270 8 Mechanical b ehaviour of polymers 0.00 0.05 0.10 0.15 0.20 0.25 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 · ε/s −1 σ/T MPa/K 21.5 °C 40 °C 60 °C 80 °C 100 °C 120 °C 140 °C Fig. 8.9. Eyring plot of polycarb onate (after [97]) Figure 8.10(a) schematically shows the temperature dependence of Young’s modulus in an amorphous thermoplastic, measured at a typical, constant load- ing time (for example, one second). Increasing the loading time would cause a reduction of Young’s modulus. As can be seen from the figure, the stiffness strongly decreases at temp eratures close to the glass temperature. The elastic behaviour will therefore be discussed separately in the temperature regimes below and above the glass temperature. Energy elasticity The elasticity of thermoplastics below their glass temperature is mainly due to the energy needed to displace atoms from their equilibrium position. On unloading, the atoms return to their original position which has the lowest energy. For this reason, this behaviour is called energy elasticity. It is mostly the weak, intermolecular van der Waals, dipole, or hydrogen bonds that are strained. The covalent bonds do not contribute significantly to the elastic properties. Their stiffness is so large th at they nearly cannot be strained elas- tically as long as the other bonds can deform. Only if the chain molecules are aligned in parallel, as in polymer fibres like aramid (kevlar), the covalent bonds determine Young’s modulus which can then take very large values of up to 440 GPa. We already saw in section 2.6 that Young’s modulus is approximately pro- portional to the melting temperature and thus to the binding energy. For amorphous polymers, the relevant temperature is the glass transition temper- ature because this is the temperature where the bonds melt. The rather low values of the glass temperature (listed in table 1.3) thus also explain why Young’s modulus of polymers is smaller than for the other material classes. The strong decrease of Young’s modulus at the glass temperature (see figure 8.10(a)) will be discussed in the next section. More interesting in the 8.3 Elastic properties of polymers 271 E (log) glass transistion secondary transition energy elastic entropy elastic T g T 0 viscous (a) Amorphous thermoplastics glass transition secondary transition energy elastic amorphous regions: entropy elastic, crystalline regions: energy elastic T g T m T 0 viscous E (log) (b) Semi-crystalline thermoplastics glass transition energy elastic entropy elastic T g T 0 E (log) (c) Elastomers glass transition energy elastic T g T 0 E (log) (d) Duromers Fig. 8.10. Temperature dependence of Young’s modulus in different types of poly- mers (after [19]). Because a logarithmic scale is used, the reduction of Young’s modulus appears to be smaller than it is in reality. More explanations in the text present context is the fact th at even below the glass temperature, there may be temperature values at which Young’s modu lus decreases markedly by about a factor of approximately 2. These so-called secondary transitions are caused by relaxation processes which enable a limited mobility of the chain molecules and thus cause a stress relaxation by movement of molecule segments. Because such rearrangements always require overcoming some activation energy, they become more probable if the loading time increases. They are responsible for the viscoelastic behaviour of polymers. As the activation energies of different relaxation processes differ, their relaxation time also differs. This is the reason why the simple spring-and- dashpot model from section 8.2.1 cann ot be used to make quantitative predic- tions. This would require coupling several such elements [97] with relaxation times chosen to fit their respective processes. We already saw in section 8.1.1 that the activation energy of some relax- ation processes is so low that it can be overcome by thermal activation already at temperatures as low as a few kelvin. At room temperature, their relaxation 272 8 Mechanical b ehaviour of polymers FF Fig. 8.11. Elastic deformation of a polymer above the glass temperature. The molecules are straightened be tween the entanglement points times are thus very short (of the order of 10 −8 s). Relaxation is almost ins tan- taneous so that these processes contribute to the initial deformation of the polymer when the load is applied. If the stressed polymer has deformed viscoelastically by relaxation, the deformed configuration has a higher energy than the initial one. Upon unload- ing, the molecules return to their initial positions. This process again requires thermal activation and is therefore time-dependent as well. Entropy elasticity If the temperature exceeds the glass temperature, Young’s modulus strongly decreases. From what has been said so far, it could be surmised that the poly- mer should deform like a viscous liquid if heated beyond the glass temperature, exhibiting viscosity, but no elasticity. This, however, is not the case. The reason for this is the strong entanglement of the chain molecules. As discussed in section 1.4.2, the chain molecules