- Email: [email protected]
Notched strength of long- and short-fibre reinforced polyamide
Composites Science and Technology 45 (1992) 43-54
Notched strength of long - and short-fibre reinforced polyamide Staffan Toll & Carl-Gustaf Aronsson Department of Aeronautical Structures and Materials, Royal Institute of Technology, S-IO0 44 Stockholm, Sweden (Received 31 May 1991; revised version received 2 August 1991; accepted 3 September 1991)
A test programme has been conducted to evaluate the tensile strength of injection moulded plates with machined and moulded-in notches. The longfibre compound, Verton, is consistently less notch sensitive than the short-fibre compound, Maranyl. Moulded-in holes appear to have several advantages over machined holes: they yield up to 25% higher strength and less sensitivity to plate thickness and injection speed. Five existing laminate fracture models are applied to the machine-notched plates; the average stress criterion and the damage zone model give the most accurate predictions.
Keywords: composite, injection moulding, strength, fracture, holes, cracks
1 INTRODUCTION
products are known to have generally better and less anisotropic mechanical properties than their short-fibre counterparts? This development has increased the interest in injection moulded composites for structural applications, making detailed knowledge about their elastic response and fracture behaviour increasingly important. Injection moulded composites have a more complicated mechanical behaviour than most engineering materials. Since the mechanical properties depend on the local microstructure, they vary throughout a component and are seldom known at all points. Some workers 3,5-7 have approached fracture and fatigue of both long- and short-fibre reinforced polyamide by using fracture mechanics. Microscopic fracture mechanisms have been studied and modelled for short-fibre compounds 8"9 and long-fibre compounds. ~0Sato et al. 9 concluded that fracture initiates as interfacial cracks at the fibre ends, followed by propagation of these cracks along the fibres and, finally, into the matrix. Friedrich xl has demonstrated a fractographic approach, dividing the fracture surface into areas of three different types, each associated with one basic fracture mechanism: fibre fracture, matrix fracture or yielding, and interface debonding; he could correlate the
Injection moulded composites potentially combine good mechanical performance with economical manufacturing and flexible design. The mechanical performance is, however, limited by factors like wetting, fibre dispersion, and fibre length. For example, an increase in fibre length from that of conventional short-fibre injection moulding compounds can improve both strength and toughness.~-3 A few years ago, a major breakthrough was made through ICI's introduction of a family of 'long-fibre' injection moulding grades, Verton. The new development was parallel to that of the Aromatic Polymer Composite (APC), and based on a similar pultrusion impregnation process. The Verton process is able to impregnate high loadings of fibres with improved wetting and dispersion, and without breaking fibres. The granules are cut from a pultruded rod to typically 10 mm lengths, which correspondingly becomes the uniform starting length of the fibres, compared to a distribution between, typically, 0-2 and 0.6 mm for short-fibre grades. The final Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 43
44
Staffan Toll, Carl-Gustaf Aronsson
fracture toughness of the material with the division of fracture surface among these three types, as measured from SEM photographs. Little work has been reported on the notched strength of injection moulded composites, yet fracture initiation at notches is a major cause of structural failure. Notches occur in the form of holes, cut-outs, corners, etc., and are introduced into an injection moulded structure in two basic ways: either by machining of the moulded component, or through the inherent shape of the mould, so-called moulded-in notches. The microstructure in the vicinity of the notch is generally very different in these two cases, t2 and so the mechanical behaviour is also different. Vanderschuren 12 investigated the notched strength of an injection-moulded, glass-fibrereinforced thermoset (phenolic) with moulded-in holes. In this work, the tensile strength of notched injection moulded plates has been tested. The programme involved two Nylon/glass fibre compounds, one long-fibre (LF) c o m p o u n d and one short-fibre (SF) compound; moulded-in circular holes and machined holes and cracks of various sizes; two injection speeds; and two plate thicknesses. A subset of these plates was subjected to microstructural studies, measurements of elastic moduli and unnotched strength, and modelling of notched strength using five existing fracture models.
~Runner
Two grades of glass-fibre filled polyamide 66 were used: a long-fibre (LF) c o m p o u n d , Verton RF7006, with 30 weight% fibre content; and a short-fibre (SF) c o m p o u n d , Maranyl A190, with 33 weight% fibre content. The fibre length prior to injection moulding was 1 0 m m for the long-fibre c o m p o u n d and less than 0 . 5 m m in average for the short-fibre compound. Moulding conditions
The mould, together with its runner system, is shown in Fig. 1. It was a single-cavity rectangular plate mould with an optional insert for central moulded-in holes of different sizes. The plate dimensions were 1 6 0 m m x 4 0 m m , and the thickness either 1.5 or 3 mm. The film gate was
Die
Cavity
Fig. I. The plate mould, here with a i0 m m
moulded-in
hole. 1 mm X 25 m m and the cross-sectional area of the runner was approximately 25 mm:. Six different plates were moulded. These, plates A to F, are presented in Table 1: two different materials and two different plate thicknesses were used. In addition, the LF (Verton) plates were processed at two different injection speeds. Each of these plates was moulded both plain and with a central mouldedin hole of three different sizes (5, 10, and 15 m m diameter). The injection moulding was carried out on a Mannesmann D e m a g D60-182NC II injection moulding machine under the conditions listed in Table 2.
Table 1. Moulded plates (160 X 40 mm)
Plate 2 MATERIALS
~
A B C D E F
Thickness Compound (mm) 3-0 3.0 3-0 1-5 1-5 1-5
LF LF SF LF LF SF
Inj. Speed High Low High Low
Table 2. Moulding Conditions
Temperatures (°C) Zone 1 and 2 Zone 3 Die Mould Injection pressure (Bar, hydraulic) Holding pressure (Bar, hydraulic) Coohng-offperiod (s) Injection speed (cm3/s) High Low Holding-pressure time (s)
LF
SF
295 295 300 95 110 65 10
275 280 285 110 110 50 10 39
16 8 8
6
Notched strength of long- and short-fibre reinforced polyamide Microstructure
°l C O
The microstructural investigation was confined to the 3 mm thick plates. The fibre orientation in the plain mouldings was investigated in recent work, ~3 and that in the mouldings with moulded-in holes was visually inspected here from micrographs taken in the 1, 2 plane. The fibre length distributions were studied in Ref. 14. The observed internal structure of the plates is schematically shown in Fig. 2: there is a surface layer having a predominant orientation parallel with the injection direction; and a core layer having a high degree of fibre orientation transverse to the injection direction. This structure can be said to be typical of injection moulded polyamide 66 composites. The fibre orientation patterns in the surface and core layers are described in Fig. 3: the angles of principal fibre orientation at different locations are indicated by the dashes. The exact meaning of principal orientation is stated in Ref. 15. In the case of the moulded-in hole, there is a knitline extending from the hole and downstream. It is seen that near any edge the principal orientation through the thickness is parallel to the edge, and near a knitline it is parallel to the knitline. This means extra fibre alignment (a) along the outer edges, (b) tangentially around a moulded-in hole, and (c) along a knitline. The average fibre length was higher in the core than in the surface layer, which is usually the case. 3
Differences between the plates. The structure of the surface layer is similar in all the plates A, B, and C (Table 1), whereas the core structure is considerably more oriented in plates A and B (LF) than in plate C (SF). The thickness of the core is approximately 1/5 of the total plate thickness in all the plates A, B, and C. Consequently, the overall degree of fibre orientation, and thus the anisotropy, is highest in ] ]
Predominant longitudinal fibre orientation Predominant transverse fibre orientation
Surface layer ~'.v.v.v::.v.'.v:.v.':::.v.v.v.'.v.v.v.':.v.v.v:.v.v.'.f~ n
,nlectian d ir e c ~// +
27
~.+[.v ,.,........,,.....v .............. .............. j~
Hole edge region
Edge region
Fig. 2. Layer structure of injection moulded plate.
C
Surface
Core
'"',',
iY---
45 Surface
Core
I% I I " '+ iii
IiI
I I i + iI
u nlnlll I nI I I I I [
,~--
I
ll'n I I I I ,nl I I I I IIi I i i i i i i i Ii ii
l l f - ~ - - ~.- '~_-'--_'Z_ ....
(o)
,
Ii,",";'" , ,'P' Ill,
2 L;
I iI I ~
,,11 al,l,'l
(b)
Fig. 3. Schematic of fibre orientation pattern. (a) Plain moulding; (b) Mouldingwith moulded-inhole. plate C. The edge region indicated in Fig. 2 is approximately 3 mm wide in plates A (LF, high injection speed) and C (SF), and about 4.5 mm in plate B (LF, low injection speed). The volume average fibre length in the central part of the plain mouldings was approximately 1-2 mm for the LF mouldings and 0-27mm for the SF mouldings.
3 EXPERIMENTAL Machine notched plates for the testing of notched fracture loads along with cut-out specimens for the testing of unnotched strength and elastic moduli were machined from the plain mouldings. Prior to testing, all specimens were conditioned to saturation in air at 50% relative humidity and 80oc for 20 days. All tensile tests were carried out on an Instron 6025 universal testing machine, at approximately 23°C (room temperature) and 40% relative humidity. Elastic moduli and unnotched strengths For the purpose of fracture modelling, engineering moduli and unnotched tensile strengths were measured on the 3 m m thick plates. The measurements were made in the central part of the plates. Each reported result is the average of results for between two and five specimens. Orthotropy was assumed. The elastic moduli Eu, E22, and E12 were measured on 14mm wide rectangular cut-out specimens. The displacement rate was i mm per minute. The moduli were defined as the secant moduli between 0.02% and 0.2% normal strain, i.e. (o,),,=0.02 e + = (Ej)E;=o.o -- (E;),,=o 2
(1)
Staffan Toll, Carl-Gustaf Aronsson
46
Table 3. Elastic moduli and unnotched strengths Plate A B C
Ell (GPa)
E22 (GPa)
EI2 (GPa)
VI2
4-8 4.3 3.2
6.1 6-1 6.6
-19 -19 -18
0.25 0-23 0-17
GI 2 (GPa)
Sl (MPa)
S2 (MPa)
2.1 2.0 1.8
86 90 73
118 121 126
ff
o
2
2
o
2e
2R
223
(i,j= 1, 2). obtained as
The
Poisson's
ratio,
v~2, was o
o"
(o)
Ell
v,2 -
........
