L U L E A I U N I V E R S I T Y ^ I k ^ ,
OF T E C H N O L O G Y
2002:12
Ji s V ^ _ > / V _ > J L V / X V . / J L J L - / -JL X X X J O J L O
Toughness of short fiber composites
- an approach based on crack-bridging
Patrik Fernberg
Department of Applied Physics and Mechanical Engineering Division of Polymer Engineering
Doctoral thesis 2002:12
Toughness of short fiber composites -
An approach based on crack-bridging
by
Patrik Fernberg
Division of Polymer Engineering
Department of Applied Physics and Mechanical Engineering Luleå University of Technology
971 87 Luleå, Sweden
Akademisk avhandling
Som med vederbörligt tillstånd av Tekniska Fakultetsnämnden vid Luleå Tekniska Universitet för avläggande av teknologie doktorsexamen kommer att offentligt försvaras i sal E246 vid Luleå Tekniska Universitet, tisdagen den 21 :a maj, kl 13.00. Fakultetsopponent är Senior Lecturer Ton Peijs, Dept. of Materials, Queen Mary and Westfield College, University of London. Avhandlingen försvaras på engelska.
LULEA U N I V E R S I T Y
OF TECHNOLOGY
Toughness of short fiber composites - An approach based on crack-bridging
Patrik Fernberg Division of Polymer Engineering
Department of Applied Physics and Mechanical Engineering Luleå University of Technology, SE-971 87, Luleå, Sweden
Luleå 2002
Abstract
The presented work considers how to properly characterize fracture properties of short fiber composites (SFC). Associated with fracture of SFC is the creation of a comparably widespread fracture process zone. This zone develops since a number of inelastic failure mechanisms (e.g. debonding, microcracking, fiber failure and fiber pull-out) take place in the vicinity of an advancing crack. In the present approach, a bridging law (or cohesive zone law) approach is adopted in order to characterize the fracture toughness of the material. Conventional fracture toughness measures, such as KIC were in most cases found not to be applicable. This was because fundamental small-scale yielding geometry requirements could not be fulfilled in experiments.
The bridging law approach captures previously mentioned mechanisms in terms of a closure stress (bridging stress). This stress acts between two fictitious crack planes. The relation between crack opening displacement and bridging stress is governed by the bridging law.
Parts of the presented work consider determination of bridging laws from experiments (Paper I and Paper III). Different experimental configurations, double cantilever beam (DCB) specimens loaded with pure moments and double edge notched tension (DENT) specimens, were used in the two studies. A main conclusion from Paper I is that the large differences in fracture characteristics between two sheet molding compound (SMC) composites could be explained on the basis of bridging laws and their influence on fracture energy. Similar observations were made in Paper III. In Paper III, it also was evident that the intrinsic non-linearity of bulk SMC material has to be considered separately in the data reduction of experimental results, in order to capture the bridging law.
Bilinear approximations of decreasing bridging laws were obtained as a result from the smdy.
A closer investigation on the mechanical behavior of SMC with varied composition was performed in Paper II. Various mechanical tests, including tension, compression, in situ studies, DCB and stiffness degradation measurements through quasi-static cyclic loading-unloading experiments, were employed. The purpose was to characterize and understand observed differences between conventional and toughened SMC with low density additives.
The applicability of the proposed bridging law approach is confirmed by the work presented in Paper IV and Paper V. In these papers, the previously measured (Paper I and Paper III) bridging relations are used as a constitutive
predict and explain the change in notch-sensitivity observed on SMC DENT- specimens with varied geometry. A comparably simple analytical route (neglecting non-linear bulk behavior and shape of bridging law) is employed with satisfactory results. In Paper V, the use of the finite element method (FEM) in conjunction with measured bilinear bridging laws, allows reconstruction of experimentally measured compact tension (CT) specimen load vs. displacement curves with good accuracy. Three different CT specimen geometries are considered. Modeling and experimental results from Paper V also shows that compression failure often is of equal importance as tensile, in real structures and loading conditions.
Preface
The work presented in the thesis was carried out at Luleå University of Technology, Division of Polymer Engineering. It was financially supported by the Swedish Foundation for Strategic Research (SSF) through the Integral Vehicle Structure (IVS) research school.
There are a number of people who have been important for the work and deserve my sincere gratitude. First of all, I would like to thank my supervisor Professor Lars Berglund whose support and guidance has been most valuable.
He is also gratefully acknowledged for his positive influence on the ambience among coworkers. This ambience contributed strongly to make my time as a student forever associated with large numbers of nice, personal memories.
In the course of my studies, I have had the privilege of gaining and sharing useful knowledge through fruitful collaborations and discussions with research colleagues such as: Dr Bent Sørensen, Dr Johan Lindhagen, Dr Kristofer Gamstedt, Dr Per Synnergren, Dr Mikael Sjödahl, Dr Normunds Jékabsons, and Tech.Lic Angelica Svanbro, Tech.Lic Magnus Oldenbo and Tech.Lic Greger Nilsson.
I would also like to thank all members of the Division of Polymer Engineering, past and present, for their contribution to the ever present, nice atmosphere. Finally, I would like to thank my family: Linda, Matilda and Johannes for their support and presence at all moments.
April 2002, Luleå, Patrik Fernberg
List of appended papers
The thesis contains an introduction and the following appended papers:
I : Fernberg SP, Berglund LA. Bridging law and toughness characterization of CSM and SMC composites, Composites Science and Technology 61 (2001) p. 2445-2454.
II: Oldenbo M , Fernberg SP, Berglund LA. Mechanical behavior of SMC composites with toughening and low density additives, submitted to Composites part A.
