STRENGTH OF POLYMERS

Transcription

1 University of Nottingham Department of Mechanical, Materials and Manufacturing Engineering POLYMER ENGINEERING STRENGTH OF POLYMERS 1. Introduction A plastic component can fail to meet its mechanical requirements in one of two ways: (i) (ii) Excessive deformation this is governed generally by the space envelope available to a component. For example a component might deform to such an extent that it interferes with another part of a machine. As already noted, deformation in polymers is generally time dependent, and can be analysed using sophisticated viscoelastic models or the simple pseudo-elastic method. Failure this represents permanent deformation and/or fracture of a component. This can be either ductile, where failure is preceded by yielding, or brittle, where failure occurs suddenly at the peak load (see Fig. 1). As will be discussed, the type of failure depends on the material and the loading conditions (in particular rate, temperature and hydrostatic pressure). Figure 1: Brittle and ductile failure behaviour (after McCrum) Strength of Polymers Page 1 of 19

2 . General Nature of Yielding in Polymers A normal tensile test on a polymer produces a stress-strain curve similar to that of a metal (see Fig.. As the strain is increased, the material passes through a recoverable elastic region which in contrast to metals is usually non-linear. The slope of the curve decreases until it reaches a peak value in stress which can be used to define the yield stress, σ y. The value of the yield strain, ε y, for a polymer is of the order 5-10% compared with 0.1% for metals. Strictly speaking, the yield point of the material should be described as the point at which permanent set takes place. This is very difficult to define in polymers as it is sometimes possible to recover apparent plastic deformation beyond the yield point by raising the temperature of the material. In fact, the distinction between recoverable and irrecoverable deformation depends on the time scale and temperature of the experiment. However, despite this, one can still define a yield stress as the peak value reached by the nominal stress, as for Stress, y metals. y Strain, Figure : Typical stress-strain curve for a ductile material The main problem in characterising the plastic deformation of polymers, unlike metals, is that the yield point is very sensitive to the experimental conditions and, in particular, to the following variables: (i) (ii) (iii) Rate of strain Temperature Hydrostatic pressure Figure 3: Variation in yield stress with temperature for PMMA at various strain rates (Langford et al) The effect of strain rate and temperature on the yield stress of PMMA is shown in Fig. 3. As a result of the time-temperature superposition principle, an increase in temperature has the same effect as reducing the strain rate, ie the yield stress reduces. The yield stress approaches zero as the glass transition temperature is approached as expected for an amorphous polymer. Semicrystalline polymers show similar behaviour except that the yield stress falls to zero at the melting temperature of the crystals instead of the T g. Reducing temp-erature or increasing the strain rate has the effect of increasing the yield stress until such a point is reached Strength of Polymers Page of 19

3 where the material no longer fails in a ductile manner but exhibits a brittle type failure. This point is termed the ductile-brittle transition for the material and will be discussed in more detail along with fracture criteria in a separate section. The ductile-brittle transition is clearly illustrated in a series of stress-strain curves of PMMA at different temperatures in Fig. 4. Figure 4: Effect of temperature on yielding of PMMA (after Andrews) An increase in hydrostatic pressure has the effect of increasing both the modulus and yield stress of a polymer, the effects being greater for amorphous materials. The effect of pressure is to reduce the free volume or density of packing in crystal regions which leads to a rise in the yield stress and a more brittle material. However, this effect is counterbalanced by the pressure tending to reduce the influence of cracks or flaws thus increasing the brittle strength. In some cases large increases in ductility can be attained by an increase in pressure. Fig. 5 shows results for PP and PE up to 6,000 atmospheres of pressure. These pressures would show no effect on the yield behaviour of metal. Figure 5: Effect of pressure on yield stress for PP & PE (after Mears et al) 3. Necking and Cold Drawing As with metals, polymers can show localised necking and cold drawing after yield. Necking is first observed as a local reduction in cross sectional area and corresponds to the drop in load observed at the yield point. If the neck is stable, the load will remain at a steady value and the neck will traverse the length of the specimen, ie cold drawing is observed. When the whole specimen has necked, strain hardening will take place and the load rises until fracture intervenes. The amount of cold drawing is referred to as the draw ratio of the material. In some polymers a stable neck does not form but the site of the original neck continues to get thinner until fracture occurs. This is termed an unstable neck. The three types of deformation are illustrated in Fig. 6. Strength of Polymers Page 3 of 19

