- Email: [email protected]
On the contact of a spherical indenter and a thin composite laminate
Composite Structures 26 (1993) 77-82
On the contact of a spherical indenter and a thin composite laminate Andreas P. Christoforou Department of Mechanical Engineering, Kuwait University, PO Box 5969, Safat, 13060 Kuwait An analytical solution for the contact between a rigid sphere and a thin composite laminate supported on a rigid substrate is presented. The contact force-deformation, including damage effects, is obtained from an axisymmetric formulation of the contact problem. The material is assumed to be transversely isotropic and following an elastic-perfectly plastic stress-strain law. Damage is predicted using a maximum shear failure criterion. Experimental data previously reported in the literature agree well with the analysis.
used to predict permanent indentations in an empirical unloading law, similar to yielding in metals, supports the idea of using plastic deformation principles to incorporate damage effects in the indentation response of composites. More recently Swanson and Rezaee 7 have conducted experiments where thin laminates resting on a rigid substrate were indented by a rigid sphere. In the initial stages the data seem to follow the Hertzian contact law, but at higher loads softening associated with significant observed damage and permanent deformation is quite evident in the force-deformation response. The purpose of this paper is to present a simple analytical model for studying the contact force-deformation between a thin laminate supported by a rigid substrate and a frictionless rigid sphere including damage effects. The solution is obtained from an axisymmetric formulation of the contact problem assuming the material is transversely isotropic. A further simplification is that the stresses along the thickness of the laminate are taken to be constant by assuming that the radius of the indenter is large compared to the thickness of the laminate. Furthermore the material is assumed to be elastic before damage and perfectly plastic after damage occurs. On the basis of experimental data reported by Poe 8 and Poe and Illg, 9 the damage region is predicted using a maximum fiber shear failure criterion. The resulting expressions of the contact force-deformation appear to be in good agreement with the experimental results reported in Ref. 7.
INTRODUCTION It is well known that fiber-reinforced composites show high strength properties when subjected to in-plane loads but show poor resistance to out-ofplane loads such as lateral impact. The residual strength of composite structures can be reduced as much as 50% as a result of impact damage. Depending on the characteristics of the impact event, impactor mass, size, velocity, target material and structural properties, impact damage is caused by either bending stresses or local contact stresses. Recent work has been directed toward separating the local contact effects from the dynamic problem such that analysis or data from static indentation laws may be used in analyzing the impact response. 1-4 The contact force-deformation relationship produced by a spherical indenter has been shown to follow a law similar to that of isotropic materials,5,6 from a Hertz analysis. There are some uncertainties, however, when a Hertzian-type law of contact based upon the contact of a sphere and a half-space is applied to thin laminates. During the initial stages of loading, when the contact radius is small compared to the laminate thickness, the Hertzian contact analysis should be adequate. However, as higher loads cause extensive damage in the contact zone the effects of damage should be a major factor of the response. Experimental work carded out by Yang and Sun, 6 where a critical indentation (acr) was 77
Composite Structures 0263-8223/93/S06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain
A. P. Christoforou
78 ANALYSIS
Boundary conditions
Governing equations
Figure 1 shows the geometry of the thin laminate resting on a rigid substrate, indented by a frictionless rigid sphere of radius R. The boundary conditions are as follows:
In the case of axisymmetric problems, the governing stress equilibrium equations in the absence of body forces are given by 4,~° Oar+ i)O____~-t or -- Oo
Or
Oz
0
r
(1)
OOz+O,z=o Or
Oz
r
The constitutive equations for a cylindrically orthotropic material are
{rI S12131/1r =/312 Ez LSI3
322 823
gO
(2)
323[ O0 833J orz
( 1 ) On the free surface z = 0:
az(r)=O
r>a
orz(r)= 0
r>0
2 t" w(r)=a--2R
O<_r
(5)
where a is the contact radius, the profile of the sphere is described by the parabolic approximation, and a is the indentation depth at r = 0. (2) On the surface z = h:
o,=(r)=O
(6)
w(r) = 0 }Prz = 3440rz
where
Sij
is the engineering compliance matrix.