I l l : Fernberg SP, Jékabsons N . Determination of bridging laws of SMC materials from DENT tests. To be submitted.
IV: Fernberg SP, Berglund LA. Notched strength of SMC composites - A bridging law approach. To be submitted.
V: Jékabsons N , Fernberg SP. Prediction of progressive fracture of SMC by application of bridging laws. To be submitted.
The following journal papers and conference proceedings were written and published during the period of postgraduate studies at LTU but not included in the thesis:
Fernberg SP, Berglund LA. Effects of glass fiber size composition (film former type) on transverse cracking in cross-ply laminates. Composites part A 31 (2000) p. 1083-1090.
Nilsson G, Fernberg SP, Berglund LA. Strain field inhomogeneities and stiffness changes in GMT containing voids. Composites part A 33 (2002) p. 75- 85.
Andersson A, Fernberg P, Sjödahl M . Optical methods to study fracture of notched glass mat composites. International Conference on trends in Optical Nondestructive Testing, 3-6 May 2000, Lugano, Switzerland.
Fernberg SP, Berglund LA, Sørensen BS. Fracture toughness characterisation of short-fibre composites in terms of bridging laws. 6th International Conference, Deformation and Fracture of Composites, 4-5 April 2001, Manchester, UK.
Contents
Abstract i Preface i i i List of appended papers v
Introduction 1 Fiber reinforced polymer composites 1
Sheet molding compounds (SMC) 1 Fracture characteristics of short fiber composites (SFC) 6
Concept of bridging laws 7 Micromechanical modeling and tailoring of bridging laws 9
Summary of papers 24
References 26 Paper I 33 Paper I I 57 Paper I I I 83 Paper IV 109 Paper V 127
Introduction
Fiber reinforced polymer composites
Fiber reinforced polymer composites (FRPC) is a class of materials which during the last decades has become well established engineering materials. Their comparably light weight in combination with high stiffness and high corrosion resistance make them attractive alternatives to steel and aluminum in many structural applications [1,2]. A large advantage with FRPC is that there is a great potential to tailor the mechanical properties of the material to suite special loading conditions. By reinforcing the material with continuous fibers, one obtains a stiffer material compared to a material with discontinuous fibers.
Furthermore it is possible to have continuous fibers oriented in a desired direction within a composite. This implies that a degree of anisotropy can be designed into the material i f so is desired. Commonly used fibers in FRPC are glass, carbon and aramid. Out of these, carbon fibers offer by far the best reinforcing effect in terms of stiffness. Associated with the numerous types of FRPC is also a large number of manufacturing techniques [3]. A somewhat simplified but in most cases true statement is that there is a correlation between mechanical properties and cost for FRPC. Higher quality FRPC e.g. epoxy based composites with continuous carbon fiber reinforcement is more expensive as compared to short fiber composites (SFC) with glass fiber reinforcement both due to higher cost of constituents and a more time-consuming manufacturing process.
Although SFCs are incapable of competing with continuous fiber composites in terms of mechanical performance there is a large number of applications where these materials are found suitable. In fact, the volume by which compression molded SFC are produced and used exceeds by far the volume of continuous FRPC in e.g. the automotive industry [3]. The large use of SFC is mainly driven by the potential to produce large series at comparably low costs through compression molding techniques.
Sheet molding compounds (SMC)
A major part of the work presented in the thesis is devoted to characterize and understand the fracture properties of compression molded SMC reinforced by discontinuous glass fibers. SMC is recognized as a material that, due to esthetical, electrical and mechanical properties, can be applied to a large number of applications.
An example is e.g. the use of SMC in external car and truck panels which has increased in recent years. This is exemplified in Figure 1. The deck lid of Volvo V70 is manufactured from molding compounds such as SMC and bulk molding compound (BMC). Both SMC and BMC are based on a thermoset polymer matrix. Molding compounds based on thermoplastic matrix, e.g.
polypropylene, are also used extensively in the industry under the term glass mat thermoplastic (GMT).
Figure 1 Volvo V70 with deck lid from SMC/BMC.
Manufacturing
SMC production technology mainly comprises two manufacturing steps;
compounding and molding. Indicated in Figure 2 is the mold in which forming, consolidation and curing of an SMC component is performed. The charge, a compound mainly constituting of fibers soaked with uncured thermoset resin, is positioned in a cavity between two mold halves. The cavity has the same shape as the final product. The compound fills the entire volume when the two halves are closed and pressure is applied. Curing is initiated at this stage, due to the temperature increase of the compound. The cured component can be removed from the mold after about 70 to 180 s of curing at elevated temperature (130 —
165°C) within the mold [4]. The route for producing GMT components by compression molding is similar as for SMC, with the difference that GMT sheets are preheated to facilitate material flow during closure of the mold. The GMT material is solidified by cooling the mold.
Matched shear edges
Heat and pressure
T
Closing CAVITY SMC charge Y J ~ | _
CORE - Matched shear
r—'fLT
Guide blocks J
edges
1 i
Heat and pressure
Figure 2 Main components of the SMC mold (Source: [4]).
Constituents
The SMC compound consists of several constituents. The fibers, usually glass fibers of grade "E" are essential to achieve desired dimensional stability and mechanical properties of the material. They are usually chopped and distributed with random orientation within the material during the compound production stage, see Figure 3. A commonly used fiber length for commercial SMC is 25 mm [5].
Haste
Figure 3 SMC compound production line (Source: [4]).