E2,
(2)
(b)
Fig. 5. Notch types and loading conditions. (a) Centre-Hole (CH), moulded-in or drilled; (b) Centre-Crack (CC).
and the shear modulus, G,z, was estimated according to Huber's approximation: 16
WrE11E2z G,2 ~ 2(1
+
v12~/E22/E,,)
(3)
The results are given in Table 3. Unnotched tensile strengths were measured on small hour-glass-shaped specimens, illustrated in Fig. 4. These were cut out from the centre of the plate, parallel and perpendicular to the injection direction. The stress concentration factor, ce, at the net-section of the specimen was calculated by a two-dimensional finite element model, from the previously measured in-plane elastic moduli of each plate. The unnotched tensile strength, S, was approximated as the average of the ultimate mean stress, Pu/ht, and the calculated maximum stress, cePu/ht, i.e. S
Pu c r + l 2
ht
(4)
where P, is the ultimate load and ht is the net cross-sectional area of the specimen, Fig. 4. The displacement rate was chosen so that the strain rate in the gauge region was approximately 1% per minute. The experimental unnotched strengths perpendicular ($1) and parallel ($2) to the injection direction are given in Table 3. Note that [mm]
O
---
t=3
the longitudinal strength, $2, and modulus, E22 , were highest for the SF plate, whereas the transverse strength, $1, and modulus, El,, were highest for the LF plate; thus the SF plate possessed a higher degree of anisotropy.
Measurement of notched strength The plates were tested in tension using three notch types: 1. moulded-in centre holes (MCH) 2. drilled centre holes (DCH) 3. cut centre cracks (CC) The notch geometries and loading conditions are illustrated in Fig. 5. Both holes and cracks were made to three different sizes: 2R, 2a = 5, 10, and 15 mm. The gauge length was 100 mm and the displacement rate 1 mm per minute.
4 EXPERIMENTAL RESULTS All the tested plates exhibited net-section failure through the notch, visibly initiating at the notch tip. The experimental notched strengths, c~Nxp, are given in Table 4 for the thick plates (3 mm thick) and Table 5 for the thin plates (1.5 mm thick). These results are illustrated in Fig. 6, where ~Nxv is plotted versus the notch length-towidth ratio, 2a/w or 2R/w, for all the different materials and notch types.
Effects of fibre length and injection speed. -
-
~
10
Fig. 4. Specimen for the measurement of unnotched strength. The shaded areas are those covered by the grips.
Figure 6 shows that the LF compound had a higher notched strength than the SF compound in all cases. It shows a clear effect of injection speed only for the thin plates with machined notches;
Notched strength of long- and short-fibre reinforced polyamide Table 4. E x p e r i m e n t a l n o t c h e d s t r e n g t h s ¢7~v xp for the thick plates Plate 2a/w,
MCH
DCH
CC
47
difference in strength between MCH and DCH plates was 15-25% at 3 m m thickness, and 6-19% at 1.5 mm thickness (Tables 4 and 5).
2R/w
A
B
C
0.125 0.250 0.375 0.125 0.250 0.375 0.125 0-250 0.375
o~ p (MPa)
Specimens
o~vXp (MPa)
Specimens
o~,Xp (MPa)
Specimens
944-3 734-3 62 4-2 974-2 75 4- 3 634-1 83 4-1 69 4- 2 59 4- 2
10 10 10 10 10 10 10 10 10
754-3 614-2 53+1 784-2 65 4- 1 534-1 67+1 58 4- 2 48 4- 1
7 7 7 7 7 6 6 6 6
70-t-2 564-2 46+1 704-1 57 4- 3 464-1 63+2 51 4- 2 43 4- 1
7 7 7 6 6 6 6 6 6
4- ---standard deviation.
5 MODELLING
An attempt was made to model the notched strength for a subset of the plates by using five existing fracture models: the inherent flaw model (IFM), Waddoups et al. ;~7 the point and average stress criteria (PSC and ASC, respectively), Whitney & Nuismer; ~8'~9 the damage zone criterion (DZC), Eriksson & Aronsson; 2° and the damage zone model (DZM), Aronsson and B~cklund.
the higher injection speed resulted in a higher strength.
Thickness effects. It can be seen from Fig. 7 that the plates with moulded-in holes exhibited no significant thickness effect; and that the plates with machined holes and cracks had slightly higher strengths for the thinner plates; but the effect was small for the LF compound, low injection speed. The thickness effects for machined holes and machined cracks were very similar to each other. Notch effects. As illustrated by Fig. 8, the plates with moulded-in holes were invariably stronger than those with machined holes, which in turn were stronger than those with cracks of the same notch length. This relative behaviour was obtained for all six plates (A-F). The notch effect was more pronounced for the thick plates than for the thin ones; for example, the Table 5. E x p e r i m e n t a l n o t c h e d s t r e n g t h s a ~ for the t h i n plates Plate
D
E
F
2a/w, 2R/w
0.125 0.250 0.375 0.125 0.250 0.375 0.125 0.250 0.375
MCH
DCH
CC exp
ON
~N~p (MPa)
Specimens
o~ p (MPa)
Specimens
(MPa)
Specimens
93 + 2 74 + 3 61 + 2 91 + 3 75 + 2 62 4- 1 83 4- 1 70 4- 2 58 4- 1
10 10 10 10 10 10 10 10 10
83 + 3 70 + 2 56 + 1 77+2 63 4- 2 52 4- 2 73 4- 1 60 + 2 51 4- 1
7 7 7 7 6 6 6 6 6
79 4- 3 61 + 2 48 4- 2 70+2 55 + 1 45 + 1 68 + 3 55 4- 2 45 4- 1
7 7 7 6 6 7 6 6 6
+ -----standard deviation.
21-23
All of these are two-parameter models for the prediction of the notched tensile strengths of laminated composites. The DZM numerically computes the complete load/displacement relation to failure, by using a finite element model of the structure, and predicts failure when the ultimate external load is reached. The other models predict failure when certain criteria are fulfilled by the stress distribution ahead of the notch, resulting in closed form expressions. The IFM, PSC, and ASC are based on linear elastic stress distributions, whereas the DZC and DZM are based on stress distributions resulting from the development of an inelastic damage zone. The performance of these models when applied to laminated composites has, except for DZC, become fairly well established. 22-25 The analyses are described in the Appendix. The modelling was performed for the 3 mm thick plain mouldings with machined holes and cracks. The in-plane property variations were thus comparatively small; and the properties measured in the central region (Table 3) were assumed to be valid throughout the plates. The fracture parameters (see Appendix) for each fracture model and each plate were calculated from the experimental notched strength for the short crack, a = 2 - 5 m m , and the unnotched strength, $2. These parameters are given in Table 6. The notched strengths were then predicted according to the fracture models.
Comment on finite width correction. The DZM and the DZC are so constructed that the finite width of the specimen is automatically accounted for. In the IFM, ASC, and PSC a finite width can only be accounted for by correcting the assumed
Staffan Toll, Carl-Gustaf Aronsson
48 • v ÷
LF, fast injection LF, slow injection SF Thick plates
Thin plates
100
d
75
50
MCH
25
75
N
50
25
~
100
50
25
0
0.125
0.250
0
0.125
0.250
0.375
Notch length-to-width ratio 2a/w or 2R/w
Fig.
6. E x p e r i m e n t a l n o t c h e d s t r e n g t h vs. n o t c h l e n g t h - t o - w i d t h r a t i o
stress distribution. For a crack, the assumed stress distribution is singular, and finite width correction factors can be used; but for a hole there is no such simple solution. Therefore the predictions made for holes from these three models assume infinitely wide specimens (w = ~). This may cause an additional error, especially for the largest holes, which should be kept in mind when interpreting the results.
Results. The predicted notched strengths are presented in Table 7. All the models predicted the crack size effect rather well, but the transition to holes seems unreliable for the D Z C , IFM, and PSC. The A S C and D Z M performed well on the
2a/w o r 2R/w.
whole, and in nearly all cases the best correlation with experimental results was obtained with the ASC. Practically all predictions were underestimates.
6 DISCUSSION
Failure modes. All the plates in the reported test programme failed along the net-section through the notch, as depicted in Fig. 9(a). However, it should be mentioned that a few plates with moulded-in holes were tested without being conditioned, i.e., with a lower moisture content in the matrix (unreported results). These
Notched strength of long- and short-fibre reinforced polyamide
49
v Thin plates • Thick plates
MCH
LF
LF
fast inj.
slow inj.
SF
25
~
75
~
50
0.125
0.250
O.125
0.250
0.125
0.250
Notch length-to-widthratio 2ahv or 2R/w
Fig. 7. Effects of plate thickness on notched strength.
~,
• MCH O DCH
100
~
75
¢..
5O
0
0.125
0.250
0.375
Notch length-to-width ratio 2a/w or 2R/w
Fig. 8. Effect of notch type on the notched strength of Plate A (LF, 3 mm thick, high injection speed). All the other plates show a similar behaviour.
Table 6. Fracture parameters Plate
$2 (MPa)
d wM (mm)
d Psc (mm)
d Asc (mm)
d Dzc (mm)
G~ (kJ/m z)
A B C
118 121 126
1.39 1.29 0.85
0.60 0.56 0.40
2-71 2.52 1-72
0.67 0.63 0.42
12 12 8
plates failed along a path circumscribing the hole, as shown in Fig. 9(b), leaving the hole edge intact. A possible explanation is as follows. In the unconditioned case, failure initiates either somewhere in the tangentially-oriented hole edge region or between the hole edge and the outer edge (see Fig. 3(b)), owing to the microstructure induced by the hole insert; in the conditioned case, the moisture softens and toughens the matrix, by the action of water as a plasticizer, to such a degree that the hole edge becomes critical instead. This phenomenon of competing failure modes was also encountered by Vanderschuren 12 in injection moulded short-fibre reinforced phenolic (thermoset).