For this problem, the material is also assumed to be transversely isotropic in the r- 0 plane. In terms of the engineering constants the compliance coefficients are given as SII = $22
1
1
E,
Eo
-- ~rO
-- VOr
L"r
E 0
S12
SI 3=$23= -
1 $33 = - E=
In the following approximate stress analysis, it is assumed that the laminate thickness is small compared to the contact radius (a/h ,> 1 ); in such contact problems arz is assumed to be equal to zero throughout the layer thickness) °,l~ Hence eqn (1) becomes
l
$44- G~
Vrz =
Solution procedure
(3)
(7)
0
r
The compressive vertical strain in the laminate is given by the geometry of deformation, i.e.
-- YOz
Er
dr
Eo
For small deformations under the assumption of axisymmetry the strain-displacement equations are given as Ou
e,
Or
Ow
e.
Oz (4)
u r
~" 0 = - -
0u 0w "~ - Oz Or
~2 rz = - -
where the displacements assume the form (u, 0, w) in cylindrical coordinates (r, 0, z) and the partial derivative with respect to 0 is identically zero.
Thinlaminateof ~
*, t"
| ¢x
,l , , I,, ), ! , / / / / / × / / / /,,, Rigidsubstrate / / / / / I ~ / ~
// / /
z Fig. 1.
Geometry of the contact between a sphere and a thin laminate supported by a rigid substrate.
Contact of spherical indenter and thin composite laminate
Owa (
ez
Oz
2Rh
1-
(8)
and the sphere indentation a at r = 0 is given by
a2 a =w(0) = 2R
(9)
Algebraic manipulation of eqns (2), (3), (4), (7) and (8) yields d2u
1 du
dr 2 +
r dr
u (1 OI- ]lrO) 'l)zr r2 (1-VrzVzr) Rh r
(10)
The solution of the differential equation (10) is given by u=Ar~
(11)
(l +'RrO)T'zr r 3 8(1 ltrz'llzr) Rh
79
it is very important that damage effects on the response are included in the analysis. A general analysis in the damage zone would be extremely difficult, and a further simplification is necessary. Following the procedure of Ref. 11, from eqn (13) it is easy to determine that the maximum stress az(r= 0), is a lot larger than the corresponding maximum values for ar and ao. Thus, it is reasonable to apply the failure criterion to a z only. Considering experimental evidence reported by Poe and Illg,9 which suggest fiber failure in shear, in the type of loading under consideration, a maximum fiber shear failure criterion is used. When damage is predicted, the material in :the composite laminate is assumed to have two distinct regions shown in Fig. 1. If the material in the damage region is assumed to be perfectly plastic while the outside is elastic, Fig. 2, it follows that
(Oz)v = 2Sf
r_< b
-
Ez a2 (az)~-2( 1 - ~ z r ) R h
The boundary condition az(a ) = 0 yields A= -
(1 + v~o) Vzr 2 a 4( 1 - V,.zVzr) Rh
(12)
Finally the expressions of the stress distribution are obtained: _
vzrEra 2
Or 411-
zrlgh
( r2) 1 - ~a 2
vzrEra2
(
0 0 - 4 ( 17v~v~r)Rh
za2
az=2(l__vrzVzr) R h
1
3r2 / 2a2]
=
(13)
(1r2)
b2 = a2 _ 4Rh(1
(1-VrzVzr) h
- VrzVzr) Sf Ez
(17)
a.-
2h( 1 - VrzV~) Sf
(18)
Ez
The total contact force is then
--a5
f f az(r)(2orr)dr=
~EzR
b<_r<_a
where Sf is the fiber shear strength. At the interface r = b, from eqn (16)
grgza4
4(1 - vzy,~ ) Rh
F(a)=
a
2
2Se(2=r)dr+
az)e(2~r)dr
(19)
(14)
Substituting eqn (9) into eqn (14) yields F(a)
1
The indentation at the onset of damage can be computed by setting b = 0 in eqn (17):
The contact force F is obtained from the contact stress distribution, F(a)=
(16)
__as
2Sf
/
1
(15)
I
The contact force-deformation relationship of eqn (15) is valid in the elastic range, assuming no permanent deformation or damage is present. However, when damage is present, as in this case,
/i
I 0
I
tcr
Strain ~z
Fig. 2.
