Knots in high strength fibres, including dyneema, can reduce the strength of the fibre by more than 50, thus a rope will tend to break where it comes out of a knot. Previous research showed that the properties of dyneema fibres change according to the tensile stress applied. So the objective of this project is to identify the structural changes within dyneema fibres due to loaded knots, and use them as a measure of the internal strain within a knot (2).
According to the Longman dictionary, a knot is a fastening of string, rope, wire etc by connecting together the ends of a piece or pieces of them (3). All knots weaken the breaking strength of ropes by about fifty percent except for the existence of defects. This was demonstrated by Richards (1). The reason for this weakness is considered to be complicated. The dissipation of strength could be affected by many factors such as strain rate, size and material of the fibres, even the tightness of the knot (1) (4).
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Figure 2 demonstrates the reduction in strength among different types of knots and materials. As far as I am concerned, the graph is not clear enough since there is no relationship between the various kinds of knots in the strength loss. So the line chart is inappropriate and should be converted into bar chart.
Vogel pointed out that a certain knot has a certain percentage of strength of the material known as knot efficiency, in the form of breaking strength of a knotted rope in proportion to the average strength of an unknotted rope (2) (5). It can be used to estimate the extent of reduction in tensile strength in the presence of a tied knot. According to Vogel, knot efficiency ranges from 100% when no failure at the knot occurs to about 50% in 5.5 Spectra/Titan cords (5).
Also, breaking strength refers to the applied load that a rope is able to withstand before rupture. Therefore, high breaking strength is a necessity and prerequisite for the design of climbing slings, fundamental tools used to make an anchor (6).
The effect of knots on tensile breaking strength
According to Uehara et al., knot geometry (crossing number) has a significant effect on the breaking mechanism of polymeric monofilaments (7). Crossing number represents the number of strands that cross. The simplest overhand knot, for instance, has a crossing number of 3 shown in Figure 3. Experiments performed by them revealed that with the increase in crossing number, there showed a steady decrease in tensile strength up to the crossing number of 9, followed by a level off at approximately 500MPa (7).
Figure 3 Appearance of thumb knot (7) (1, 2, 3 represents the crossing number)
crossing number vs.breaking strength.jpg
Figure 4 Tensile strength versus crossing number for various knots of PVDF monofilament, the value for the unknotted filament is indicated by the dotted line(7)
Breaking mechanism of knots
The reasons put forward by Uehara et al. are squeezing and rotation, characteristic deformation modes for a certain knot (7). They determine whether the breaking position is inside or outside the knot. Experimental results indicated that at low crossing number, knots were prone to be squeezed predominantly, causing the rope to break at the shoulder position within the knot. In comparison, knots with large crossing number tended to break at the entrance of the knot, due to the rotation of internal filament within knot. This rotation gives rise to a rupture before full squeezing for the knots. For the overhand knot, the breaking position is the knot entrance (7).
Figure 5 Tensile breaking model for rotated-type knots, the applied load gradually increases with (a) knot squeezing, (b) initial breaking, (c) progressive deformation and (d) complete fracturing (7).
Although the material these researchers used in the experiment was poly (vinylidene fluoride) (PVDF), commercial fishing line rather than dyneema fibres, there still exists something to be learned from this work. It could be interesting to investigate whether dyneema behaves in the same manner.
Knot for the experiment
We can see that various kinds of knots have been used in the previous experiments. For example, figure eight knot, water knot, fisherman's knot, bowline, overhand knot (1) (4). From assorted knots, we will choose the overhand knot or thumb knot, the simplest of all knots to conduct the experiment since this kind of knot is the easiest to reproduce. It is well known that repeatability is the key point in experiment. (1)
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Figure 6 schematic diagram of overhand knot (8)
Hypotheses of knot effects
Other hypotheses were also raised in terms of the effect of knots. Bates asserted that knots would act as stress concentration sites in the rope (2). Saitta & Klein used molecular simulations to show that when undergoing the stretching-induced solidification, the knot served as a nucleation site for the crystallization of the sample (9). This work considered individual chains only and so is not relevant to this report.
Figure 7 Side view of the simulation supercell at the nucleation of the solid phase around the knotted chain (larger and darker atoms). The right side shows the average hexatic order of the chains farthest from the defect (9).
A bundle of twisted or braided fibres is the instrument of ropes (3), cords or slings, which are the basic climbing equipments. In ancient times, they were made of natural fibres, generated from either cellulose or protein (10). Nylon, as a new synthetic fibre, is used in modern ropes to improve the properties. Ropes are widely utilized in maritime industry and find some applications in rock climbing. Since high strength, low elongation, light weight and abrasion resistance are of vital importance (4), there is an increasing trend that dyneema fibres are replacing nylon to become the most broadly used polymer fibres in the climbing field, especially for the slings.
