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Greenstick fracture in composite pultruded rods
Composites Part B 110 (2017) 106e115
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Composites Part B journal homepage: www.elsevier.com/locate/compositesb
Greenstick fracture in composite pultruded rods G. Vargas*, F. Mujika ‘Materials þ Technologies’ Group, Department of Mechanical Engineering, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Plaza n, Spain Europa 1, 20018. Donostia-San Sebastia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 August 2016 Received in revised form 21 October 2016 Accepted 1 November 2016 Available online 4 November 2016
The paper analyses the greenstick fracture mode on anisotropic beams subjected to flexural loading. This fracture mode can be seen in plant branches, young mammal bones or reinforced plastic rods. Two analytical models have been studied: a straight beam model and a uniform curved beam model. Circumferential normal stress distribution has been determined considering the classical beam theory, and radial and shear stresses have been predicted by using the Airy's stress function. Experimental work has been implemented on initially straight pultruded rods with longitudinally oriented fibres. Three and four-point bending tests adopting several span-to-diameter ratios have been performed. Experimental conditions that promote a greenstick fracture mode in composite pultruded rods have been considered. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Fracture Analytical modeling Mechanical testing Pultrusion
1. Introduction Considering a solid cross-section curved beam subjected to flexural loading, the plane stress state is composed by circumferential normal stress sq, radial normal stress sr, and shear stress trq, as presented in Fig. 1. In beams composed of isotropic materials radial stresses are negligible compared with circumferential ones, and they do not affect the mechanism of failure. Therefore, traditional analysis of isotropic beams do not take into account these radial stresses. Nevertheless, radial stresses must be considered when beams composed of anisotropic materials are studied. Unidirectional fibre reinforced composites, both natural and man-made, are stronger and stiffer longitudinally than transversely, and consequently radial stresses due to bending can produce catastrophic failure. Thus, such radial stresses could enlighten the greenstick fracture behaviour of some anisotropic beams such as green branches and twigs, bones of young mammals, and uniaxially reinforced composite beams. A greenstick fracture is a partial, angled fracture where the convex side ruptures (without the fracture line traversing the beam), and the concave side bends [1]. On Fig. 2 a greenstick fracture of a green branch is presented.
Abbreviations: CBT, Classical beam theory; 1/3, One-third span; 1/2, One-half span; 3-PB, Three-point bending; 4-PB, Four-point bending. * Corresponding author. E-mail address: [email protected] (G. Vargas). http://dx.doi.org/10.1016/j.compositesb.2016.11.001 1359-8368/© 2016 Elsevier Ltd. All rights reserved.
Wooden branches and trunks have greater longitudinal than transverse strength due to the alignment of the tracheids (fibre cells) along the stem axis. The wood fibres are oriented in the longitudinal direction; their cell wall structure is optimized to provide sufficient mechanical properties regarding the main loading conditions of the living tree (e.g. bending of the stem due to wind loads). In both the radial and tangential directions (i.e. the transverse plane) the mechanical properties of wood are an order of magnitude lower compared to the longitudinal direction [2]. In trees, secondary vascular tissues on the inner side are wood elements, mostly tracheids in gymnosperms, but also vessel elements, vesselassociated cells, axial parenchyma and fibers in dicotyledons. On the outer side, secondary vascular tissues are phloem cells, that is, sieve tubes, and, in dicotyledons, companion cells, axial parenchyma, and fibers [3]. Bone is mechanically and structurally anisotropic with oriented collagen fibrils and nanometer-sized mineral particles aggregating into lamellar or woven bone [4,5]. In particular, cortical bone is a naturally occurring composite material comprising mineral (essentially hydroxyapatite) and organic constituents (collagen and bone cells): within the collagen fibrils, the mineral is embedded in the form of thin hydroxyapatite platelets. Their long axis is mainly parallel to the long axis of the collagen fibrils. At a higher level of the hierarchy, several collagen molecules form fibrils, and the interface layer between neighboring fibrils is filled with extrafibrillar matrix containing noncollagenous proteins [6]. The mineralized collagen fibril represents the basic building block of bone, and can be arranged in different ways, forming fiber bundles
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Nomenclature A E F I k M N P Q r S z z
cross-sectional area modulus reaction force at supports of a simply-supported beam moment of inertia with respect to the middle plane constant bending moment normal load applied load shear force radial coordinate, radius of curvature span out-of-plane coordinate location of the neutral axis
Greek symbols d deflection d_ displacement rate
ε
107
strain Airy's stress function in polar coordinates circumferential coordinate/arc angle radius of the circular-cross section normal stress shear stress
f q r s t
Subscripts 1 longitudinal direction CBT classical beam theory G centroidal axis, evaluated at z’ ¼ 0 i inner (between loading points) LP loading points max maximum MS mid-span NA neutral axis, evaluated at z ¼ 0 o outer (between supporting points) r radial direction q circumferential direction
ratios as large as 1:15 (wet bone) and 1:7 (dry bone) in tensile strength between orientations perpendicular and parallel to the main collagen fiber orientation. Man-made composite unidirectional fibre reinforced plastics are also highly anisotropic materials with longitudinal-to-transverse strength ratios up to 40:1 [9]. In the present study, analytical approaches and experimental work for studying greenstick fracture mode of anisotropic beams subjected to flexural loading is presented. Analytical approach is a geometrically nonlinear solution for a deflecting beam, which is initially straight, that considers all stress components: normal radial and shear stresses, that have been predicted by using Airy's stress function, previously determining circumferential normal stresses by classical beam theory. Mechanical testing has been carried out by both three-point bending and four-point bending tests on circular solid cross-section pultruded rods considered as transversely isotropic along the longitudinal axis.
2. Analytical approach 2.1. Classical beam theory Fig. 1. Stress components in polar coordinates r,q in the case of plane stress: circumferential stress sq radial stress sr, and, and shear stress trq.
with varying degrees of organization. On the one hand, in adult human cortical bone, the main microstructural features are the osteons, which are near-cylindrical structures formed by concentric lamellae. These lamellae structures surround the Harversian canals, with the outer boundary separating the osteons from an interstitial matrix called cement line [7]. On the other hand, woven bone is characterized by randomly oriented fibril arrays. This is typically found in quickly formed bone, for example, in embryonic or young bones and at repair sites after fractures. It is a less mineralized and mechanically weakest bone type, and often substituted during remodeling by other more organized bone types [5]. Fig. 3 displays greenstick incomplete fractures of young bones: radius and ulna [8]. Seto et al. [4] have measured the mechanical anisotropy of the mineralized fiber bundles in fibrolamellar bovine bone in the periosteal region (outer surface) of long bones, finding anisotropy
Circumferential normal stress sq has been considered for both three-point bending (3-PB) and four-point bending (4-PB). In Fig. 4 the loading configuration for 4-PB is presented as well as the shear force Q and the bending moment M diagrams, considering an applied load P, a circular-cross section specimen with radius r, a support span So, and a loading span Si. Shear force Q and bending moment M are:
Q¼
P 2
(1a)
M¼
P ðSo Si Þ 4
(1b)
Three bending configurations have been analyzed: 3-PB set-up achieved for Si ¼ 0, 4-PB one-third span set-up completed for Si ¼ So/3, and 4-PB one-half span set-up attained for Si ¼ So/2. According to classical beam theory (CBT), the maximum circumferential stress sqmax is:
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Fig. 2. The so-called greenstick fracture of a branch: a break on the convex side before splitting down the middle, with large longitudinal cracks along the centre line.
Fig. 3. Frontal (a) and lateral (b) greenstick fractures of the radius and ulna of a 5-yearold child. Fully broken cortex (black arrows) on one side, and intact bone on the other side (white arrows). From Lee et al. [4] with permission.
sqmax ¼ k1
P So
p r3
(2)
where k1 depends on bending set-up, as presented in Table 1.