The diameter of a single fiber filament is typically about 14 urn. This diameter is, however, in most cases not representing the smallest reinforcing unit of the material. A surface treatment (sizing) is applied to the fibers during production of the continuous roving which eventually is to be chopped. The sizing contributes to the formation of bundles consisting of several hundreds of filaments. Once the fibers are chopped, bundles will have random distribution
within the material whereas the filaments will be aligned within the bundle [6].
As a consequence, matrix impregnated bundles rather then single fiber filaments are the main reinforcing units in SMC. Unsaturated polyester in combination with styrene is the most frequently used matrix material. Thermoplastic polymers may often be blended in to the matrix to provide additional toughness (toughening additives) of the matrix but also to reduce detrimental effects from shrinkage (low-profile additives) during cross-linking of the thermoset matrix [7]. Inorganic filler materials such as Calciumcarbonate (CaCÜ3) are widely used in SMC, partially in order to improve properties but mainly to reduce cost.
Commercial SMC compositions may have filler content of about 3 (MIO % by weight. Lower density SMC can be produced if some of the filler is replaced by hollow glass or ceramic microspheres [8,9]. Microspheres are available in densities ranging from 0.15 to 1.1 g/cm3. Other constituents of the SMC compound are/can be; pigments applied to obtain a desired color of the component; carbon black added to increase electrical conductivity and thereby facilitate spray painting directly on the detail; peroxide catalyst to initiate cross- linking of the matrix and a thickening agent to promote viscosity increase during storage of the compound.
Table 1 Typical properties of various SMC composites
Property SMC-R25 SMC-R50 SMC-R65
Fiber wt.% 25 50 65
Density 1.83 1.87 1.82
Tensile strength [MPa] 82.4 164 227
Tensile modulus [GPa] 13.2 15.8 14.8
Poisson's ratio 0.25 0.31 0.26
Strain to failure [%] 1.34 1.73 1.63
Compressive strength [MPa] 183 225 241
In-plane shear strength [MPa] 79 62 128
In-plane shear modulus [GPa] 4.48 5.94 5.38
Source: Heimbuch and Sanders [10].
Mechanical properties
Typical static mechanical properties of several SMC composites are given in Table 1. Not reported in the table is the often substantial variation in measured properties of SMC and other compression molded composites [11,12]. The
technique, inherent difficulties to control fiber distribution during the compression molding operation.
20 30 40 50 60 70
x (mm)
Figure 4 Measured strain field variations on 50 mm wide GMT test coupon.
Average strain in loading direction is 0.45 % (source: Nilsson et al. [13]).
An example of stiffness inhomogeneity caused by non-uniform fiber distribution within a swirled mat GMT composite is illustrated in Figure 4. The figure shows the large local strain variations on the surface of a GMT specimen subjected to tensile test as detected by digital speckle photography (DSP). A typical stress-strain diagram of SMCs is presented in Figure 5. The tensile modulus is highly nonlinear over the full interval. A bilinear characteristic with a corresponding "knee" is evident. The "knee" is associated with onset of microcracking at fibers oriented with transverse direction to loading [14,15].
The tensile failure strain is usually between 1 and 2 %.
JOO L
R65
STRAIN (%)
Figure 5 Typical tensile stress-strain curves for SMC (Source: [10])
Fracture characteristics of short fiber composites (SFC)
Due to the random nature of SFC a large number of mechanisms are active and contributes to the overall fracture characteristics. These mechanisms contribute in distributing the fracture process over a comparably large region in the vicinity of the advancing crack [16,17]. This is exemplified in Figure 6, showing the fracture surface of an SMC specimen. Fiber and fiber bundles are pulled out from the material. Some of the fibers were fractured prior to complete pull-out whereas others (those with short embedment length) were completely pulled out without any fracture. During the progression of the fracture no obvious separation of the material in two distinct pieces separated by a single crack is evident at an early stage. It is rather a multitude of small cracks that develops in a fracture process zone, thus gradually degrading the material at the crack.
Figure 6 Opposite SMC fracture surfaces along a crack from a DCB test.
Plausible inelastic mechanisms associated with progression of fracture are illustrated in Figure 7. Interfacial failure such as debonding may take place at locations where shear stresses are exceeding the strength of the interface. This mechanism may be initiated at discontinuities such as fiber ends, fiber fracture locations or at the crack face. Nonlinearities of the polymeric matrix may also influence and contribute to the onset of microdamage at the fracture site. Fiber fracture occurs when/if the bridging stress exceeds the strength of the bridging unit. The presence of fibers with oblique orientation (most often SFC fibers have random distribution) with respect to the macroscopic crack is adding complexity
recalling that reinforcing units (and thereby also bridging units) of SMC can be divided in two scales, one on fiber bundle level and one on fiber filament level.
Fibre breaks
Yielding & viscoelastic deformation of matrix
Fibre bridging & pull-out
F
Debonding
Oblique fibre orientation
Figure 7 Possible active mechanism during fracture of random SFC
The numerous abovementioned mechanisms are interacting and contributing to the macroscopic fracmre behavior of an SFC. One may from this realize that a measure of the resistance to macroscopic crack growth, the fracmre toughness of the material, consequently must incorporate contributions from all mechanisms. Conventional single scalar fracture toughness values KIC, obtained by a linear elastic fracture mechanics (LEFM) approach, does this with great success provided that the fracture process zone is restricted to the very close vicinity of the tip of an advancing crack (for e.g. brittle ceramics, polymers and metals). Since this requirement in most cases is not fulfilled [18,19,20] for SFCs (unless unrealistically large specimens are considered) a different concept has to be considered.
Concept of bridging laws
The presented thesis considers that the mode I toughness of short fiber composites is governed by the processes within the fracture process zone.
