I~
UTTE E I N
RWORTH E M A N
Composites 26 (1995) 803-814 9 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0010-4361/95/$10.00
N
Approximate elastic-plastic analysis of the static and impact behaviour of polymer composite sandwich beams R. A. W. Mines* and Norman Jones Impact Research Centre, Department of Mechanical Engineering, University of Liverpool, PO Box 147, Liverpool L69 3BX, UK (Received 29 March 1995; revised 16 June 1995) An elastic-plastic beam bending model has been developed to simulate the post-upper skin failure energy absorption behaviour of polymer composite sandwich beams under three-point bending. The beam skins consist of woven and chopped strand glass, while the core is a resin impregnated non-woven polyester material known as Coremat. A polyester resin was used for the construction. The theoretical model consists of a central hinge dominated by a crushing core and tensile elastic strains in the lower skin. Experimental measurements of the non-linear force-deflection characteristics for the beam are compared to the theoretical predictions from the model, and it is shown that the shear crushing of the core has an important effect on the behaviour of the beam. The model shows that the most important material properties are the lower skin tensile failure strain and the core crushing strength. Dynamic effects are included in the model in the form of a strain rate dependence of the core crushing stress and the strain rate dependence of the failure strain in the lower skin. The increase in material strength with strain rate gives rise to an improved energy absorption capacity for the beam under impact loading. (Keywords: elastic-plastic analysis; sandwich beam; impact behaviour)
INTRODUCTION Structural members using sandwich construction are designed usually in terms of flexural stiffness and strength. However, some structures must be capable of sustaining large overloads, e.g. foreign object impact 1 or impact with solid objects 2. In these cases, the non-linear (post-failure) behaviour of the structure is important and the parameters of interest are the integrity of the structure and the energy absorbing capabilities of the structure before total collapse. Given the complexity of structural response in the general case, it was decided to study a simple structural element (e.g. a beam) in the first instance. Previous experimental work 3 has examined the behaviour of selected polymer composite sandwich beams under static and dynamic loading in a three-point bend configuration. A number of different failure modes were found which depend on the beam construction and span. In general, up to eight possible modes of failure may occur for sandwich beams 4, including upper skin wrinkling, upper skin compression failure, lower skin tensile failure and core shear. For thin skinned and large spanned beams, the first three failure modes are the most likely, and, for thick skinned and short spanned beams, * T o w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d
the last failure mode is the most likely. Ref. 3 compared constructions having different polymer composite skins and with two core materials, namely aluminium honeycomb and Coremat. The latter is a proprietary material and is a resin impregnated non-woven polyester with a binder and 50% microspheres. Upper skin compression failure in the vicinity of the central loading point of the beam was the most common failure mode observed in this experimental work and this failure mode gave stable energy absorbing behaviour after initial skin failure. The beam materials and geometries selected for study are typical of transport and lightweight marine applications, viz. thin skins and intermediate values of beam span-todepth ratio. The elastic analysis of sandwich beam construction is well documented 4-6. However, little post-failure, non-linear, theoretical work for composite beams has been published except for aramid-reinforced laminated beams. Zweben 7, Reedy s and Fischer and Marom 9 used simple beam models in which the compression plasticity of aramid fibre gives rise to an elastic-plastic hinge at the loading point of a three-point bend specimen. In the case of sandwich construction, the non-linear, crushing behaviour of the core contributes to the behaviour of the elastic-plastic hinge at the loading point. The crushing behaviour of foams has been studied in some detail 1~ and the failure of foams under multi-axial
COMPOSITES Volume 26 Number 12 1995
803
Elastic-plastic analysis of sandwich beams: R. A. W. Mines and N. Jones stresses is important for cases in which foams are used as structural materials, e.g. in sandwich beams. This paper focuses on a polyester impregnated Coremat core 3, although the multi-axial behaviour of this material has not yet been reported ~3. Hence, it is assumed that standard ductile metal plasticity analyses, with the appropriate material properties, are approximately applicable. It is worth noting that Coremat has a high density (an order of magnitude higher than standard foams) and, therefore, has a high crushing strength. Thus, the localized crushing due to the concentrated force may be neglected, which allows the use of a simplified beam analysis. A phenomenological approach is taken here. In other words, simple strength of materials models are developed in order to gain some insight into the relevant phenomena and to provide a straightforward approach to optimizing beam construction from the viewpoint of energy absorption. THEORY The sandwich beam geometry selected for study is shown in Figure 1. The beam was loaded by a 20 mm diameter steel roller and the beam was supported on 20 mm diameter fixed steel rollers. A schematic of a typical experimental beam response is shown in Figure 2. Point (i) coincides with compression failure of the upper skin, region (ii) corresponds to the crushing of the core at the loading point and total beam failure occurs at (iii). A photograph of a deformed beam after upper skin failure is shown in Figure 3. It can be seen that a severely deformed region is concentrated at the central loading point and it is proposed that this area can be idealized
cI
g ( L
-I
Figure 1 Sandwich beam geometry. Note that upper skin is not modelled. Each skin has two plies
Force (F)
Figure 3 Photograph of deformed beam (beam C) after upper skin failure. Drop height = 1.5 m, drop mass = 4.3 kg
