Es , 0 s
Figure 3 (a) Substrate specimen; (b) adhesive-bondedspecimen
a constant or calibration factor t3 is proposed for expressing the ratio of the stress intensity factors:
Ka
13- K -
/[J.Ea(1-Ves ) ]
(5)
The calibration factor 13 is an empirical correction factor to take account of the effects of the adhesiveto-substrate material properties modelled in the finiteelement result. Therefore this correction factor allows us to use the established solutions for the substrate specimen. By multiplying the correction factor, the solution for the adhesive joint is obtained: K~ = 13Ks
(6)
This provides a useful relationship, which forms the basis for the finite-element calibration factors for the compact mixed mode (CMM) adhesive-bonded joint specimen. The adhesive-bonded CMM specimen consists of aluminium alloy parts bonded with epoxy adhesives. The elastic modulus and Poisson's ratio for aluminium are 71 000 N mm -2 and 0.33. This was used for the substrate specimen, and stress intensity factors were compared with available closed-form solutions for the CMM specimen. For subsequent analysis of the adhesive joint, the properties of the aluminium and those of the epoxy adhesive (3300 N mm -2 and 0.34) were used respectively. FINITE-ELEMENT ANALYSIS
98 mm
+
+ 84
Figure 2 Test specimen
414 Fatigue, 1994, Vol 16, August
mm
The finite-element models consists of plane strain, eight-noded quadratic elements. The crack tip stress singularities were modelled by employing collapsed eight-noded elements with the mid-side nodes moved to the quarter-point 3. This approach was applied for the first ring of elements surrounding the crack tip. Stress intensity factors for the respective mode I, mode II and combined mode (I and II) were calculated using the classical solutions from the crack tip displacements as shown in Figure 4. A virtual crack extension based J-integral method by Parks 4 was also used in the M A R C finite-element program. The elastic Jintegral result intrinsically consists of the mode I and mode II components, and the separation of the individual modes is shown in Figure 5. This method was used in a benchmark study reported by the author 5, where the ratio of the crack face mode I to mode II displacements was used to separate the
Mixed-mode fracture in adhesive-bonded joints: H.L.J. Pang
l ~'o (1 + v) LTJ
~
K'Lo ~
g~)
IT]
v
_ 6.691(at
f(0) = (2g + 1)sin02 - sin 30
IK
Y
g(0) = -(2K + 3)sin 0 - sin ~30
,
3 --V
K =
3 -
fn ( a ) = 1.006-0. 313(a) + 3. 344(a) 2
4v for plane strain; K = ~
,
r
for plane stress
Figure 4 Displacementsubstitution method Mixed m o d e fracture 2u
G =
E'
(assuming
K,,
Au
K,
Av
RESULTS AND DISCUSSION
Separate stress intensity factors
[ GE' ] in K, = [ ~ j Figure 5
and P is the applied load; a is the crack length; w is the width of the specimen and t is the thickness. For a/w = 0.5, Equations (7) and (8) give fl = 2.7167 and fii = 1.2022, and this is used as a benchmark solution for comparison with the finite-element result. Finite-element analysis of the compact mixed mode (CMM) substrate specimen and subsequently for the adhesive joint specimen with inter-layer cracks was modelled. These specimens were loaded in pure mode I, pure mode II, and mixed-mode I and II loading, as shown in Figure 6. A nominal adhesive layer thickness of 0.5 mm was modelled and an inter-layer crack with depth to width ratio a/w = 0.5 was located at the centre line of the adhesive layer.
K,. = O)
Using displacement along crack faces R:,
+,.649(at4
and K,, = .,K,
,_
I
-,
Separation of mixed-mode stress intensity factors ([ and
II)
corresponding stress intensity factors. This procedure gives an approximate separation of the mode I to mode II stress intensity factors from the J-integral value, so that a check can be made in comparison with the displacement substitution result. Only the displacement substitution result was used in deriving the calibration factors. Finite-element calibration for the CMM fracture specimen was carried out to determine the stress intensity solution for adhesive-bonded joints according to equations (5) and (6). Several finite-element meshes were modelled for the case of a homogeneous substrate material consisting of the aluminium CMM specimen. Mesh refinement studies were conducted and compared with the stress intensity factor solution provided by Banks-Sills and Arcan 2.