Furthermore it is considered that the behavior of the process zone can be described by a simplified approach. In this approach, the entire fracture process zone is lumped on to a line extending along the fracture path as indicated in Figure 8a and b. Thus instead of considering that fracture occurs in an infinitely small volume at the crack tip (small scale yielding case (SSY)) we consider that the process zone has infinitely small height but finite length LPZ. The fracture process zone may then be characterized by a stress-displacement relation,
designated as the bridging law (or cohesive zone law). A hypothetical bridging law relation is indicated in Figure 8c.
The bridging law describes the relation between cohesive stresses ab (applied by previously described mechanisms) and opening displacement S between the two fictitious crack faces in Figure 8b.
Figure 8 Schematic illustration of bridging law concepts: a) Fracture process zone in real material, b) bridging law governed model, c) hypothetical bridging law.
The fundamental ideas behind the concept was introduced in the early sixties by Barenblatt [21] and Dugdale [22] to represent nonlinear processes located at the front of a pre-existing crack. In the original work, Dugdale assumed that a constant cohesive stress, corresponding to the yield strength of the material, was applied along the crack line of on advancing crack. Hillerborg et al. [23] extended the concept in the late seventies, in work applied to concrete materials, mainly by assuming that the cohesive stress transferred through the crack was a function of the crack opening i.e. ah = f ( S ) . Since then, the cohesive crack or bridging law approach has been applied to a number of different materials exhibiting significant fracture process zones e.g. short fiber composites [16,24,25,26]; cross-over bridging during delamination of continuous fiber composites [27,28]; ceramic matrix composites [29]; paper [30]; adhesive glues [31]; foams [32] and polymers [33]. A general description of the concept is given by Bazant and Planas [34] whereas Suo [35] and Sørensen et al. [36] provides general discussions on its applicability to fibrous composites.
A strong argument in favor of the proposed concept is that it eliminates
a) b) c)
fracture mechanics (LEFM). LEFM has proven to be a versatile tool to characterize crack growth resistance of materials and structures [37]. A prerequisite for the applicability of LEFM is that the actual fracture occurs in a very small volume in the close vicinity of the crack tip. The theory then gives that a l/-Jr stress singularity is present at the crack tip. I f fracture occurs in the close vicinity of the tip then this stress singularity is completely determining the stress state at the tip. The stress field can then be weighted to macroscopic factors such as remote loading cr. and specimen geometry by the stress intensity factor K. However, i f the fracture process zone length LPZ is comparable in dimension to other characteristic dimensions in the problem, as in almost all cases with comparably large fracture process zones of SFCs, then these also will influence the stress state at the fracture location. This implies, as indicated earlier, that the critical stress intensity factor KC, is not a valid material properly, since it is geometry dependent [19,20].
If large scale bridging (LSB) assumptions are fulfilled i.e. the process zone height extension is negligible and LPZ extends beyond the zone within which the stress singularity obtained with LEFM determines the stress state, then the bridging law becomes the fundamental material property that governs macroscopic fracture toughness of the material [38]. Since the bridging law approach considers that all inelastic mechanisms involved during fracture contribute to the response of the bridge, this implies that the bridging law provides a link between microscopic fracture events and macroscopic behavior.
Thus, provided that the link between material composition and microscopic fracture is established, this can be used in efforts to tailor materials for a desired macroscopic fracmre behavior through the bridging law.
Micromechanical modeling and tailoring of bridging laws
Micromechanical models may serve as guidelines and tools in designing the microstructure for a desired macroscopic behavior. Such a model is bound to become rather complex since the nature of polymer matrix SFCs involves a multitude of contributing and interacting inelastic mechanisms. McCartney [39], Hutchinson et al. [40] and Marshall et al. [41] presented bridging models for the case of debonding during crack opening of composites with continuous fibers oriented parallel to the crack face. McCartney found that
where V is volume fraction, E is modulus, R is fiber radius and r is interfacial shear stress (assumed constant). Subscripts "f and "m" denote fiber and matrix respectively. We may note that (1) predicts a continuously increasing bridging stress. This is because the models do not consider fiber fracture. Bao et al. [42]
derived a model in which slip-dependent interfacial stresses also were taken in account. These were found only to influence the resulting bridging stresses for large values of 8. Continuously increasing bridging laws were still predicted.
Li et al. [43,44,45] published several papers where they developed and analyzed a micromechanical model for prediction of bridging laws of short fiber composites. Fibers were assumed to have deterministic strengths. Their primary interest was to model reinforced concrete. Models somewhat similar to Li's models were also developed by Wetherhold et al. [46,47]. Maalej later extended the model developed by L i so that variability in fiber strength properties also was included [48]. We will use Maalej's model to estimate the influence of some microstructural parameters on fracture characteristics of polymer SFCs.
However, some general differences between polymer matrix SFCs and reinforced concrete may call for caution in the interpretation of the results (concrete composites usually have lower Vf than polymer SFCs , concrete is extremely brittle in tension whereas polymer matrices often are rather ductile etc.).
Bridging law modeling of SMC
Maalej's model is based on the following relationship [43] between debond lengthy and stress ad in a single fiber with diameter df, see Figure 9
z in (2) denotes frictional bond stress whereas V denotes volume fraction and E
y (2)
where
7 = (3)
debonding are considered. They assume uniform fiber orientation distribution.
Non-deterministic fiber strength is incorporated in the model by Weibull statistics. It follows from their derivation that the bridging relation can be divided in two parts. One at the initial stage of the bridging curve denoted debonding and fracture branch. The other part prevails at the later stage and is denoted pull-out branch.