as a plastic hinge. It should be noted that the elastic behaviour of the beam can be modelled easily 5 and ref. 3 discussed the rate dependence of the compression failure in the upper skin and, hence, the rate dependence of beam initial failure loads and energies assuming quasistatic beam behaviour. When the upper skin fails, the elastic bending energy in the beam instantaneously converts to plastic hinge work (~50%) and to skin-core debonding (~ 50%). Subsequent loading of the beam gives rise to plastic deformation in the hinge and it is this mode of deformation that is the subject of this paper. The elastic-plastic model is shown in Figure 4. The upper skin has failed (point (i) in Figure 2) and hence only the core and the lower skin are modelled. It is assumed that the lower skin remains elastic at all times. The controlling parameter for the elastic-plastic beam analysis is the upper surface compression strain ern. The stress o-in the direction along the beam is termed the axial stress in the following analysis. Figure 5 gives the non-linear compression crushing characteristic of the resin impregnated Coremat core in terms of engineering stress and strain. These experimental results were obtained from 20 • 20 mm specimens with a thickness of 5 mm. The specimen dimensions were selected as a result of the capacity of the compression testing machine and due to the fact that the Coremat was supplied in a 5 mm thick sheet. Three repeat tests were conducted at a displacement rate of 0.02 mm s-~ and the average curve of these results is shown in Figure 5. Crushing initiated at a compressive stress of 12 MPa and a compression strain of 0.025. The equation for this
j(i)
Em
/
s
(ii) A
(iii) It
Deflection Figure 2 Force~teflectionschematicfor the sandwich beam. (i) Upper skin compressionfailure,(ii) core crushing, (iii) lowerskin tensilefailure
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COMPOSITES Volume 26 Number 12 1995
'~c~ (o)
Y
N
Ay L ek,134 (b)
Figure 4 Elastic-plasticmodel for loading point hinge. (a) Strain distribution, (b) axial stress distribution (NA = neutral axis)
Elastic-plastic analysis o f s a n d w i c h b e a m s : R. A. W. M i n e s and N. J o n e s (r tMPQ)
=
n
-
(7)
-104
Es is the skin modulus and ef = -0.025 (from Figure 5), and the beam dimensions and nomenclature are given in Figures 1 and 4. The sum of al, o~2, a3 and o~4 should be zero as there is no axial force on the beam, and hence z is obtained. Note that z is a function of the upper core surface strain era. Equation (6) assumes that the skin is much thinner than the core. Once the position of the neutral axis (z) is defined, then the elastic-plastic bending m o m e n t of the crosssection can be derived. Again, there are four contributions, namely J~l, ~2, ]~3 and /~4 (see Figure 4):
-78
-52
=o.s -2~
0~=-12 t"l Po 0
-o~ -o-o2s
-02
-03
-o.4
-o.5
-06
~:
Figure 5 Elastic-plastic compression data for polyester impregnated Coremat core. a, Experimental curve; b, analysis curve for r/vf = 0; c, analysis curve for ~/Tr = 0.5
+~-( 1- EmJ Ef12(2gEm+2b ~3 3 +-~-)
bilinear characteristic in compression is: o-=ae
for-ef
(1)
fore<-ef
(2)
o-=b+ge
fl, = 2(1- Ef ff SfZl(gEm Em )
where, from Figure 5, a = Ec = 0.48 GPa, b = -9.65 MPa, g = 94.2 MPa and ef = -0.025. Ec is the elastic stiffness of the core, er is the initial strain for core crushing and g is the 'work hardening' index of the core. In the following it is assumed that the modulus of the core is the same in tension as in compression. In reality, the tensile modulus of the core can be up to two times the compression modulus. However, given the small contribution of the core tensile strains in the model (see below), it was decided to neglect this complication. I f the beam is subject to pure bending (plane sections remain plane) then the strain distribution across the section is linear. It is assumed that there is continuity of strain across the lower skin-core bondline. The position of the neutral axis (z) in Figure 4 is unknown and has to be derived from the axial force equilibrium requirements of the cross-section. There are four contributions to the force equilibrium, i.e. al, a2, a3 and an, where:
af(erzl 2
(C --
133 = a
(8)
(9)
Z) 3
em
3Z
134 = ( d - z) 2 e---~mtE s
(10)
(11)
z
where the fl values are evaluated for a unit beam width. Hence, the m o m e n t of the section is given by: M ( g m ) = (ill + f12 + f13 + f14) X B
(12)
where B is the beam width. The next step in the analysis requires relating the top surface strain of the beam •m to the overall deflection 6 of the beam. F r o m Figure 3, it can be seen that the plastic deformation is concentrated within a small region of
(3)
a' 2 = 0.5(z - y)o-r = 0.5 er zo'r
(4)
Em
O~3 = - 0 . 5 (c - z ) 2 z
o:4 _
(d-
gma
z) emEst
X (5)
(6)
z
and
Figure 6 Deformation at hinge in vicinity of central load (WXYZ = plastic hinge, NA = neutral axis)
COMPOSITES Volume 26 Number 12 1995 805
Elastic-plastic analysis of sandwich beams: R. A. W. Mines and N. Jones length l0 (see Figure 6). It is convenient to call this region a 'hinge' despite the fact that idealized hinges may form only in perfectly plastic materials. The hinge length 10 is unknown and is measured from experimental results. For a given experimental 'hinge', the radius of curvature of the hinge R and the included angle of the hinge 0 can be estimated from photographs of a failed specimen (see Figures 3 and 6), and hence: (13)
It = RO = lo (1 +em)
However, neither em nor 10 are known for this specific case. Nevertheless, the hinge length 10 can be expressed in two ways, namely in terms of z or em: lo = (R + z)O
(14)
RO lo = - (1 + em)
(15)
or
from equation (13). These two equations should give the same value of l0 for the given experimental result (e.g. see Figure 3). For a given beam construction, Z(em) is known from equations (3)-(7), and hence this function can be substituted into equations (14) and (15), and when the same values of l0 are obtained, then the value of em is known and l0 is derived. It is assumed that l0 remains constant during loading (see below). The base variable em is used for calculating R and 0 as the beam deforms and z has been derived from the axial equilibrium of the section. From equation (13), It can be derived and hence: R-
zlt
(16)
to-l,
Therefore, R can be calculated for a given em and from equation (15): 0 -- ~ ( l + E m )
(17)
R
and hence 0 can be calculated for a given em. The beam supports are fixed rollers and the beam slides over these as it deflects so that the transverse displacement 6 of the beam can then be predicted for a given em (the base variable) using the simplified displacement profile in Figure 7:
H
" "- ""T
L I-
Figure 7 Forceequilibrium for one-halfof a deflectedbeam
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COMPOSITES Volume 26 Number 12 1995
,
(18)
2
