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Modelling the behaviour of plastics for design under impact
Polymer Testing 20 (2001) 677–683 www.elsevier.nl/locate/polytest
Property Modelling
Modelling the behaviour of plastics for design under impact G. Dean, B. Read Centre for Materials Measurement and Technology, National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK Received 9 October 2000; accepted 19 December 2000
Abstract Computer methods based on finite element analysis are able to predict the performance of plastics under impact loading. Although the accuracy of results depends on the model used to describe the deformation behaviour of the polymer, whichever model is used, the analysis requires stress/strain data over a wide range of strain rate. These data are most conveniently measured in tension, but procedures are currently not available for determining results at high strain rates. ISO standards for tensile property measurement are applicable for strain rates up to around 0.1 s⫺1. To simulate behaviour under impact, data are required at rates that are 3 or 4 orders of magnitude higher than this. For accurate data acquisition at these higher rates, attention needs to be paid to apparatus design in order to minimise contributions from transient forces arising from resonances and the propagation of shock waves in the apparatus. In addition, procedures and extensometers are not routinely available for determining strains at the higher rates of deformation. This paper illustrates the acquisition of data over a wide range of strain rates through a combination of measurements at low and moderate strain rates with extrapolation of these data to higher rates. In order to maximise accuracy at moderate strain rates, suitable designs need to be selected for the transducers, the test specimen geometry and the test assembly. Extrapolation is achieved by the use of mathematical functions to model the stress/strain curves and their rate dependence. Reference is also made to the development of a new materials model for describing the deformation behaviour of toughened plastics at large strains. 2001 NPL. Published by Elsevier Science Ltd. All rights reserved. Keywords: Measurement; Plastics; Modelling; Impact; Materials; FEA
1. Computer aided design Most modern grades of plastics are very tough materials that are capable of sustaining large strain levels prior to failure, making them an attractive option for many applications. Such applications include housings for domestic and electronic appliances, which may have to sustain accidental impact without showing signs of damage, and interior components of motor vehicles, which may be impacted by occupants in a crash and must minimise damage to the human body part involved. In these applications and others, it is important that the performance and the safe operating limits of the component under impact are known. Substantial savings can be made in both time and cost if this can be done through
competent design rather than the moulding and testing of prototypes. Computer methods are available based on finite element analysis that enable predictions to be made of performance under realistic loading situations. There are two factors that limit the wider use of these computer methods. Firstly, a lack of recognised and routine test methods for measuring some of the data required by the analysis, and secondly, limited knowledge of the validity for plastics of the materials models employed by the computer analysis, and thus uncertainty in the accuracy of derived results. This paper discusses the experimental issues associated with data measurement, and a procedure for determining properties by modelling and extrapolation is proposed. The limitations of elastic–plastic materials
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models for describing non-linear deformation in plastics are also discussed and the basis for a new model is explained.
2. Determination of data required for impact analysis 2.1. Materials models and data Non-linear stress/strain behaviour of plastics is generally associated with an enhanced stress–relaxation, or creep arising from an increase in molecular mobility induced by elevated stresses. Much of the strain is recoverable at levels below the yield strain. Satisfactory models of non-linear viscoelasticity have yet to be developed, and most FE packages consider material non-linearity in terms of elastic–plastic models that were developed for metals and adapted for use with plastics and other materials. With elastic–plastic models, calculations of stress and strain distributions at low strains (ⱕ0.01) are based on the theory of linear elasticity using, for isotropic materials, values of the Young’s modulus E and elastic Poisson’s ratio ne. The onset of non-linearity in a stress– strain curve is then ascribed to plastic deformation and occurs at a stress level regarded as the first yield stress. The subsequent increase in yield stress with strain is associated with the effects of strain hardening. The hardening behaviour is required by the computer analysis and is most commonly defined by a plot of tensile yield stress against the tensile plastic strain. This is derived from a measured tensile stress/strain curve by subtracting the elastic component of strain at each stress level. Since plastics are viscoelastic materials, the hardening curves are dependent upon the strain rate associated with the applied load. 2.2. Experimental methods Much of the material property data required by a finite element analysis of the performance of a component under impact loading can be derived from a series of tensile stress/strain curves measured over a range of strain rates. For many applications, the strain rate should reach or exceed 100 s⫺1. Standard test methods for measuring tensile behaviour are confined to strain rates below 0.1 s⫺1. Using servo-hydraulic test machines, smaller specimens and light weight extensometers, reliable results can generally be obtained to strain rates around 1 s⫺1. Above this value experimental difficulties, especially with strain measurement, can significantly reduce the accuracy of results. The development of high-speed extensometers that operate at higher strain rates has been reported, although these devices are not used routinely for data acquisition.