are strongly folde d. Different chain molecules are thus ‘tied together’ like a knot in many places. On load- ing, the molecules are straightened. Directly above the glass temperature, they cannot slide past each other because this movement is hamp ered by the sur- rounding molecules (see figure 8.11). The molecules thus straighten between their entanglement points. During sliding, energy barriers have to be overcome because the chain molecules are straightened, rotate, and because side groups have to move. Due to the higher temperature and the larger distance between the molecules, this process is much easier at temperatures above the glass temperature. The deformation of the material is still time-dependent due to the required thermal activation. If the load is removed, there is no force on the s traightened molecules, so there seems to be no reason why they should return to their initial posi- tion. Because of the stochastic thermal movements of the molecules in the polymer, the molecule will probably return from the straightened to a folded geometry because there are a lot more possibilities for a folded molecule than for a straightened one. The arbitrary thermal collisions with the surrounding molecules thus fold up the molecule again. Thus, there is a thermodynamic driving force because the entropy of the molecule is larger in the folded than 8.3 Elastic properties of polymers 273 in the straightened state. This behaviour is therefore called entropy elasticity. The entanglement points between the chain molecules remain in a fixed posi- tion on the molecules during elastic deformation so that the molecule returns to its initial shape. In contrast to the deformation below the glass tempera- ture, it is not the smaller energy of the initial configuration that drives the return to this form, but its larger entropy. As before, the movement of the molecules is time-dependent. 3 The viscoelasticity of amorphous polymers is most pronounced near the glass temperature in the transition regime between energy-elastic and entropy- elastic behaviour. At lower temperatures, only smaller parts of the molecules can slide past each other as explained above. As we approach the glass tem- perature, more and more sliding processes become possible. As the sliding processes can be more e asily thermally activated the higher the temperature becomes, the relaxation time decreases. At temperatures well above the glass temperature, relaxation times are small, and the system returns quickly to its initial state. So far, we only considered amorphous thermoplastics. Semi-crystalline ther- moplastics show a different behaviour as shown in figure 8.10(b). Due to the stronger intermolecular bonds in the crystalline regions, their elastic stiffness is usually larger than that of amorphous polymers. The decrease in Young’s modulus on reaching the glass temperature is smaller because only the amor- phous regions become entropy-elastic, whereas the crystalline regions remain in the energy-elastic state. The cohesion between the crystalline and the amor- phous regions is ensured because most chain molecules extend over several crystalline and amorphous regions. 8.3.2 Elastic properties of elastomers and duromers Elastomers and duromers are characterised by additional covalent cross-links between the chain molecules. In the energy-elastic regime, these additional bonds do not influence the elastic properties significantly; Young’s modulus only increase s slightly. At temperatures above the glass temperature, the additional bonds become important. Elastomers are entropy-elastic at these temperatures. The covalent bonds between the molecules increase the linking between them compared to thermoplastics where molecules are linked by geometric entanglement only. These additional links cannot be broken during sliding of the molecules and thus increase the effect of entropy elasticity. With increasing number of cross- links, the covalent bonds are loaded more heavily during elastic deformation so that Young’s modulus increases with the cross-linking density as can be seen 3 Ab ove the glass temperature, there is always plastic deformation as well. If the loading time is sufficiently short, the plastic strain rate is small enough to be neglected; at larger loading times, it has to be taken into account (see also sec- tion 8.4.1). 274 8 Mechanical b ehaviour of polymers from figure 8.10(c). Because the restoring f orce in entropy-elastic deformation is the entropy, which becomes more important the larger the temperature is (see equation (C.3)), Young’s modulus of elastomers often increases with increasing temperature. Contrary to metals and ceramics, the elastic strains in elastomers can become very large and attain values of several hundred percent. The reason is that the molecules are straightened during deformation, but the cross-links prevent the molecules from sliding past each other and thus inhibit plastic deformation. Upon unloading, entropy-elasticity completely restores the initial arrangement of the molecules. This behaviour is called hyperelasticity. During deformation of hyperelastic materials, large