Stress-strain response assumed for the laminate material.
A. P. Christoforou
80
Integrating and combining with eqns (17) and (18) yields
F(a)=2~RSf(2a-acr)
O~cr~a -< O~m (20)
During the unloading phase the vertical compressive stress is given by
[ E~ (a-ao) o-=~ h(1-VrzV:~) " ] E_:a2_~_[1 r: I t2Rh(1-v~zv.~) k a 2]
0 <_r<_bm
(21) b m -< r<
a
Therefore
oz(r)(2arr)dr
F(a)= 0
yields
F(a)(1
~rRE_ - vr v ,)h
( a 2 - a~)
(22)
where
a. = b2/2R
= o~m -- O~cr
(23)
is the permanent indentation.
RESULTS AND DISCUSSION
Calculations were performed on a [0/+68/0]~ IM7/55A carbon/epoxy laminate to compare with experimental data reported by Swanson and Rezaee. 7 The laminated plates had a nominal thickness of 2.4 mm, were supported by a hardened steel block, and were subjected to lateral loads by a hardened steel spherical indenter with a diameter of 15.88 mm. The ply properties of the Hercules carbon/epoxy system are given in Table 1. The contact force-deformation response predicted by the analysis was found to be more dependent on the through-thickness properties than the in-plane properties. The result is consistent with previous findings. 3,4.9 Thus, the transversely isotropic axisymmetric model seems to be a reasonable approximation.
Damage was found to have a dominant effect on the overall force-deformation response. The damage zone was predicted using a maximum fiber shear failure criterion. Since such experimental data were not available for this material system, the fiber shear strength St = 310 MPa s f o r the Hercules AS4/55A, a similar Hercules carbon/epoxy system, was used. The onset of damage and the subsequent response were found to be directly proportional to the shear strength of the fibers. The critical indentation, a . , at the onset of damage that is used to predict permanent deformations, does not seem to be a material property as suggested in Ref. 3. Besides the shear strength and through-thickness modulus, O~crgiven by eqn (18) also depends on the laminate thickness. Thus a more appropriate material property seems to be the critical normal strain, ecr, which is the critical indentation divided by the thickness. This observation is consistent with fundamental materials behavior under simple loading, and findings that suggest fiber failures are best modeled by maximum strain failure criteria. However, this finding depends on an approximate analysis and more experimental data are needed. Figure 3 shows the comparison of the finite element solution of the same problem 7 and eqn (15) of the present analysis. The agreement between the two methods is excellent. This important result gives credence to the analysis since the finite element in this case is a better approximation of the real set-up, and in view of the greatly simplifying assumption of constant through-thickness stresses used in the analysis. As can be seen in Fig. 4, in the elastic range any analysis -- finite element, the present one, or a Hertzian type -- will be adequate in predicting the
100
t
r
t
8O
60
o
4o
m
Finite Elemt~t
Soluti~ (tqef.7) Pre, c~,nt A n a l y s i s
Equ.(lS)
20
Table 1. Material properties for IM7/55A carbon/epoxy Fiber-direction Young's modulus, E ~ Transverse Young's modulus, E22 In-plane shear modulus, G~ 2 Major Poisson's ratio, v 12
169 GPa 6.72 GPa 5'26 GPa 0.266
0
0.5
1.0
I 1.5
210
2.5
Indentation aclmm]
Fig. 3.
Contact force-indentation response of a thin supported laminate.
Contact of spherical indenter and thin composite laminate
response. At higher load levels when damage effects are quite evident in the response, linear elastic analyses give poor predictions while the present analysis including damage effects provides a better estimate of the response. This can also be seen in Figs 5 and 6, where the complete loading histories at two different load levels are shown. It is very encouraging that the simple model presented here gives reasonably good predictions when compared with available experimental data. The data are very limited, however, and more experimental work is needed.