Two different fiber groups contribute to bridging tractions during progression and development of the debonding and fracture branch. In the first group are fibers that eventually will debond completely. These are the surviving fibers that later undergo frictional pull-out. A fraction of these fibers are at each value of S in a debonding mode.
t I I t I
I I 1 1 I
Figure 9 Single fiber bridging a plane crack.
The other fraction (fibers with short embedded length) is either in pull-out mode or has completely pulled out of the matrix. The second group consists of fibers that rupture during the process. These fibers are either in a debonding mode or in pull-out mode. Maalej's model shows that the pull-out contribution of ruptured fibers to the bridging stress is very small compared to that of surviving fibers. Disregarding the contribution from pull-out of ruptured fibers the, following relation is obtained within the debonding and fracture branch
+ x*r[2a,t0 ( 4 ' , (s)]-år[a,t0 [s\tt (S)]
\ l / 2
m + 1 1/2 - m + 1
1 -
1 - " * - T a , / ,r0
) J
(4)
for J < S , and where
;S = 2TL,
LF /2 (1 + 7])Efdf s W
m + \)\ 2
1 . ÅeS . f 1
m + l I// 2 Vx
f l
m+l
' s )
*
(*)• exp
r r /- ? v / n
t / , d t
4 r ' 4 r ( l + 7 )
cr
0m(l + w)
2nd fk
i_
m+l
(5)
Fiber strength statistics is incorporated in the model through the scale parameter <r0 (or characteristic strength) and the shape parameter m (or Weibull modulus) of a Weibull distribution. This distribution is given by
t A v/™
(6)
where o and L are the strength and fiber length under consideration. A0 is the surface area of a fiber with a given reference length and diameter. The function
r [ a , x , , x2 ] in (4) is the generalized incomplete gamma function, defined by
r[a, X j , x2 ] = Jf2 r° 'e ' dt. (7)
During later stages of bridging (for 5 > S *) the debonding mechanism ceases to operate and pull-out of fibers becomes the only contributing mechanism. This is the pull-out branch. Again two different groups of fibers contribute to this branch. The first group consists of intact fibers that were completely debonded during the debonding branch and are in pull-out mode.
The second group contains fibers that fractured during debonding and are also in pull-out mode. After dropping the contribution from the second group (by the same argument as in the case of pull-out of ruptured fibers in the debonding branch) one obtains
<rc = £ * [ * * ] r[3aj\t](o)}+xtr[2aj\t](S)]-dr[a,t* J^d)] (8)
for the bridging stress within the pull-out branch.
Modeling of SMC bridging laws - input values
We will use Maalej's model to estimate the influence of some constituent parameters on the fracture energy GJC of polymer SFCs. GJC is calculated by integrating bridging laws obtained from the model. The default input parameter used for calculation of the normalizing fracture energy G0 is presented in Table 2. The estimated values are based on typical microstructural features of SMC. A complicating factor in the modeling of SMC is the observation [49] that two different bridging units of different scale are operative during fracture. One is on the scale of bundles and the other is on the individual fiber scale. Mechanisms on both scales contribute to the overall bridging. As a consequence, the choice of input parameters to Maalej's model becomes somewhat cumbersome.
The 25 mm fiber (and bundle) length is typical for SMC materials.
Results from single fiber tests on glass fibers [50] gives typical single fiber values of Weibull parameters m and o&. These tests are however performed on pristine continuous fibers whereas the fibers within SMC have undergone chopping and pressing during manufacturing of the composite. It is likely that the impact from processing may change the strength statistics slightly.
The default values of Vf, Ef, a0 are all estimated based on the assumption that fiber bundles are the main contributors to bridging. They are calculated based on an assumption that the local volume fraction of fibers within the bundle is 0.5 and that the bundle performs as a fiber with properties of a unidirectional composite. The actual fiber volume fraction of the composite is 0.18 (calculated from manufacturer's data). Simple rule of mixture estimations of Vf, Ef and rx0 then gives values according to Table 2. The chosen matrix (in reality a mix of polymer matrix and inorganic filler) stiffness value Em was obtained from ref. [51] where this value was found to give accurate micromechanical predictions of macroscopic SMC stiffness. Values of T and df were to some extent determined by fit of model results to experimental results.
Table 2 Default input values to micromechanical model
I/[mm] df[\aa\ Vf £>[GPa] Em [GPa] r[MPa] a0 [MPa] m G0 [kJ/m2] 25 32 0.36 37 5 5 250 2 20.0
Value reported corresponds to volume fraction of bundles (based on a true glass filament volume content of 0.18)
Modeling of SMC bridging laws - parametric study
The predicted bridging response by default input values is presented in Figure 10. Also indicated in the figure is an experimentally determined bilinear estimation of the bridging law for a commercial SMC material. Both model and experiments show an initial region which corresponds to the previously described debonding and fracture branch where debonding, fiber fracture and pull-out govern the response. It is evident that the model underestimates the distance over which the debonding branch prevails. This is most likely due to an oversimplified single crack plane model whereas bridging in real SMC materials occurs over some finite volume. This can be observed in Figure 11 where an in situ image of an advancing crack is presented. A main crack is observed on the left side of the image. We also see that this crack has several branches. Thus debonding, of the type that the single crack plane model assumes, often occurs at several locations along a single bridging unit. The total Sin a real material can thus be considered to have contributions from not only one but several crack planes and is therefore shifted towards larger Sin Figure 10.
140
120
100 debonding and
fracture pull-out
0 0.01 0.02 0.03 0.04 0.05 8 [mm]
experimental model
1 1.5
S [mm]
2.5
Figure 10 Predicted bridging law with default input values, comparison with experimental curve from [52]. Inserted graph shows details of initial predictions.