This equation does not take into account any elastic transverse shear deformation, which is less than 5% of the total deflection at the maximum load of the beam. The beam moment M(em) is also known [see equations (8)-(12)], and from equilibrium of one-half of the beam: 4M F = -L
(19)
Using equations (18) and (19), the transverse forcedeflection characteristic of the beam can be derived in terms of the base variable em. Other parameters of interest are the average tensile elastic strain in the lower skin e2, which is given by: d-z
and the strain at the lower core-skin bondline, given by: C--Z
The above analysis assumes that the axial core flow stress or is a result of bending compression loading only. In reality, the stress state in the plastic hinge (Figure 6) is complex - compression stresses result from the concentrated external beam load and beam transverse shear stresses also occur. Hence, the axial core flow stress is modified as a result of the multi-axial stress state. Gibson et al. 1~ and Triantafillou and Gibson 14 model the local plastic yielding of cellular foams and honeycombs with densities of less than 300 kg m -3 using a modified von Mises criterion. Schreyer et al. 15 studied cellular foams and honeycombs with densities less than 320 kg m -3 and they included large formations (up to 70%) in their model and hence the initiation of cell 'lockup' where the cell walls start to touch. Their yield function included hydrostatic pressure effects. However, these authors did not apply their criteria to cases in which shear stresses are dominant. Coremat is a hybrid material, which consists of resin impregnated non-woven polyester with 50% microspheres. Hence, the material does not constitute a cellular material as defined by Gibson and Ashby 4, viz. a relative density of less than 0.3, where the relative density is the ratio of the density of the core to that of the solid material from which the core is made. The density of the resin impregnated Coremat is 640 kg m -a, which is markedly higher than standard foams. Also, a feature of the crushing behaviour of the Coremat core is the dominance of strain hardening (see Figure 5) and this behaviour should be compared to the negligible strain hardening for standard foams 4'15. Given the lack of a theoretical model for the multiaxial crushing of Coremat, a number of simple yield criteria based on the plasticity of metals ~6-18 and polymers 19 were considered. In this, it is assumed that resin
Elastic-plastic analysis of sandwich beams: R. A. W. Mines and N. Jones impregnated Coremat is isotropic - a reasonable assumption given the random nature of the non-woven polyester. It should be noted that after the initial failure of the core, i.e. ~rr = 12 MPa, not only is the core yielding but also damage is occurring, e.g. fibres are pulling out and microspheres are crushing. Hence, the core behaviour is not pure material flow. However, in this paper it is assumed that core crushing may be modelled by the standard theories of plasticity, e.g. material flow is governed by the invariants of the deviatoric stress system, the von Mises yield criterion is applicable ~9 and isotropic hardening occurs 17. Another issue was the fact that large strains were encountered in core crushing, i.e. 40-60%. Zyczkowski 17 briefly discusses the effects of finite strains in metal plasticity. Large strain effects will modify the assessment of strain values and hence will modify the effects of work hardening on the core crush flow stress. However, given the other approximations in the elastic-plastic beam model, it was felt that neglecting large strain effects was reasonable in the first instance. A further simplification to the models described here was the neglecting of the effect of core compression due to the concentrated load (see general discussion). Hence, the interaction model is due to axial compression and shear only. The transverse core shear stress vwas calculated from: F "r = - 2Bc
(22)
where F is the force at the mid-span, B is the beam width and c is the core depth. It is assumed that the transverse shear force is supported by the core only and that the distribution of the shear stress across the core is linear. The latter is true for shear rigid cores 5 but, as the Coremat core contributes to bending, the assumption is an approximation. Model 1 used the yon Mises yield criterion including strain hardeningl8: 2
+
=1
(23)
where o-e and re are the effective stresses and where oand ~ are related by:
o" = "r-~-
(24)
Hence, the experimentally measured uniaxial core crushing stress-strain curve can be regarded as a 'master' equivalent stress-strain curve. Model 2 assumed that no strain hardening occurred for the transverse shear stress component and hence:
~-e
t.rf
)
(25)
where vf is the shear strength of the core, i.e. the shear stress at initial crush. Model 3 assumed a linear relationship between the direct and shear stress components:
CT{MPa) 60
ab
- 50
c
-
-tO
-30
9
e
d -20
-0.1 0.222
-0.2 0315
-0.3 0405
-04 -0.5 0-1,95 0.587
Em 7"['[sfF0r
Beam A
Figure 8 G r a p h of core compression flow stress ~ against upper core surface strain em assuming shear stresses taken for beam A. Curve a, no shear; b, Model 1; c, Model 2; d, Model 3; e, Model 4
(26)
The basis for this is the simplified interaction relation between the bending moment and transverse shear forces which has been used for studying the behaviour of ductile metal beams 16-18. Interaction criteria have been developed for rectangular sections assuming elastic perfectly plastic material behaviour and these criteria have been shown to be elliptic, circular and parabolic 17. In the case of the Coremat beam, the bending stress distribution is asymmetric (see Figure 4) and there is extensive work hardening (see Figure 5). However, the linear interaction criterion should provide a lower bound to the actual curve. Model 4 consisted of a hybrid interaction criterion in which the von Mises criterion was assumed to apply up to the initial crush condition [equation (23) with o-e = O'f and ve = Vr] and, thereafter, a linear relationship held [equation (26)]. The basis for comparison of each model was in terms of the variation of axial flow stress ~r in terms of the core upper surface strain em. The results for one of the beams considered in this paper (beam A) are shown in Figure 8, and included on this curve are the r/~'f values for each value of er~. Also shown is the effective stress O'e, assuming no shear. From Figure 8, it can be seen that Model 1 gives a negligible modification to the flow stress as a result of transverse shear stresses and Model 2 gives only a modest modification to the flow stress. The linear models, i.e. models 3 and 4, predict a marked reduction in the flow stress for the values of shear stress given, and these two models gave beam force-displacement results closest to the experimental results. In the absence of any experimental data, Model 4 is used in the following. The corresponding modification to the core crushing stress-strain curve using Model 4 is shown in Figure 5 for T/Tf--- 0.0 and 0.5. In the full beam analysis, the force-deflection characteristic was derived assuming no shear and this effect was then included to derive the modified flow stress and