4o~
~
i
w
i
1
I--
zo
g 10
f
li I I
~' ' Hertzi~n Anaiy~is~ ' ' (Ref. 7) \ PresentAnalysis~
I
30
/ -J
80
®
81
0
/ I ] I !
J9
Experimentaldata" c,.,.7)
----- Present Analysis _ (tqus ls.z0,zz ) ®- Elastic Analysis -"
0.5 1.o 1.5 2.0 2.5 Indentation 0~ ( m m)
3.0
Fig. 6. Force-indentation response for the contact between a spherical indenter and a thin laminate supported by a rigid substrate.
,-Z 6o U. 4o
O
,.~ 2o
/
/
/ ~ ~"
[--
J t~."
I
¢
I
data
ExPerimental
( Ref. ? )
I
1.0
I
I
2.0
3.0
Indentation o~(mm) Fig. 4.
Comparison of analytical predictions and experimental data for a thin supported laminate.
20
,
,
,
,
°I,'
,_.IS Z
/
"5 "°1°
i/
tO//
i
5
/ J ,
°o
,
o.~
/
Experimental data ( Ref. 7 ) --P r e s e n t Analysis [Eque 15,20,22) 0 ElastlcAnalysis I
0.4 o.~ 0.8 Indentation o~(mrn)
Finally the results of this simple analytical model can be used to study the impact response of composite laminates with similar support conditions. Furthermore the damage zone predicted by the analysis can be used to estimate the loss of strength in the laminate due to the lateral contact loads. The advantage of such analytical models is that they usually give significant physical insight for the problem in hand. CONCLUSIONS A simple analytical model has been presented for describing the contact force-deformation response between a rigid spherical indenter and a thin laminate supported on a rigid substrate. The analysis agrees well with experimental data reported in the literature. Damage is predicted to occur at low load levels and has a major effect on the force-deformation response. The model provides useful information for studying impact damage effects in composite laminates. REFERENCES
I
1.o 1.2
Fig. 5. Force-indentation response for the contact between a spherical indenter and a thin laminate supported by a rigid substrate.
1. Sun, C. T. & Chattopadhyay, S., Dynamic response anisotropic laminates under initial stress to impact mass. J. Appl. Mech., 42 (1975) 693-8. 2. Sun, C. T., An analytical method for evaluation impact damage energy of laminated composites.
of of of In
82
3. 4. 5. 6.
7.
A. P. Christoforou Composite Materials: Testing and Design (Fourth Conference), ASTM STP 617, ASTM, Philadelphia, PA, 1977, pp. 427-40. Tan, T. M. & Sun, C. T., Use of statistical indentation laws in the impact analysis of laminated composite plates. J. Appl. Mech., 52 (1985) 6-12. Cairns, D. S. & Lagace, P. A., Thick composite plates subjected to lateral loading. J. Appl. Mech., 54 (1987) 611-16. Willis, J. R., Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids, 14 (1966) 163-76. Yang, S. H. & Sun, C. T., Indentation law for composite laminates. In Composite Materials: Testing and Design (Sixth Conference), ASTM STP 787, ASTM, Philadelphia, PA, 1982, pp. 425-49. Swanson, S. R. & Rezaee, H. G., Strength loss in corn-
8. 9.
10. 11.
posites from lateral contact loads. (bmp. Sci. 7~'ch., 38 (1990) 43-54. Poe, C. C., Jr, Simulated impact damage in a thick graphite/epoxy laminate using spherical indenters. NASA TM 100539 (1988). Poe, C. C., Jr & Illg, W., Strength of a thick graphite/ epoxy rocket motor case after impact by a blunt object. In Test Methods for Design Allowables for Fibrous Composites: 2nd Volume, ASTM STP 1003, ASTM, Philadelphia, PA, 1989, pp. 150-79. Jaffar, M. J., Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation. Int. J. Mech. Sci., 31 (3)(1989)229-35. Conway, H. D., Lee, H. C. & Bayer, R. G., The impact between a rigid sphere and a thin layer. J. Appl. Mech., 37 (1970) 159-62.