Fig. 4. Four-point bending configuration, and the shear force Q and the bending moment M diagrams.
fibre reinforced composite beam with transverse isotropy along the longitudinal axis is assumed. Circumferential strain in q-direction εq, is a linear function of the radial coordinate taken from the neutral axis z (see Fig. 5), and
2.2. Curved beam The theoretical analysis, that considers an initially straight beam, is a geometrically nonlinear solution for a deflecting curved beam that considers the following assumptions: The layers of the beam are made of a linearly elastic and orthotropic material whose principal axes of orthotropy coincide with the considered coordinate system axes. In particular, a
Table 1 Constants related to analytical approach for bending configurations: three-point bending (3-PB), four-point bending one-third span (4-PB-1/3), and four-point bending one-half span (4-PB-1/2). Bending set-up
k1
k2
k3
k4
k5
k6
3-PB 4-PB-1/3 4-PB-1/2
1 2/3 1/2
1/48 23/1296 11/768
5/18 5/27 5/36
1/48 5/324 1/96
1/12 5/81 1/24
10/3 3 10/3
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depends on the radius of curvature of the neutral axis rNA (evaluated at z ¼ 0). It can be expressed as:
εq ¼
z rNA
(3)
Mechanical and thermal properties, into the range considered, are temperature-independent. Free-edge effects are not considered. Normal stresses in the direction perpendicular to plane r,q (i.e. z,q) are negligible.
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calculated as function of both the flexure modulus of elasticity E1 and the radius of curvature rNA:
MðqÞ ¼
p E1 4 rNA
r4
(7)
Applying Hooke's law, and equating Eq. (3) and Eq. (7), the circumferential stress sq can be expressed as function of the radial coordinate r, taking into account that z ¼ r e rNA (see Fig. 5).
sq ¼
4 MðqÞ
p r4
ðr rNA Þ
(8)
Due to the loading condition of the beam, the neutral axis does not coincide with the cross section centroid axis. The radial coordinate taken from the centroid line z’, is adopted as reference [10], considering the following relation according to Fig. 5:
To determine the location of the neutral axis z, defined as the line in the cross-section of the beam where the circumferential strain is zero (εq ¼ 0), Eqs. (5a) and (5b) must be equated, finding respectively that:
z ¼ z0 z
1
(4)
where z is the distance between the neutral axis and the centroid line. Due to the vector system equivalence, normal load N(q) and bending moment M(q), are resultant force and resultant moment of normal stresses, respectively, both applied at the centroidal axis G (at z’ ¼ 0). This equivalence, considering Eq. (3) and in combination with the Hooke's law, can be expressed as:
Z A
Z
Z z sq dA ¼ NðqÞ0 E1 dA ¼ NðqÞ rNA Z
A
A
z 0 z dA ¼ MðqÞ rNA
1
NðqÞ p E1 z r2
¼
r
4 MðqÞ p E1 r4
(9a)
(9b)
From Eqs. (9a) and (9b) can be inferred that:
z¼
r2 NðqÞ 4 MðqÞ
(10)
For a simply-supported beam subjected to transverse point loads, as in the case of 3-PB and 4-PB, the following system of internal forces are generated [10]:
(5b)
NðqÞ ¼ F Sin q
(11a)
Q ðqÞ ¼ F Cos q
(11b)
MðqÞ ¼ rNA F Sin q
(11c)
where E1 is the circumferential modulus of elasticity. From Eq. (5b) the radius of curvature of the neutral axis rNA can be derived:
1 MðqÞ ¼ rNA E1 I
¼
(5a)
A
sq z0 dA ¼ MðqÞ0 E1
r
(6)
where I is the moment of inertia with respect to the middle plane. Moreover, from Eq. (6) the bending moment M(q) can be
where N(q) is the normal load, Q(q) is the shear load, M(q) is the bending moment, q is the arc angle, rNA is the radius of curvature of the centroid line, and F is the reaction force at supports. Consequently, replacing Eqs. (11a) and (11c) in Eq. (10) the location of the neutral axis is obtained:
Fig. 5. Radial and circumferential variables of the beam. (a) r,q plane. (b) Section A-A, transverse cross-section.
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r2
(12)
4 rNA
In both loading cases, 3-PB and 4-PB, the bending moments at the mid-span MMS and at the loading points MLP can be determined considering a curved beam, as depicted in Fig. 6. The curved beam is assumed to have a constant curvature equal to the curvature of the mid-span point, where the bending moment M is maximum and the shear load Q is zero. MMS and MLP are given by:
MMS ¼
P P ðSo Si Þ þ ½d Tan qo ðdMS dLP ÞTan qi 4 2 MS
(13a)
MLP ¼
P P ðSo Si Þ þ ðdLP Tan qo Þ 4 2
(13b)
where qo is the curved beam outer angle, measured from the midspan to the supporting point, and qi is the curved beam inner angle, measured from the mid-span to the loading point, as depicted in Fig. 6. In addition, the first term in Eq. (13) coincides with Eq. (1b) which corresponds to the CBT model considering the beam as a straight element, i.e. the bending due to the span. Besides, the second term in Eq. (13) indicates the bending contribution of the load considering the beam as a curved element, i.e. the bending due to the deflection. Moreover, in Eq. (13) can be noticed that bending moments MMS and MLP are equal when qo ¼ qi, as shown in Fig. 4.
2.3. Radial and shear stresses Airy's stress function f(r,q) in polar coordinates relates three components of stress: circumferential stress sq, radial stress sr, and shear stress trq, as depicted in Fig. 1. The stress components obtained from this stress function satisfy equilibrium conditions in the considered domain [11], and they can be expressed as follows:
1 v4 1 v2 4 þ 2 2 sr ¼ r vr r vq
sq ¼
v2 4 vr 2
(14b)
trq ¼
1 v4 1 v2 4 r 2 vq r vr vq
(14c)
As the circumferential stresses sq can be expressed as function of the radial coordinate r and circumferential coordinate q, see Eq. (8), it is possible to determine the Airy's stress function f in polar coordinates from Eq. (14b), accomplishing several integrations. Afterwards sr and trq stresses can also be determined from Eq. (11a) and Eq. (14c), respectively. According to the presented analytical model, once the beam is deformed, it features a uniform curvature which is equal to the curvature value obtained in the mid-span. Subsequently, the bending moment M does not vary with q, and, as will be presented in Section 4.4, the bending moments MMS and MLP can be considered as equivalent. Therefore, the Airy's stress function depends only on the radial variable r, i.e. f(r) and the stress components obtained from f are:
sr ¼
1 d4 r dr
(15a)
sq ¼
d2 4 dr 2
(15b)
trq ¼ 0 Therefore from Eqs. (15a) and (15b) can be inferred that:
sr ¼ (14a)
(15c)
1 r
Z
sq dr
(16)
A
After equating Eq. (16) the following equation to determine radial stress is obtained:
Fig. 6. Four-point bending configuration of the curved beam considering a constant curvature.
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2 MðqÞ ðrNA z þ rÞ ðrNA 3z rÞ þ r 2ðr sr ¼ 2zÞ NA r p r4 (17)
Table 2 Testing parameters for bending tests: three-point bending (3-PB), four-point bending one-third span (4-PB-1/3), and four-point bending one-half span (4-PB-1/ 2). Bending set-up
So (mm)
Si (mm)
So/(2r) ()
d_ LP (mm/min)
3-PB
64 96 128 200 72 96 128 200 64 96 128 200
0 0 0 0 24 32 42.67 66.67 32 48 64 100
8 12 16 25 9 12 16 25 8 12 16 25
0.533e0.747 1.200e1.680 2.133e2.987 5.208e7.292 0.750e1.050 1.333e1.867 2.370e3.319 5.787e8.102 0.533e0.747 1.200e1.680 2.133e2.987 5.208e7.292
3. Experimental work 4-PB-1/3
3.1. Material and apparatus Unidirectional fibre reinforced pultruded rods with solid crosssection were considered for experimental work due to the highly anisotropic mechanical behaviour. Moreover, circular section has been selected in order to approach the failures observed in Nature, which have been pointed out in Section 1. Composite circular solid cross-section rods, consisted of vinylester resin reinforced with longitudinally oriented E-glass fibre, were fabricated and kindly provided by Abeki Composites, S.L. Pultruded rods had a nominal diameter of 8 mm. A universal testing machine MTS - Insight 10 with a 10 kN load cell and standard bending test fixtures was used.