COMPOSITES Volume 26 N u m b e r 12 1995
807
Elastic-plastic analysis of sandwich beams: R. A. VV. Mines and N. Jones Table 1
Beam construction, dimensions and plastic hinge sizes (see Figures 1, 4 and 6 for nomenclature)
Beam A (woven glass) Skin (gm2): Ply 1~ Ply 2 Loading t (ram) c (ram) d (ram) L (ram)
L/D
R~xp(ram) Ocxp ( r a d )
l0 (mm)
Io/c
Beam B (woven/ CSM~ glass)
Beam C (woven/ CSM~ glass)
360 360
450 360
450 360
Static 0.48 9.34 9.58 165 16.0 47.3 0.31 17.30 1.85
Static 0.80 8.70 9.10 185 18.0 30.9 0.31 12.22 1.40
Impact 0.80 8.70 9.10 180 17.5 14.5 0.35 8.10 0.93
Adjacent to core hence the modified force-deflection characteristic. The effect of shear on the elastic and plastic beam deformation has not been considered here, and so the calculated beam deflections remain unaffected by the effects of shear. This assumption will become invalid for low beam span-to-depth ratios and for core materials with a relatively low stiffness and strength. The elastic shear deformation of Coremat cored beams has been discussed in ref. 3, and it was shown that for the beams considered here, the elastic shear deflection is less than 12.5% of the total elastic deflection. From the study of failed beam specimens, it was concluded that plastic shear deformation in the central beam hinge is negligible. C O M P A R I S O N O F T H E O R Y W I T H PREVIOUS EXPERIMENTS Three beams of similar span-to-depth ratios were selected for study. Table 1 gives the three constructions and more detailed information is given in ref. 3. The fibre orientation for the woven roving in the skins was 0/90 ~. All beams had polyester impregnated Coremat cores and the beams were made by the hand lay-up process. The polyester resin used was Scott Bader Crystic 272. Average static tensile stress-strain curves for the lower skin are
o'(M Fb) ~75
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IQ
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i
I
2oo
Z
;
i
1'0
~z
s
;6
I',
10
h Exl0"3
~=00195 ~00208
Figure 9 Statictensile stress-strain curves for woven glass skin (WR) (curve a) and woven and CSM glass skin (WR/CSM) (curve b)
808
COMPOSITES Volume 26 Number 12 1995
shown in Figure 9. These data were derived from tensile tests conducted to the BS 2782 Part 10 Standard 2~ Five repeat tests were conducted for each skin case. A fourply specimen was tested in the case of the woven roving (WR). In the case of the mixture of woven roving and chopped strand mat (CSM) (see ref. 3), the specimen lay-up was WR/CSM/CSM/WR. The specimens were strain gauged in order to obtain values for the stiffness and strain. Beam A had woven glass skins and was loaded statically. The experimentally measured transverse forcedeflection data are shown in Figure lOb. Beam failure occurred due to upper skin compression failure and subsequent loading occurred at a constant force. Figure lOa gives the relation between em and z for beam A according to equations (3)-(7). The effect of transverse shear [equations (25) and (26)] increases z for a given era. Note that z cannot be larger than c, which equals 9.34 mm. Figure lOb compares the theoretical and experimental force-deflection characteristics for beam A. The theoretical curve is plotted for Tr = 4 MPa, which was taken from manufacturer's data zl. Selected values of em and r/rr are indicated on the theoretical curve. This curve becomes inaccurate after s = --0.45, given that the equation for core crushing becomes inaccurate (see Figure 5) and that values o f z are close to the value o f e (see Figure lOa). The latter suggests that the effect of shear [equation (26)] may be too 'strong'. The two effects indicated above should tend to cancel each other out and this suggests that the beam force-deflection characteristic should be flat above a strain [em[ of 0.5. This is shown in Figure lOb as a dotted line. Note the large strains em at the failure deflection of the beam. Figure lOb also shows the beam force-deflection characteristic assuming no transverse shear effects. Figure lOc gives the variation of the lower skin tensile strain e2 with respect to beam deflection. The failure strain from Table 2 is indicated on the graph and is attained at a beam deflection of only 15 mm, although it can be seen that ez is fairly constant above this deflection. The skin for beam A was made from two plies of woven roving, each of 0.24 mm thickness, and the values of inner ply strain are shown in the figure. The distance from the neutral axis has changed by only 0.12 mm and, hence, it can be concluded that the predicted lower skin strains are very sensitive to the distance from the neutral axis. This has implications for the accuracy of beam section property measurement and the reproducibility of beam section dimensions (see below). Figure 11 compares the failed tensile specimen with the lower skin of a failed beam. It can be seen from the photograph that there is more extensive damage in the case of the tensile test, suggesting different micromechanisms of damage. More detailed study of the relation between the micromechanics of failure of the lower skin with that from skin tensile tests is required. Beams B and C both had the same skin construction, namely woven roving outer ply and chopped strand mat adjacent to the core 3. Beam B was loaded statically, whereas beam C was loaded by the impact of a 4.3 kg weight which was dropped from a height of 1.5 m (ref. 22). The loading geometries for the static and impact cases were the same (see ref. 3 for further details). Figure 12a gives the growth of z with em for beam B. Again, the effect of shear is to increase z for a given era,
Elastic-plastic analysis of sandwich beams: R. A. VV. Mines and N. Jones Beam A
Z (mm)
",O
,o
9.4,
[ : 9-3r mm
9-0 -Q
Woven skin
Woven/ CSM skin
Coremat core
Eu(initial) (Pa)
18 x 109
12.5 x 109
0.48 x 109
EH at e= 0.1 (Pa)
11.8 x 10 9 275 X 10 6 369 x 106 0.0208 (0.0313) (413 X 10 6) (0.03) _ -
11.1 x 10 9 217 • 10 6 149 x 10 6 0.0195 (0.0134) (326 x 106)
or(-)(d: = 10-4 s-l) (Pa) ef(+)(e = 10-4 s i) er(-)(~: = 10 4 s 1) or(+)(s = 100 s 1) (Pa) o'f(-)(d: = 100 s ~) (Pa) er(+)(e = 100 s i) el(-)(/: = 100 s 1) ~'r(T; = 104 s I) (Pa)
!