The mesh for the CMM specimen that converged closest to the solution in Equations (7) and (8) was adopted for the study. The refined mesh modelled has element sizes in the adhesive layer down to 0.125 mm. The stress intensity factors were determine using both the displacement substitution method and the Jintegral method. The non-dimensionalized stress intensity factor for the substrate specimen is in good agreement with the solution given in Equations (7) and (8). The finite element results for the homogeneous substrate (aluminium) model where f~ = 2.7133 and fIl : 1.1455 are within - 0 . 5 % and - 4 . 7 % of the reference solution. For the adhesive joint no benchmark solution is available and is normalized with the substrate solution to derive the calibration factor according to Equation (5) for pure mode I, pure mode II and mixed-mode loading conditions (Table 1). With these calibration
mode
II
For pure mode I loading: Kx~(CMM)=
FV%ra [a\ wt f'lw)
(7) mode
where
I
f i ( a ) = 5 . 5 2 8 - 4 2 . 2 9 ( a ) + 159.8(a) z - 2 5 4 . 1 ( a ) 3 + 162.5(a) 4 For pure mode II loading:
P~/~a_ [ a \ Km(CMM) = --wt .ti,~) where
(8) Figure 6 Finite-element model for CMM specimen
Fatigue, 1994, Vol 16, August 415
Mixed-mode fracture in adhesive-bonded joints: H.L.J. Pang Table 1 Stress intensity factor and calibration factor result CMM specimen Mode I 0° Mixed mode 45° Mode II 90°
KI~ (N mm 15)
KIIa (N mm-1~)
KI~ (N mm Ls)
Kn~ (N mm-1'5)
~t
16t.6
-
620.8
-
0.26
119.2
57.8
436.8
175.2
0.27
0.33
-
82.1
-
262.1
-
0.31
[311
:1 t
Figure 7 Finite-element model for CT specimen factors, the stress intensity factor for the CMM specimen can be expressed as KI~(CMM) = [3~Kts(CMM)
(9)
Kn~(CMM) = ~HKHs(CMM)
(10)
Examining the calibration factor for mode I, 13~, it is interesting to note that there is good agreement between the pure mode I loading case (13~ = 0.26) and the corresponding result for the mixed-mode case ([31 = 0.27). Similarly, for mode II, there was good agreement between the pure mode II case (I3H = 0.31) and the corresponding result for the mixed-mode case ( ~ I I ---~ 0.33). AS a matter of interest, a compact tension (CT) specimen 6 was also analysed for comparison with the mode I condition for the CMM specimen. The finiteelement mesh for the CT specimen is shown in Figure 7. Both the substrate specimen and the adhesive joint cases were modelled as in the CMM case. The stress intensity factors were derived in the same manner. It is interesting to note that the calibration factor 13~ for the CT specimen (13t = 0.23) shows satisfactory agreement with the pure mode I case for the CMM specimen. This confirms the applicability of these empirical correction factors for the stress intensity factor solution of the CMM adhesive-bonded joint specimen.
416 Fatigue, 1994, Vol 16, August
CONCLUSIONS Mixed-mode fracture analysis of CMM and CT specimens for the homogeneous substrate and adhesivebonded specimens provided stress intensity factor calibrations that are employed in fracture toughness determination of pure mode I, pure mode II and mixed-mode fracture in adhesive-bonded joints. From the finite-element analysis result it was noted that the correction factors 13t and 13n are in good agreement for the respective modes for the CMM specimen. These calibration factors, employed in Equations (9) and (10), provide an empirical solution for the stress intensity factor of adhesive-bonded joints.