Fibre length effect. The LF plates had a significantly lower degree of anisotropy than the SF plates: the longitudinal unnotched strength, $2, and Young's modulus, E2, were actually highest in the SF plates, whereas the transverse strength and modulus, $1 and E~, were highest in the LF plates. The reason for this is that the core, which dominates the transverse properties, has a higher fibre orientation and fibre content in
50
Staffan Toll, Carl-Gustaf Aronsson Table 7. Predicted notched strengths for the 3 mm thick plates
2a/w, 2R/w
Plate
CC-notches 0-125 A B C 0.250 A B C 0.375 A B C DCH-notches 0.125 A B C 0.250 A B C 0.375 A B C
IFM ON
PSC ON
AS(" ON
DZC ON
DZM ON
AIFM
APSC
AASC
ADZ(:
ADZM
(MPa)
(MPa)
(MPa)
(MPa)
(MPa)
(%)
(%)
(%)
(%)
(%)
70.0 70.0 63-0 53.8 52.9 46.4 42-7 42.5 36.8
70-0 70.0 63.0 53.3 53.1 47.6 44.7 44-5 39.6
70.0 70.0 63.0 54.5 54-3 48.3 46.1 45.9 40-4
70.0 70.0 63.0 51.8 51.8 45.9 41.2 41.4 36.2
70.0 70.0 63.0 54.8 55.0 48.1 43.8 44-1 38-1
0.0 0.0 0.0 -5-2 -7.2 -9.0 -7.2 -7-6 -14.4
0-0 0-0 0-0 -4.8 -6.8 -6-7 -2-8 -3-3 -7.9
0-0 0.0 0.0 -2-7 -4.7 -5.3 0.0 0-0 -6.0
0.0 0.0 0.0 -7.5 -9-1 -10-0 -10.4 -10.6 -15.8
0.0 0.0 0.0 -2.0 -3.5 -5.7 -4-7 -4.3 -11.4
62.1 63.6 62"1 49.2 48.4 48-1 38.1 37.9 37.0
60.5 60.6 57.1 50-0 50.7 48.5 46.3 47.1 45-5
71.5 71.9 67.0 59"0 59.4 55.8 53.3 53-8 51.0
61.7 62-0 57.3 49.9 50.6 48.0 44-5 45.4 43.6
70.1 70.8 63.9 53-2 53.8 48.9 50-6 51.1 47-8
-17.2 -18.5 -7.3 -19-3 -25.5 -17.1 -28-1 -28.5 -22-9
-19.3 -22.3 -14.8 -18.0 -22.0 -16.4 - 12-6 -11-1 -5.2
-4.7 -7.8 0.0 -3.3 -8"6 -3.8 0.0 -1-5 6-3
-17.7 -20.5 -14.5 -18-2 -22-2 -17-2 - 16.0 -14.3 -9-2
-6-5 -9-2 -4.6 -12.8 - 17-2 -15.7 -4.5 -3-6 -0.4
A--deviation from ONe~P.
the LF plates than that in the SF plates, whereas the surface layers, which dominate the longitudinal properties, hardly differ at all. Nevertheless, the longitudinal notched strength was invariably higher for the LF plates; in other words, the LF compound exhibits a lower notch sensitivity or, with yet another term, a higher toughness than the SF compound. This higher toughness is manifested by larger values of the characteristic distances and G~ in the fracture modelling of the LF compound, Table 6. Apart from the longer fibres, other factors such as better wetting and fibre dispersion of the Verton compound may contribute to its higher toughness.
Thickness effects. The higher strength of the thinner plates can be explained by a higher ratio of surface layer thickness to core thickness, making the through-thickness degree of fibre alignment in the loading direction higher for the thin plates. It is natural that the plates with
~5
t-
(o)
(b) Fig. 9. Failure modes.
machined cracks and holes show the same thickness effect since the mouldings, and therefore the microstructure, are the same. It is also natural that the plates with moulded-in holes show less thickness effect, since the governing microstructure is probably that of the hole edge region, which has little thickness dependence since it contains no core (see Fig. 2).
Notch effects. As long as net-section failure prevails (Fig. 9(a)), it is perfectly reasonable that moulded-in holes are stronger than machined ones, because of their higher fibre alignment along the stress trajectories at the hole edge. But for other failure modes, e.g. that in Fig. 9(b), this is not obvious. Plates with holes were stronger than plates with cracks. This is not always the case for composites, and most of the models predicted the opposite behaviour for small notches. Modelling. The varying microstructure of the plain plates, especially towards the edges, makes the interpretation of the modelling results difficult. Since the edge regions have a high fibre orientation parallel to the edges (see Section 2) they stiffen the edges longitudinally, and thereby relieve the stress field in front of the notch tips. This effect should become increasingly pronounced at decreasing distance between the edge and the notch tip, thus diminishing the otherwise
Notched strength of long- and short-fibre reinforced polyamide
expected notch length effect. Since the models do not account for this, they should overestimate the notch length effect. This appears to be true for the CC predictions: the model parameters were determined for the shortest crack, and they all underestimated the notched strengths at increasing crack lengths. The interpretation of the D C H predictions is less certain: first, since the material toughness parameters were determined from a CC specimen, the notch geometry dependence of the models becomes influential; second, an infinite plate width was assumed for IFM, PSC, and ASC. The overall good performance of the A S C may be a coincidence; if the stiffened edges had been accounted for, the A S C probably had performed worse and the other models better. There are at least two important limitations in the present fracture modelling that may contribute to errors: 1. the spatial variation of material properties due to the variation of fibre orientation, fibre content, etc., is ignored; 2. linear material behaviour is assumed. Taking varying and nonlinear material properties into account can probably provide a basis for a more accurate failure analysis of injection moulded composites. Of the models tested here, the D Z M can most easily, and without modification accommodate varying material properties. 26 However, the local material properties may be difficult to determine accurately.
7 CONCLUSIONS The long-fibre c o m p o u n d was less notch sensitive than the short-fibre compound: the long-fibre c o m p o u n d had the lower unnotched strength in the injection direction, for reasons of fibre orientation, but invariably the higher notched strength. The strength in the injection direction was 6-25% higher for the plates with moulded-in holes than for those with machined holes. The influence on strength of plate thickness and injection speed was smaller for the plates with moulded-in holes, probably owing to an insensitivity of the structure around the hole to these variables. The simplified modelling of notched strength using laminate fracture models was fairly
51
successful in predicting the crack size effect, and the errors can be explained. The modelling of plates with machined holes had, however, limited success, with a fair performance only for the average stress criterion and the damage zone model.
ACKNOWLEDGMENT This work was part of a project supported by ICI Advanced Materials, UK, and the Swedish National Board for Technical D e v e l o p m e n t (STU). The injection moulding was carried out at AB Konstruktionsbakelit, Sweden.
REFERENCES 1. Bowyer, W. H. & Bader, M. G., On the re-inforcement of thermoplastics by imperfectly aligned discontinuous fibres. J. Materials Science, 7 (1972) 1315-21. 2. Bader, M. G. & Bowyer, W. H., The mechanical properties of thermoplastics strengthened by short discontinuous fibres. J. Phys. D: Appl. Phys., 5 (1972) 2215-25. 3. Karger-Kocsis, J. & Friedrich, K., Fracture behavior of injection-molded short and long glass fiber-polyamide 6.6 composites. Composites Science and Technology, 32 (1988) 293-325. 4. Marshall, D. F., Long-fibre reinforced thermoplastics. Materials & Design, 8(2) (March/April 1987) 77-81. 5. Friedrich, K., Microstructural efficiency and fracture toughness of short fiber/thermoplastic matrix composites. Composites Science and Technology, 22 (1985) 43-74. 6. Karger-Kocsis, J. & Friedrich, K., Fatigue crack propagation in short and long fibre-reinforced injectionmoulded PA 6.6 composites. Composites, 19(2) (1988) 293-325. 7. Leach, D. C. & Moore, D. R., Failure and fracture of short glass fibre-reinforced nylon composites. Composites, 16(2) (1985) 113-20. 8. Curtis, P. T., Bader, M. G. & Bailey, J. E., The stiffness and strength of a polyamide thermoplastic reinforced with glass and carbon fibres. J. Materials Science, 13 (1978) 377-90. 9. Sato, N., Kurauchi, T., Sato, S. & Kamigaito, O., Mechanisms of fracture of short glass fibre-reinforced polyamide thermoplastic. J. Materials Science, 19 (1984) 1145-52. 10. Denault, J., Vu-Khanh, T. & Foster, B., Tensile properties of injection molded long fiber thermoplastic composites. Polymer Composites, 10(5) (1989) 313-21. 11. Friedrich, K., Fracture mechanical behavior of short fiber reinforced thermoplastics. Fortschrittberichte der VDI Zeitschrifien, Reihe 18, Nr. 18, VDI Verlag, Diisseldorf, 1984. 12. Vanderschuren, J. A., Prediction of the Strength of Short Fiber Composites With Molded-in Holes. PhD thesis, CCM, Univ. of Delaware, Newark, Delaware, 1982.
52
Staffan Toll, Carl-Gustaf Aronsson
13. Toll, S. & Andersson, P.-O., Microstructure of longand short-fibre reinforced injection moulded polyamide. Polymer Composites (in press). 14. Toll, S. & Borgvaid, U., Fibre length distributions in long-fibre injection moulded polyamide. In Proc. 48th Ann. Tech. Conf., SPE, Connecticut, USA, pp. 1335-8. 1990. 15. Toil, S. & Andersson, P.-O., Microstructural characterisation of injection moulded composites using image analysis. Composites 22 (4) (1991) 298-306. 16. Gulley, T. J. & Summerscales, J., Poisson's ratios in glass fibre reinforced plastics. In The 15th Reinforced Plastics Congress 1986, Nottingham UK. The British Plastics Federation, 1986. 17. Waddoups, M. E., Eisenmann, J. R. & Kaminski, B. E., Macroscopic fracture mechanics of advanced composite materials. J. Composite Materials, 5(4) (1971) 446-54. 18. Whitney, J. M. & Nuismer, R. J., Stress fracture criteria for laminated composites containing stress concentrations. J. Composite Materials, 8(2) (1974) 253-65. 19. Nuismer, R. J. & Whitney, J. M., Uniaxial failure of composite laminates containing stress concentrations. In
32. Wennerstr6m, H., Glemberg, R. & Petersson, 1t., GENFEM-3---a computer program for general finite element analysis, users manual. Publication 79:4, Dept. of Structural Mechanics, Chalmer's university of Technology, G6teborg, 1979.