7-8
7.4,
(0.03) -
-
2 X 106 12 x 106 (0.004) 0.025 (18 • 106) (0.0375)
4 X 106
"Values in parentheses indicate derived quantities; + indicates tension and - compression
7.0 (a)
Property
O'f(+)(,~-- 10-4 S-I) (Pa)
8-6
82
Table 2 Skin and core properties"
II
0
- 02
- 0.4,
-
06 -
0.8
-
1.0 Elm
.0,
ForceIN)
Beam A
2800
2400
"-C)
tt tt
S
2000
S
,o
oU
o iii
1600 _
I "-~
tO
1200 800 -b 400
lb
(b)
io
i0
,'7 tl
0.03
sb 6o Deft ec lion (ram)
do
o
o
~
~
~
,,
tl
Beam A
I
"~. l k
S
a n d z t e n d s to the v a l u e o f c at a em v a l u e o f ~ -0.45. Figure 12b c o m p a r e s the t h e o r e t i c a l p r e d i c t i o n s a n d e x p e r i m e n t a l results for b e a m B, w h i c h has a s i m i l a r b e h a v i o u r to b e a m A. T h e l o a d levels are similar to those o f b e a m A, w i t h the m a i n differences b e i n g the skin stiffness (see Table 2 a n d Figure 9) a n d the h i n g e size (see Table 1). T h e f o r c e - d e f l e c t i o n c u r v e c a n be p l o t t e d o n l y u p to Em = --0.45 d u e to the a b o v e o b s e r v a t i o n , a n d it is p r o p o s e d t h a t the s u b s e q u e n t f o r c e - d e f l e c t i o n b e a m c h a r a c t e r i s t i c will be flat for the s a m e r e a s o n s as given for b e a m A. T h e g r o w t h o f the s t r a i n in the lower skin 62 is s h o w n in Figure 12c. T h e c u r v e is i n c o n c l u s i v e since it c a n be p l o t t e d o n l y u p to a v a l u e o f em = - 0 . 4 5 , b u t it does suggest t h a t e2 will exceed Ef f r o m (~ ----28 m m u p u n t i l failure. Figure 13a gives t h e g r o w t h o f z w i t h Em for the i m p a c t l o a d e d b e a m C with rate i n d e p e n d e n t ( R I ) m a t e r i a l p r o p e r t i e s . T h e effect o f s h e a r is s i m i l a r to t h a t for b e a m s A a n d B. Figure 13b s h o w s t h a t the t h e o r e t i c a l predict i o n s u n d e r p r e d i c t the c o r r e s p o n d i n g e x p e r i m e n t a l results for b e a m C. T h e g r o w t h in the lower skin s t r a i n is given in Figure 13c. A g a i n , given the loss o f v a l i d i t y o f results a b o v e v a l u e s [em[ = 0.4 the c u r v e is i n c o n c l u sive. N o t e t h a t in the i m p a c t e x p e r i m e n t the b e a m d i d n o t fail at the m a x i m u m deflection o f 33 m m . It h a s b e e n s h o w n t h a t the core c r u s h i n g stress for f o a m s increases with s t r a i n rate 23 a n d t h a t the i n - p l a n e s t r e n g t h o f w o v e n
l
"
w~/~"
0"01
0I
o
lo
io
3'o
4'o
6t (e) Deflection (mm) Figure 10 (a) Variation of position of neutral axis (z) with emfor beam A: curve a, no shear; b, with shear. (b) Comparison of theoretical and experimental force-deflection values for beam A: curve a, model with no shear; b, experimental results; c, model with shear. (c) Variation in lower skin tensile strain ez with beam deflection 6 for: curve a, midskin; and b, skin mid-ply nearest core for beam A. ef = failure strain from Table 2. 6f = failure deflection from experiment (Table 3)
Figure I 1 Comparison of failure of lower skin for beam A with failure from static tensile test
COMPOSITES
V o l u m e 26 N u m b e r
12 1995
809
Elastic-plastic analysis of sandwich beams: R. A. W. Mines and N. Jones
Beam B
Z(mm) 9"0 b 8"6
C: 8'70 mm
/
//
8"2
7"8
glass laminates also increases with strain rate 24'25. Hence the energy absorbing capability of the beam should increase with strain rate. A rate dependent analysis is discussed in the next section. Table 3 summarizes the comparisons between the theoretical predictions and the corresponding experimental results. It is evident that the model is not too unreasonable for the prediction of force and energy absorption but is less successful in predicting the deflections at failure of the beams. G E N E R A L DISCUSSION
7~
i
7.0
i
-o.2
0
-o-~.