4 Stress, No neck Unstable neck Stable neck Strain, Figure 6: Stress-strain curves for formation of stable, unstable or no neck The conditions for necking and cold drawing are best shown by plotting the true stress (load P divided by instantaneous cross sectional area) against the strain, ε. The relationship between nominal stress σ n (load P divided by original cross sectional area) and the true stress σ can be derived as follows: True stress σ = P/A Nominal stress σ n = P/A 0 where A 0 = original cross sectional area Assuming deformation is at constant volume, this gives: Al = A l 0 0 n A l0 = = A0 l n 1 1 = = 1+ R where R is the draw ratio. When a neck starts to form the load reaches a maximum, ie dp/dε = 0. Thus: d n d n 1 d = d dr R dr R Equating this to zero gives the condition for a neck to form. Thus: d = dr R A special construction (Considère) shows the point at which necking occurs (see Fig. 7). A tangent to the true stress-strain curve from the point ε = -1, ie R = 0, intersects the curve when the above condition is satisfied. The construction also allows the condition for a stable neck to be determined. This is the case only when a second tangent can be drawn to the curve, the point of intersection corresponding to a minimum in the stress-strain curve. The Considère construction can therefore be used as a criterion to decide whether a polymer will neck, or will neck and draw. Strength of Polymers Page 4 of 19

5 Figure 7: Three types of true stress-strain curves for polymers. (a) No neck; (b) Unstable neck; (c) Stable neck (after Vincent) The mechanisms of necking start with either a local reduction in cross sectional area or a local fluctuation in material properties which allows yielding to take place in a small element. Deformation will continue within this element because it has a lower effective stiffness and the local stress is higher. This region would develop as an unstable neck, however, the deformation may be arrested by strain hardening of the material. In the case of polymers this is thought to be due to molecular orientation (multiplication of dislocations in metals). If the strain hardening takes place to a sufficient extent, further deformation will take place in new elements of the material which appear softer, although the actual stress on the elements may be lower. In this way a stable neck can propagate along the length of the sample. The formation of a stable neck is also dependent on the dissipation of heat during the drawing process. Too much localised heat will result in a local unstable neck and consequent failure. Stable drawing is particularly important in processes such as fibre extrusion and film blowing. 4. Yield Criteria for Polymers 4.1 Traditional criteria A tensile test alone cannot give a full description of the yield behaviour for design and it is important to obtain a general yield criteria, ie a function of the stress or strain components which reaches a critical value for all forms of test and combinations of stress components. Although many yield criteria have been suggested, with varying experimental support, the two main criteria adopted for metals are those of Tresca (1864) and Von Mises (1913). With some modification these criteria can also be applicable to polymers. The criteria for isotropic materials are normally given in terms of the stress components of the stress tensor, σ ij. ij = By choosing axes parallel to the principal stress directions, ie where the shear stresses are all zero, the stress tensor becomes: ij = Strength of Polymers Page 5 of 19

6 and the criteria can be written in terms of the three principal stresses, σ 1, σ and σ 3. For isotropic materials the yield criteria must be invariant with respect to coordinate transformation. Tresca Yield Criterion This criterion proposes that yield occurs when the maximum shear stress, σ s, reaches a critical value. For σ 1 > σ > σ s = = constant In a tensile test σ 1 = applied stress and σ = σ 3 = 0 1 y s = = where σ y = tensile yield stress. The Tresca criterion is therefore: y = Von Mises Yield Criterion This criterion is based on the condition that yield occurs when the strain energy of shear reaches a critical value. It can be expressed as follows: In a tensile test this reduces to: = constant = constant The critical value for the constant is σ y where σ y = tensile yield stress. Von Mises criterion is therefore: = y In pure shear σ 1 = -σ and σ 3 = 0 and the criterion gives: 1= y 3 The shear yield stress is predicted to be 1/3 times the tensile yield stress compared with σ y / for the Tresca condition. Graphical Presentation of Tresca and Von Mises Criterion in Plane Stress Consider initially the case of plane stress, σ 3 = 0. Von Mises criterion reduces to: or = 1 y Strength of Polymers Page 6 of 19