Demand for high-performance fibres
In recent years, high performance fibres have been in great demand, and there is a lot of research into the development of polymer fibres with superlative mechanical properties. Dyneema, the trade name of ultra high molecular weight polyethylene (UHMWPE), is the product firstly introduced by DSM High Performance Fibres in Netherlands and Toyobo in Japan. Researchers Smith and Lemstra patented it in the U.S. on August 17, 1982 (11). Unlike Kevlar, Dyneema is flexible chain polymer. Without strong hydrogen bonds, high molecular weight, high degree of chain extension, and adequate chain overlap are needed to enhance the intermolecular bonds in high-performance polyethylene fibres (12) (13).
Microstructure of dyneema
Dyneema fibres are made from ultra-high molecular weight polyethylene. UHMWPE is linear polyethylene with extremely long chains (14). The space of these folded zigzag chains varies from 10 to 50nm (15), different from that of usual polyethylene shown in Figure 8.
Figure 8 Arrangement of molecular chains in a unit cell for polyethylene (14)
As seen in Figure 10, Dyneema SK60 compromises of 780 single filaments, the coarseness of which is 800 den. The filament contains approximately 150 macrofibrils (16). As for the microstructure of these macrofibrils, there exist two possible models. One is the microfibrillar model, saying macrofibril contains thousands of highly oriented chains. The other model describes a macrofibril consisting of a continuous crystalline phase with interdispersed defects (16).
Figure 9 High-resolution TEM micrograph of a gel-spun hot drawn film of UHMWPE (16)
microstructure.jpgFigure 10 Substructure of a gel-spun Dyneema SK60 fibre (16)
Dyneema fibres have a higher degree of parallel orientation (>95%) and crystallinity (>85%) than normal polyethylene (17). Despite of the high orientation, Dyneema fibres are not monocrystalline. According to Berger, a fraction of 25% non-crystalline region was found in the material. Furthermore, 85% orthorhombic and 15% monoclinic were proved to constitute the crystalline phases (16).
Figure 11 Macromolecular orientations between Dyneema and normal PE (17)
Figure 12 Two-dimensional mechanical model of UHMWPE showing the essential structural elements (16)
In Figure 12, region A and B stand for completely chain-extended crystalline areas; Region C represents the non-crystalline containing a portion of slight disorder; the grey region is referred to as intramicrofibrillar tie-molecules. Tie-molecules can transmit most of the axial stresses (16). Kausch's research cited in High performance dyneema fibres in composites demonstrated that a stress of 7.5GPa was required for a tie-molecule to be pulled out from a perfect orthorhombic crystal (16).
It is mentioned in several reports that the crystalline structure of UHMWPE exists mainly in orthorhombic form at room temperature, with a small amount of non-orthorhombic phase and a tiny portion of amorphous phase (12)(18). The orthorhombic form undergoes structure transformation under different thermal or mechanical conditions, which can be shown in Figure 13 (18).
Figure 13 Structure transformation of UHMWPE fibre
3.3.3 Deformation mechanism
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As for the deformation mechanism of Dyneema fibres, it seems that chain rupture contributes a little while the main reason lies in the chain slippage and that of fully crystalline regions (16).
Figure 14 Stages in the deformation of a semi-crystalline polymer (14).
(See below for details of the deformation stages)
The deformation of a semi-crystalline polymer undergoes several phases (14):
Two adjoining chain-folded lamellae and interlamellar amorphous material exist before deformation.
Lamellar ribbons slide past each other and chains in the amorphous region are stretched.
Lamellae are tilted, making chain folds aligned in the load direction.
Crystalline block segments are separated but maintain attached.
Block segments become orientated with the tie chains in the load direction.
Properties and applications of UHMWPE
UHMWPE has a molecular weight exceeding, usually 2 to 6 million, while its density is below one, about 0.94g/cm3 (15). It possesses the similar chemical resistance properties, electrically insulating properties, tensile impact strength and elastic modulus as high density polyethylene (HDPE) (15). What distinguishes UHMWPE from HDPE is outstanding self-lubricating properties and low temperature performance properties. In addition, the abrasion resistance of UHMWPE is better than that of steels. These properties allow numerous and versatile applications of UHMWPE, bullet-proof vests, fishing line, biomedical prostheses and so forth (14). Similar to other polyethylenes, its upper optional temperature range is below 100oC (15).
Strongest fibre on the market
Dyneema is considered as the strongest fibre in the world (19). For instance, the tensile strength of SK60 is 2.7GPa while the density is less than 1g/cm3. This results in a tenacity (specific strength) of 30 g/den (17) (g/den is the common unit used to describe the specific modulus and specific strength of manufactured fibres for the first 50 years (20)). The modulus is also very high, 87GPa and 1000g/den on a specific basis. In the future market, higher values are expected. Also, Dyneema has gained an increasing popularity in the composite field for its light weight and high performances (17).