Figure 117« situ ESEM micrograph of advancing branched crack in SMC.
The initial behavior during the debonding and fracture branch is also presented in a more detailed plot in Figure 10. At the very beginning, there is an increasing part of the bridging law (increasing bridging stress with increasing crack opening displacement S). This corresponds to initiation of debonding at an increasing number of bridging units. Any increasing bridging law relation was difficult to verify experimentally [52] because of contributions from non-linear volumetric bulk material behavior. A l l fibers are intact and contribute to the bridging stress at this initial stage. Further increase of S gradually causes fibers to rupture, debond and pull-out. At this stage, the bridging stress starts to decrease since the number of bridging units decreases.
We will in the subsequent section use Maalej's model in a parametric study of influence of constituents on bridging laws and fracture energies.
Although some deviations between model predictions and data are observed in Figure 10, we still find the general agreement encouraging. We therefore believe that the most important mechanisms are incorporated in the model.
4
Qk , , , , , , ,
0 10 20 30 40 50 60 70
Lf [mm]
Figure 12 Influence of fiber length Lf on fracture energy. Corresponding and governing bridging law trends are illustrated in the inserted graph.
The influence of fiber length Lf on SMC fracture energy G/ c is presented
to the model. This is in agreement with the findings of Li [44], Wetherhold [46]
and Kelly [53]. The governing physics is understood by considering the bridging laws in Figure 12. Two of the plotted relations are obtained for Lf shorter than the optimum whereas two are obtained by having Lf larger than optimum. No steeply decreasing part is observed for the two shortest Lf. Thus no fiber fracture occurs. All fibers have a shortest embedded length shorter than the critical fiber length required to cause fiber fracture. The critical fiber length Lc is given by
L. = af «df
Ar (9)
where ofu denotes fiber failure stress. For small Lf (when L/2 < Lc) the bridging law tail part is contributing significantly to Gic since all fibers undergo complete pull-out. For Lf shorter than optimum, increasing Lf implies that the pull-out length increases and thereby also the fracture energy. The optimum Lf is when fiber length is equal to a critical fiber length. The maximum bridging stress is continuously increasing with Z/but as L/2 > Lc, the occurrence of fiber fracture causes a steeply decreasing bridging relation. The number of bridging units contributing to the fracture energy during the pull-out stage is therefore decreased. Hence GIC starts to decrease with increasing Lf.
4
o
ISO 1 1
ISO 1
t
1
[MPa] 1 \ \ increasing Vf 1
I 1 /
^ 5 0
\=
1 1 / / / /
0 c 0
c 0.02 0.04 0.06 0.08 0 1 >
8 [mm] / / / /
-
/
^0
/
^ " "
>'——^ ! 1 ' ' 1 '
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Vf (real)
Figure 13 Influence of fiber volume fraction Vf on fracture toughness GIC. The governing bridging law response is inserted in the graph.
It is seen in Figure 13 that an increasing fiber volume fraction has a positive effect on the fracture energy. This is because increased Vf leads to increasing number of bridging units and thus generally higher bridging stresses, as is seen in the inserted graph. Note that the volume fractions presented in Figure 13 corresponds to the real Vfi.e. the volume fraction occupied by single glass filaments. The input values to the model are calculated from real Vf, assuming that volume fraction within the bundle is 0.5.
Figure 14 Response surface obtained from parametric study of influence of fiber length L/ and fiber volume fraction Pyon fracture toughness G/o
By varying both Lf and Vf in the model, one obtains the response presented in Figure 14. The considered values of Lf and Vf are
Lf e {2,4,6,...,30} and Vf G {0.04,0.08,0.12,...,0.32}. From the results in Figure 14 we recognize the trends presented in Figure 12 and Figure 13; fracture energy increases with increasing Vf, it also increases with increasing Lf until a maximum is reached. The energy decreases with increasing Lf after the maximum.
The results obtained by varying fiber diameter are presented in Figure 15.
The tendency is that fracture toughness decreases with increasing fiber diameter for the diameters considered. This is mainly because the modeled fiber strength decreases with increasing diameter. The model considers that a Weibull area
units since the strength of brittle fibers is mainly governed by the density of surface flaws.
The use of an area distribution for the case of bridging in SMC, where bundles rather than single filaments are bridging units can be a matter for further discussion. It is possible that a volume distribution could be more appropriate in the case of bundle bridging units. I f we, on the other hand, neglect influence of df on fiber strength we see from (9) that Lc increases with increasing d/. By this we expect that the number of contributing fibers during the pull-out stage should increase when df is increased. This would give increasing G/c with increasing df.
The two counteracting contributions are both taken into consideration by the model and it seems that the influence on fiber strength is dominating. Since there is some doubt regarding the applicability of the implemented fiber statistics model in the case of varying fiber diameter we are cautious with making definite conclusions based on Figure 15.
0 0.01 0.02 0.03 0.04 0.05 0.06
df [mm]
Figure 15 Influence of fiber diameter df on fracture toughness G/c- The governing bridging law response is inserted in the graph.
The influence of varying fractional shear strength is indicated in Figure 16. We see that fracture energy decreases with increasing r. This is since a larger number of fibers will fracture during the debonding branch i f ris large. It can also be explained on the basis of equation in (9). Lc decreases with increasing r and a larger number of fibers will then a have shortest embedment
length which is longer than Lc. The number of fibers surviving the debonding branch so that they can contribute to the pull-out branch will therefore decrease.
As a result we observe a decrease in fracture energy.