-o'.6
-o:e
-7-o
(a)
Em
ForcelN) / 3200F
Beam B
a
2400 c3 o 2000
"t"
~
"~
1600 un o
9
i
i
i
II
~
-
1200
8O0
fb 4O0
The approximate theoretical model developed in this paper relates the constituent material properties to the overall beam performance. The use of the model raises a number of relevant issues which are discussed in this section. The magnitude of the load in region (ii) of the idealized response shown in Figure 2 depends on the crushing strength of the core which is related to the compression strength, shear strength and work hardening characteristics of the core material. Generally speaking, the experimental results on beams exhibit a flat crushing characteristic 3, which is a result of the above three effects. It would be interesting to investigate other core materials with different compression and shear crushing strength characteristics in order to optimize the beam energy absorption. However, multi-axial crushing data [e.g. equations (25) and (26)] would have to be developed for each core material. The relative importance of compression and shear stresses will also depend on the beam geometry (e.g. span-to-depth ratio) and boundary conditions. Figure 14 shows that there is a local core crushing stress Crc in the vicinity of the loading point for a sandwich beam 26. This stress will influence the core crushing and if, as previously, von Mises yielding is assumed, then the criterion for core crushing for the three stress components (~r, O'c, ~'r) is given by27:
/O'/2 (O'/(O'......~c/.k_(O'c/2.k_/T/2
(b)
o~
lb
10
i0
4b
5'0
(27)
60
Deflecl'ionfmm) Beam B
E21
This equation can be regarded as a quadratic in (cr/~rr) and hence:
003
~rr = 2 ~ . ~ - f ) +
.~Q
0"02
o.oi "
I I
Y b'
0o
(e)
,
,
lO
20
I, 30
i
~0
50
60
5f
Deflection(ram)
Figure 12 (a) Variation of position of neutral axis (z) with emfor beam B: curve a, no shear; b, with shear. (b) Comparison of theoretical and experimental force~leflectionvalues for beam B: curve a, model with no shear; b, experimental results; c, model with shear. (c) Variation in lower skin tensile strain e2 with beam deflection 6 for: curve a, midskin; and b, skin mid-ply nearest core for beam B. ef = failure strain from Table 2. ar = failure deflection from experiment (Table 3)
810
COMPOSITES Volume 26 N u m b e r 12 1995
1-4~.~
(28)
which reduces to equation (23) for o-c = 0. Equation (28) shows that the effect of o-c is to increase the axial flow stress cr for a given loading on the beam. For typical load values for the experimental work described here, the axial flow stress would increase by ~ 10%, mainly due to the first term on the right-hand side of equation (28). This means that the beam elastic-plastic moment, and hence the beam external force, will increase slightly. However, it would be difficult to verify, experimentally, the yield criterion given by equation (28) for a given core material. It has been shown that the final failure of the beams is dependent largely on the tensile strain in the lower skins. However, assessment of this failure is problematic
Elastic-plastic analysis of sandwich beams: R. A. W. Mines and N. Jones
Beam C (RD+RII
Z(rnm)
9'0
cd
C:8.?Omm.
Force(N) Beam C
8.6
r 8.2
7.8
3600
U
3200
7.4
2800
0
(a)
m
-
012
-
04
i
-
0'-6
-
08
-
110
2~C
Em
Beam C
F--2
'
200C
0"04
b
/~ /'~Haximum / ~J ~DeflectJon
1600
d 0'03
,
"~
1200
(+)d
I
........K ~ b
0"02
I /
0
%(+)s
r
I
0"01
(c)
800
II
(h)
l
u~
10
20
30
40
50
Deflection (ram)
! !
10
2'0
3'0 ~m
~0
Deflecfion(mm)
Figure 13 (a) Variation of position of neutral axis (z) with e,, for rate independent (RI) and rate dependent (RD) material properties for beam C: curve a, no shear (Rl); b, no shear (RD); c, with shear (RI); d, with shear (RD). (b) Comparison of theoretical and experimental force-deflection values for beam C: curve a, model with no shear; b, experimental results; c, model with shear, static rate = 0.0001 s t; d, model with shear, impact rate = 100 s J. (c) Variation in lower skin tensile strain e2 with beam deflection 6 for: curve a, rate independent (R[); and b, rate dependent (RD) material properties for beam C. er(+)s = static failure strain (rate = 0.0001 s-l), er(+)a = impact failure strain (rate = 100 s 1)
due to the sensitivity of the strain calculation to the position, z, taken in the beam. Theoretical calculations indicate that strain values attain a critical value for a large portion of the beam response, hence the actual failure of the lower skin is dependent on the strain history of the skin and on a complex interaction between the skin and core damage stresses. Also, the measurement of beam cross-section dimensions is critical and this has been discussed in ref. 3. Ref. 3 shows that, for a woven carbon skinned Coremat cored beam, the maximum deflection is 5 ram. The failure strain of woven carbon is 0.01 (see ref. 28) and, hence, it is proposed that the smaller deflection of the beam at failure, as compared with the glass-reinforced beam, is a result of a lower failure strain in the bottom skin. It would be interesting to investigate
the effects of other skin materials on the magnitude of the beam deflection at failure. A skin with a high strain to failure should give a large beam deflection to failure. The magnitudes of the core crushing stress and the lower skin failure strain can be increased by impact loading, provided that other beam failure modes are avoided 3. The rate dependence of the core crushing stress of various cross-linked and linear PVC foams has been discussed in ref. 23. It was shown that the compression strength can increase by 30-100% for an increase in strain rate from 0.001 to 100 s-1. In the case of the beam test described here, for a given displacement rate at the loading point of the beam, the strain rate varies linearly across the cross-section due to the linear variation in strain (see Figure 4a).