APPENDIX: FRACTURE MODELS
Note. For consistency with the literature, unnotched tensile strength in the injection direction, denoted Sz in the main text, is here denoted a0, and the co-ordinates 1 and 2 are denoted x and y, respectively. The remote stress, o, the notch lengths, 2a and 2R, and the specimen width, w, are given in Fig. 5. For a CC specimen the linear elastic normal stress distribution, oyy(x), ahead of the crack is exactly z7
Fracture Mechanics of Composites, ASTM STP 593, 20.
21. 22. 23. 24.
25.
26.
27. 28. 29. 30.
American Society for Testing and Materials, Philadelphia, 1975, pp. 117-42. Eriksson, I. & Aronsson, C.-G., Strength of tensile loaded graphite/epoxy laminates containing cracks, open and filled holes. J. Composite Materials, 24(5) (1990) 456-82. B~icklund, J., Fracture of notched composites. Computers and Structures, 13 (1981) 145-154. B/icklund, J. & Aronsson, C.-G., Tensile fracture of laminates with holes. J. Composite Materials, 20(3) (1986) 259-86. Aronsson, C.-G. & B~icklund, J., Tensile fracture of laminates with cracks. J. Composite Materials, 20(3) (1986) 287-307. Hollmann, K., Computational Damage Mechanics for Advanced Composites. PhD thesis, Dept. of Aeronautical Structures and Materials, Royal Institute of Technology, Stockholm, 1989. Awerbuch, J. & Madhukar, M. S., Notched strength of composite laminates: predictions and experiments--a review. J. Reinforced Plastics and Composites 4(1) (1985) 3-159. Hollmann, K., Clarin, P., Aronsson, C.-G. & B~icklund, J., Damage zone fracture analysis of fibrous composites. In Proceedings of Workshop: Composites Design for Space Applications, ES-TEC, Noordwijk, October 1985, pp. 169-74. Lekhnitskii, S. G., Anisotropic Plates. Gordon and Breach, New York, 1968. Tada, H., Paris P. C. & Irwin, G. R., The Stress Analysis of Cracks Handbook. Del Research Corporation, Hellertown, PA, 1973. Konish, H. J. & Whitney, J. M., Approximate stresses in an orthotropic plate containing a circular hole. J. Composite Materials, 9 (1975) 157-66. Paris, P. C. & Sih, G. C., Stress analysis of cracks. In
Fracture Toughness Testing and its Applications, ASTM STP 381, American Society for Testing and Materials, Philadelphia, 1965, pp. 30-85. 31. Clarin, P., Hollmann, K. & Aronsson, C.-G., FRACOM--a computer program for fracture analysis of notched composites. Report No. 85-5, Dept. of Aeronautical Structures and Materials, Royal Institute of Technology, Stockholm, 1985.
x
~y(x) = OY v ~ _ a
x
2 K,~/zm(xZ_aZ )
(5)
where K~ is the mode I stress intensity factor and Y is the finite width correction factor, approximately given by 2s
Y(a/w) = [1 - 0.5(a/w) + 0.37(a/w) 2 - O.044(a/w)3]/(X/1- a/w)
(6)
For a CH specimen of infinite width, a~y(x) is approximately 29 O~ry(X)= ½0{2 + (R)2 + 3 ( R ) 4 - ( K r - 3)
where + K~ = 1 +
- AlE
~a~2)]
AliA22 -
''2
(8) is the orthotropic stress concentration factor z7 and Aij are the orthotropic in-plane stiffnesses of the laminate. Inherent flaw model (IFM) In the IFM, ~7 a crack emanating from the notch is assumed to be present, and the length of this crack, d ~w, is considered to be a material constant. The resulting singular stress field is evaluated by its stress intensity factor, KI. Failure
53
Notched strength of long- and short-fibre reinforced polyamide is assumed to occur at a critical value of Kt K,c = o o V r ~ Ira'4
(9)
where ao is the unnotched tensile strength. For a CC specimen, K~c is thus written as KIC =
a~MyX/~(a
+ d IEM)
distribution is evaluated by the average of ~y(x), eqns (5) and (7), over a characteristic distance, d Asc, ahead of the notch, and fracture is assumed to occur at a critical value of this average dASC
(10)
By equating (9) and (10), the notched tensile strength, a~TM, is obtained as dIEM a i M = ao ~ / a -~ +dlF M (11)
r
(12)
By equating (9) and (12), o~T M is obtained as o~FM=
ao
ONASC=
(18)
ao ~/4/2 d Asc Y a + d AsC
(19)
For a CH specimen, a~ sc is obtained substitution of (7) into (18), with c = R
by
s c = 2Oo(1 - r / ) / [ 2 - r / 2 - r/4
(13)
f(d'FM/R)
O~yy(X)dx = o0
For a CC specimen, the notched tensile strength, a~ sc, is obtained by substitution of (5) into (18), with c = a
For a CH specimen of infinite width, K~c is written as Kic = a l N V M ~ f ( d l F M / R )
fcC+dASC
+
3)( 16 - T/8)]
(20)
Values off(dWM/R) can be found in Ref. 30. where
Point stress criterion (PSC)
rl = R/(R + d ^sc)
In the P S C , 18A9 the linear elastic stress solution is assumed to apply. The stress distribution is evaluated by the stress component ~y(X), eqns (5) and (7), at a characteristic distance, d Psc, ahead of the notch, and fracture is assumed to occur at a critical value of this stress
~y(C + d Psc) = Oo
(14)
where o0 is the unnotched tensile strength and c is half the notch length, either a or R in Fig. 5. For a CC specimen, the notched tensile strength, o ~ c, is derived by eliminating O~y between eqns (5) and (14), with x = c + d Psc and c=a __--
arSC - OOyi l
a
- ( a + ~PSC)
Damage zone criterion (DZC) In the D Z C , 2° a damage zone having a length d is assumed to form at the notch tip. Within the damage zone, (c --- x --- c + d), the stress component Oyy(X) is assumed to be constant and equal to the unnotched tensile strength Oo
oyy(c -< x - c + d) = Oo
(22)
Ahead of the damage zone, (c + d <-x <- w/2), a modified linear elastic stress distribution is assumed
2
(15,
For a C H specimen, a Psc is derived by eliminating ~ y between eqns (7) and (14), with x = c + d esc and c = R a~ sc = 200/[2 + ~2 + 3~4 _ ( g ~ - 3)(5~ 6 -- 7~8)] (16) where
= R/(R + d PsC)
(21)
(17)
Oyy(C + d <- x <- w / 2 ) =
+ Oo - o
,(c + d )
(23) Fracture is assumed to occur at a critical value of the damage zone length d = d Dzc, and the notched tensile strength is obtained from an equilibrium consideration W
( w12
o Dzc ~- t = OodDZCt+
[O~yy(X)+ Oo dc + d DZC
Average stress criterion (ASC) The A S C , 18'19 like the PSC, assumes the linear elastic stress solution to be valid. The stress
- O~y(C+ dDZC)]t dx
(24)
For a CC specimen, the notched tensile strength, o gzc, is obtained by substitution of (5)
54
Staffan Toll, C a r l - G u s t a f A r o n s s o n
into (24), with c = a
orNDZC W
(;¢;
Oo(W/2 - a) __ a2 __ ~/dDZC(2 a + dDZC )
(a + dDZC)[ 2 -- (a + dDZC)] /
(o)
(b)
(c)
Fig. 10. The Damage Zone Model. (a) Damage zone; (b) (25) For a CH specimen, aNDzc is obtained by substitution of (7) into (24), with c = R w - 2R
UN DZC = O"0 w-m2+m3+ml
¢
-R-d
Dzc
) (26)
where
ml--2--I.-(~)2--l.-3(p~)4 -(K,~-3)[5(R)6-7(R) 8]
(27)
m2 = 2q
(28)
q
q3 + ( K ~ - 3)
R2 R4 m3 = 2p
P
p3 + ( K r - 3)
[R~
e_~] (29)
and p = R + d Dzc
(30)
q = w/2
(31)
Damage zone model (DZM) In the D Z M , 21-24'26 a damage zone is represented by an equivalent crack with cohesive stresses acting on the crack surfaces, Fig. I0. Within the damage zone, a linear relationship between cohesive stress, Oyy(X), and crack opening, v(x), is assumed (linear softening), Fig. 10(c); outside the damage zone the material is assumed to be
Assumed equivalent cohesive crack and resulting stress distribution; (c) Linear relationship between stress and crack surface displacement in the damage zone.
linear elastic. The area'below the (~yy/~ c u r v e is equal to the fracture energy, G~, the total energy dissipated by the various fracture mechanisms; and the maximum stress equals the unnotched tensile strength, a0, of the material. Damage initiation is assumed to take place when the normal stress, ayy, reaches the unnotched strength, o0, at the notch tip. At increasing external load the damage zone is assumed to grow along a predetermined path so that the normal stress, ayy, at the tip of the equivalent crack is kept at O'yy= O0 (see Fig. 10(b)). As the damage zone grows, the stresses within the damage zone reduce and those ahead of it redistribute to maintain equilibrium. By using a condensed finite element stiffness matrix of the structure, along with the linear relationship between the cohesive stress and crack opening for each node point along the equivalent crack, the external force may be computed for any chosen location of the crack tip. The entire process of damage initiation and growth may be simulated by calculating the external load and the corresponding displacement for a series of crack lengths. This results in a non-linear record of external load versus displacement. The notched tensile strength, aNDzM, is then taken to be the peak value of the external load. In this work the DZM computations were carried out using the computer code F R A C O M . 3~ The condensed stiffness matrices were computed using the finite-element code GENFEM. 32
Notched strength of long - and short-fibre reinforced polyamide Staffan Toll & Carl-Gustaf Aronsson Department of Aeronautical Structures and Materials, Royal Institute of Technology, S-IO0 44 Stockholm, Sweden (Received 31 May 1991; revised version received 2 August 1991; accepted 3 September 1991)
A test programme has been conducted to evaluate the tensile strength of injection moulded plates with machined and moulded-in notches. The longfibre compound, Verton, is consistently less notch sensitive than the short-fibre compound, Maranyl. Moulded-in holes appear to have several advantages over machined holes: they yield up to 25% higher strength and less sensitivity to plate thickness and injection speed. Five existing laminate fracture models are applied to the machine-notched plates; the average stress criterion and the damage zone model give the most accurate predictions.