It can be noticed that Maalej [48] reports that the fracture energy relations presented in Figure 15 and Figure 16 also are exhibiting a maximum, similar to our results for Lf presented in Figure 12. However, within the range we have chosen for realistic parameter vales, no local maximum in G/c is observed.
0i , , , , , , ^
0 2 4 6 8 10 12 14 X [MPa]
Figure 16 Influence of frictional shear strength ron fracture toughness G1C-
1.5
Ü
0.5
increasing m
0 0.02 0.04 0.06 0.08 0.1 8 [mm]
Increasing variability in fiber strength has a positive effect on G/c- This is observed in Figure 17 since G/c decreases with increasing Weibull modulus m (or Weibull shape parameter). By increasing m one is approaching a more deterministic value of fiber strength since large m implies a narrow fiber strength distribution. The fracture energy becomes large for small m since the fiber fracture process during the debonding and fracture branch occurs in a more progressive manner when fiber strengths are widely distributed.
2.5
O
1.5
1
0.5
0
1 1 1
150
1 100
° 50
\ \ , increasing cjQ
/ / t / /
/ /
< 1 *
0 C 0
C 0.05 0.1 0.1 6 [mm]
5 0.2 s
- é
s s
ø s -
—i — — i i
0 50 100 150 200 250 300 350 400 450
% [MPa]
Figure 18 Influence of Weibull scale parameter o0 on fracture toughness GIC.
From Figure 18 we see that G/c increases with increasing values of the Weibull scale parameter aQ. Thus an increase in inherent strength of the fiber has a positive effect on the fracture toughness of the composite. An increase of fiber strength will reduce the number of fractured fibers in the debonding and fracture branch leaving a large number of bridging units capable to contribute in the dissipation of energy during the pull-out stage.
Microstructural tailoring
The preceding results illustrate the potential to use bridging laws to understand the coupling between fundamental microstructural parameters and macroscopic fracture behavior through the intrinsic bridging law relation.
It was obvious that an efficient way to increase fracture energy is to increase fiber volume fraction. The beneficial effect of high Vf on fracture toughness of SFCs is supported by the results in refs [54,55]. An advantageous side effect also obtained by increasing I / i s increased stiffness of the material. A disadvantage is that cost of the SMC composite increases since an increase of fiber content also is associated with a relative decrease of the cheaper mineral filler constituent.
We saw that SMC materials are expected to exhibit decreasing G/c with increasing Lf (when fibers are longer than about 6 mm). A similar trend was experimentally observed by Hitchen et al. [56] who studied fracture of carbon fiber/epoxy SFCs with three different fiber lengths (Z/= 1, 5 and 15 mm). The highest fracture toughness value was obtained for Lf = 5 mm. There are also experimental studies that report that fracture toughness of SFCs reaches a steady state level once a critical fiber length is exceeded [55].
Since most commercial SMC have Lf = 25 mm [5] there is a possibility that fiber length can be decreased in order to increase G!C. It must be noted that the micromechanical model under consideration is likely to overestimate the detrimental effect of fiber fractures on the fracture energy. This is since the model considers that bridging units ceases to contribute to bridging tractions once they rupture. Due to the statistical nature of failure of brittle fibers in real materials, the fracture may however occur away from the major crack face leaving bridging segments of the fiber available for further contribution to bridging stresses and energy dissipation through pull-out.
An increase in G/c can also be obtained by decreasing the fiber diameter df. This is likely achieved if fibers are better dispersed within the material so that individual fibers rather than fiber bundles are the major bridging units. Our micromechanical modeling as well as experiments [49,57] shows that an improvement in fracture toughness sense is obtained i f interfacial adhesion is decreased.
The model considers a brittle matrix whereas polymer matrices often are ductile and may thus yield significantly prior to fracture. It is possible that lowered yielding strength of a polymer matrix influences the fracture toughness in similar manner as decreased interfacial adhesion in our model. Instead of the debonding stage, an initial stage where matrix yielding will precede fiber fractures and pull-out. Toughening of the SMC matrix e.g. by various additives that increase matrix ductility may thus have positive effects on SMC fracture toughness.
It was also demonstrated that G/c can be increased by increasing the
the number of surface flaws is kept at a minimum, since brittle fiber fracmre usually is initiated at surface flaws. This implies that any mechanical treatment such as e.g. chopping of the fibers is likely to result in a decrease of crø. Yet, the detrimental effect on ao of mechanical treatment does not necessarily mean that G/c decreases. This is because the increased flaw density also is likely to broaden the fiber strength distribution. Weibull modulus m will thus decrease.
As illustrated previously, such a decrease is associated with an increase in G/c- The coupling between the two statistical parameters (Weibull parameters) in the model may serve as an illustration of a problem often encountered in reality. A change of one microstructural parameter may change one or several other parameters. This of course, makes tailoring of real materials more complicated than what is indicated in the preceding numerical study, where one parameter at a time may be changed.
Summary of papers
Paper I: Bridging law and toughness characterisation of CSM and SMC composites (Fernberg SP, Berglund LA)
An experimental investigation of fracture properties of three different short-fiber reinforced composites (one chopped strand mat (CSM) and two sheet molding compound (SMC) materials) was performed. The tests were performed on double cantilever beam (DCB) specimens loaded with pure bending moments.
This experimental setup was in previous work found to allow estimations of bridging laws directly from measurements. From the experimental results, the coupling between microstructure and fracture behavior is discussed through the measured bridging laws. The beneficial effect (in terms of fracture energy) of increasing tendency for pull-out was confirmed. This increasing tendency for pull-out was also observed to shift the bridging law towards larger crack openings.