COMPOSITES Volume 26 Number 12 1995 811
Elastic-plastic analysis of sandwich beams: R. A. W. Mines and N. Jones Table 3
Comparison of theoretical predictions and experimental results
Theoretical Force at maximum experimental deflection (shear effects included), F~h (N)
Beam A (WR)
Beam B (WR/CSM)
1460
1280
Beam C (WR/CSM/impact)
1350(RI)~ 1440(RD) ~
Maximum deflection, 8th, from experimentally measured er b (mm)
15
28
17
Energy from upper skin failure to maximum deflection, Eth (J)
73
60
18(RI) 19(RD)
Experimental Force at maximum deflection, Fexp (N) Maximum deflection, 8exp(mm) Energy from upper skin failure to maximum deflection, Eexp (J)
1220
900
1900
60
60
32
62
45
23
1.20 0.25 1.18
Fth/Fexp
8,dSexp E,h/Ecxp
1.42 0.47 1.33
0.71 0.53 0.78(RI) 0.84(RD)
u RI = rate independent; RD -- rate dependent h Strain measured at mid-skin
,
I
',.T.,, Figure 14 axis)
"l
_
I a
Local stresses in beam near loading point (NA = neutral
Also the loading point displacement rate in a drop weight impact test varies during the test. For beam C, the total loading time up to the maximum deflection of 32 mm is 8.14 ms. If it is assumed that the maximum upper core surface strain near failure is ~ 0.6 (see Figure 13b) then the average compression strain for the section is ~ 0.3. This means that the average core strain rate is 36 sq. Assuming that the rate dependence of Coremat is similar to PVC foams and taking an average value of increase in core crush stress from ref. 23 then, for beam C in Figure 13b, the crushing stress should increase by 50% to 18 MPa. This increase in flow stress has been included in the elastic-plastic analysis [equations (1)-(26)] and the rate dependent (RD) results are shown in Figures 13a, b and c. Note that the behaviour of the beam is assumed to be quasi-static, i.e. no vibrations occur. The effect of shear on the position of the neutral axis has been reduced and the load levels in the beam have increased by 10%. No effects due to the rate dependence in zr has been shown in Figures 13a, b and c. If zf increases with the strain rate 3~ (see ref. 18 for metals) then the level of beam load will increase further. One feature of the experimental results for impact loading is the large oscillation on remote force measurements after upper skin failure 3'29. This makes the experimental verification of the theoretical model for the rate dependence of the plastic hinge difficult to achieve. The use of a sensor on the surface, or embedded in the beam, in the
812
COMPOSITES Volume 26 Number 12 1995
vicinity of the plastic hinge could provide more dependable data. However, such a sensor would have to register strains of up to 50%. In the case of the skins, for beam C, the average strain rate is about 100 s-l. Harding and W e l s h 24,25 have shown a 50% increase in the uniaxial tensile failure strain for fine plain weave glass/epoxy specimens from static loading (e -- 0.0001 s-I) to a rate of 100 s-~. However, they also showed a 150% increase in the tensile failure strain for static loading (t~ = 0.0001 s4) to a strain rate of 22 s-~ for a satin weave glass/polyester composite. The skins in beam C were a mixture of woven roving and chopped strand mat 3. Rate dependent data were unavailable for chopped strand mat. It is proposed that the rate dependence of the failure strain in the skins in beam C will be similar to that of the plain weave woven roving mentioned above and, hence it can be expected that the lower skin failure strain increases from 0.0195 [for the static case (see Table 2)] to ~ 0.03 (for the impact case). Thus, it is likely that larger strains can occur in the lower skin for the impact case as compared with the static case. For the case given in Figure 13e, the lower skin showed no large scale damage after impact even though the predicted lower skin strains are above the static material value. It can also be concluded that the type of weave can be selected to maximize the strain to failure and, hence, the magnitude of the beam deflection to complete failure. Some preliminary experimental work has been conducted on these types of beams with clamped end conditions 3~ The mode of beam failure changes to core shear and lower skin debonding, although the shape of the transverse force-deflection characteristic is similar to the simply supported case. Tensile stresses are induced along the beam and these will modify the crushing of the core 16q8 and the failure of the lower skin. An important assumption incorporated into the theoretical model is a constant hinge size 10, which is independent of the deformation era. This aspect has been investigated using high speed photography, and Figure 15 shows that there is little variation of the hinge size l0 with the deflection of beam C. Error bars show
Elastic-plastic analysis of sandwich beams: R. A. VV. Mines and N. Jones Io(mm) 16 15
11 10
z~
i~,
16
fi
3'0
h
17"2
19-7
20-2
21"7
23'2
Z4"7
e (Degreel Deflection(rural
Figure 15 Variation of hinge size/0 with beam deflection 6 and angle 0 for beam C from high speed photography results
the possible variation in measurements from the photographs of the deformed beam specimens. The use of the theoretical model requires the measurement of hinge size at a given deflection. It has been shown that the hinge size is dependent on the nature of the skin (beams A and B), and on the rate of loading (beams B and C). In the first case, the skin changed from woven to a mixture of woven and chopped strand mat and lo/c changed from 1.85 to 1.40. The reason for this is that a thicker skin (beam B) gives rise to a more localized crushing. In the second case, the difference was in the rate of loading and lo/c changed from 1.40 to 0.93. Impact loading increases the effect of crush localization. An important issue is the sensitivity of the beam deformation to the hinge size. The average value of l0 for beams A and B is 14.7 and if this value was taken for beam A, then the predicted force for a given deflection would change by 5%. The formation of the hinge is a complex two-dimensional problem in which compression and shear crushing contribute and in which crack formation and skin-core debonding may well play a role. Ref. 31 considers the effect of the quality of beam construction on predictions from the elastic-plastic model. Ref. 32 applies the model to Coremat cored beams with a span-to-depth ratio of 38, and good agreement is shown between the model and experimental results. It should be noted that, from the structural design point of view, standard sandwich panel cores, with densities typically of 100 kg m -3, would not give such a large energy absorption capability for the structural behaviour described in this paper. In other words, if high energy absorption was required in a localized area of a given structure, then special high density core construction would be required specifically in that area. CONCLUSIONS A simple 'strength of materials' model has been developed for the post-failure energy absorbing behaviour of Coremat cored sandwich beams subjected to static and impact loads. The aims have been to gain understanding of the effects of the constituent material properties on the overall behaviour of a beam and to provide a basis for optimizing the impact energy absorption of a beam prior to any detailed (i.e. finite element) analysis. The main assumptions and limitations in the model are as follows. A specific beam response has been assumed, viz. upper skin compression failure and subsequent deformation due to core crushing in the central
plastic hinge. Standard plasticity methods have been used as the basis for the model, i.e. flow theory for the crushing core, and simplifying assumptions concerning stress and strain distributions across the beam have been made. These assumptions have been justified and reasonable agreement has been obtained between the model and experimental results. The use of the model requires the measurement of the plastic hinge size for each construction and there is a need to predict hinge size. Two approaches are possible, namely either the development of empirical curves for various materials and structural cases or the development of a model for the formation of the hinge. The latter may require consideration of the effects of core-skin debonding. Another potential improvement to the model concerns the assessment of the flow stress in the core. It has been shown that this is a result of multiaxial stresses in the core and there is a requirement to experimentally measure this. Such experimental multiaxial crushing tests would be complex as simultaneous compression and shear measurements would be required. Other, less critical, issues that need to be addressed are as follows. A bilinear core crush stress characteristic has been taken here, and in reality a trilinear characteristic would be more suitable. Small strain plasticity theory has been used, whereas core crush strains of 40-60% were encountered, and so finite strain effects could be included in the model. Strain rate effects have been included in core crushing characteristic using average strain rate values. The inclusion of rate effects would be improved by a more accurate assessment of strain rates within the beam and by measuring the dynamic properties for beam materials. From the above, it can be seen that the detailed analysis of the behaviour of these beams requires large scale effort in material modelling and in the measurement of material properties. It is felt that the development of simplified, phenomenological models allows the investigation of the complete structural problem whilst retaining the essentials of the wide ranging and complex nature of possible material response. One issue that needs to be addressed concerns the applicability of the model to other beam configurations, viz. increased beam sizes, different skin thickness to core thickness ratios, different boundary conditions and other core materials, e.g. foams. These different configurations may give other modes of failure and may invalidate some of the assumptions made in the model. Hence, further phenomenological models are required to cover all possible modes of beam behaviour. Finally, the application of the model requires predictable and reproducible beam material properties and beam section dimensions. The critical parameters are skin thickness, for the assessment of lower skin tensile failure strains, and the variation of properties across the core, for assessment of the core flow stress. ACKNOWLEDGEM ENTS The authors are indebted to Dr A.M. Roach, of the Impact Research Centre at the University of Liverpool, who conducted the tensile tests for the beam skin materials. Dr. R. Cox, of Lantor BV, supplied much useful information on the mechanical properties of Coremat.