On the contact of a spherical indenter and a thin composite laminate Andreas P. Christoforou Department of Mechanical Engineering, Kuwait University, PO Box 5969, Safat, 13060 Kuwait An analytical solution for the contact between a rigid sphere and a thin composite laminate supported on a rigid substrate is presented. The contact force-deformation, including damage effects, is obtained from an axisymmetric formulation of the contact problem. The material is assumed to be transversely isotropic and following an elastic-perfectly plastic stress-strain law. Damage is predicted using a maximum shear failure criterion. Experimental data previously reported in the literature agree well with the analysis.
used to predict permanent indentations in an empirical unloading law, similar to yielding in metals, supports the idea of using plastic deformation principles to incorporate damage effects in the indentation response of composites. More recently Swanson and Rezaee 7 have conducted experiments where thin laminates resting on a rigid substrate were indented by a rigid sphere. In the initial stages the data seem to follow the Hertzian contact law, but at higher loads softening associated with significant observed damage and permanent deformation is quite evident in the force-deformation response. The purpose of this paper is to present a simple analytical model for studying the contact force-deformation between a thin laminate supported by a rigid substrate and a frictionless rigid sphere including damage effects. The solution is obtained from an axisymmetric formulation of the contact problem assuming the material is transversely isotropic. A further simplification is that the stresses along the thickness of the laminate are taken to be constant by assuming that the radius of the indenter is large compared to the thickness of the laminate. Furthermore the material is assumed to be elastic before damage and perfectly plastic after damage occurs. On the basis of experimental data reported by Poe 8 and Poe and Illg, 9 the damage region is predicted using a maximum fiber shear failure criterion. The resulting expressions of the contact force-deformation appear to be in good agreement with the experimental results reported in Ref. 7.
INTRODUCTION It is well known that fiber-reinforced composites show high strength properties when subjected to in-plane loads but show poor resistance to out-ofplane loads such as lateral impact. The residual strength of composite structures can be reduced as much as 50% as a result of impact damage. Depending on the characteristics of the impact event, impactor mass, size, velocity, target material and structural properties, impact damage is caused by either bending stresses or local contact stresses. Recent work has been directed toward separating the local contact effects from the dynamic problem such that analysis or data from static indentation laws may be used in analyzing the impact response. 1-4 The contact force-deformation relationship produced by a spherical indenter has been shown to follow a law similar to that of isotropic materials,5,6 from a Hertz analysis. There are some uncertainties, however, when a Hertzian-type law of contact based upon the contact of a sphere and a half-space is applied to thin laminates. During the initial stages of loading, when the contact radius is small compared to the laminate thickness, the Hertzian contact analysis should be adequate. However, as higher loads cause extensive damage in the contact zone the effects of damage should be a major factor of the response. Experimental work carded out by Yang and Sun, 6 where a critical indentation (acr) was 77
Composite Structures 0263-8223/93/S06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain
A. P. Christoforou
78 ANALYSIS
Boundary conditions
Governing equations
Figure 1 shows the geometry of the thin laminate resting on a rigid substrate, indented by a frictionless rigid sphere of radius R. The boundary conditions are as follows:
In the case of axisymmetric problems, the governing stress equilibrium equations in the absence of body forces are given by 4,~° Oar+ i)O____~-t or -- Oo
Or
Oz
0
r
(1)
OOz+O,z=o Or
Oz
r
The constitutive equations for a cylindrically orthotropic material are
{rI S12131/1r =/312 Ez LSI3
322 823
gO
(2)
323[ O0 833J orz
( 1 ) On the free surface z = 0:
az(r)=O
r>a
orz(r)= 0
r>0
2 t" w(r)=a--2R
O<_r
(5)
where a is the contact radius, the profile of the sphere is described by the parabolic approximation, and a is the indentation depth at r = 0. (2) On the surface z = h:
o,=(r)=O
(6)
w(r) = 0 }Prz = 3440rz
where
Sij
is the engineering compliance matrix.