7 1 y + y 1 - = 1 y y which describes an ellipse in principal stress space as shown in Fig. 8. Figure 8: Tresca and Von Mises yield criteria for plane stress The Tresca criterion in plane stress depends on the sign of the applied stress components. For σ 1 and σ with the same sign (both either tensile or compressive): σ 1 = σ y or σ = σ y For σ 1 and σ with opposite signs (one tensile and the other compressive): σ 1 - σ = σ y or σ - σ 1 = σ y It can be seen from Fig. 8 that the Tresca criterion inscribes the Von Mises ellipse. Von Mises predicts the possibility of an applied stress larger than the tensile yield stress in some stress states. This is not the case for Tresca. 4. Modified Criteria for Polymers The applicability of either of the above criteria can be tested by performing experiments with different states of stress, eg plane stress, pure shear, biaxial tension etc. Modifications of the criteria have been found necessary for polymers. Firstly it was found that compressive strengths are greater than tensile strengths, having the effect of shifting the yield ellipse or hexagon to the more compressive values. Hydrostatic pressure has also been shown to have a pronounced effect on the yield behaviour, a fact clearly related to the increased compressive strengths. Von Mises and Tresca criteria have therefore been modified to take into account the effect of hydrostatic stress. In most cases polymers tend to follow a pressure dependent Von Mises criterion. The effect of hydrostatic stress can be introduced into Von Mises criterion through an additional hydrostatic term: A B = 1 Strength of Polymers Page 7 of 19

8 where A and B are constants. A and B can be defined in terms of the tensile and compressive yield stresses, σ YT and σ YC, since in these cases σ = σ 3 = 0. Thus: YC - YT A= YC YT and 1 B= YC YT Fig. 9 shows the modified Von Mises criterion fitted to polystyrene data. modified Tresca helix based on Coulomb's work on failure in soils is shown. In addition, a 5. Crazing in Polymers Figure 9: Yield criteria fitted to data for PS (after Whitney & Andrews) Crazing is a phenomenon found in most amorphous glassy plastics and some semi-crystalline plastics. It shows itself particularly in transparent materials as a local whitening or loss of clarity and can often be mistaken for the growth of a network of small internal cracks. The structure of a craze comprises localised regions of highly drawn material called fibrils, adjacent to microscopic holes from which the material is drawn (Fig. 10). Crazes initiate at regions of high stress concentration such as crack tips, at inclusions in the material and at surface defects. Although a craze can look similar to a crack it differs in one important respect: loads can be supported across the body of a craze through the highly drawn, oriented fibrils. Because of the holes present in a craze, its density can be about half that of the base polymer. Although crazing is a result of high stress, an external stress may not be required to form it. Internal stresses resulting from shrinkage in the mould can cause the initiation of crazes over a period of time. Crazing can also be caused by the absorption of solvents. External stress causes craze voids to open. These are susceptible to the absorption of solvents, which in turn promotes further growth of the craze. Although in themselves crazes are not a sign of failure, craze voids can coalesce to form cracks which can lead eventually to brittle failure. Crazing is therefore often the precursor to crack growth. Solvent crazing, also called environmental stress cracking, can also lead to Strength of Polymers Page 8 of 19

9 catastrophic failure in some materials. Figure 10: Craze formation (after Powell) 6. Fracture in Polymers It has been shown that the load-elongation curve at constant strain rate depends on the temperature of the material. This is illustrated schematically in Fig. 11. Load, P (a) (b) (c) (d) Strain, (a) (b) Figure 11: Effect of temperature on stress-strain relationship for polymers Low temperature - brittle failure. Increased temperature - yield point, load falls before failure, sometimes necking, strains ~ 10-0%, ductile failure. (c) Higher temperature - yield point, stabilised neck and cold drawing, strains ~1,000%. (d) Above glass transition - rubber-like behaviour. Increasing the strain rate has the same effect as lowering the temperature. Brittle behaviour is defined as taking place when the specimen fails at its maximum load and at low strains (typically <0%). The distinction between brittle and ductile failure is also manifested in two further ways: (i) Energy dissipated is lower for a brittle failure. (ii) Fracture surface is rougher for a brittle failure. Strength of Polymers Page 9 of 19