strains of 100% or more can occur. The material behaviour is strongly non-linear. There- fore, the theory of large deformations has to be used to describe the material b ehaviour (see section 3.1). The basis of the description is the energy of the deformation: Be- cause it is elastic (i. e., reversible), energy is stored in the material and can be regained on unloading. Hyperelastic materials can therefore be described by specifying the energy density as a function of strain. The stress in the material can be calculated as the derivative of the energy density with respect to the strain. This description is useful for two reasons: On the one hand, the energy density in the material can be calculated using methods of thermodynamics, on the other hand, it en- sures that the stored energy does not depend on the material history, but only on the current state of deformation. This is necessary because hyperelastic processes do not dissipate energy; it would be difficult to accomplish by defining a stress-dependent Young’s modulus. If the cross-linking density of a polymer is increased further, the entropy- elastic behaviour vanishes nearly completely because the large number of cross- links prevent the straightening of the molecules. For this reason, duromers show only a small decrease of Young’s modulus with temperature (see fig- ure 8.10(d)) caused by relaxation processes. They are energy elastic even above the glass temperature. Table 8.1 lists the magnitude of Young’s modulus for the different polymer groups as a function of their cross-linking density. This quantity is normalised by assigning a value of 1 to diamond in which all atoms contribute to the cross-linking. 8.4 Plastic behaviour Polymer elasticity is determined by the reversible deformation of the chain molecules as we saw in the previous section. Polymers can also deform plas- tically, with chain molecules sliding past each other over large distances as 8.4 Plastic behaviour 275 Table 8.1. Cross-linking density and Young’s modulus of different types of polymers (cf. section 1.4.2) type of material cross-linking density E/GPa thermoplastics 0 0.1 . . . 5 (for T < T g ) elastomers 10 −4 . . . 10 −3 0.001 . . . 0.1 (for T > T g ) duromers 10 −2 . . . 10 −1 1 . . . 10 diamond 1 1 000 sketched in figure 8.4 on page 261. The plastic behaviour of polymers strongly depends on the temperature because obstacles have to be overcome by ther- mal activation and because the size of the ‘tunn els’ in which the molecules move is determined by the specific volume (see section 8.1.2). As in the previous section, we start by discussing amorphous thermoplas- tics and afterwards discuss how things change in semi-crystalline thermoplas- tics. Elastomers and duromers only allow for a small amount of plastic defor- mation because the cross-links prevent molecule sliding as explained above. Elastomers used above their glass temperature can be deformed with large elastic strains instead; duromers are brittle, with the covalent bonds between the chain molecules breaking in brittle failure. 8.4.1 Amorphous thermoplastics We start this section by discussing the plastic behaviour of amorphous ther- moplastics. The stated temperature regions are, due to the time-dep e nde nce of plastic deformation, valid for rather large strain rates (with testing times of a few seconds). Increasing the testing time i. e., decreasing the strain rate, is e quivalent to increasing the temperature (see section 8.2). Far below the glass temperature At temperatures lower than about 80% of the glass temperature T g , the bonds between the molecules are so strong and the specific volume is so small that chain molecules cannot move by sliding. On loading, the molecules are straight- ened viscoelastically. If the load is raised further, as sketched in figure 8.12(a), brittle failure ensues, mainly breaking the intermolecular bonds. Slightly below the glass temperature At temperatures of about 80% of the glass temperature T g , amorphous ther- moplastics have a limited ductility (see figure 8.12(b)). At these higher temper- atures, the mean distance between the chain molecules is larger and enables 276 8 Mechanical b ehaviour of polymers ¾ " brittle fracture elastic (a) T < 0.8T g limited plastic deformation ¾ " elastic (b) T ≈ 0.8T g Fig. 8.12. Stress-strain diagrams of an amorphous thermoplastic at different tem- p eratures [9] them to partially overcome the binding forces, giving the molecules a limited mobility. In contrast to metals, polymers do not work-harden because no new ob- stacles are created when the molecules slide past each other. Heat generated during deformation causes a local increase in temperature, further easing plas- tic deformation. This res ults in a local s often ing of the material, similar to a metal with an apparent yield point (see section 6.4.3). Only if the plas- tic strain becomes larger do es some hardening occur because the molecules become aligned in the direction of the applied stress. A typical microstructure of an amorphous thermoplastic loaded in tension slightly below the glass temperature is shown in figu re 8.13(a). There