COMPOSITES Volume 26 Number 12 1995
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Elastic-plastic analysis of sandwich beams: R. A. VV. Mines and N. Jones REFERENCES 1
2
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
814
De Kalbermatten, T. and Reif, G. Cab in 'GRP/Foam Sandwich Design for Modern High Speed Locomotives: Sandwich Constructions 2' (Ed. D. Weissmann-Berman and K.A. Olsson), EMAS, Cradley Heath, 1992, pp. 873-886 Haug, E. and De Rouvray, A. in 'Crash Response of Composite Structures: Structural Crashworthiness and Failure' (Eds Norman Jones and T. Wierzbicki), Elsevier, London, 1993, pp. 237-294 Mines, R.A.W., Worrall, C.M. and Gibson, A.G. Composites 1994, 25, 95 Gibson, L.J. and Ashby, M.F. 'Cellular Solids - Structure and Properties', Pergamon Press, Oxford, 1988 Allen, H.G. 'Analysis and Design of Structural Sandwich Panels', Pergamon Press, Oxford, 1969 Plantema, F.J. 'Sandwich Construction', Wiley, Chichester, 1966 Zweben, C. ,L Compos. Mater. 1978, 12, 422 Reedy, E.D.J. Compos. Mater. 1988, 22, 955 Fischer, S. and Marom, G. Fibre Sci. Technol. 1984, 20, 91 Gibson, L.J., Ashby, M.F., Zhang, J. and Triantafillou, T.C. Int. ,L Mech. Sci. 1989, 31,635 Triantafillou, T.C., Zhang, J., Shercliff, T.L., Gibson, L.J. and Ashby, M.F. Int. J. Mech. Sci. 1989, 31,665 Rothschild, Y., Echtermeyer, A.T. and Arnesen, A. Composites 1994, 25, 111 Cox, R.J. Personal communication, Lantor BV, Veenendaal, Sept. 1994 Triantafillou, T.C. and Gibson, L.J.J. Eng. Mech. 1990, 116, 2772 Schreyer, H.L., Zuo, Q.H. and Maji, A.K.J. Eng. Mech. 1994, 120, 1913 Horne, M.R. 'Plastic Theory of Structures', Thomas Nelson, London, 1971 Zyczkowski, M. 'Combined Loadings in the Theory of Plasticity', Polish Scientific Publishers, Warsaw, 1981
COMPOSITES Volume 26 Number 12 1995
18 19 20
21 22 23
24 25 26 27 28 29 30 31
32
Jones, N. 'Structural Impact', Cambridge University Press, 1989 Williams, J.G. 'Stress Analysis of Polymers' (2nd edn), Ellis Horwood, Chichester, 1980 BS 2782 Part 10 Method 1003, 'Determination of Tensile Properties of Glass Reinforced Plastics', British Standards Institution, 1977 Firet Coremat Data Sheet, Lantor BV, Veenendaal, The Netherlands, 1993 Worrall, C.M. Unpublished data, University of Liverpool, 1987 Von Gellhorn, E. and Reif, G. in 'Think Dynamic - Dynamic Test Data for the Design of Dynamically Loaded Structures: Sandwich Constructions 2', Vol. 2 (Eds D. Weissmann-Bermann and K.A. Olsson), EMAS, Cradley Heath, 1992, pp. 541-557 Harding, J. and Welsh, L.M.J. Mater. Sci. 1983, 18, 1810 Welsh, L.M. and Harding, J. J. de Physique Colloque 5 1985, Supplement au No. 8 (DYMAT85), C5405 Johnson, K.L. 'Contact Mechanics', Cambridge University Press, 1985 Mendelson, A. 'Plasticity: Theory and Applications', Macmillan, New York, 1968 Hancox, N.L. and Mayer, R.M. 'Design Data for Reinforced Plastics', Chapman and Hall, London, 1994 Mines, R.A.W., Worrall, C.M. and Gibson, A.G. in 'Proc. ECCM-3', Bordeaux, France, 1989 AbduI-Ghani, A., Impact performance of GRP sandwich beams, B Eng Final Year Project Report, University of Liverpool, 1994 Mines, R.A.W., and Jones, N., An approximate elastic-plastic analysis of the static and impact behaviour of polymer composite sandwich beams, Impact Research Centre Report No. 124/95, The University of Liverpool, 1995 Mines, R.A.W. 'The Static and Impact Elastic-Plastic Behaviour of Polymer Composite Sandwich Beams: Sandwich Constructions 3' (Ed. H. G. Allen), EMAS, Cradley Heath, 1995