10 6.1 Brittle-Ductile Transition By lowering the temperature or increasing the strain rate, the mode of failure changes from ductile to brittle at the brittle-ductile transition. It is assumed that brittle fracture and plastic flow (or yield) are separate processes which have different dependence on temperature. Typical curves for brittle fracture stress σ B and yield stress σ Y as a function of temperature at constant strain rate are shown in Fig. 1 (continuous lines). At a particular temperature, failure will occur by whichever process has the lower stress. Thus the intersection of the σ B - σ Y curves defines the brittle-ductile transition. The effect of increasing strain rate is to shift each curve to higher stress levels as shown by the dashed lines in Fig. 1. The shift in σ Y is more pronounced resulting in a higher transition temperature and hence a greater likelihood of brittle failure at the higher strain rate. Stress Brittle stress Yield stress Temperature 6. Notch Sensitivity Figure 1: Variation in brittle fracture and yield stress with temperature. Dashed line shows effect of increasing strain rate. The most important factor affecting the brittle-ductile transition is that of notch sensitivity. The presence of a notch, crack or small flaw within a component can change what is generally a ductile material into a brittle material resulting in premature failure. For a flat plate, width W, containing an edge crack (or notch) of length a, the conditions for failure due to yield and fracture are as follows: Yield when: where σ y =yield stress of the material. = y (W - a) W Fracture when: K IC = Y a where K IC = critical stress intensity factor, Y = geometric factor. Strength of Polymers Page 10 of 19

11 Failure stress a w Yield Fracture a Crack length, a Figure 13: Yield and fracture stresses versus crack length The failure stress at yield is a linear function of crack length and the failure stress at fracture varies as 1/a (see Fig. 13). The crack or notch size therefore dictates whether yield or fracture will occur first. For short cracks, the transition from one failure mode to the other is shown by the intersection at a crack length of a'. Thus: Yield, ie ductile failure when Fracture, ie brittle failure when a < a' a > a' The principles of fracture mechanics are equally applicable to polymers as they are to metals and are, in fact, necessary for designing against such failures Fracture Mechanics in Design a a Figure 14: Stress field near an idealized crack of length a (after Ward & Hadley) Consider a crack of length a within a large thin plate of linear elastic material (Fig. 14). A remote nominal stress is applied to the plate and the effect of the crack is to concentrate this stress around the region of the crack surface. It can be shown that the local stress distribution near the crack is given by the following equations: Strength of Polymers Page 11 of 19

12 xx yy = = K r K r I I 1 1 cos 1 - sin sin cos 1+ sin sin 3 3 where K I = stress intensity factor for opening (mode I) stresses. K I zz = cos (1) 1 (r ) The distribution of the stresses due to the crack is found by substituting appropriate values for r and θ into equations (1) and is independent of component shape, crack geometry and the applied stresses. The stress intensity factor, K I, however, depends on all these parameters in the following way: =Y K I a () Y is a geometric factor which depends on the crack and plate geometry (Fig. 15). For a small crack in an infinite plate Y = 1. Through equations (1) and (), the external stress (σ) can be related to the crack tip stresses (σ xx, σ yy and σ zz ), and the magnitude of the stresses clearly depend on the value of K I. Figure 15: Typical crack configurations (after Crawford) Strength of Polymers Page 1 of 19

13 6.4 Critical Stress Intensity Factor - Fracture Toughness Brittle failure will occur within a component when the crack tip stresses reach some critical value, ie when the stress intensity factor K I reaches some critical value called the critical stress intensity factor K IC. K IC is a material constant and is a measure of the fracture toughness of a material. It is obtained experimentally by testing a specimen which has a geometry for which the relationship between external loading and stress intensity factor is known. Factors such as loading rate, temperature and environment can all affect the value of K IC. Units are MN/m 3/ and some typical values are given in Table 1. The fracture toughness of a material can also be measured in terms of the critical strain energy release rate, G IC, required to grow a crack. It can be shown to be related to K IC as follows: K G IC = E where E is the material modulus. Its advantage as a toughness parameter is that it can be directly measured in a series of impact tests by monitoring the energy to break. Units are J/m, ie energy to break per unit fracture surface area. IC 6.5 Design for Strength Critical flaw size The analysis shows that if K I < K IC then brittle fracture will not occur. Thus the condition for failure is: K I =Y a K IC (3) For a fixed geometry (Y) it is necessary to ensure that either the design stress (σ) or crack size (a) is sufficiently small to prevent this condition. A critical flaw size is the size of flaw which will cause the above condition to be reached for a specific geometry and applied stress. It is possible to process plastics and check the quality to ensure that the critical flaw size is not surpassed. However, flaws below the critical size can grow under both constant loading and cyclic loading conditions over a period of time until they reach the critical size to cause failure. In order to design against this, it is necessary to understand the nature of slow crack growth under such loading conditions. Constant load For many polymers the crack length, a, increases at a rate which is related to the stress intensity factor, K I, at time, t, as follows: da n = C 1 K I (4) dt Strength of Polymers Page 13 of 19