are mi- croscopically small, lens-shaped cavities, called crazes. They have a thickness of about 1 µm to 10 µm and a diameter of about 10 µm to 1000 µm and are bridged by fibrils. The fibrils comprise several chain molecules and have a diameter of approximately 10 nm to 100 nm. Their volume fraction within the craze is between 10% and 50%. Although the crazes do look crack-like, the strength of the material is only slightly reduced in this region compared to the strength of the undeformed material since the chain molecules within the fibrils are straightened and thus can bear a higher load. The thickness of a craze is almost independent of the applied stress, but it increases with increas- ing temperature. If the applied stress is large, a large number of small crazes form, if it is small, their number is smaller. Usually, crazes are initiated at surfac e defects, for example scratches or impurities. Plastic deformation starts in these regions due to the slight stress concentration caused by these defects. Because the material s often s as ex- plained above, plastic deformation concentrates in this region, resulting in a slight local necking. This, in turn, causes the stress state to become tri- axial and increases the hydrostatic tension. Small cavities with a diameter of a few nanometres form (figure 8.14). Because of the stress concentration, the material between the cavities is heavily loaded and deforms plastically, 8.4 Plastic behaviour 277 1 µm– 10 µm 10 µm–1000 µm 10 nm–100 nm10 nm–100 nm ¾ ¾ (a) Cross sectional view of a craze. The lens-shaped cavity and the fibrils can be observed (b) Scaled partial view of the edge of a craze Fig. 8.13. Microstructure of a craze (after [82, 128]) ¾ ¾ 11 ¾ 11 ¾ 22 ¾ 22 x 1 x 2 ¾ plastically deformed area (a) Local necking ¾ ¾ (b) Formation of cavities ¾ ¾ (c) Extension of cavities Fig. 8.14. Development of a craze by formation of cavities (after [82]) straightening the molecules in this region. Fibrils between the cavities emerge and a craze is formed. Despite the load-be aring capacity of the fibrils, there is a stress concentra- tion near the edges of a craze, easing its further growth. The growth mech- anism is a so-called meniscus instability: Near the edge of the craze, finger- shaped extensions evolve and contract, forming new fibrils (figure 8.15). Fibrils within the crazes initially elongate further by drawing other chain molecules from the bulk material. Cross-links between the fibrils may form if opposite 278 8 Mechanical b ehaviour of polymers (a) (b) (c) Fig. 8.15. Growth of a craze by a meniscus instability (after [82]) ¾ " F F F F F F linear elastic b b a a c c formation of shear bands shear band Fig. 8.16. Deformation of an amorphous thermoplastic under compressive load by formation of shear bands (after [9]) ends of a chain molecule are drawn into neighbouring fibrils (see figure 8.13(b)). Finally, fibrils in the centre of the craze break. The craze then grows contin- uously at constant load, rendering the plastic deformation time-d ependent. This growth can eventually cause fracture of the polymer. A polymer can deform not only by crazing, but also by forming shear bands, created at an angle between 45° and 60° [44, 82, 132] to the loading direction (figure 8.16). Formation of shear bands is especially important under com- pressive loads. Within the shear bands, large localised plastic deformations of 100% or more can occur, whereas the deformation is very small outside of them. Shear band formation has not been studied as closely as crazing. A sim- ple mechanical model is based on the shearing of chain molecules (figure 8.17). The shear stress component causes the chain molecules to either straighten or to form two kinks, resulting in a region with aligned chain molecules. If sev- eral shear bands converge, a crack can be initiated if one shear band reaches the already straightened molecules. Because these cracks are now loaded un- der shear where, according to section 5.1.1, the fracture toughness is larger (K IIc  K Ic ), the fracture s train is significantly larger than under tensile loading. Several factors determine whether a polymer deforms by shear bands or crazing. The crucial factor is that crazes, which are initiated by cavitation, 8.4 Plastic behaviour 279 ¿ ¿ ⇒ ¿ ¿ Fig. 8.17. Formation of a shear band by local stretching and contracting of the molecule chains (after [35,132]) ¾ 1 ¾ 2 crazes plastic flow ¾ m > 0 ¾ m < 0 Fig. 8.18. Yield surface of an amorphous thermoplastic that can fail by crazing or formation of shear bands. If the hydrostatic tensile s tress is sufficiently large, crazing o ccurs before shear bands (after [92,132]) can only form under hydrostatic tensile stress. 4 The larger the hydrostatic tensile stress is, the stronger is the tendency for crazing. Figure 8.18 shows the yield surface of a polymer in plane stress, illustrating this. The yield strength of polymers generally depends on hydrostatic stress because hydrostatic compression decreases the specific volume and thus ham- pers sliding of the molecules. This was already discussed phenomenologically in section 3.3.3. However, the criteria discussed there did not take crazing into account. Apart from the multiaxiality of the stress state, the temperature and the loading time also play a role in determining the deformation mechanism. Large strain rates (and small temperatures) make shear band formation more diffi- cult, thus favouring crazing. 