Table 1 Physical properties of Dyneema SK60, SK75, SK76 (12)
Table 2 Basic properties of important fibres (17)
Ethylene (CH2=CH2) is the simplest and most commonly used monomer. Branching always occurs during the polymerization of ethylene (12). To acquire linear polyethylene, Zieglar-Natta catalyst is employed to the reaction system. According to Ruhrchemie AG, the polymerization of UHMWPE takes place in an inert gas environment, at 65-85oC and under the pressure of 0.5-2MPa (12).
Figure 15 Repeat unit of polyethylene (n represents the degree of polymerization)
Spinning is extensively employed to produce filaments, fibres and yarns. Molten thermoplastic polymer is forced through a spinnerette, which is a die containing many tiny holes for the extrusion of polymer melt (21). A fibre is formed with the rotation of this die. During the process, polymer chains are aligned extensively in the fibre axis, giving rise to the increase in strength (21).
Based on the principle of spinning process, high strength and stiffness polyethylene fibres are usually produced by gel spinning process. With respect to DSM's Dyneema, ultra-high molecular weight PE (2-3) is first dissolved in a volatile solvent. In the heating chamber, part of the solvent is removed. Then the fibres are drawn (disentangled) with decreasing solvent concentration and increasing temperature. Finally, a high degree of orientation is achieved and retained upon cooling (12).Figure 16 and Figure 17 show clearly the processing route.
Figure 16 Gel-spinning process schematic diagram (12)
Figure 17 Drawing process schematic diagram (12)
It is reported that the mechanical properties of UHMWPE can be elevated by one order of magnitude when using gel-spinning process (12). The most important reason is ascribed to the drawing process, during which the gel fibre is substantially elongated (12). Based on the Griffith Theory, tensile strength is related to the fibre diameter:
Where is the tensile strength, represents the strength of a flawless fibre or the theoretical strength a perfect polyethylene fibre, is the fibre diameter, stands for the diameter of a flawless fibre (this is considered to be zero in practice), is the energy needed for the creation of a crack with critical dimensions, is the Young's Modulus and a is a constant (22).
According to Amornsakchai et al, a linear plot of against d1/2 could be achieved (22) (23). That means the smaller the diameter, the stronger the fibre.
Tensile test machine
Tensile test machine is widely used to measure the mechanical properties of materials such as tensile strength, Young's modulus and elongation. Breaking strength of Dyneema fibres, either knotted or unknotted has been tested under different conditions by many researchers listed as follows: Richards (1), Harmston (4), Kromm (24), Parsons (19) and Chocron et al (25). After the experiments, corresponding stress-strain curves were plotted.
Most of the researchers performed the tensile test using different crosshead speeds: 39"/minute (1), medium (1s-1) and high (1000s-1) strain rates (25), 0.04-160mm/min (24) and 200mm/min (19). Furthermore, some of them conducted the tensile test under different temperature: 22oC+6oC (4), ranging from 70oC to 140oC (25). Besides, Parsons used 12, 24 and 36 yarn length Dyneema fibres to conduct the test (19).
Scanning Electron Microscope (SEM)
Scanning Electron Microscope (SEM) has been utilized to observe the surface structure of the sample at the fracture point before and after the breaking (7). (See Figure 5)
X-ray diffraction has been used to analyse the structure of crystalline polymer fibres as well as the fracture morphology. After the tensile test, fibres could be cut into small fragments, each of which was examined under the XRD equipment to identify changes like the crystal morphology and microstress.
Figure 18 Resolved wide-angle X-ray diffraction trace of the UNMWPE fibres at room temperature (18)
Differential Scanning Calorimetry (DSC)
In Parson's work, Differential Scanning Calorimetry (DSC) has been employed to map out the changes in thermal properties of the small fragments of Dyneema fibres. DSC is a thermal analytical technique, in which the difference of heat flow is measured as a function of temperature.
Figure 19 Comparison of stretched and un-stretched UHMWPE film (19)
From the obtained traces, we can see that with the external tensile stress, a shift in melting temperature occurred correspondingly (19). That is to say, tensile properties of the material brought about a microstructure change in the fibre, which was then reflected on the macroscopic change in thermal properties.
It is obvious that with the existence of a knot, the extent of the shift cannot be the same as that in unknotted samples. Therefore, we can understand the effect of knot in the Dyneema fibres better in a quantitative way.
In general, knots deteriorate the strength of fibres invariably, to an extent of about 50%. This strength loss can be influenced by many factors like the type of materials or knots, the tightness of a knot and the strain rate applied. The breaking position might be relevant to the squeezing or rotation mechanism within a knot. For dyneema fibres the high performance properties are significantly influenced by the fibre microstructure along with the processing route. Therefore, the tensile test machine as well as the DSC can be employed to investigate the effect of knots on polymer fibres. A qualitative or quantitative result can be obtained by comparing the shift of DSC peaks from knotted and unknotted fragments.