14 C 1 and n are constants which depend on the material and test conditions. The time taken to grow a crack from some initial size a 0 to the critical size a c is found by integrating equation 4. Substituting equation for the stress intensity factor into the above gives: The integration gives: da n n = C1(Y ) ( a ) dt 1 n 1-n a0 - ac t c= n (n - )C1(Y ) (5) The critical flaw size a c is found by re-arranging equation 3: K IC ac = Y 1 (6) Therefore from equation 5 one can determine either the design stress to ensure a particular service life or the service life for an applied stress. Material data required includes K IC and the size of initial flaws, a 0, ie the intrinsic defect size (see Table ). Cyclic loading Crack growth also occurs under cyclic fatigue loading conditions. In this case, the rate of crack growth follows a modified Paris Law as follows: da = C m dn where C and m are constants. λ is a function of the stress intensity factor, thus: (7) = I max I min I mean I K - K = K Equation 7 can be integrated with respect to the number of cycles N. To simplify the integration it is assumed loading is in tension only (ie no compression) such that K I min = 0. The solution in this case is: Procedure for design calculations N = a - a 1m 1m 0 c K m - 1 C Y m (8) (i) (ii) From the known value of K IC (or G IC ) and geometry factor Y, calculate the critical flaw size, a c, for the design stress σ (equation 6). From the known value of intrinsic defect size or initial flaw size, a 0, and crack growth parameters (equations 4 or 7), calculate the time (equation 5) or number of cycles (equation 8) to reach the critical flaw size and hence failure. Strength of Polymers Page 14 of 19

15 References/Further Reading: R J Crawford (1998) Plastics Engineering, Butterworth-Heinemann. Chapter Mechanical behaviour of plastics. N G McCrum, C P Buckley & C B Bucknall (1997) Principles of Polymer Engineering, Oxford Science Publications. Chapter 5 Yield and fracture. P C Powell, A Jan Ingen Housz (1998) Engineering with polymers, Stanley Thornes (Publishers) Ltd. Chapter 6 Strength of polymer products. I M Ward, D W Handley (1993) Mechanical Properties of Solid Polymers, John Wiley & Sons. Chapter 11 The yield behaviour of polymers & Chapter 1 Breaking phenomena. R J Young, P A Lovell (1991) Introduction to Polymers, Chapman & Hall. Chapter 5 Mechanical properties. Strength of Polymers Page 15 of 19

16 Table 1 Some values for plane strain fracture toughness (ie critical stress intensity factor K IC ) at 0 o C. Material K IC MN/m 3/ Epoxy 0.6 Polystyrene (PS) 1.0 Acrylic sheet (PMMA) 1.6 Polycarbonate (PC). upvc pipe compound.3 HDPE pipe compound 3.0 PA66.75 PP 4.0 POM 4.0 LDPE 6.0 Table Intrinsic Defect Size (a 0 ) upvc PMMA Nylon 66 (dry) Acetal Copolymer (POM) LDPE 430μm 146μm 19μm 70μm 36μm Polymer Engineering\Strength\Strength.doc Strength of Polymers Page 16 of 19

17 Worked Example - Brittle failure A wide sheet of polycarbonate contains a central sharp crack 5 mm in length. The sheet fails under an external stress of 1 MPa. Determine :- (a) (b) The fracture toughness, K IC, of the material. The failure stress for a wide sheet of the same material containing a 50 mm crack. Strength of Polymers Page 17 of 19

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19 Worked Example - Fatigue failure A wide sheet of polystyrene contains an internal flaw (a o ), 1mm in size. Using the following crack growth data, determine the fatigue life (no. of cycles to failure) of the sheet when it is subjected to a cyclic stress of 0-10 MPa. K IC for polystyrene = 10 6 N / m 3/ Crack growth data da/dn m/cycle ΔK I N/m 3/ Strength of Polymers Page 19 of 19