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4-PB-1/2
3.2. Fibre volume fraction The fibre volume fraction of the laminates has been determined through the ignition loss method (or burn-out technique) according to ASTM D 2584 - 11 [12]. The determination of fibre volume fraction requires the measurement of the composite density, which was measured in accordance with ASTM D 792 - 08 [13], considering that the void content is negligible. Experimental results reveal that fibre volume fraction and density of composite rods were 58.6% and 2.02 g/cm3, respectively. 3.3. Bending test Three-point bending (3-PB), four-point bending one-third span (4-PB-1/3), and four-point bending one-half span (4-PB-1/2) tests have been performed in initially straight beams, considering different support span-to-diameter ratios. The nominal support span So, the nominal loading span Si, and support span-to-diameter ratios So/(2r) considered are consigned in Table 2, as depicted in Fig. 4. For determining circumferential strength Xq Eq. (2) has been considered:
Xq ¼ k1
Pmax So
p r3
(18)
Five specimens were loaded until first-ply failure defined as the first deviation from linear load-deflection curve, as presented in Fig. 7, where failure load corresponds with maximum load Pmax. For all strength tests a support span So ¼ 96 mm has been taken. In composite pultruded rods the flexural load-deflection curves are typical linear and elastic [14] up to the first breakage of the outer fibers in tension [15]. A minimum overhang of 10% of the support span at each extreme fixture, in accordance with ASTM D 4476 - 09 [16], must be taken into account. The load was applied at a stress rate of 250e350 MPa/min in the outer fibres of the specimen as recommended by ACI 440 for tensile tests of FRP rods [17]. The ranges of loading point displacement rates d_ LP are presented in Table 2. For determining circumferential flexure modulus of elasticity E1, five specimens were loaded considering as strain limits 0.1% and 0.3%. Regarding the stiffness of circular-cross section composite pultruded rods, the deflection at the mid-span dMS and the
Fig. 7. Typical load-displacement curves (L-d) obtained for each bending set-up: 3-PB (continuous thin line), 4-PB-1/3 (continuous thick line), and 4-PB-1/2 (dashed thick line). Support span So ¼ 96 mm.
deflection at loading points dMS, can be derived assuming that CBT applies [11] considering the effect of both bending moment and shear force, having a shear correction factor for circular-cross sections equal to 10/9 [18]:
P S3o P So þ k3 E1 I GA
(19)
P S3o P So þ k3 E1 I GA
(20)
dMS ¼ k2
dLP ¼ k4
where k2, k3, and k4 depends on bending set-up, as displayed in Table 1, E1 is the flexure modulus of elasticity on the fibre direction (circumferential), G is the in-plane shear modulus, I is the moment of inertia with respect to the middle plane, and A is the crosssectional area. According to Eq. (20) it is possible to obtain experimentally the circumferential modulus E1 and the in-plane shear modulus G, considering two different support spans (solving a 2 2 system of equations) based on the Applied load vs. Loading point deflection curve. For circular-cross section beams, the slope dLP/P can be derived from Eq. (20) as follows:
dLP P
¼
k5 S3o
p r4
"
1 E1
2 # r 1 þ k6 G So
(21)
where k5 and k6 vary for each flexure set-up, as given in Table 1.
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mainly due to the bending moment, and that the influence of the axial load, considered by the beam curvature, is negligible. As a consequence, it would be possible to consider that the neutral axis coincides with the centroidal axis (i.e. z ¼ 0, z ¼ z’).
4. Discussion and results 4.1. Circumferential strength Experimental results of circumferential strength Xq, given at first rupture of outer fibres, are consigned in Table 3. In spite of different testing set-up was used for determining circumferential strength, 3-PB, 4-PB-1/3, and 4-PB-1/2, results reveal that there is no appreciable difference between those experimental conditions (i.e. support span So ¼ 96 mm) according to CBT, without considering shear effects and taking into account an initially straight beam. 4.2. Circumferential stiffness Circumferential flexure modulus E1 experimental results, consigned in Table 3, reveal that there is no appreciable difference between the considered test bending conditions (i.e. 3-PB, 4-PB 1/3, 4-PB 1/2). Hence, it can be concluded that the analysis carried out by CBT, taking into account a straight beam, can be considered suitable. However, shear modulus G experimental results (Table 3) exhibit a considerable difference (nearly 20%) between the three test bending conditions. This fact is due to shear effects. Contrasting shear force diagrams (Fig. 4), it can be inferred that as loading span increases shear effects decrease. It can also be noticed comparing shear deflection with bending deflection, the two terms on Eqs. (19) and (20). On that sense, shear effects are higher on three-point bending and they are lower in four-point bending one-half span. Regarding this deflection of the composite beams, expressed on Eqs. (19) and (20), there are two components: one due to bending moment and other one due to shear force. Shear deflection, compared to bending one, can be considered negligible for a support span-to-diameter ratio equal to 25 and a flexure modulus-toshear modulus ratio higher than 40 [19]. After that, the beam deflection is mainly due to bending load. 4.3. Curved beam As presented in section 2.3. the bending curvature of the initially straight beam has been considered for calculating radial stresses. In order to study the deformed curved beam, some results are presented in Table 4: the maximum displacement at loading points-tospan ratio (dLPmax/So), and the maximum displacement at midspan-to-span ratio (dMSmax/So). Those results indicate that most of the tests have been performed under small displacements condition which requires a dmax/So ratio less than 10% [20]. Nevertheless, the 4PB-1/2 tests with So/(2r) ¼ 25 have been carried out under great displacements condition (dMSmax/So ¼ 11.67%). In a bending situation under great displacements the radial stress has a significant influence. Furthermore, geometrical results that deal with the curved beam model, as the location of the neutral axis z, and the radius of curvature of the neutral axis-to-diameter ratio rNA/(2r) are also displayed in Table 4. As presented, the values of z are three order of magnitude lesser than the thickness (diameter, 2r ¼ 8 mm) of the beam. From a mechanical point of view, it means that stresses are
Table 3 Experimental results related to straight beam: mechanical properties. Bending set-up
Xq (MPa)
E1 (MPa)
G (MPa)
3-PB 4-PB-1/3 4-PB-1/2
814 ± 7 866 ± 24 790 ± 14
50292 ± 479 50898 ± 361 51070 ± 110
1221 ± 64 1148 ± 45 1005 ± 14
4.4. Bending moments For studying the difference between the bending moments at the mid-span MMS and at the loading points MLP, presented in Eq. (13), which would confirm the constant curvature hypothesis, the ratio MMS/MLP has been calculated and consigned in Table 4. Results point out that differences between MMS and MLP are not significant, i.e. up to 1%, except in the 4PB-1/2 tests condition with So/(2r) ¼ 25, where the tests have been performed under a great displacements situation. Besides, the bending moments are compared considering both the CBT model (straight beam) and the curved beam model. The bending moment at the mid-span MMS, as presented in Eq. (13a), is compared with the maximum bending moment MCBT calculated with the CBT, as stated in Eq. (1b). Bending moment results point out that in most of the loading cases both moments are similar, because the contribution of the load considering the beam as a curved element is negligible due to the small displacements condition. Nevertheless, in the 4PB-1/2 tests condition with So/ (2r) ¼ 25 the bending moment MMS of the curved beam model is 50% greater than the CBT bending moment MCBT, because the bending caused by the reaction load due to the beam deflection is significant. 4.5. Stress components The stress components calculated with the curved beam model,
sq and sr, based on Eqs. (8) and (17) respectively, depends on the bending moment M. As in the curved beam model the bending moment is maximum at the mid-span point, the circumferential and the radial normal stresses are also maximum at that point. Consequently the failure of the beam starts at that point. Furthermore, at the mid-span point the shear load Q is zero. On that sense, the distribution along the diameter of the beam of this two stress components, sq (Figs. 8 and 9) and sr (Figs. 10 and 11) is presented. Those stress results are exhibited for two testing conditions: 3-PB, Figs. 8 and 10, and 4-PB one-half span, Figs. 9 and 11, taking into account that the 4-PB one-third span set-up is an intermediate testing condition. Those results correspond to two support span-todiameter ratios, So/(2r) ¼ 16 and 25, in order to compare a testing set-up with small displacements, and another one with a great displacements condition, where radial stresses have greater influence. In the case of circumferential stress sq a comparison between straight beam, Eq. (2), and curved beam, Eq. (8), is displayed. The variation of circumferential stress sq through the diameter of the beam, i.e. considering dimensionless radial coordinate (z’/r), for both the 3-PB set-up and the 4-PB-1/2 span set-up is shown in Figs. 8 and 9, respectively. In order to compare circumferential stresses, the sq/sqCBT ratio, i.e. circumferential stress for curved beam model -to- circumferential stress for straight beam model ratio, has been calculated (not presented). This stress ratio is almost constant along the beam thickness (diameter). Circumferential stress differences at outer fibres (i.e. z’ ¼ ±r) are small, sq/sqCBT ratio up to 1.09, except for the 4-PB-1/2 span condition with So/(2r) ¼ 25, with sq/sqCBT ratio up to 1.51, due to the difference on bending moment stated in Section 4.4. Concerning sr stress, the curved beam model proposed in this work, Eq. (17), is compared with the model proposed by Ennos and van Casteren [1]:
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Table 4 Experimental results related to curved beam: displacements, bending moments and geometrical parameters. Bending set-up
So/(2r) ()
dLPmax/So (%)
dMSmax/So (%)
MMS/MLP ()
MMS/MCBT ()
z (mm)
rNA/(2r) ()
3-PB
16 25 16 16 25
5.8% 8.9% 6.6% 5.4% 8.6%
5.8% 8.9% 7.5% 7.3% 11.7%
1.01 1.00 1.01 1.01 1.16
1.03 1.08 1.07 1.08 1.50
0.0161 0.0220 0.0187 0.0174 0.0276
30.97 22.77 26.73 28.81 18.08
4-PB-1/3 4-PB-1/2
Fig. 8. Circumferential stress sq distribution along the diameter of the beam for the 3PB set-up: curved beam model (continuous lines), straight beam model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
Fig. 10. Radial stress sr (in compression) distribution along the diameter of the beam for the 3-PB set-up: curved beam model (continuous lines), Ref [1] model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
Fig. 9. Circumferential stress sq distribution along the diameter of the beam for the 4PB one-half span set-up: curved beam model (continuous lines), straight beam model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
Fig. 11. Radial stress sr (in compression) distribution along the diameter of the beam for the 4-PB one-half span set-up: curved beam model (continuous lines), Ref [1] model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
4 MðqÞ 2 02 sr ¼ r z 3p r 4 r G
(22)
For both models, radial stresses are compressive when the bending tends to increase curvature, as in the case of both 3-PB and 4-PB tests. The distributions along the diameter of the beam for 3PB and 4-PB-1/2 span configurations indicate that sr is maximum at neutral axis, and that is zero at inner and outer fibre, as presented in Figs. 10 and 11. Comparing curved beam and Ref [1] beam models, the differences between point-to-point stress distributions are similar through the entire beam diameter. In most cases, sr stresses with the curved beam model are two orders of magnitude greater compared to the radial stresses obtained with the model proposed in Ref. [1]. On that sense, the proposed curved beam model gives greater importance to the influence of compressive radial stresses
for predicting a greenstick fracture mode. Those differences are due to different assumptions. The model proposed by Ennos and van Casteren [1] takes a curved beam bent to increase its curvature, considering that the tension in the outer half of the beam develops inward pressure on the material inside it, and the compression in the inner half establishes outward pressure on the material outside it. The model proposed in this work, instead, calculates stress components from a stress function that satisfy equilibrium conditions in the considered domain.
4.6. Failure mode Figs. 12e14 show the two flexure failure modes presented in FRP rods after bending tests. Fig. 12 displays a brittle failure observed in all 3-PB tests for all support spans, having dMSmax/So ratios up to
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Fig. 12. Brittle failure mode of pultruded rods under 3-PB, So/(2r) ¼ 25.
Fig. 13. Brittle failure mode of pultruded rods under 4-PB one-half span, So/(2r) ¼ 16.
Fig. 14. Ductile failure mode of pultruded rods under 4-PB one-half span, So/(2r) ¼ 25.
8.9%. In the case of 4-PB with one-half span the failure mode was brittle in the case of low support span (see Fig. 13), i.e. So/(2r) ¼ 16, and ductile for the high support span, i.e. So/(2r) ¼ 25, as presented in Fig. 14. The ductile mode was characterized by dMSmax/So ratios up to 11.7%, a feature that reveal a great displacements situation. The brittle failure of the specimen starts with rupture of fibres on the convex side of the loading section; subsequently such rupture extended into a vertical crack toward the concave side. On ductile failure mode the tensile stress on the convex side develops the rupture of fibres; afterwards broken fibres delaminate promoted by the combined effect of tensile circumferential stresses sq and compressive radial stresses sr. This state of stresses originates a significant shear stress oriented 45 with respect to circumferential direction, as presented in Fig. 15. Shear stress is greater on the outer (convex) half of the beam than on the inner (concave) half, as graphically explained in Fig. 15. Outer shear stress, with compressive radial and tensile circumferential stresses, is greater than inner shear stress, with both compressive stresses: radial and circumferential. In fact, delamination takes place on outer (convex) half of the beam, as can be seen in Figs. 2, 3, 13 and 14. The ductile failure mode is known as greenstick fracture: limited fibre ruptures on the convex tensile side followed by large longitudinal cracks along the centre line [1].
5. Conclusions In the present contribution analytical approaches an experimental work for studying beams of unidirectional fibre reinforced plastic subjected to flexural loading have been presented, considering two models: an initially straight beam model, based on classical beam theory, and a proposed curved beam model. The aim of the paper was to evaluate the greenstick fracture mode presented in some anisotropic beams such as plant branches, young mammal bones and composite pultruded rods with longitudinally oriented fibres. The analytical approach initially deals with the study of the circumferential normal stress sq. Subsequently, the curved beam model assuming a constant curvature is presented. Finally, the Airy's stress function for obtaining the radial stress sr, and the shear stress trq is offered, bearing in mind that stress components obtained from this stress function satisfy equilibrium conditions in the considered domain. Experimental work has been performed under three-point and four-point bending conditions, and several support span-todiameter ratios have been considered. Brittle failures, characterized by a rupture of fibres that starts on the bottom side of the loading section and quickly develops into a vertical crack toward the upper side, have been presented in all tested specimens under 3-PB loading. 4-PB one-half span specimens presented two failure
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References
Fig. 15. Radial sr and circumferential sq stresses. (a) Stress distributions through the beam thickness (diameter). (b) Inner (concave) half of the beam subjected to compressive radial and compressive circumferential stresses. (c) Outer (convex) half of the beam subjected to tensile radial and compressive circumferential stresses.
modes: brittle mode for a small support span, So/(2r) ¼ 16, and ductile one for a high support span, So/(2r) ¼ 25. That ductile failure was obtained under a great displacements condition, where shear stresses oriented 45 with respect to circumferential direction is developed by the combination of tensile circumferential and compressive radial stresses. Such ductile failure is known as greenstick fracture, where limited fibre ruptures on the convex tensile side are followed by large longitudinal cracks along the centre line.
[1] Ennos AR, van Casteren A. Transverse stresses and modes of failure in tree branches and other beams. Proc Roy Soc Lond B Biol Sci 2010;277:1253e8. [2] Reiterer A, Burgert I, Sinn G, Tschegg S. The radial reinforcement of the wood structure and its implication on mechanical and fracture mechanical properties - a comparison between two tree species. J Mat Sci 2002;37(5):935e40. [3] Plomion C, Leprovost G, Stokes A. Wood formation in trees. Plant Physiol 2001;127:1513e23. [4] Seto J, Gupta HS, Zaslansky P, Wagner HD, Fratzl P. Tough lessons from bone: extreme mechanical anisotropy at the mesoscale. Adv Funct Mater 2008;18: 1905e11. [5] Dunlop JWC, Fratzl P. Biological composites. Annu Rev Mater Res 2010;40: 1e24. [6] Wagermaier W, Klaushofer K, Fratzl P. Fragility of bone material controlled by internal interfaces. Calcif Tissue Int 2015;97:201e12. [7] Silva FGA, de Moura MFSF, Dourado N, Xavier J, Pereira FAM, Morais JJL, et al. Mixed-mode IþII fracture characterization of human cortical bone using the Single Leg Bending test. J Mech Behav Biomed Mater 2016;54:72e81. [8] Lee P, Hunter TB, Taljanovic M. Musculoskeletal colloquialisms: how did we come up with these names? RadioGraphics 2004;24(4):1009e27. [9] Daniel IM, Ishai O. Engineering mechanics of composite materials. second ed. New York: Oxford University; 2006. [10] Vargas G, Arrese A, Carbajal N, Mujika F. Analysis of in-plane and out-of-plane thermo-mechanical stresses in un-symmetric cross-ply curved laminated strips. J Compos Mater 2009;43:3157e84. [11] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill; 1951. [12] ASTM. ASTM Standard D2584e11: standard test method for ignition loss of cured reinforced resins. West Conshohocken (PA): ASTM International; 2011. [13] ASTM. ASTM Standard D792e08: standard test methods for density and specific gravity (relative density) of plastics by displacement. West Conshohocken (PA): ASTM International; 2008. [14] Chaallal O, Benmokrane B. Fiber-reinforced plastic rebars for concrete applications. Compos B 1996;27B:245e52. [15] Zhang B, Masmoudi R, Benmokrane B. New method for testing fiberreinforced polymer rods under flexure. J Test Eval 2006;35:1e6. [16] ASTM. ASTM Standard D4476e09: standard test method for flexural properties of fiber-reinforced pultruded plastic rods. West Conshohocken (PA): ASTM International; 2009. [17] ACI. ACI 440K: guide test methods for fiber-reinforced plastic (FRP) rods and sheets. Farmington Hills (MI): American Concrete Institute; 2004. [18] Pilkey WD. Formulas for stress, strain, and structural matrices. second ed. Hoboken: Wiley; 2005. [19] Mujika F. On the difference between flexural moduli obtained by three-point and four-point bending tests. Polym Test 2006;25:214e20. [20] ISO. ISO 14125:1998: fibre-reinforced plastic composites, Determination of flexural properties. Geneva (Switzerland): International Organization for Standardization; 1998.