Keywords: composite, injection moulding, strength, fracture, holes, cracks
1 INTRODUCTION
products are known to have generally better and less anisotropic mechanical properties than their short-fibre counterparts? This development has increased the interest in injection moulded composites for structural applications, making detailed knowledge about their elastic response and fracture behaviour increasingly important. Injection moulded composites have a more complicated mechanical behaviour than most engineering materials. Since the mechanical properties depend on the local microstructure, they vary throughout a component and are seldom known at all points. Some workers 3,5-7 have approached fracture and fatigue of both long- and short-fibre reinforced polyamide by using fracture mechanics. Microscopic fracture mechanisms have been studied and modelled for short-fibre compounds 8"9 and long-fibre compounds. ~0Sato et al. 9 concluded that fracture initiates as interfacial cracks at the fibre ends, followed by propagation of these cracks along the fibres and, finally, into the matrix. Friedrich xl has demonstrated a fractographic approach, dividing the fracture surface into areas of three different types, each associated with one basic fracture mechanism: fibre fracture, matrix fracture or yielding, and interface debonding; he could correlate the
Injection moulded composites potentially combine good mechanical performance with economical manufacturing and flexible design. The mechanical performance is, however, limited by factors like wetting, fibre dispersion, and fibre length. For example, an increase in fibre length from that of conventional short-fibre injection moulding compounds can improve both strength and toughness.~-3 A few years ago, a major breakthrough was made through ICI's introduction of a family of 'long-fibre' injection moulding grades, Verton. The new development was parallel to that of the Aromatic Polymer Composite (APC), and based on a similar pultrusion impregnation process. The Verton process is able to impregnate high loadings of fibres with improved wetting and dispersion, and without breaking fibres. The granules are cut from a pultruded rod to typically 10 mm lengths, which correspondingly becomes the uniform starting length of the fibres, compared to a distribution between, typically, 0-2 and 0.6 mm for short-fibre grades. The final Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 43
44
Staffan Toll, Carl-Gustaf Aronsson
fracture toughness of the material with the division of fracture surface among these three types, as measured from SEM photographs. Little work has been reported on the notched strength of injection moulded composites, yet fracture initiation at notches is a major cause of structural failure. Notches occur in the form of holes, cut-outs, corners, etc., and are introduced into an injection moulded structure in two basic ways: either by machining of the moulded component, or through the inherent shape of the mould, so-called moulded-in notches. The microstructure in the vicinity of the notch is generally very different in these two cases, t2 and so the mechanical behaviour is also different. Vanderschuren 12 investigated the notched strength of an injection-moulded, glass-fibrereinforced thermoset (phenolic) with moulded-in holes. In this work, the tensile strength of notched injection moulded plates has been tested. The programme involved two Nylon/glass fibre compounds, one long-fibre (LF) c o m p o u n d and one short-fibre (SF) compound; moulded-in circular holes and machined holes and cracks of various sizes; two injection speeds; and two plate thicknesses. A subset of these plates was subjected to microstructural studies, measurements of elastic moduli and unnotched strength, and modelling of notched strength using five existing fracture models.
~Runner
Two grades of glass-fibre filled polyamide 66 were used: a long-fibre (LF) c o m p o u n d , Verton RF7006, with 30 weight% fibre content; and a short-fibre (SF) c o m p o u n d , Maranyl A190, with 33 weight% fibre content. The fibre length prior to injection moulding was 1 0 m m for the long-fibre c o m p o u n d and less than 0 . 5 m m in average for the short-fibre compound. Moulding conditions
The mould, together with its runner system, is shown in Fig. 1. It was a single-cavity rectangular plate mould with an optional insert for central moulded-in holes of different sizes. The plate dimensions were 1 6 0 m m x 4 0 m m , and the thickness either 1.5 or 3 mm. The film gate was
Die
Cavity
Fig. I. The plate mould, here with a i0 m m
moulded-in
hole. 1 mm X 25 m m and the cross-sectional area of the runner was approximately 25 mm:. Six different plates were moulded. These, plates A to F, are presented in Table 1: two different materials and two different plate thicknesses were used. In addition, the LF (Verton) plates were processed at two different injection speeds. Each of these plates was moulded both plain and with a central mouldedin hole of three different sizes (5, 10, and 15 m m diameter). The injection moulding was carried out on a Mannesmann D e m a g D60-182NC II injection moulding machine under the conditions listed in Table 2.
Table 1. Moulded plates (160 X 40 mm)
Plate 2 MATERIALS
~
A B C D E F
Thickness Compound (mm) 3-0 3.0 3-0 1-5 1-5 1-5
LF LF SF LF LF SF
Inj. Speed High Low High Low
Table 2. Moulding Conditions
Temperatures (°C) Zone 1 and 2 Zone 3 Die Mould Injection pressure (Bar, hydraulic) Holding pressure (Bar, hydraulic) Coohng-offperiod (s) Injection speed (cm3/s) High Low Holding-pressure time (s)
LF
SF
295 295 300 95 110 65 10
275 280 285 110 110 50 10 39
16 8 8
6
Notched strength of long- and short-fibre reinforced polyamide Microstructure
°l C O
The microstructural investigation was confined to the 3 mm thick plates. The fibre orientation in the plain mouldings was investigated in recent work, ~3 and that in the mouldings with moulded-in holes was visually inspected here from micrographs taken in the 1, 2 plane. The fibre length distributions were studied in Ref. 14. The observed internal structure of the plates is schematically shown in Fig. 2: there is a surface layer having a predominant orientation parallel with the injection direction; and a core layer having a high degree of fibre orientation transverse to the injection direction. This structure can be said to be typical of injection moulded polyamide 66 composites. The fibre orientation patterns in the surface and core layers are described in Fig. 3: the angles of principal fibre orientation at different locations are indicated by the dashes. The exact meaning of principal orientation is stated in Ref. 15. In the case of the moulded-in hole, there is a knitline extending from the hole and downstream. It is seen that near any edge the principal orientation through the thickness is parallel to the edge, and near a knitline it is parallel to the knitline. This means extra fibre alignment (a) along the outer edges, (b) tangentially around a moulded-in hole, and (c) along a knitline. The average fibre length was higher in the core than in the surface layer, which is usually the case. 3
Differences between the plates. The structure of the surface layer is similar in all the plates A, B, and C (Table 1), whereas the core structure is considerably more oriented in plates A and B (LF) than in plate C (SF). The thickness of the core is approximately 1/5 of the total plate thickness in all the plates A, B, and C. Consequently, the overall degree of fibre orientation, and thus the anisotropy, is highest in ] ]
Predominant longitudinal fibre orientation Predominant transverse fibre orientation
Surface layer ~'.v.v.v::.v.'.v:.v.':::.v.v.v.'.v.v.v.':.v.v.v:.v.v.'.f~ n
,nlectian d ir e c ~// +
27
~.+[.v ,.,........,,.....v .............. .............. j~
Hole edge region
Edge region
Fig. 2. Layer structure of injection moulded plate.
C
Surface
Core
'"',',
iY---
45 Surface
Core
I% I I " '+ iii
IiI
I I i + iI
u nlnlll I nI I I I I [
,~--
I
ll'n I I I I ,nl I I I I IIi I i i i i i i i Ii ii
l l f - ~ - - ~.- '~_-'--_'Z_ ....
(o)
,
Ii,",";'" , ,'P' Ill,
2 L;
I iI I ~
,,11 al,l,'l
(b)
Fig. 3. Schematic of fibre orientation pattern. (a) Plain moulding; (b) Mouldingwith moulded-inhole. plate C. The edge region indicated in Fig. 2 is approximately 3 mm wide in plates A (LF, high injection speed) and C (SF), and about 4.5 mm in plate B (LF, low injection speed). The volume average fibre length in the central part of the plain mouldings was approximately 1-2 mm for the LF mouldings and 0-27mm for the SF mouldings.
3 EXPERIMENTAL Machine notched plates for the testing of notched fracture loads along with cut-out specimens for the testing of unnotched strength and elastic moduli were machined from the plain mouldings. Prior to testing, all specimens were conditioned to saturation in air at 50% relative humidity and 80oc for 20 days. All tensile tests were carried out on an Instron 6025 universal testing machine, at approximately 23°C (room temperature) and 40% relative humidity. Elastic moduli and unnotched strengths For the purpose of fracture modelling, engineering moduli and unnotched tensile strengths were measured on the 3 m m thick plates. The measurements were made in the central part of the plates. Each reported result is the average of results for between two and five specimens. Orthotropy was assumed. The elastic moduli Eu, E22, and E12 were measured on 14mm wide rectangular cut-out specimens. The displacement rate was i mm per minute. The moduli were defined as the secant moduli between 0.02% and 0.2% normal strain, i.e. (o,),,=0.02 e + = (Ej)E;=o.o -- (E;),,=o 2
(1)
Staffan Toll, Carl-Gustaf Aronsson
46
Table 3. Elastic moduli and unnotched strengths Plate A B C
Ell (GPa)
E22 (GPa)
EI2 (GPa)
VI2
4-8 4.3 3.2
6.1 6-1 6.6
-19 -19 -18
0.25 0-23 0-17
GI 2 (GPa)
Sl (MPa)
S2 (MPa)
2.1 2.0 1.8
86 90 73
118 121 126
ff
o
2
2
o
2e
2R
223
(i,j= 1, 2). obtained as
The
Poisson's
ratio,
v~2, was o
o"
(o)
Ell
v,2 -
........