For this problem, the material is also assumed to be transversely isotropic in the r- 0 plane. In terms of the engineering constants the compliance coefficients are given as SII = $22
1
1
E,
Eo
-- ~rO
-- VOr
L"r
E 0
S12
SI 3=$23= -
1 $33 = - E=
In the following approximate stress analysis, it is assumed that the laminate thickness is small compared to the contact radius (a/h ,> 1 ); in such contact problems arz is assumed to be equal to zero throughout the layer thickness) °,l~ Hence eqn (1) becomes
l
$44- G~
Vrz =
Solution procedure
(3)
(7)
0
r
The compressive vertical strain in the laminate is given by the geometry of deformation, i.e.
-- YOz
Er
dr
Eo
For small deformations under the assumption of axisymmetry the strain-displacement equations are given as Ou
e,
Or
Ow
e.
Oz (4)
u r
~" 0 = - -
0u 0w "~ - Oz Or
~2 rz = - -
where the displacements assume the form (u, 0, w) in cylindrical coordinates (r, 0, z) and the partial derivative with respect to 0 is identically zero.
Thinlaminateof ~
*, t"
| ¢x
,l , , I,, ), ! , / / / / / × / / / /,,, Rigidsubstrate / / / / / I ~ / ~
// / /
z Fig. 1.
Geometry of the contact between a sphere and a thin laminate supported by a rigid substrate.
Contact of spherical indenter and thin composite laminate
Owa (
ez
Oz
2Rh
1-
(8)
and the sphere indentation a at r = 0 is given by
a2 a =w(0) = 2R
(9)
Algebraic manipulation of eqns (2), (3), (4), (7) and (8) yields d2u
1 du
dr 2 +
r dr
u (1 OI- ]lrO) 'l)zr r2 (1-VrzVzr) Rh r
(10)
The solution of the differential equation (10) is given by u=Ar~
(11)
(l +'RrO)T'zr r 3 8(1 ltrz'llzr) Rh
79
it is very important that damage effects on the response are included in the analysis. A general analysis in the damage zone would be extremely difficult, and a further simplification is necessary. Following the procedure of Ref. 11, from eqn (13) it is easy to determine that the maximum stress az(r= 0), is a lot larger than the corresponding maximum values for ar and ao. Thus, it is reasonable to apply the failure criterion to a z only. Considering experimental evidence reported by Poe and Illg,9 which suggest fiber failure in shear, in the type of loading under consideration, a maximum fiber shear failure criterion is used. When damage is predicted, the material in :the composite laminate is assumed to have two distinct regions shown in Fig. 1. If the material in the damage region is assumed to be perfectly plastic while the outside is elastic, Fig. 2, it follows that
(Oz)v = 2Sf
r_< b
-
Ez a2 (az)~-2( 1 - ~ z r ) R h
The boundary condition az(a ) = 0 yields A= -
(1 + v~o) Vzr 2 a 4( 1 - V,.zVzr) Rh
(12)
Finally the expressions of the stress distribution are obtained: _
vzrEra 2
Or 411-
zrlgh
( r2) 1 - ~a 2
vzrEra2
(
0 0 - 4 ( 17v~v~r)Rh
za2
az=2(l__vrzVzr) R h
1
3r2 / 2a2]
=
(13)
(1r2)
b2 = a2 _ 4Rh(1
(1-VrzVzr) h
- VrzVzr) Sf Ez
(17)
a.-
2h( 1 - VrzV~) Sf
(18)
Ez
The total contact force is then
--a5
f f az(r)(2orr)dr=
~EzR
b<_r<_a
where Sf is the fiber shear strength. At the interface r = b, from eqn (16)
grgza4
4(1 - vzy,~ ) Rh
F(a)=
a
2
2Se(2=r)dr+
az)e(2~r)dr
(19)
(14)
Substituting eqn (9) into eqn (14) yields F(a)
1
The indentation at the onset of damage can be computed by setting b = 0 in eqn (17):
The contact force F is obtained from the contact stress distribution, F(a)=
(16)
__as
2Sf
/
1
(15)
I
The contact force-deformation relationship of eqn (15) is valid in the elastic range, assuming no permanent deformation or damage is present. However, when damage is present, as in this case,
/i
I 0
I
tcr
Strain ~z
Fig. 2.