Contents lists available at ScienceDirect
Composites Part B journal homepage: www.elsevier.com/locate/compositesb
Greenstick fracture in composite pultruded rods G. Vargas*, F. Mujika ‘Materials þ Technologies’ Group, Department of Mechanical Engineering, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Plaza n, Spain Europa 1, 20018. Donostia-San Sebastia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 August 2016 Received in revised form 21 October 2016 Accepted 1 November 2016 Available online 4 November 2016
The paper analyses the greenstick fracture mode on anisotropic beams subjected to flexural loading. This fracture mode can be seen in plant branches, young mammal bones or reinforced plastic rods. Two analytical models have been studied: a straight beam model and a uniform curved beam model. Circumferential normal stress distribution has been determined considering the classical beam theory, and radial and shear stresses have been predicted by using the Airy's stress function. Experimental work has been implemented on initially straight pultruded rods with longitudinally oriented fibres. Three and four-point bending tests adopting several span-to-diameter ratios have been performed. Experimental conditions that promote a greenstick fracture mode in composite pultruded rods have been considered. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Fracture Analytical modeling Mechanical testing Pultrusion
1. Introduction Considering a solid cross-section curved beam subjected to flexural loading, the plane stress state is composed by circumferential normal stress sq, radial normal stress sr, and shear stress trq, as presented in Fig. 1. In beams composed of isotropic materials radial stresses are negligible compared with circumferential ones, and they do not affect the mechanism of failure. Therefore, traditional analysis of isotropic beams do not take into account these radial stresses. Nevertheless, radial stresses must be considered when beams composed of anisotropic materials are studied. Unidirectional fibre reinforced composites, both natural and man-made, are stronger and stiffer longitudinally than transversely, and consequently radial stresses due to bending can produce catastrophic failure. Thus, such radial stresses could enlighten the greenstick fracture behaviour of some anisotropic beams such as green branches and twigs, bones of young mammals, and uniaxially reinforced composite beams. A greenstick fracture is a partial, angled fracture where the convex side ruptures (without the fracture line traversing the beam), and the concave side bends [1]. On Fig. 2 a greenstick fracture of a green branch is presented.
Abbreviations: CBT, Classical beam theory; 1/3, One-third span; 1/2, One-half span; 3-PB, Three-point bending; 4-PB, Four-point bending. * Corresponding author. E-mail address: [email protected] (G. Vargas). http://dx.doi.org/10.1016/j.compositesb.2016.11.001 1359-8368/© 2016 Elsevier Ltd. All rights reserved.
Wooden branches and trunks have greater longitudinal than transverse strength due to the alignment of the tracheids (fibre cells) along the stem axis. The wood fibres are oriented in the longitudinal direction; their cell wall structure is optimized to provide sufficient mechanical properties regarding the main loading conditions of the living tree (e.g. bending of the stem due to wind loads). In both the radial and tangential directions (i.e. the transverse plane) the mechanical properties of wood are an order of magnitude lower compared to the longitudinal direction [2]. In trees, secondary vascular tissues on the inner side are wood elements, mostly tracheids in gymnosperms, but also vessel elements, vesselassociated cells, axial parenchyma and fibers in dicotyledons. On the outer side, secondary vascular tissues are phloem cells, that is, sieve tubes, and, in dicotyledons, companion cells, axial parenchyma, and fibers [3]. Bone is mechanically and structurally anisotropic with oriented collagen fibrils and nanometer-sized mineral particles aggregating into lamellar or woven bone [4,5]. In particular, cortical bone is a naturally occurring composite material comprising mineral (essentially hydroxyapatite) and organic constituents (collagen and bone cells): within the collagen fibrils, the mineral is embedded in the form of thin hydroxyapatite platelets. Their long axis is mainly parallel to the long axis of the collagen fibrils. At a higher level of the hierarchy, several collagen molecules form fibrils, and the interface layer between neighboring fibrils is filled with extrafibrillar matrix containing noncollagenous proteins [6]. The mineralized collagen fibril represents the basic building block of bone, and can be arranged in different ways, forming fiber bundles
G. Vargas, F. Mujika / Composites Part B 110 (2017) 106e115
Nomenclature A E F I k M N P Q r S z z
cross-sectional area modulus reaction force at supports of a simply-supported beam moment of inertia with respect to the middle plane constant bending moment normal load applied load shear force radial coordinate, radius of curvature span out-of-plane coordinate location of the neutral axis
Greek symbols d deflection d_ displacement rate
ε
107
strain Airy's stress function in polar coordinates circumferential coordinate/arc angle radius of the circular-cross section normal stress shear stress
f q r s t
Subscripts 1 longitudinal direction CBT classical beam theory G centroidal axis, evaluated at z’ ¼ 0 i inner (between loading points) LP loading points max maximum MS mid-span NA neutral axis, evaluated at z ¼ 0 o outer (between supporting points) r radial direction q circumferential direction
ratios as large as 1:15 (wet bone) and 1:7 (dry bone) in tensile strength between orientations perpendicular and parallel to the main collagen fiber orientation. Man-made composite unidirectional fibre reinforced plastics are also highly anisotropic materials with longitudinal-to-transverse strength ratios up to 40:1 [9]. In the present study, analytical approaches and experimental work for studying greenstick fracture mode of anisotropic beams subjected to flexural loading is presented. Analytical approach is a geometrically nonlinear solution for a deflecting beam, which is initially straight, that considers all stress components: normal radial and shear stresses, that have been predicted by using Airy's stress function, previously determining circumferential normal stresses by classical beam theory. Mechanical testing has been carried out by both three-point bending and four-point bending tests on circular solid cross-section pultruded rods considered as transversely isotropic along the longitudinal axis.
2. Analytical approach 2.1. Classical beam theory Fig. 1. Stress components in polar coordinates r,q in the case of plane stress: circumferential stress sq radial stress sr, and, and shear stress trq.
with varying degrees of organization. On the one hand, in adult human cortical bone, the main microstructural features are the osteons, which are near-cylindrical structures formed by concentric lamellae. These lamellae structures surround the Harversian canals, with the outer boundary separating the osteons from an interstitial matrix called cement line [7]. On the other hand, woven bone is characterized by randomly oriented fibril arrays. This is typically found in quickly formed bone, for example, in embryonic or young bones and at repair sites after fractures. It is a less mineralized and mechanically weakest bone type, and often substituted during remodeling by other more organized bone types [5]. Fig. 3 displays greenstick incomplete fractures of young bones: radius and ulna [8]. Seto et al. [4] have measured the mechanical anisotropy of the mineralized fiber bundles in fibrolamellar bovine bone in the periosteal region (outer surface) of long bones, finding anisotropy
Circumferential normal stress sq has been considered for both three-point bending (3-PB) and four-point bending (4-PB). In Fig. 4 the loading configuration for 4-PB is presented as well as the shear force Q and the bending moment M diagrams, considering an applied load P, a circular-cross section specimen with radius r, a support span So, and a loading span Si. Shear force Q and bending moment M are:
Q¼
P 2
(1a)
M¼
P ðSo Si Þ 4
(1b)
Three bending configurations have been analyzed: 3-PB set-up achieved for Si ¼ 0, 4-PB one-third span set-up completed for Si ¼ So/3, and 4-PB one-half span set-up attained for Si ¼ So/2. According to classical beam theory (CBT), the maximum circumferential stress sqmax is:
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Fig. 2. The so-called greenstick fracture of a branch: a break on the convex side before splitting down the middle, with large longitudinal cracks along the centre line.