E2,
(2)
(b)
Fig. 5. Notch types and loading conditions. (a) Centre-Hole (CH), moulded-in or drilled; (b) Centre-Crack (CC).
and the shear modulus, G,z, was estimated according to Huber's approximation: 16
WrE11E2z G,2 ~ 2(1
+
v12~/E22/E,,)
(3)
The results are given in Table 3. Unnotched tensile strengths were measured on small hour-glass-shaped specimens, illustrated in Fig. 4. These were cut out from the centre of the plate, parallel and perpendicular to the injection direction. The stress concentration factor, ce, at the net-section of the specimen was calculated by a two-dimensional finite element model, from the previously measured in-plane elastic moduli of each plate. The unnotched tensile strength, S, was approximated as the average of the ultimate mean stress, Pu/ht, and the calculated maximum stress, cePu/ht, i.e. S
Pu c r + l 2
ht
(4)
where P, is the ultimate load and ht is the net cross-sectional area of the specimen, Fig. 4. The displacement rate was chosen so that the strain rate in the gauge region was approximately 1% per minute. The experimental unnotched strengths perpendicular ($1) and parallel ($2) to the injection direction are given in Table 3. Note that [mm]
O
---
t=3
the longitudinal strength, $2, and modulus, E22 , were highest for the SF plate, whereas the transverse strength, $1, and modulus, El,, were highest for the LF plate; thus the SF plate possessed a higher degree of anisotropy.
Measurement of notched strength The plates were tested in tension using three notch types: 1. moulded-in centre holes (MCH) 2. drilled centre holes (DCH) 3. cut centre cracks (CC) The notch geometries and loading conditions are illustrated in Fig. 5. Both holes and cracks were made to three different sizes: 2R, 2a = 5, 10, and 15 mm. The gauge length was 100 mm and the displacement rate 1 mm per minute.
4 EXPERIMENTAL RESULTS All the tested plates exhibited net-section failure through the notch, visibly initiating at the notch tip. The experimental notched strengths, c~Nxp, are given in Table 4 for the thick plates (3 mm thick) and Table 5 for the thin plates (1.5 mm thick). These results are illustrated in Fig. 6, where ~Nxv is plotted versus the notch length-towidth ratio, 2a/w or 2R/w, for all the different materials and notch types.
Effects of fibre length and injection speed. -
-
~
10
Fig. 4. Specimen for the measurement of unnotched strength. The shaded areas are those covered by the grips.
Figure 6 shows that the LF compound had a higher notched strength than the SF compound in all cases. It shows a clear effect of injection speed only for the thin plates with machined notches;
Notched strength of long- and short-fibre reinforced polyamide Table 4. E x p e r i m e n t a l n o t c h e d s t r e n g t h s ¢7~v xp for the thick plates Plate 2a/w,
MCH
DCH
CC
47
difference in strength between MCH and DCH plates was 15-25% at 3 m m thickness, and 6-19% at 1.5 mm thickness (Tables 4 and 5).
2R/w
A
B
C
0.125 0.250 0.375 0.125 0.250 0.375 0.125 0-250 0.375
o~ p (MPa)
Specimens
o~vXp (MPa)
Specimens
o~,Xp (MPa)
Specimens
944-3 734-3 62 4-2 974-2 75 4- 3 634-1 83 4-1 69 4- 2 59 4- 2
10 10 10 10 10 10 10 10 10
754-3 614-2 53+1 784-2 65 4- 1 534-1 67+1 58 4- 2 48 4- 1
7 7 7 7 7 6 6 6 6
70-t-2 564-2 46+1 704-1 57 4- 3 464-1 63+2 51 4- 2 43 4- 1
7 7 7 6 6 6 6 6 6
4- ---standard deviation.
5 MODELLING
An attempt was made to model the notched strength for a subset of the plates by using five existing fracture models: the inherent flaw model (IFM), Waddoups et al. ;~7 the point and average stress criteria (PSC and ASC, respectively), Whitney & Nuismer; ~8'~9 the damage zone criterion (DZC), Eriksson & Aronsson; 2° and the damage zone model (DZM), Aronsson and B~cklund.
the higher injection speed resulted in a higher strength.
Thickness effects. It can be seen from Fig. 7 that the plates with moulded-in holes exhibited no significant thickness effect; and that the plates with machined holes and cracks had slightly higher strengths for the thinner plates; but the effect was small for the LF compound, low injection speed. The thickness effects for machined holes and machined cracks were very similar to each other. Notch effects. As illustrated by Fig. 8, the plates with moulded-in holes were invariably stronger than those with machined holes, which in turn were stronger than those with cracks of the same notch length. This relative behaviour was obtained for all six plates (A-F). The notch effect was more pronounced for the thick plates than for the thin ones; for example, the Table 5. E x p e r i m e n t a l n o t c h e d s t r e n g t h s a ~ for the t h i n plates Plate
D
E
F
2a/w, 2R/w
0.125 0.250 0.375 0.125 0.250 0.375 0.125 0.250 0.375
MCH
DCH
CC exp
ON
~N~p (MPa)
Specimens
o~ p (MPa)
Specimens
(MPa)
Specimens
93 + 2 74 + 3 61 + 2 91 + 3 75 + 2 62 4- 1 83 4- 1 70 4- 2 58 4- 1
10 10 10 10 10 10 10 10 10
83 + 3 70 + 2 56 + 1 77+2 63 4- 2 52 4- 2 73 4- 1 60 + 2 51 4- 1
7 7 7 7 6 6 6 6 6
79 4- 3 61 + 2 48 4- 2 70+2 55 + 1 45 + 1 68 + 3 55 4- 2 45 4- 1
7 7 7 6 6 7 6 6 6
+ -----standard deviation.
21-23
All of these are two-parameter models for the prediction of the notched tensile strengths of laminated composites. The DZM numerically computes the complete load/displacement relation to failure, by using a finite element model of the structure, and predicts failure when the ultimate external load is reached. The other models predict failure when certain criteria are fulfilled by the stress distribution ahead of the notch, resulting in closed form expressions. The IFM, PSC, and ASC are based on linear elastic stress distributions, whereas the DZC and DZM are based on stress distributions resulting from the development of an inelastic damage zone. The performance of these models when applied to laminated composites has, except for DZC, become fairly well established. 22-25 The analyses are described in the Appendix. The modelling was performed for the 3 mm thick plain mouldings with machined holes and cracks. The in-plane property variations were thus comparatively small; and the properties measured in the central region (Table 3) were assumed to be valid throughout the plates. The fracture parameters (see Appendix) for each fracture model and each plate were calculated from the experimental notched strength for the short crack, a = 2 - 5 m m , and the unnotched strength, $2. These parameters are given in Table 6. The notched strengths were then predicted according to the fracture models.
Comment on finite width correction. The DZM and the DZC are so constructed that the finite width of the specimen is automatically accounted for. In the IFM, ASC, and PSC a finite width can only be accounted for by correcting the assumed
Staffan Toll, Carl-Gustaf Aronsson
48 • v ÷
LF, fast injection LF, slow injection SF Thick plates
Thin plates
100
d
75
50
MCH
25
75
N
50
25
~
100
50
25
0
0.125
0.250
0
0.125
0.250
0.375
Notch length-to-width ratio 2a/w or 2R/w
Fig.
6. E x p e r i m e n t a l n o t c h e d s t r e n g t h vs. n o t c h l e n g t h - t o - w i d t h r a t i o
stress distribution. For a crack, the assumed stress distribution is singular, and finite width correction factors can be used; but for a hole there is no such simple solution. Therefore the predictions made for holes from these three models assume infinitely wide specimens (w = ~). This may cause an additional error, especially for the largest holes, which should be kept in mind when interpreting the results.
Results. The predicted notched strengths are presented in Table 7. All the models predicted the crack size effect rather well, but the transition to holes seems unreliable for the D Z C , IFM, and PSC. The A S C and D Z M performed well on the
2a/w o r 2R/w.
whole, and in nearly all cases the best correlation with experimental results was obtained with the ASC. Practically all predictions were underestimates.
6 DISCUSSION
Failure modes. All the plates in the reported test programme failed along the net-section through the notch, as depicted in Fig. 9(a). However, it should be mentioned that a few plates with moulded-in holes were tested without being conditioned, i.e., with a lower moisture content in the matrix (unreported results). These
Notched strength of long- and short-fibre reinforced polyamide
49
v Thin plates • Thick plates
MCH
LF
LF
fast inj.
slow inj.
SF
25
~
75
~
50
0.125
0.250
O.125
0.250
0.125
0.250
Notch length-to-widthratio 2ahv or 2R/w
Fig. 7. Effects of plate thickness on notched strength.
~,
• MCH O DCH
100
~
75
¢..
5O
0
0.125
0.250
0.375
Notch length-to-width ratio 2a/w or 2R/w
Fig. 8. Effect of notch type on the notched strength of Plate A (LF, 3 mm thick, high injection speed). All the other plates show a similar behaviour.
Table 6. Fracture parameters Plate
$2 (MPa)
d wM (mm)
d Psc (mm)
d Asc (mm)
d Dzc (mm)
G~ (kJ/m z)
A B C
118 121 126
1.39 1.29 0.85
0.60 0.56 0.40
2-71 2.52 1-72
0.67 0.63 0.42
12 12 8
plates failed along a path circumscribing the hole, as shown in Fig. 9(b), leaving the hole edge intact. A possible explanation is as follows. In the unconditioned case, failure initiates either somewhere in the tangentially-oriented hole edge region or between the hole edge and the outer edge (see Fig. 3(b)), owing to the microstructure induced by the hole insert; in the conditioned case, the moisture softens and toughens the matrix, by the action of water as a plasticizer, to such a degree that the hole edge becomes critical instead. This phenomenon of competing failure modes was also encountered by Vanderschuren 12 in injection moulded short-fibre reinforced phenolic (thermoset).