4 Even in a uniaxial stress state, there is a hydrostatic stress according to equa- tion (3.25): σ hyd = σ/3. [...]... done in polypropylene and polyvinyl chloride, the mechanical properties improve accordingly as already discussed 290 8 Mechanical behaviour of polymers Table 8. 2 Typical properties of several polymers (after [44, 98] ) Since the properties also depend on the degree of polymerisation and additives, the values specified only serve as guidelines material low-density polyethylene high-density polyethylene... regions of crystalline into blocks regions F (e) Formation of microscopic fibres (microfibrils) Fig 8. 21 Stages of plastic deformation of a semi-crystalline thermoplastic (after [44, 82 ]) larvae of the silk moth Bombyx mori [144] Silks are made of protein fibres, spun to strings with a diameter between about 2 µm and 10 µm in spider silk and 10 µm and 50 µm in silk of the silk moth Proteins are polymers. .. fibrils [83 ] This was already discussed in section 8. 1.1 for the side groups in pmma  286 8 Mechanical behaviour of polymers Fig 8. 23 Spatial structure of a ptfe chain molecule The strong repulsion between the fluorine atoms results in a twisted and rigid molecule 126℃ The large electron affinity of fluorine causes the fluorine atoms to be partially negatively charged They thus repel each other and cause... and crystalline regions is therefore weak and cracks may initiate there Nature frequently uses polymers for load-bearing applications as well (see also section 9.4.4) One particularly interesting example for a biological polymer is the silk of spiders or some insects, for example the 5 This will be explained in detail in section 9.3.2 for the case of fibre composites  282 8 Mechanical behaviour of polymers. .. intent of increasing the ductility This is called internal plasticisation.10 Figure 8. 27 shows the temperature dependence of Young’s modulus of two polymers and their copolymers One example of a copolymer is polybutadiene styrene, made of the monomers of polybutadiene and polystyrene If the copolymerisation is alternating or random, the glass temperature is between that of the constituting polymers. .. rule of mixtures (equation (9.2)) to calculate the stress in the composite: 304 σ 9 Mechanical behaviour of fibre reinforced composites fracture fibres of the fibre composite matrix yield strength of the matrix fracture of the matrix ε Fig 9.3 Schematic stress-strain diagram of a fibre-reinforced polymer (after [9]) σ = σf ff + σm (1 − ff ) , (9.7) with σf and σm being the stresses in fibre and matrix, and. .. moulding more difficult 8. 5 Increasing the thermal stability 287 8. 5.2 Increasing the crystallinity The strength of semi-crystalline polymers is larger than that of amorphous polymers If the crystallinity of a polymer can be increased, the mechanical properties are improved accordingly The crystallinity of a polymer can be changed by the manufacturing process and by the structure of the chain molecules... section of the material and oriented perpendicularly to the applied load (figure 9.2(a)) In the serial connection, the balance of forces must hold at each fibrematrix interface: 302 9 Mechanical behaviour of fibre reinforced composites F E Ef Em fibre matrix F 0 1 ff (a) Arrangement of fibres and matrix (b) Dependence of Young’s modulus on the fibre volume fraction Fig 9.2 In-series connection of fibre and matrix... 280 8 Mechanical behaviour of polymers F moleculs oriented d ¾ F F F drawing of the moleculs b c F F a elastic a b F c F d " Fig 8. 19 Stress-strain curve of an amorphous thermoplastic closely below the glass temperature (after [9]) Fig 8. 20 Configuration of a drawn thermoplastic made of fibre bundles (after [35]) Close to the glass temperature... 65 70 75 80 E/GPa 0.3 1.0 1.5 3.0 3.0 3.3 2.3 3.5 /(g/cm3 ) 0.92 0.96 0.91 1.4 1.3 1.2 1.2 1.2 (a) Alternating (b) Random (c) Block copolymer (d) Graft copolymer Fig 8. 26 Different types of copolymers (after [19, 31]) One method of improving the strength or stiffness is particularly important in polymers: Combining them with other materials to form composites This is the subject of chapter 9 8. 7 Increasing . over large distances as 8. 4 Plastic behaviour 275 Table 8. 1. Cross-linking density and Young’s modulus of different types of polymers (cf. section 1.4.2) type of material cross-linking density E/GPa thermoplastics. sectional view of a craze. The lens-shaped cavity and the fibrils can be observed (b) Scaled partial view of the edge of a craze Fig. 8. 13. Microstructure of a craze (after [82 , 1 28] ) ¾ ¾ 11 ¾ 11 ¾ 22 ¾ 22 x 1 x 2 ¾ plastically deformed area (a). larger and enables 276 8 Mechanical b ehaviour of polymers ¾ " brittle fracture elastic (a) T < 0.8T g limited plastic deformation ¾ " elastic (b) T ≈ 0.8T g Fig. 8. 12. Stress-strain