Fig. 3. Frontal (a) and lateral (b) greenstick fractures of the radius and ulna of a 5-yearold child. Fully broken cortex (black arrows) on one side, and intact bone on the other side (white arrows). From Lee et al. [4] with permission.
sqmax ¼ k1
P So
p r3
(2)
where k1 depends on bending set-up, as presented in Table 1.
Fig. 4. Four-point bending configuration, and the shear force Q and the bending moment M diagrams.
fibre reinforced composite beam with transverse isotropy along the longitudinal axis is assumed. Circumferential strain in q-direction εq, is a linear function of the radial coordinate taken from the neutral axis z (see Fig. 5), and
2.2. Curved beam The theoretical analysis, that considers an initially straight beam, is a geometrically nonlinear solution for a deflecting curved beam that considers the following assumptions: The layers of the beam are made of a linearly elastic and orthotropic material whose principal axes of orthotropy coincide with the considered coordinate system axes. In particular, a
Table 1 Constants related to analytical approach for bending configurations: three-point bending (3-PB), four-point bending one-third span (4-PB-1/3), and four-point bending one-half span (4-PB-1/2). Bending set-up
k1
k2
k3
k4
k5
k6
3-PB 4-PB-1/3 4-PB-1/2
1 2/3 1/2
1/48 23/1296 11/768
5/18 5/27 5/36
1/48 5/324 1/96
1/12 5/81 1/24
10/3 3 10/3
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depends on the radius of curvature of the neutral axis rNA (evaluated at z ¼ 0). It can be expressed as:
εq ¼
z rNA
(3)
Mechanical and thermal properties, into the range considered, are temperature-independent. Free-edge effects are not considered. Normal stresses in the direction perpendicular to plane r,q (i.e. z,q) are negligible.
109
calculated as function of both the flexure modulus of elasticity E1 and the radius of curvature rNA:
MðqÞ ¼
p E1 4 rNA
r4
(7)
Applying Hooke's law, and equating Eq. (3) and Eq. (7), the circumferential stress sq can be expressed as function of the radial coordinate r, taking into account that z ¼ r e rNA (see Fig. 5).
sq ¼
4 MðqÞ
p r4
ðr rNA Þ
(8)
Due to the loading condition of the beam, the neutral axis does not coincide with the cross section centroid axis. The radial coordinate taken from the centroid line z’, is adopted as reference [10], considering the following relation according to Fig. 5:
To determine the location of the neutral axis z, defined as the line in the cross-section of the beam where the circumferential strain is zero (εq ¼ 0), Eqs. (5a) and (5b) must be equated, finding respectively that:
z ¼ z0 z
1
(4)
where z is the distance between the neutral axis and the centroid line. Due to the vector system equivalence, normal load N(q) and bending moment M(q), are resultant force and resultant moment of normal stresses, respectively, both applied at the centroidal axis G (at z’ ¼ 0). This equivalence, considering Eq. (3) and in combination with the Hooke's law, can be expressed as:
Z A
Z
Z z sq dA ¼ NðqÞ0 E1 dA ¼ NðqÞ rNA Z
A
A
z 0 z dA ¼ MðqÞ rNA
1
NðqÞ p E1 z r2
¼
r
4 MðqÞ p E1 r4
(9a)
(9b)
From Eqs. (9a) and (9b) can be inferred that:
z¼
r2 NðqÞ 4 MðqÞ
(10)
For a simply-supported beam subjected to transverse point loads, as in the case of 3-PB and 4-PB, the following system of internal forces are generated [10]:
(5b)
NðqÞ ¼ F Sin q
(11a)
Q ðqÞ ¼ F Cos q
(11b)
MðqÞ ¼ rNA F Sin q
(11c)
where E1 is the circumferential modulus of elasticity. From Eq. (5b) the radius of curvature of the neutral axis rNA can be derived:
1 MðqÞ ¼ rNA E1 I
¼
(5a)
A
sq z0 dA ¼ MðqÞ0 E1
r
(6)
where I is the moment of inertia with respect to the middle plane. Moreover, from Eq. (6) the bending moment M(q) can be
where N(q) is the normal load, Q(q) is the shear load, M(q) is the bending moment, q is the arc angle, rNA is the radius of curvature of the centroid line, and F is the reaction force at supports. Consequently, replacing Eqs. (11a) and (11c) in Eq. (10) the location of the neutral axis is obtained:
Fig. 5. Radial and circumferential variables of the beam. (a) r,q plane. (b) Section A-A, transverse cross-section.
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z¼
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r2
(12)
4 rNA
In both loading cases, 3-PB and 4-PB, the bending moments at the mid-span MMS and at the loading points MLP can be determined considering a curved beam, as depicted in Fig. 6. The curved beam is assumed to have a constant curvature equal to the curvature of the mid-span point, where the bending moment M is maximum and the shear load Q is zero. MMS and MLP are given by:
MMS ¼
P P ðSo Si Þ þ ½d Tan qo ðdMS dLP ÞTan qi 4 2 MS
(13a)
MLP ¼
P P ðSo Si Þ þ ðdLP Tan qo Þ 4 2
(13b)
where qo is the curved beam outer angle, measured from the midspan to the supporting point, and qi is the curved beam inner angle, measured from the mid-span to the loading point, as depicted in Fig. 6. In addition, the first term in Eq. (13) coincides with Eq. (1b) which corresponds to the CBT model considering the beam as a straight element, i.e. the bending due to the span. Besides, the second term in Eq. (13) indicates the bending contribution of the load considering the beam as a curved element, i.e. the bending due to the deflection. Moreover, in Eq. (13) can be noticed that bending moments MMS and MLP are equal when qo ¼ qi, as shown in Fig. 4.
2.3. Radial and shear stresses Airy's stress function f(r,q) in polar coordinates relates three components of stress: circumferential stress sq, radial stress sr, and shear stress trq, as depicted in Fig. 1. The stress components obtained from this stress function satisfy equilibrium conditions in the considered domain [11], and they can be expressed as follows:
1 v4 1 v2 4 þ 2 2 sr ¼ r vr r vq
sq ¼
v2 4 vr 2
(14b)
trq ¼
1 v4 1 v2 4 r 2 vq r vr vq
(14c)
As the circumferential stresses sq can be expressed as function of the radial coordinate r and circumferential coordinate q, see Eq. (8), it is possible to determine the Airy's stress function f in polar coordinates from Eq. (14b), accomplishing several integrations. Afterwards sr and trq stresses can also be determined from Eq. (11a) and Eq. (14c), respectively. According to the presented analytical model, once the beam is deformed, it features a uniform curvature which is equal to the curvature value obtained in the mid-span. Subsequently, the bending moment M does not vary with q, and, as will be presented in Section 4.4, the bending moments MMS and MLP can be considered as equivalent. Therefore, the Airy's stress function depends only on the radial variable r, i.e. f(r) and the stress components obtained from f are:
sr ¼
1 d4 r dr
(15a)
sq ¼
d2 4 dr 2
(15b)
trq ¼ 0 Therefore from Eqs. (15a) and (15b) can be inferred that:
sr ¼ (14a)
(15c)
1 r
Z
sq dr
(16)
A
After equating Eq. (16) the following equation to determine radial stress is obtained:
Fig. 6. Four-point bending configuration of the curved beam considering a constant curvature.