A simpler approach to determining strain in such tests is to derive values from cross-head movement or the change in grip separation. An apparent or average strain is obtained from the ratio of the change in grip separation to the length of the specimen between the grips prior to load application. [Note, if the measurement of cross-head movement is used then a correction will, in general, be necessary arising from the compliance of the test assembly.] Since the test specimen is usually waisted in the gauge region, this average strain will differ from actual strain in the gauge region of the specimen, where the corresponding stress is calculated. It is possible to estimate the actual strain values from the average strain using a correlation of these quantities determined from lower speed tests, where actual strain can be measured with extensometers. The correlation obtained will be specific to a particular test machine, the specimen size and geometry and the strain level in the specimen when this is above the limit for linear behaviour. There will also be a small dependence upon material properties and hence test speed. There remains some concern over the use of this approach for determining strain in materials where the strain is not uniform in the gauge section of the specimen beyond the peak in the stress/strain curve. Materials that show a visible neck region are an extreme example of this. In this way, it is generally possible to obtain valid tensile data up to test speeds of around 1 m/s which will correspond to strain rates of typically 10 s⫺1. At test speeds above this, additional problems arise. Owing to the finite time for the hydraulic ram to accelerate to the set speed, the test will not be performed under constant strain rate. In general, this is not a major problem since in the initial part of the test where the behaviour is linear, the strain rate is less than in the latter part where plastic deformation is dominant. Hence, elastic and plastic behaviour should be analysed using different values of strain rate. A more serious problem arises with the rapid accelerations associated with load application that give rise to transient forces. Such forces are recorded by the force transducer and superimposed onto the measured force on the specimen. These transient forces arise from excitation of a resonance of the force transducer and from multiple reflections of the acoustic pulse generated at one end of the specimen by the sudden load application. The influence of these forces on the accuracy of measurements can be minimised by using components of low mass in the test assembly and a high stiffness force transducer and by filtering or smoothing the force signal. 2.3. Data acquisition by materials modelling The measurement of data characterising rate-dependent behaviour suitable for impact analyses is laborious
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and prone to errors unless suitable precautions are taken. An alternative approach to obtaining plastic strain hardening behaviour at high strain rates is to use mathematical models of stress/strain curves measured at low and intermediate strain rates to extrapolate behaviour to higher rates where experimental difficulties arise. The advantages of this approach are: 1. Data acquisition is achieved with the use of minimal testing effort and expensive apparatus. The testing procedures required for data extrapolation are relatively straightforward. 2. Extensive materials property data can be readily calculated from a mathematical function together with values for the material parameters in this function that have been derived from selected tests. This is a convenient and efficient way of accessing materials properties in a package that allows a user to define the expression for rate-dependent hardening. 3. If valid functions are available for modelling ratedependent behaviour, then materials properties at high strain rates derived by extrapolation will be more accurate than measured values. This may be particularly relevant for those materials in which the strain distribution in the gauge length of the specimen is not uniform after the peak load has been reached. Measured stresses and strains are then average values which can depart significantly from true stress and strain in regions of strain localisation. To address this latter problem, true stress and strain measurements can be made using a video extensometer and multiple, short gauge regions along the specimen. By measuring lateral strain as well as longitudinal strain, in these discrete regions in the tensile specimen, values for Poisson’s ratio and thus true stresses can be calculated. Video extensometers can only be used at low or moderate strain rates but by extrapolating precise measurements at these rates to higher rates, it should be possible to achieve higher accuracy in true stress and strain values than is possible by measurement of average values. The realisation of these objectives depends critically on the validity of the model functions used to describe strain and strain-rate dependence. Some suitable functions are proposed below and their application illustrated using extensive property data measured on a propylene– ethylene copolymer containing 8% ethylene. 2.4. Modelling plastic strain hardening Fig. 1 shows true stress/true strain curves for the propylene copolymer measured in tension at strain rates ranging from 3×10−4 s⫺1 to 90 s⫺1. These strain rates refer to strain levels of 0.03 and above where plastic
Fig. 1. Tensile stress–strain curves for a propylene–ethylene copolymer measured at different test speeds. The values for the plastic strain rate for each test are shown with each curve.