Stress-strain response assumed for the laminate material.
A. P. Christoforou
80
Integrating and combining with eqns (17) and (18) yields
F(a)=2~RSf(2a-acr)
O~cr~a -< O~m (20)
During the unloading phase the vertical compressive stress is given by
[ E~ (a-ao) o-=~ h(1-VrzV:~) " ] E_:a2_~_[1 r: I t2Rh(1-v~zv.~) k a 2]
0 <_r<_bm
(21) b m -< r<
a
Therefore
oz(r)(2arr)dr
F(a)= 0
yields
F(a)(1
~rRE_ - vr v ,)h
( a 2 - a~)
(22)
where
a. = b2/2R
= o~m -- O~cr
(23)
is the permanent indentation.
RESULTS AND DISCUSSION
Calculations were performed on a [0/+68/0]~ IM7/55A carbon/epoxy laminate to compare with experimental data reported by Swanson and Rezaee. 7 The laminated plates had a nominal thickness of 2.4 mm, were supported by a hardened steel block, and were subjected to lateral loads by a hardened steel spherical indenter with a diameter of 15.88 mm. The ply properties of the Hercules carbon/epoxy system are given in Table 1. The contact force-deformation response predicted by the analysis was found to be more dependent on the through-thickness properties than the in-plane properties. The result is consistent with previous findings. 3,4.9 Thus, the transversely isotropic axisymmetric model seems to be a reasonable approximation.
Damage was found to have a dominant effect on the overall force-deformation response. The damage zone was predicted using a maximum fiber shear failure criterion. Since such experimental data were not available for this material system, the fiber shear strength St = 310 MPa s f o r the Hercules AS4/55A, a similar Hercules carbon/epoxy system, was used. The onset of damage and the subsequent response were found to be directly proportional to the shear strength of the fibers. The critical indentation, a . , at the onset of damage that is used to predict permanent deformations, does not seem to be a material property as suggested in Ref. 3. Besides the shear strength and through-thickness modulus, O~crgiven by eqn (18) also depends on the laminate thickness. Thus a more appropriate material property seems to be the critical normal strain, ecr, which is the critical indentation divided by the thickness. This observation is consistent with fundamental materials behavior under simple loading, and findings that suggest fiber failures are best modeled by maximum strain failure criteria. However, this finding depends on an approximate analysis and more experimental data are needed. Figure 3 shows the comparison of the finite element solution of the same problem 7 and eqn (15) of the present analysis. The agreement between the two methods is excellent. This important result gives credence to the analysis since the finite element in this case is a better approximation of the real set-up, and in view of the greatly simplifying assumption of constant through-thickness stresses used in the analysis. As can be seen in Fig. 4, in the elastic range any analysis -- finite element, the present one, or a Hertzian type -- will be adequate in predicting the
100
t
r
t
8O
60
o
4o
m
Finite Elemt~t
Soluti~ (tqef.7) Pre, c~,nt A n a l y s i s
Equ.(lS)
20
Table 1. Material properties for IM7/55A carbon/epoxy Fiber-direction Young's modulus, E ~ Transverse Young's modulus, E22 In-plane shear modulus, G~ 2 Major Poisson's ratio, v 12
169 GPa 6.72 GPa 5'26 GPa 0.266
0
0.5
1.0
I 1.5
210
2.5
Indentation aclmm]
Fig. 3.
Contact force-indentation response of a thin supported laminate.
Contact of spherical indenter and thin composite laminate
response. At higher load levels when damage effects are quite evident in the response, linear elastic analyses give poor predictions while the present analysis including damage effects provides a better estimate of the response. This can also be seen in Figs 5 and 6, where the complete loading histories at two different load levels are shown. It is very encouraging that the simple model presented here gives reasonably good predictions when compared with available experimental data. The data are very limited, however, and more experimental work is needed.