Paper II: Mechanical behavior of SMC composites with toughening and low density additives (Oldenbo M, Fernberg SP, Berglund LA)
The mechanical behavior of SMC materials developed for automotive exterior body panels was investigated. Toughened, low-density SMCs (Flex-SMCs) containing hollow glass spheres and thermoplastic toughening additives were studied. A conventional SMC (Std-SMC) was used as a reference material. The materials were tested in monotonic and cyclic tension, monotonic compression and DCB experiments. The influence of progressive damage with increasing load was investigated through the measured stiffness degradation obtained from cyclic tension tests. In-situ microscopy studies were also employed to reveal the governing mechanisms responsible for observed differences in macroscopic performance between toughened and standard materials.
Paper III: Determination of bridging laws for SMC materials from DENT tests (Fernberg SP, Jékabsons N)
A bridging law (or cohesive zone law) approach was employed to evaluate the fracture of double edge notched tensile (DENT) specimen from two SMC
was used to separate volumetric body contributions and true crack opening from measured displacements over the cracked region. We found that the non-linear material response gave a significant contribution to measured displacements. By applying corrections for volumetric displacements in the data reduction scheme, we were able to estimate the bridging laws of the two SMCs respectively.
Paper IV: Notched strength of SMC composites - A bridging law approach (Fernberg SP, Berglund LA)
The influence of specimen geometry on the strength of notched sheet molding compound (SMC) specimen was studied. In the tests double edge notched tension (DENT) with various dimensions and a/w-ratios were considered. Two SMC materials, both predicted to exhibit notch-ductile response for the geometries considered were tested. A notch sensitivity model that takes the influence of the fracmre process zone into account was used to predict notched strengths. A very good agreement between predicted and experimental strengths was obtained even though a very simplified bridging law shape was considered.
The proposed model was found suitable for first order approximations of the notched strength of SMC structures.
Paper V: Prediction of progressive fracture of SMC by application of bridging laws (Jékabsons N, Fernberg SP)
Experimentally obtained load vs. displacement curves from compact tension tests (CT) of two different SMC materials were analyzed. Progressive fracmre was attained in all tests. This gave rise to a long post-peak tail part in the load vs. displacement curve. By implementing bridging laws and volumetric stiffness degradation of bulk SMC in an FEM model we were able to reproduce the two larger geometries considered with high accuracy. Discrepancy between model predictions and experiments for the smallest geometry considered was observed.
This was due to premature compressive failure on the side opposite to the CT specimen precrack. The successful use of bridging laws suggested that they are intrinsic material properties governing the fracture behavior of SMC materials.
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PAPER I
Bridging law and toughness characterisation of C S M and S M C composites
Paper I
Bridging law and toughness characterisation of CSM and SMC
composites
S.P. Fernberg and L.A. Berglund
Division of Polymer Engineering, Luleå University of Technology, SE-971 87 Luleå, Sweden
In: Composites Science and Technology, Vol. 61 (2002) p. 2445-2454
Abstract
This work presents an experimental investigation of fracmre properties of three different short-fiber reinforced composites (one chopped strand mat (CSM) and two sheet molding compound (SMC) materials). Fracmre tests are performed on double cantilever beam (DCB) specimens loaded with pure bending moments.
In this experimental configuration, the bridging law for the material can be derived directly from measurements. No significant dependency of specimen height was observed in our results. The determined bridging laws can therefore be considered as material properties. The coupling between microstructure and fracture behaviour is discussed through the measured bridging laws. The beneficial effect (in terms of fracture energy) of increasing tendency for pull-out is confirmed for one SMC termed Flex-SMC, showing remarkably high fracmre energy, Jc=56.0 kJ/m2, compared to a standard SMC termed Std-SMC, 7C=25.9 kJ/m2. This increasing tendency for pull-out is observed to shift the bridging law towards larger crack openings. Based on our observations we find the concept of characterising the failure behaviour in terms of bridging laws attractive since it can be used as a tool for tailoring of microstructure towards desired fracmre behaviour.
Introduction
The use of short-fiber reinforced polymer composite materials in the transportation industry in general and automotive industry in particular is increasing. A major advantage of these materials is that cost-efficient processing is possible. The processing also allows production of components with complex geometries in a single process. As a consequence, the use of these materials in external panels for cars and trucks has increased in recent years. One important property of an external panel is its capacity to withstand impact. In this context, improved understanding of crack growth and toughening mechanisms of the material is of great interest.
Linear elastic fracture mechanics (LEFM) is widely used both in design and in characterisation of material fracmre properties. In this theory it is found that the normal stress approaches infinity at the crack tip. This is called singular field or K-field. I f the yield or damage zone at the crack tip is assumed to be small, the microscopic details of the damage zone can be neglected. All information related to load and overall geometry is communicated to the crack tip through the K-field. In this case the fracture property of the material can quantified by a single scalar parameter, Kc, fracture toughness.
Because of the nature of the damage process in short-fiber composites, these small scale damage zone requirements are often not fulfilled in experimental tests, the damage zone is simply too large. This also means that valid fracture toughness data can not be measured in experiments, unless specimen dimensions are sufficiently large compared with the damage zone size.
There are several studies in the literature [1,2,3,4] presenting fracture toughness data for materials similar to the materials of our interest, using comparably small compact tension (CT) and single edge notched specimens in bending (SENB) and in tension (SENT), with free ligament lengths of less than 30 mm. To use such small specimen and use LEFM concepts in the interpretation of results is highly questionable since the damage zone length could be in excess of 50 mm [5]. The increased fracture toughness in larger specimens is caused by the larger damage zone, which increases the resistance to crack growth. The explanation in [1] is not correct.