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2 MðqÞ ðrNA z þ rÞ ðrNA 3z rÞ þ r 2ðr sr ¼ 2zÞ NA r p r4 (17)
Table 2 Testing parameters for bending tests: three-point bending (3-PB), four-point bending one-third span (4-PB-1/3), and four-point bending one-half span (4-PB-1/ 2). Bending set-up
So (mm)
Si (mm)
So/(2r) ()
d_ LP (mm/min)
3-PB
64 96 128 200 72 96 128 200 64 96 128 200
0 0 0 0 24 32 42.67 66.67 32 48 64 100
8 12 16 25 9 12 16 25 8 12 16 25
0.533e0.747 1.200e1.680 2.133e2.987 5.208e7.292 0.750e1.050 1.333e1.867 2.370e3.319 5.787e8.102 0.533e0.747 1.200e1.680 2.133e2.987 5.208e7.292
3. Experimental work 4-PB-1/3
3.1. Material and apparatus Unidirectional fibre reinforced pultruded rods with solid crosssection were considered for experimental work due to the highly anisotropic mechanical behaviour. Moreover, circular section has been selected in order to approach the failures observed in Nature, which have been pointed out in Section 1. Composite circular solid cross-section rods, consisted of vinylester resin reinforced with longitudinally oriented E-glass fibre, were fabricated and kindly provided by Abeki Composites, S.L. Pultruded rods had a nominal diameter of 8 mm. A universal testing machine MTS - Insight 10 with a 10 kN load cell and standard bending test fixtures was used.
111
4-PB-1/2
3.2. Fibre volume fraction The fibre volume fraction of the laminates has been determined through the ignition loss method (or burn-out technique) according to ASTM D 2584 - 11 [12]. The determination of fibre volume fraction requires the measurement of the composite density, which was measured in accordance with ASTM D 792 - 08 [13], considering that the void content is negligible. Experimental results reveal that fibre volume fraction and density of composite rods were 58.6% and 2.02 g/cm3, respectively. 3.3. Bending test Three-point bending (3-PB), four-point bending one-third span (4-PB-1/3), and four-point bending one-half span (4-PB-1/2) tests have been performed in initially straight beams, considering different support span-to-diameter ratios. The nominal support span So, the nominal loading span Si, and support span-to-diameter ratios So/(2r) considered are consigned in Table 2, as depicted in Fig. 4. For determining circumferential strength Xq Eq. (2) has been considered:
Xq ¼ k1
Pmax So
p r3
(18)
Five specimens were loaded until first-ply failure defined as the first deviation from linear load-deflection curve, as presented in Fig. 7, where failure load corresponds with maximum load Pmax. For all strength tests a support span So ¼ 96 mm has been taken. In composite pultruded rods the flexural load-deflection curves are typical linear and elastic [14] up to the first breakage of the outer fibers in tension [15]. A minimum overhang of 10% of the support span at each extreme fixture, in accordance with ASTM D 4476 - 09 [16], must be taken into account. The load was applied at a stress rate of 250e350 MPa/min in the outer fibres of the specimen as recommended by ACI 440 for tensile tests of FRP rods [17]. The ranges of loading point displacement rates d_ LP are presented in Table 2. For determining circumferential flexure modulus of elasticity E1, five specimens were loaded considering as strain limits 0.1% and 0.3%. Regarding the stiffness of circular-cross section composite pultruded rods, the deflection at the mid-span dMS and the
Fig. 7. Typical load-displacement curves (L-d) obtained for each bending set-up: 3-PB (continuous thin line), 4-PB-1/3 (continuous thick line), and 4-PB-1/2 (dashed thick line). Support span So ¼ 96 mm.
deflection at loading points dMS, can be derived assuming that CBT applies [11] considering the effect of both bending moment and shear force, having a shear correction factor for circular-cross sections equal to 10/9 [18]:
P S3o P So þ k3 E1 I GA
(19)
P S3o P So þ k3 E1 I GA
(20)
dMS ¼ k2
dLP ¼ k4
where k2, k3, and k4 depends on bending set-up, as displayed in Table 1, E1 is the flexure modulus of elasticity on the fibre direction (circumferential), G is the in-plane shear modulus, I is the moment of inertia with respect to the middle plane, and A is the crosssectional area. According to Eq. (20) it is possible to obtain experimentally the circumferential modulus E1 and the in-plane shear modulus G, considering two different support spans (solving a 2 2 system of equations) based on the Applied load vs. Loading point deflection curve. For circular-cross section beams, the slope dLP/P can be derived from Eq. (20) as follows:
dLP P
¼
k5 S3o
p r4
"
1 E1
2 # r 1 þ k6 G So
(21)
where k5 and k6 vary for each flexure set-up, as given in Table 1.
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G. Vargas, F. Mujika / Composites Part B 110 (2017) 106e115
mainly due to the bending moment, and that the influence of the axial load, considered by the beam curvature, is negligible. As a consequence, it would be possible to consider that the neutral axis coincides with the centroidal axis (i.e. z ¼ 0, z ¼ z’).
4. Discussion and results 4.1. Circumferential strength Experimental results of circumferential strength Xq, given at first rupture of outer fibres, are consigned in Table 3. In spite of different testing set-up was used for determining circumferential strength, 3-PB, 4-PB-1/3, and 4-PB-1/2, results reveal that there is no appreciable difference between those experimental conditions (i.e. support span So ¼ 96 mm) according to CBT, without considering shear effects and taking into account an initially straight beam. 4.2. Circumferential stiffness Circumferential flexure modulus E1 experimental results, consigned in Table 3, reveal that there is no appreciable difference between the considered test bending conditions (i.e. 3-PB, 4-PB 1/3, 4-PB 1/2). Hence, it can be concluded that the analysis carried out by CBT, taking into account a straight beam, can be considered suitable. However, shear modulus G experimental results (Table 3) exhibit a considerable difference (nearly 20%) between the three test bending conditions. This fact is due to shear effects. Contrasting shear force diagrams (Fig. 4), it can be inferred that as loading span increases shear effects decrease. It can also be noticed comparing shear deflection with bending deflection, the two terms on Eqs. (19) and (20). On that sense, shear effects are higher on three-point bending and they are lower in four-point bending one-half span. Regarding this deflection of the composite beams, expressed on Eqs. (19) and (20), there are two components: one due to bending moment and other one due to shear force. Shear deflection, compared to bending one, can be considered negligible for a support span-to-diameter ratio equal to 25 and a flexure modulus-toshear modulus ratio higher than 40 [19]. After that, the beam deflection is mainly due to bending load. 4.3. Curved beam As presented in section 2.3. the bending curvature of the initially straight beam has been considered for calculating radial stresses. In order to study the deformed curved beam, some results are presented in Table 4: the maximum displacement at loading points-tospan ratio (dLPmax/So), and the maximum displacement at midspan-to-span ratio (dMSmax/So). Those results indicate that most of the tests have been performed under small displacements condition which requires a dmax/So ratio less than 10% [20]. Nevertheless, the 4PB-1/2 tests with So/(2r) ¼ 25 have been carried out under great displacements condition (dMSmax/So ¼ 11.67%). In a bending situation under great displacements the radial stress has a significant influence. Furthermore, geometrical results that deal with the curved beam model, as the location of the neutral axis z, and the radius of curvature of the neutral axis-to-diameter ratio rNA/(2r) are also displayed in Table 4. As presented, the values of z are three order of magnitude lesser than the thickness (diameter, 2r ¼ 8 mm) of the beam. From a mechanical point of view, it means that stresses are
Table 3 Experimental results related to straight beam: mechanical properties. Bending set-up
Xq (MPa)
E1 (MPa)
G (MPa)
3-PB 4-PB-1/3 4-PB-1/2
814 ± 7 866 ± 24 790 ± 14
50292 ± 479 50898 ± 361 51070 ± 110
1221 ± 64 1148 ± 45 1005 ± 14
4.4. Bending moments For studying the difference between the bending moments at the mid-span MMS and at the loading points MLP, presented in Eq. (13), which would confirm the constant curvature hypothesis, the ratio MMS/MLP has been calculated and consigned in Table 4. Results point out that differences between MMS and MLP are not significant, i.e. up to 1%, except in the 4PB-1/2 tests condition with So/(2r) ¼ 25, where the tests have been performed under a great displacements situation. Besides, the bending moments are compared considering both the CBT model (straight beam) and the curved beam model. The bending moment at the mid-span MMS, as presented in Eq. (13a), is compared with the maximum bending moment MCBT calculated with the CBT, as stated in Eq. (1b). Bending moment results point out that in most of the loading cases both moments are similar, because the contribution of the load considering the beam as a curved element is negligible due to the small displacements condition. Nevertheless, in the 4PB-1/2 tests condition with So/ (2r) ¼ 25 the bending moment MMS of the curved beam model is 50% greater than the CBT bending moment MCBT, because the bending caused by the reaction load due to the beam deflection is significant. 4.5. Stress components The stress components calculated with the curved beam model,