4o~
~
i
w
i
1
I--
zo
g 10
f
li I I
~' ' Hertzi~n Anaiy~is~ ' ' (Ref. 7) \ PresentAnalysis~
I
30
/ -J
80
®
81
0
/ I ] I !
J9
Experimentaldata" c,.,.7)
----- Present Analysis _ (tqus ls.z0,zz ) ®- Elastic Analysis -"
0.5 1.o 1.5 2.0 2.5 Indentation 0~ ( m m)
3.0
Fig. 6. Force-indentation response for the contact between a spherical indenter and a thin laminate supported by a rigid substrate.
,-Z 6o U. 4o
O
,.~ 2o
/
/
/ ~ ~"
[--
J t~."
I
¢
I
data
ExPerimental
( Ref. ? )
I
1.0
I
I
2.0
3.0
Indentation o~(mm) Fig. 4.
Comparison of analytical predictions and experimental data for a thin supported laminate.
20
,
,
,
,
°I,'
,_.IS Z
/
"5 "°1°
i/
tO//
i
5
/ J ,
°o
,
o.~
/
Experimental data ( Ref. 7 ) --P r e s e n t Analysis [Eque 15,20,22) 0 ElastlcAnalysis I
0.4 o.~ 0.8 Indentation o~(mrn)
Finally the results of this simple analytical model can be used to study the impact response of composite laminates with similar support conditions. Furthermore the damage zone predicted by the analysis can be used to estimate the loss of strength in the laminate due to the lateral contact loads. The advantage of such analytical models is that they usually give significant physical insight for the problem in hand. CONCLUSIONS A simple analytical model has been presented for describing the contact force-deformation response between a rigid spherical indenter and a thin laminate supported on a rigid substrate. The analysis agrees well with experimental data reported in the literature. Damage is predicted to occur at low load levels and has a major effect on the force-deformation response. The model provides useful information for studying impact damage effects in composite laminates. REFERENCES
I
1.o 1.2
Fig. 5. Force-indentation response for the contact between a spherical indenter and a thin laminate supported by a rigid substrate.
1. Sun, C. T. & Chattopadhyay, S., Dynamic response anisotropic laminates under initial stress to impact mass. J. Appl. Mech., 42 (1975) 693-8. 2. Sun, C. T., An analytical method for evaluation impact damage energy of laminated composites.
of of of In
82
3. 4. 5. 6.
7.
A. P. Christoforou Composite Materials: Testing and Design (Fourth Conference), ASTM STP 617, ASTM, Philadelphia, PA, 1977, pp. 427-40. Tan, T. M. & Sun, C. T., Use of statistical indentation laws in the impact analysis of laminated composite plates. J. Appl. Mech., 52 (1985) 6-12. Cairns, D. S. & Lagace, P. A., Thick composite plates subjected to lateral loading. J. Appl. Mech., 54 (1987) 611-16. Willis, J. R., Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids, 14 (1966) 163-76. Yang, S. H. & Sun, C. T., Indentation law for composite laminates. In Composite Materials: Testing and Design (Sixth Conference), ASTM STP 787, ASTM, Philadelphia, PA, 1982, pp. 425-49. Swanson, S. R. & Rezaee, H. G., Strength loss in corn-
8. 9.
10. 11.
posites from lateral contact loads. (bmp. Sci. 7~'ch., 38 (1990) 43-54. Poe, C. C., Jr, Simulated impact damage in a thick graphite/epoxy laminate using spherical indenters. NASA TM 100539 (1988). Poe, C. C., Jr & Illg, W., Strength of a thick graphite/ epoxy rocket motor case after impact by a blunt object. In Test Methods for Design Allowables for Fibrous Composites: 2nd Volume, ASTM STP 1003, ASTM, Philadelphia, PA, 1989, pp. 150-79. Jaffar, M. J., Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation. Int. J. Mech. Sci., 31 (3)(1989)229-35. Conway, H. D., Lee, H. C. & Bayer, R. G., The impact between a rigid sphere and a thin layer. J. Appl. Mech., 37 (1970) 159-62.