REFERENCES Kinloch, A.J. 'Adhesion and Adhesives: Science and Technology', Chapman & Hall, London, 1987 Banks-Sills, L. and Arcan, M. In Fracture mechanics vol 17, ASTM STP 905, American Society for Testing and Materials, 1986, pp 347-363 Barsoum, R.S. Int. J. Numer. Methods Eng 1977, 11, 85 Parks D.M. Int. J. Fracture 1974, 10, 487 Pang, H.L.J. 'NAFEMS Benchmarks: 2D Test Cases in Linear Elastic Fracture Mechanics', NAFEMS report, 1990 BS 5447, 'K~c Fracture toughness testing', British Standards Institution, 1977
The strength of adhesive-bonded joints is a major concern when they are used in structural applications. Adhesive joints usually fail by initiation and propagation of flaws, and under these circumstances fracture mechanics is a practical approach for determining the toughness of adhesive joints. Kinloch I has documented the application of fracture mechanics concepts to adhesive joints, and provided a range of fracture toughness data for different adhesive joint designs. Of particular interest, the determination of mixed mode (mode I and II) fracture toughness was reported for scarf joints, but there is still no standard testing procedure for mixed-mode fracture toughness determination and hence there is scope for further research. This study reports an alternative adhesive joint design for mixed-mode fracture testing. A compact mixed mode (CMM) fracture specimen is proposed. It is capable of determining the complete range of fracture toughness under pure mode I, pure mode II and mixed mode (I and II) loading conditions. The experimental set-up for the CMM fracture specimen is shown in Figure 1, where different modes of loading are indicated on the loading frame. The single-edge cracked specimen shown in Figure 2 is inserted into the loading frame and secured by six loading pins. The concept for the CMM fracture specimen was modified from the compact mode II fracture specimen reported earlier by Banks-Sills and Arcan 2. The development of the CMM fracture specimen provides a comprehensive procedure for determining the fracture toughness of adhesive bonds under pure and mixed modes I and I I of fracture using this unique specimen. This test procedure will provide the toughness data for characterizing the fracture criterion of adhesive bonds subjected to mixed mode I and II loading. This paper will deal only with the finiteelement work. The experimental work is still in progress and will be reported later. 0142-1123/94/060413-04 (~ 1994 Butterworth-Heinemann Ltd
T
/
\;.,o
mixed mode
m o d e II
Figure I C M M
load frame
FRACTURE MECHANICS ANALYSIS In linear elastic fracture mechanics, the mode I fracture toughness parameters such as the stress intensity factor K~c, strain energy release rate Gic and J-integral Jl~ are well established for metallic materials. For a crack in a homogeneous material, a simple relationship exists between these parameters: (gic) 2
JI~=GI~=
(1)
E'
where E' = E for plane stress and E'a -
Ea ( 1 - v 2) for
plane strain. Fatigue, 1994, Vol 16, August
413
Mixed-mode fracture in adhesive-bondedjoints: H.L.J. Pang For a crack at the centre of an adhesive layer, relatively distant from any interface, it can be assumed that the above expression is still valid, and the appropriate value of elastic properties for the adhesive, E~ and Va, are substituted to give Jic(joint) = Gic(joint) where E'~
= Ea
[Klc(joint)] 2 E'a
for plane stress and E'~
ftt
lt
Es , Os
Es , Os
A
(2) Ea (l_v2)
Lt ta
Ea , 9a
for
plane strain. In practice, the adhesive layer thickness is in the range of 0.1-1.0 mm for adhesive-bonded fracture toughness test specimens. A predetermined crack size is placed in the centre of the adhesive layer. This is often a Teflon tape inserted during the bonding process. Although the crack tip is within the adhesive layer, the close proximity of the interfaces (i.e. substrate surfaces) will result in complications, and the stress intensity factor solution must take into account the interaction effects of the adhesive and substrate material properties. There are no clearly established relationships for this problem, and the author is proposing an empirical solution derived from finite-element calibrations described in the procedure below. Consider an edge crack in a homogeneous material of a substrate specimen given in Figure 3a and the corresponding case with a crack at the centre of an adhesive joint shown in Figure 3b. Employing finiteelement analysis for both these models and taking account of the appropriate material properties, it is possible to compute the corresponding stress intensity factors and J-integrals. The relationship of the stress intensity factor to J-integral for the substrate (Ks and Js) and adhesive joint (Ka and J,) under plane strain conditions is given by