sq and sr, based on Eqs. (8) and (17) respectively, depends on the bending moment M. As in the curved beam model the bending moment is maximum at the mid-span point, the circumferential and the radial normal stresses are also maximum at that point. Consequently the failure of the beam starts at that point. Furthermore, at the mid-span point the shear load Q is zero. On that sense, the distribution along the diameter of the beam of this two stress components, sq (Figs. 8 and 9) and sr (Figs. 10 and 11) is presented. Those stress results are exhibited for two testing conditions: 3-PB, Figs. 8 and 10, and 4-PB one-half span, Figs. 9 and 11, taking into account that the 4-PB one-third span set-up is an intermediate testing condition. Those results correspond to two support span-todiameter ratios, So/(2r) ¼ 16 and 25, in order to compare a testing set-up with small displacements, and another one with a great displacements condition, where radial stresses have greater influence. In the case of circumferential stress sq a comparison between straight beam, Eq. (2), and curved beam, Eq. (8), is displayed. The variation of circumferential stress sq through the diameter of the beam, i.e. considering dimensionless radial coordinate (z’/r), for both the 3-PB set-up and the 4-PB-1/2 span set-up is shown in Figs. 8 and 9, respectively. In order to compare circumferential stresses, the sq/sqCBT ratio, i.e. circumferential stress for curved beam model -to- circumferential stress for straight beam model ratio, has been calculated (not presented). This stress ratio is almost constant along the beam thickness (diameter). Circumferential stress differences at outer fibres (i.e. z’ ¼ ±r) are small, sq/sqCBT ratio up to 1.09, except for the 4-PB-1/2 span condition with So/(2r) ¼ 25, with sq/sqCBT ratio up to 1.51, due to the difference on bending moment stated in Section 4.4. Concerning sr stress, the curved beam model proposed in this work, Eq. (17), is compared with the model proposed by Ennos and van Casteren [1]:
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Table 4 Experimental results related to curved beam: displacements, bending moments and geometrical parameters. Bending set-up
So/(2r) ()
dLPmax/So (%)
dMSmax/So (%)
MMS/MLP ()
MMS/MCBT ()
z (mm)
rNA/(2r) ()
3-PB
16 25 16 16 25
5.8% 8.9% 6.6% 5.4% 8.6%
5.8% 8.9% 7.5% 7.3% 11.7%
1.01 1.00 1.01 1.01 1.16
1.03 1.08 1.07 1.08 1.50
0.0161 0.0220 0.0187 0.0174 0.0276
30.97 22.77 26.73 28.81 18.08
4-PB-1/3 4-PB-1/2
Fig. 8. Circumferential stress sq distribution along the diameter of the beam for the 3PB set-up: curved beam model (continuous lines), straight beam model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
Fig. 10. Radial stress sr (in compression) distribution along the diameter of the beam for the 3-PB set-up: curved beam model (continuous lines), Ref [1] model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
Fig. 9. Circumferential stress sq distribution along the diameter of the beam for the 4PB one-half span set-up: curved beam model (continuous lines), straight beam model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
Fig. 11. Radial stress sr (in compression) distribution along the diameter of the beam for the 4-PB one-half span set-up: curved beam model (continuous lines), Ref [1] model (dashed lines). So/(2r) ¼ 16 (thin lines), So/(2r) ¼ 25 (thick lines).
4 MðqÞ 2 02 sr ¼ r z 3p r 4 r G
(22)
For both models, radial stresses are compressive when the bending tends to increase curvature, as in the case of both 3-PB and 4-PB tests. The distributions along the diameter of the beam for 3PB and 4-PB-1/2 span configurations indicate that sr is maximum at neutral axis, and that is zero at inner and outer fibre, as presented in Figs. 10 and 11. Comparing curved beam and Ref [1] beam models, the differences between point-to-point stress distributions are similar through the entire beam diameter. In most cases, sr stresses with the curved beam model are two orders of magnitude greater compared to the radial stresses obtained with the model proposed in Ref. [1]. On that sense, the proposed curved beam model gives greater importance to the influence of compressive radial stresses
for predicting a greenstick fracture mode. Those differences are due to different assumptions. The model proposed by Ennos and van Casteren [1] takes a curved beam bent to increase its curvature, considering that the tension in the outer half of the beam develops inward pressure on the material inside it, and the compression in the inner half establishes outward pressure on the material outside it. The model proposed in this work, instead, calculates stress components from a stress function that satisfy equilibrium conditions in the considered domain.
4.6. Failure mode Figs. 12e14 show the two flexure failure modes presented in FRP rods after bending tests. Fig. 12 displays a brittle failure observed in all 3-PB tests for all support spans, having dMSmax/So ratios up to
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Fig. 12. Brittle failure mode of pultruded rods under 3-PB, So/(2r) ¼ 25.
Fig. 13. Brittle failure mode of pultruded rods under 4-PB one-half span, So/(2r) ¼ 16.
Fig. 14. Ductile failure mode of pultruded rods under 4-PB one-half span, So/(2r) ¼ 25.
8.9%. In the case of 4-PB with one-half span the failure mode was brittle in the case of low support span (see Fig. 13), i.e. So/(2r) ¼ 16, and ductile for the high support span, i.e. So/(2r) ¼ 25, as presented in Fig. 14. The ductile mode was characterized by dMSmax/So ratios up to 11.7%, a feature that reveal a great displacements situation. The brittle failure of the specimen starts with rupture of fibres on the convex side of the loading section; subsequently such rupture extended into a vertical crack toward the concave side. On ductile failure mode the tensile stress on the convex side develops the rupture of fibres; afterwards broken fibres delaminate promoted by the combined effect of tensile circumferential stresses sq and compressive radial stresses sr. This state of stresses originates a significant shear stress oriented 45 with respect to circumferential direction, as presented in Fig. 15. Shear stress is greater on the outer (convex) half of the beam than on the inner (concave) half, as graphically explained in Fig. 15. Outer shear stress, with compressive radial and tensile circumferential stresses, is greater than inner shear stress, with both compressive stresses: radial and circumferential. In fact, delamination takes place on outer (convex) half of the beam, as can be seen in Figs. 2, 3, 13 and 14. The ductile failure mode is known as greenstick fracture: limited fibre ruptures on the convex tensile side followed by large longitudinal cracks along the centre line [1].
5. Conclusions In the present contribution analytical approaches an experimental work for studying beams of unidirectional fibre reinforced plastic subjected to flexural loading have been presented, considering two models: an initially straight beam model, based on classical beam theory, and a proposed curved beam model. The aim of the paper was to evaluate the greenstick fracture mode presented in some anisotropic beams such as plant branches, young mammal bones and composite pultruded rods with longitudinally oriented fibres. The analytical approach initially deals with the study of the circumferential normal stress sq. Subsequently, the curved beam model assuming a constant curvature is presented. Finally, the Airy's stress function for obtaining the radial stress sr, and the shear stress trq is offered, bearing in mind that stress components obtained from this stress function satisfy equilibrium conditions in the considered domain. Experimental work has been performed under three-point and four-point bending conditions, and several support span-todiameter ratios have been considered. Brittle failures, characterized by a rupture of fibres that starts on the bottom side of the loading section and quickly develops into a vertical crack toward the upper side, have been presented in all tested specimens under 3-PB loading. 4-PB one-half span specimens presented two failure
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References
Fig. 15. Radial sr and circumferential sq stresses. (a) Stress distributions through the beam thickness (diameter). (b) Inner (concave) half of the beam subjected to compressive radial and compressive circumferential stresses. (c) Outer (convex) half of the beam subjected to tensile radial and compressive circumferential stresses.
modes: brittle mode for a small support span, So/(2r) ¼ 16, and ductile one for a high support span, So/(2r) ¼ 25. That ductile failure was obtained under a great displacements condition, where shear stresses oriented 45 with respect to circumferential direction is developed by the combination of tensile circumferential and compressive radial stresses. Such ductile failure is known as greenstick fracture, where limited fibre ruptures on the convex tensile side are followed by large longitudinal cracks along the centre line.
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