Fibre length effect. The LF plates had a significantly lower degree of anisotropy than the SF plates: the longitudinal unnotched strength, $2, and Young's modulus, E2, were actually highest in the SF plates, whereas the transverse strength and modulus, $1 and E~, were highest in the LF plates. The reason for this is that the core, which dominates the transverse properties, has a higher fibre orientation and fibre content in
50
Staffan Toll, Carl-Gustaf Aronsson Table 7. Predicted notched strengths for the 3 mm thick plates
2a/w, 2R/w
Plate
CC-notches 0-125 A B C 0.250 A B C 0.375 A B C DCH-notches 0.125 A B C 0.250 A B C 0.375 A B C
IFM ON
PSC ON
AS(" ON
DZC ON
DZM ON
AIFM
APSC
AASC
ADZ(:
ADZM
(MPa)
(MPa)
(MPa)
(MPa)
(MPa)
(%)
(%)
(%)
(%)
(%)
70.0 70.0 63-0 53.8 52.9 46.4 42-7 42.5 36.8
70-0 70.0 63.0 53.3 53.1 47.6 44.7 44-5 39.6
70.0 70.0 63.0 54.5 54-3 48.3 46.1 45.9 40-4
70.0 70.0 63.0 51.8 51.8 45.9 41.2 41.4 36.2
70.0 70.0 63.0 54.8 55.0 48.1 43.8 44-1 38-1
0.0 0.0 0.0 -5-2 -7.2 -9.0 -7.2 -7-6 -14.4
0-0 0-0 0-0 -4.8 -6.8 -6-7 -2-8 -3-3 -7.9
0-0 0.0 0.0 -2-7 -4.7 -5.3 0.0 0-0 -6.0
0.0 0.0 0.0 -7.5 -9-1 -10-0 -10.4 -10.6 -15.8
0.0 0.0 0.0 -2.0 -3.5 -5.7 -4-7 -4.3 -11.4
62.1 63.6 62"1 49.2 48.4 48-1 38.1 37.9 37.0
60.5 60.6 57.1 50-0 50.7 48.5 46.3 47.1 45-5
71.5 71.9 67.0 59"0 59.4 55.8 53.3 53-8 51.0
61.7 62-0 57.3 49.9 50.6 48.0 44-5 45.4 43.6
70.1 70.8 63.9 53-2 53.8 48.9 50-6 51.1 47-8
-17.2 -18.5 -7.3 -19-3 -25.5 -17.1 -28-1 -28.5 -22-9
-19.3 -22.3 -14.8 -18.0 -22.0 -16.4 - 12-6 -11-1 -5.2
-4.7 -7.8 0.0 -3.3 -8"6 -3.8 0.0 -1-5 6-3
-17.7 -20.5 -14.5 -18-2 -22-2 -17-2 - 16.0 -14.3 -9-2
-6-5 -9-2 -4.6 -12.8 - 17-2 -15.7 -4.5 -3-6 -0.4
A--deviation from ONe~P.
the LF plates than that in the SF plates, whereas the surface layers, which dominate the longitudinal properties, hardly differ at all. Nevertheless, the longitudinal notched strength was invariably higher for the LF plates; in other words, the LF compound exhibits a lower notch sensitivity or, with yet another term, a higher toughness than the SF compound. This higher toughness is manifested by larger values of the characteristic distances and G~ in the fracture modelling of the LF compound, Table 6. Apart from the longer fibres, other factors such as better wetting and fibre dispersion of the Verton compound may contribute to its higher toughness.
Thickness effects. The higher strength of the thinner plates can be explained by a higher ratio of surface layer thickness to core thickness, making the through-thickness degree of fibre alignment in the loading direction higher for the thin plates. It is natural that the plates with
~5
t-
(o)
(b) Fig. 9. Failure modes.
machined cracks and holes show the same thickness effect since the mouldings, and therefore the microstructure, are the same. It is also natural that the plates with moulded-in holes show less thickness effect, since the governing microstructure is probably that of the hole edge region, which has little thickness dependence since it contains no core (see Fig. 2).
Notch effects. As long as net-section failure prevails (Fig. 9(a)), it is perfectly reasonable that moulded-in holes are stronger than machined ones, because of their higher fibre alignment along the stress trajectories at the hole edge. But for other failure modes, e.g. that in Fig. 9(b), this is not obvious. Plates with holes were stronger than plates with cracks. This is not always the case for composites, and most of the models predicted the opposite behaviour for small notches. Modelling. The varying microstructure of the plain plates, especially towards the edges, makes the interpretation of the modelling results difficult. Since the edge regions have a high fibre orientation parallel to the edges (see Section 2) they stiffen the edges longitudinally, and thereby relieve the stress field in front of the notch tips. This effect should become increasingly pronounced at decreasing distance between the edge and the notch tip, thus diminishing the otherwise
Notched strength of long- and short-fibre reinforced polyamide
expected notch length effect. Since the models do not account for this, they should overestimate the notch length effect. This appears to be true for the CC predictions: the model parameters were determined for the shortest crack, and they all underestimated the notched strengths at increasing crack lengths. The interpretation of the D C H predictions is less certain: first, since the material toughness parameters were determined from a CC specimen, the notch geometry dependence of the models becomes influential; second, an infinite plate width was assumed for IFM, PSC, and ASC. The overall good performance of the A S C may be a coincidence; if the stiffened edges had been accounted for, the A S C probably had performed worse and the other models better. There are at least two important limitations in the present fracture modelling that may contribute to errors: 1. the spatial variation of material properties due to the variation of fibre orientation, fibre content, etc., is ignored; 2. linear material behaviour is assumed. Taking varying and nonlinear material properties into account can probably provide a basis for a more accurate failure analysis of injection moulded composites. Of the models tested here, the D Z M can most easily, and without modification accommodate varying material properties. 26 However, the local material properties may be difficult to determine accurately.
7 CONCLUSIONS The long-fibre c o m p o u n d was less notch sensitive than the short-fibre compound: the long-fibre c o m p o u n d had the lower unnotched strength in the injection direction, for reasons of fibre orientation, but invariably the higher notched strength. The strength in the injection direction was 6-25% higher for the plates with moulded-in holes than for those with machined holes. The influence on strength of plate thickness and injection speed was smaller for the plates with moulded-in holes, probably owing to an insensitivity of the structure around the hole to these variables. The simplified modelling of notched strength using laminate fracture models was fairly
51
successful in predicting the crack size effect, and the errors can be explained. The modelling of plates with machined holes had, however, limited success, with a fair performance only for the average stress criterion and the damage zone model.
ACKNOWLEDGMENT This work was part of a project supported by ICI Advanced Materials, UK, and the Swedish National Board for Technical D e v e l o p m e n t (STU). The injection moulding was carried out at AB Konstruktionsbakelit, Sweden.
REFERENCES 1. Bowyer, W. H. & Bader, M. G., On the re-inforcement of thermoplastics by imperfectly aligned discontinuous fibres. J. Materials Science, 7 (1972) 1315-21. 2. Bader, M. G. & Bowyer, W. H., The mechanical properties of thermoplastics strengthened by short discontinuous fibres. J. Phys. D: Appl. Phys., 5 (1972) 2215-25. 3. Karger-Kocsis, J. & Friedrich, K., Fracture behavior of injection-molded short and long glass fiber-polyamide 6.6 composites. Composites Science and Technology, 32 (1988) 293-325. 4. Marshall, D. F., Long-fibre reinforced thermoplastics. Materials & Design, 8(2) (March/April 1987) 77-81. 5. Friedrich, K., Microstructural efficiency and fracture toughness of short fiber/thermoplastic matrix composites. Composites Science and Technology, 22 (1985) 43-74. 6. Karger-Kocsis, J. & Friedrich, K., Fatigue crack propagation in short and long fibre-reinforced injectionmoulded PA 6.6 composites. Composites, 19(2) (1988) 293-325. 7. Leach, D. C. & Moore, D. R., Failure and fracture of short glass fibre-reinforced nylon composites. Composites, 16(2) (1985) 113-20. 8. Curtis, P. T., Bader, M. G. & Bailey, J. E., The stiffness and strength of a polyamide thermoplastic reinforced with glass and carbon fibres. J. Materials Science, 13 (1978) 377-90. 9. Sato, N., Kurauchi, T., Sato, S. & Kamigaito, O., Mechanisms of fracture of short glass fibre-reinforced polyamide thermoplastic. J. Materials Science, 19 (1984) 1145-52. 10. Denault, J., Vu-Khanh, T. & Foster, B., Tensile properties of injection molded long fiber thermoplastic composites. Polymer Composites, 10(5) (1989) 313-21. 11. Friedrich, K., Fracture mechanical behavior of short fiber reinforced thermoplastics. Fortschrittberichte der VDI Zeitschrifien, Reihe 18, Nr. 18, VDI Verlag, Diisseldorf, 1984. 12. Vanderschuren, J. A., Prediction of the Strength of Short Fiber Composites With Molded-in Holes. PhD thesis, CCM, Univ. of Delaware, Newark, Delaware, 1982.
52
Staffan Toll, Carl-Gustaf Aronsson
13. Toll, S. & Andersson, P.-O., Microstructure of longand short-fibre reinforced injection moulded polyamide. Polymer Composites (in press). 14. Toll, S. & Borgvaid, U., Fibre length distributions in long-fibre injection moulded polyamide. In Proc. 48th Ann. Tech. Conf., SPE, Connecticut, USA, pp. 1335-8. 1990. 15. Toil, S. & Andersson, P.-O., Microstructural characterisation of injection moulded composites using image analysis. Composites 22 (4) (1991) 298-306. 16. Gulley, T. J. & Summerscales, J., Poisson's ratios in glass fibre reinforced plastics. In The 15th Reinforced Plastics Congress 1986, Nottingham UK. The British Plastics Federation, 1986. 17. Waddoups, M. E., Eisenmann, J. R. & Kaminski, B. E., Macroscopic fracture mechanics of advanced composite materials. J. Composite Materials, 5(4) (1971) 446-54. 18. Whitney, J. M. & Nuismer, R. J., Stress fracture criteria for laminated composites containing stress concentrations. J. Composite Materials, 8(2) (1974) 253-65. 19. Nuismer, R. J. & Whitney, J. M., Uniaxial failure of composite laminates containing stress concentrations. In
32. Wennerstr6m, H., Glemberg, R. & Petersson, 1t., GENFEM-3---a computer program for general finite element analysis, users manual. Publication 79:4, Dept. of Structural Mechanics, Chalmer's university of Technology, G6teborg, 1979.