K ~ = ,~/ (\J~, ]Es t
(3)
Ka = ~ / \ 1 - ~ /
(4)
Using the finite-element result for these two cases,
Es , 0 s
Figure 3 (a) Substrate specimen; (b) adhesive-bondedspecimen
a constant or calibration factor t3 is proposed for expressing the ratio of the stress intensity factors:
Ka
13- K -
/[J.Ea(1-Ves ) ]
(5)
The calibration factor 13 is an empirical correction factor to take account of the effects of the adhesiveto-substrate material properties modelled in the finiteelement result. Therefore this correction factor allows us to use the established solutions for the substrate specimen. By multiplying the correction factor, the solution for the adhesive joint is obtained: K~ = 13Ks
(6)
This provides a useful relationship, which forms the basis for the finite-element calibration factors for the compact mixed mode (CMM) adhesive-bonded joint specimen. The adhesive-bonded CMM specimen consists of aluminium alloy parts bonded with epoxy adhesives. The elastic modulus and Poisson's ratio for aluminium are 71 000 N mm -2 and 0.33. This was used for the substrate specimen, and stress intensity factors were compared with available closed-form solutions for the CMM specimen. For subsequent analysis of the adhesive joint, the properties of the aluminium and those of the epoxy adhesive (3300 N mm -2 and 0.34) were used respectively. FINITE-ELEMENT ANALYSIS
98 mm
+
+ 84
Figure 2 Test specimen
414 Fatigue, 1994, Vol 16, August
mm
The finite-element models consists of plane strain, eight-noded quadratic elements. The crack tip stress singularities were modelled by employing collapsed eight-noded elements with the mid-side nodes moved to the quarter-point 3. This approach was applied for the first ring of elements surrounding the crack tip. Stress intensity factors for the respective mode I, mode II and combined mode (I and II) were calculated using the classical solutions from the crack tip displacements as shown in Figure 4. A virtual crack extension based J-integral method by Parks 4 was also used in the M A R C finite-element program. The elastic Jintegral result intrinsically consists of the mode I and mode II components, and the separation of the individual modes is shown in Figure 5. This method was used in a benchmark study reported by the author 5, where the ratio of the crack face mode I to mode II displacements was used to separate the
Mixed-mode fracture in adhesive-bonded joints: H.L.J. Pang
l ~'o (1 + v) LTJ
~
K'Lo ~
g~)
IT]
v
_ 6.691(at
f(0) = (2g + 1)sin02 - sin 30
IK
Y
g(0) = -(2K + 3)sin 0 - sin ~30
,
3 --V
K =
3 -
fn ( a ) = 1.006-0. 313(a) + 3. 344(a) 2
4v for plane strain; K = ~
,
r
for plane stress
Figure 4 Displacementsubstitution method Mixed m o d e fracture 2u
G =
E'
(assuming
K,,
Au
K,
Av
RESULTS AND DISCUSSION
Separate stress intensity factors
[ GE' ] in K, = [ ~ j Figure 5
and P is the applied load; a is the crack length; w is the width of the specimen and t is the thickness. For a/w = 0.5, Equations (7) and (8) give fl = 2.7167 and fii = 1.2022, and this is used as a benchmark solution for comparison with the finite-element result. Finite-element analysis of the compact mixed mode (CMM) substrate specimen and subsequently for the adhesive joint specimen with inter-layer cracks was modelled. These specimens were loaded in pure mode I, pure mode II, and mixed-mode I and II loading, as shown in Figure 6. A nominal adhesive layer thickness of 0.5 mm was modelled and an inter-layer crack with depth to width ratio a/w = 0.5 was located at the centre line of the adhesive layer.
K,. = O)
Using displacement along crack faces R:,
+,.649(at4
and K,, = .,K,
,_
I
-,
Separation of mixed-mode stress intensity factors ([ and
II)
corresponding stress intensity factors. This procedure gives an approximate separation of the mode I to mode II stress intensity factors from the J-integral value, so that a check can be made in comparison with the displacement substitution result. Only the displacement substitution result was used in deriving the calibration factors. Finite-element calibration for the CMM fracture specimen was carried out to determine the stress intensity solution for adhesive-bonded joints according to equations (5) and (6). Several finite-element meshes were modelled for the case of a homogeneous substrate material consisting of the aluminium CMM specimen. Mesh refinement studies were conducted and compared with the stress intensity factor solution provided by Banks-Sills and Arcan 2.
The mesh for the CMM specimen that converged closest to the solution in Equations (7) and (8) was adopted for the study. The refined mesh modelled has element sizes in the adhesive layer down to 0.125 mm. The stress intensity factors were determine using both the displacement substitution method and the Jintegral method. The non-dimensionalized stress intensity factor for the substrate specimen is in good agreement with the solution given in Equations (7) and (8). The finite element results for the homogeneous substrate (aluminium) model where f~ = 2.7133 and fIl : 1.1455 are within - 0 . 5 % and - 4 . 7 % of the reference solution. For the adhesive joint no benchmark solution is available and is normalized with the substrate solution to derive the calibration factor according to Equation (5) for pure mode I, pure mode II and mixed-mode loading conditions (Table 1). With these calibration
mode
II
For pure mode I loading: Kx~(CMM)=
FV%ra [a\ wt f'lw)
(7) mode
where
I
f i ( a ) = 5 . 5 2 8 - 4 2 . 2 9 ( a ) + 159.8(a) z - 2 5 4 . 1 ( a ) 3 + 162.5(a) 4 For pure mode II loading:
P~/~a_ [ a \ Km(CMM) = --wt .ti,~) where
(8) Figure 6 Finite-element model for CMM specimen
Fatigue, 1994, Vol 16, August 415
Mixed-mode fracture in adhesive-bonded joints: H.L.J. Pang Table 1 Stress intensity factor and calibration factor result CMM specimen Mode I 0° Mixed mode 45° Mode II 90°
KI~ (N mm 15)
KIIa (N mm-1~)
KI~ (N mm Ls)
Kn~ (N mm-1'5)
~t
16t.6
-
620.8
-
0.26
119.2
57.8
436.8
175.2
0.27
0.33
-
82.1
-
262.1
-
0.31
[311
:1 t
Figure 7 Finite-element model for CT specimen factors, the stress intensity factor for the CMM specimen can be expressed as KI~(CMM) = [3~Kts(CMM)
(9)
Kn~(CMM) = ~HKHs(CMM)
(10)
Examining the calibration factor for mode I, 13~, it is interesting to note that there is good agreement between the pure mode I loading case (13~ = 0.26) and the corresponding result for the mixed-mode case ([31 = 0.27). Similarly, for mode II, there was good agreement between the pure mode II case (I3H = 0.31) and the corresponding result for the mixed-mode case ( ~ I I ---~ 0.33). AS a matter of interest, a compact tension (CT) specimen 6 was also analysed for comparison with the mode I condition for the CMM specimen. The finiteelement mesh for the CT specimen is shown in Figure 7. Both the substrate specimen and the adhesive joint cases were modelled as in the CMM case. The stress intensity factors were derived in the same manner. It is interesting to note that the calibration factor 13~ for the CT specimen (13t = 0.23) shows satisfactory agreement with the pure mode I case for the CMM specimen. This confirms the applicability of these empirical correction factors for the stress intensity factor solution of the CMM adhesive-bonded joint specimen.
416 Fatigue, 1994, Vol 16, August
CONCLUSIONS Mixed-mode fracture analysis of CMM and CT specimens for the homogeneous substrate and adhesivebonded specimens provided stress intensity factor calibrations that are employed in fracture toughness determination of pure mode I, pure mode II and mixed-mode fracture in adhesive-bonded joints. From the finite-element analysis result it was noted that the correction factors 13t and 13n are in good agreement for the respective modes for the CMM specimen. These calibration factors, employed in Equations (9) and (10), provide an empirical solution for the stress intensity factor of adhesive-bonded joints.
REFERENCES Kinloch, A.J. 'Adhesion and Adhesives: Science and Technology', Chapman & Hall, London, 1987 Banks-Sills, L. and Arcan, M. In Fracture mechanics vol 17, ASTM STP 905, American Society for Testing and Materials, 1986, pp 347-363 Barsoum, R.S. Int. J. Numer. Methods Eng 1977, 11, 85 Parks D.M. Int. J. Fracture 1974, 10, 487 Pang, H.L.J. 'NAFEMS Benchmarks: 2D Test Cases in Linear Elastic Fracture Mechanics', NAFEMS report, 1990 BS 5447, 'K~c Fracture toughness testing', British Standards Institution, 1977