APPENDIX: FRACTURE MODELS
Note. For consistency with the literature, unnotched tensile strength in the injection direction, denoted Sz in the main text, is here denoted a0, and the co-ordinates 1 and 2 are denoted x and y, respectively. The remote stress, o, the notch lengths, 2a and 2R, and the specimen width, w, are given in Fig. 5. For a CC specimen the linear elastic normal stress distribution, oyy(x), ahead of the crack is exactly z7
Fracture Mechanics of Composites, ASTM STP 593, 20.
21. 22. 23. 24.
25.
26.
27. 28. 29. 30.
American Society for Testing and Materials, Philadelphia, 1975, pp. 117-42. Eriksson, I. & Aronsson, C.-G., Strength of tensile loaded graphite/epoxy laminates containing cracks, open and filled holes. J. Composite Materials, 24(5) (1990) 456-82. B~icklund, J., Fracture of notched composites. Computers and Structures, 13 (1981) 145-154. B/icklund, J. & Aronsson, C.-G., Tensile fracture of laminates with holes. J. Composite Materials, 20(3) (1986) 259-86. Aronsson, C.-G. & B~icklund, J., Tensile fracture of laminates with cracks. J. Composite Materials, 20(3) (1986) 287-307. Hollmann, K., Computational Damage Mechanics for Advanced Composites. PhD thesis, Dept. of Aeronautical Structures and Materials, Royal Institute of Technology, Stockholm, 1989. Awerbuch, J. & Madhukar, M. S., Notched strength of composite laminates: predictions and experiments--a review. J. Reinforced Plastics and Composites 4(1) (1985) 3-159. Hollmann, K., Clarin, P., Aronsson, C.-G. & B~icklund, J., Damage zone fracture analysis of fibrous composites. In Proceedings of Workshop: Composites Design for Space Applications, ES-TEC, Noordwijk, October 1985, pp. 169-74. Lekhnitskii, S. G., Anisotropic Plates. Gordon and Breach, New York, 1968. Tada, H., Paris P. C. & Irwin, G. R., The Stress Analysis of Cracks Handbook. Del Research Corporation, Hellertown, PA, 1973. Konish, H. J. & Whitney, J. M., Approximate stresses in an orthotropic plate containing a circular hole. J. Composite Materials, 9 (1975) 157-66. Paris, P. C. & Sih, G. C., Stress analysis of cracks. In
Fracture Toughness Testing and its Applications, ASTM STP 381, American Society for Testing and Materials, Philadelphia, 1965, pp. 30-85. 31. Clarin, P., Hollmann, K. & Aronsson, C.-G., FRACOM--a computer program for fracture analysis of notched composites. Report No. 85-5, Dept. of Aeronautical Structures and Materials, Royal Institute of Technology, Stockholm, 1985.
x
~y(x) = OY v ~ _ a
x
2 K,~/zm(xZ_aZ )
(5)
where K~ is the mode I stress intensity factor and Y is the finite width correction factor, approximately given by 2s
Y(a/w) = [1 - 0.5(a/w) + 0.37(a/w) 2 - O.044(a/w)3]/(X/1- a/w)
(6)
For a CH specimen of infinite width, a~y(x) is approximately 29 O~ry(X)= ½0{2 + (R)2 + 3 ( R ) 4 - ( K r - 3)
where + K~ = 1 +
- AlE
~a~2)]
AliA22 -
''2
(8) is the orthotropic stress concentration factor z7 and Aij are the orthotropic in-plane stiffnesses of the laminate. Inherent flaw model (IFM) In the IFM, ~7 a crack emanating from the notch is assumed to be present, and the length of this crack, d ~w, is considered to be a material constant. The resulting singular stress field is evaluated by its stress intensity factor, KI. Failure
53
Notched strength of long- and short-fibre reinforced polyamide is assumed to occur at a critical value of Kt K,c = o o V r ~ Ira'4
(9)
where ao is the unnotched tensile strength. For a CC specimen, K~c is thus written as KIC =
a~MyX/~(a
+ d IEM)
distribution is evaluated by the average of ~y(x), eqns (5) and (7), over a characteristic distance, d Asc, ahead of the notch, and fracture is assumed to occur at a critical value of this average dASC
(10)
By equating (9) and (10), the notched tensile strength, a~TM, is obtained as dIEM a i M = ao ~ / a -~ +dlF M (11)
r
(12)
By equating (9) and (12), o~T M is obtained as o~FM=
ao
ONASC=
(18)
ao ~/4/2 d Asc Y a + d AsC
(19)
For a CH specimen, a~ sc is obtained substitution of (7) into (18), with c = R
by
s c = 2Oo(1 - r / ) / [ 2 - r / 2 - r/4
(13)
f(d'FM/R)
O~yy(X)dx = o0
For a CC specimen, the notched tensile strength, a~ sc, is obtained by substitution of (5) into (18), with c = a
For a CH specimen of infinite width, K~c is written as Kic = a l N V M ~ f ( d l F M / R )
fcC+dASC
+
3)( 16 - T/8)]
(20)
Values off(dWM/R) can be found in Ref. 30. where
Point stress criterion (PSC)
rl = R/(R + d ^sc)
In the P S C , 18A9 the linear elastic stress solution is assumed to apply. The stress distribution is evaluated by the stress component ~y(X), eqns (5) and (7), at a characteristic distance, d Psc, ahead of the notch, and fracture is assumed to occur at a critical value of this stress
~y(C + d Psc) = Oo
(14)
where o0 is the unnotched tensile strength and c is half the notch length, either a or R in Fig. 5. For a CC specimen, the notched tensile strength, o ~ c, is derived by eliminating O~y between eqns (5) and (14), with x = c + d Psc and c=a __--
arSC - OOyi l
a
- ( a + ~PSC)
Damage zone criterion (DZC) In the D Z C , 2° a damage zone having a length d is assumed to form at the notch tip. Within the damage zone, (c --- x --- c + d), the stress component Oyy(X) is assumed to be constant and equal to the unnotched tensile strength Oo
oyy(c -< x - c + d) = Oo
(22)
Ahead of the damage zone, (c + d <-x <- w/2), a modified linear elastic stress distribution is assumed
2
(15,
For a C H specimen, a Psc is derived by eliminating ~ y between eqns (7) and (14), with x = c + d esc and c = R a~ sc = 200/[2 + ~2 + 3~4 _ ( g ~ - 3)(5~ 6 -- 7~8)] (16) where
= R/(R + d PsC)
(21)
(17)
Oyy(C + d <- x <- w / 2 ) =
+ Oo - o
,(c + d )
(23) Fracture is assumed to occur at a critical value of the damage zone length d = d Dzc, and the notched tensile strength is obtained from an equilibrium consideration W
( w12
o Dzc ~- t = OodDZCt+
[O~yy(X)+ Oo dc + d DZC
Average stress criterion (ASC) The A S C , 18'19 like the PSC, assumes the linear elastic stress solution to be valid. The stress
- O~y(C+ dDZC)]t dx
(24)
For a CC specimen, the notched tensile strength, o gzc, is obtained by substitution of (5)
54
Staffan Toll, C a r l - G u s t a f A r o n s s o n
into (24), with c = a
orNDZC W
(;¢;
Oo(W/2 - a) __ a2 __ ~/dDZC(2 a + dDZC )
(a + dDZC)[ 2 -- (a + dDZC)] /
(o)
(b)
(c)
Fig. 10. The Damage Zone Model. (a) Damage zone; (b) (25) For a CH specimen, aNDzc is obtained by substitution of (7) into (24), with c = R w - 2R
UN DZC = O"0 w-m2+m3+ml
¢
-R-d
Dzc
) (26)
where
ml--2--I.-(~)2--l.-3(p~)4 -(K,~-3)[5(R)6-7(R) 8]
(27)
m2 = 2q
(28)
q
q3 + ( K ~ - 3)
R2 R4 m3 = 2p
P
p3 + ( K r - 3)
[R~
e_~] (29)
and p = R + d Dzc
(30)
q = w/2
(31)
Damage zone model (DZM) In the D Z M , 21-24'26 a damage zone is represented by an equivalent crack with cohesive stresses acting on the crack surfaces, Fig. I0. Within the damage zone, a linear relationship between cohesive stress, Oyy(X), and crack opening, v(x), is assumed (linear softening), Fig. 10(c); outside the damage zone the material is assumed to be
Assumed equivalent cohesive crack and resulting stress distribution; (c) Linear relationship between stress and crack surface displacement in the damage zone.
linear elastic. The area'below the (~yy/~ c u r v e is equal to the fracture energy, G~, the total energy dissipated by the various fracture mechanisms; and the maximum stress equals the unnotched tensile strength, a0, of the material. Damage initiation is assumed to take place when the normal stress, ayy, reaches the unnotched strength, o0, at the notch tip. At increasing external load the damage zone is assumed to grow along a predetermined path so that the normal stress, ayy, at the tip of the equivalent crack is kept at O'yy= O0 (see Fig. 10(b)). As the damage zone grows, the stresses within the damage zone reduce and those ahead of it redistribute to maintain equilibrium. By using a condensed finite element stiffness matrix of the structure, along with the linear relationship between the cohesive stress and crack opening for each node point along the equivalent crack, the external force may be computed for any chosen location of the crack tip. The entire process of damage initiation and growth may be simulated by calculating the external load and the corresponding displacement for a series of crack lengths. This results in a non-linear record of external load versus displacement. The notched tensile strength, aNDzM, is then taken to be the peak value of the external load. In this work the DZM computations were carried out using the computer code F R A C O M . 3~ The condensed stiffness matrices were computed using the finite-element code GENFEM. 32