deformation is dominant. At strain rates between 0.0003 s⫺1 and 0.03 s⫺1 measurements were obtained using the ISO multipurpose test specimen and cross-head speeds of 1, 10 and 100 mm/min. Strain measurements at low values (ⱕ0.01) were made using contacting extensometers to measure both the longitudinal and transverse strains. At higher strains, a contacting lateral extensometer was used for transverse strain measurements, and a video extensometer was used to obtain longitudinal strains. From these measurements, Poisson’s ratio was derived over the strain range and used to calculate values for the true tensile stress. No significant variation of Poisson’s ratio with strain rate was discernible from these tests. Data at strain rates between 0.2 and 30 were obtained using tensile specimens having the same geometry as the standard multipurpose specimen but scaled to half the size. A servohydraulic test machine was used to achieve cross-head speeds in the range 10 to 1000 mm/s. Contacting extensometers were used to measure longitudinal strain up to a strain of about 0.1 in tests carried out at speeds of 10 and 100 mm/s. At higher strains, and in tests at the higher speed of 1 m/s, longitudinal strains were derived from measurements of cross-head movement. Data at the highest strain rate were obtained from falling weight impact tests at a speed at contact of 4 m/s. Lateral strain measurements were not made in tests at strain rates above 0.03 s⫺1. Since tests at lower rates showed that any variation of Poisson’s ratio with strain rate was small, true stresses in the higher rate tests were calculated using Poisson’s ratio data obtained at the lower rates. Values for tensile modulus E were obtained at each strain rate from measurements below a strain level of 0.005. These were used to derive curves of stress s against plastic strain ep based on the assumption that the
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parameters b and q are essentially rate independent for this material and can be assumed to take average values of 0.69 and 0.6 respectively without any loss of predictive accuracy. The parameter eps decreases with strain rate and reflects a shift in the strain at which the peak in stress occurs with increasing strain rate as observed in Fig. 2. This dependence can be either modelled as shown below or assumed constant for convenience, if the associated loss of precision is acceptable. 2.5. Modelling rate-dependent behaviour
Fig. 2. True tensile stress/true tensile plastic strain curves derived from the data in the figure.
total strain e is the sum of elastic and plastic components. Thus ep⫽e⫺
s E
(1)
The resulting curves are shown in Fig. 2 and are the ratedependent hardening data required by a finite element analysis. Each of these curves can be accurately modelled by a function of the form s⫽[so⫹(sf⫺so)(1⫺exp⫺(ep/eps)b)](1⫺qep)
(2)
With this curve, stress values range between so (the first yield stress) corresponding to zero plastic strain and a value of around sf (the initial flow stress) corresponding approximately to the plateau stress. The parameter eps represents some mean strain within the strain range over which the stress rises rapidly between so and sf, and the parameter b influences the width of that range. The term containing the parameter q describes a small decrease of stress with plastic strain beyond the strain at which the peak in stress occurs. Values for these parameters giving the best fit of Eq. (2) to each of the curves in Fig. 2 are shown in Table 1. It is evident from Table 1 that the main strain rate dependence occurs in the parameters so and sf. The
The variation of the yield stress of polymers with strain rate is generally described using the Eyring equation [1]. In its most simple form, this can be written sy⫽A⫹B log e˙ p
(3)
where sy is usually the peak stress in a tensile test carried out at the strain rate e˙ p and A and B are temperaturedependent materials parameters. The rate-dependence of the parameters so and sf in Table 1 have accordingly been modelled using the expressions sf⫽sfo⫹b log e˙ p
(4a)
and so⫽soo⫹c log e˙ p
(4b)
Plots of sf and so taken from Table 1 against log e˙ p are shown in Fig. 3. Eq. (4) gives a satisfactory fit to
Fig. 3. Plots of the parameters sf, so and eps shown in Table 1 against log plastic strain rate.
Table 1 Values for the parameters in Eq. (2) giving the fits shown to the data in Fig. 2 e˙ p (s⫺1) so (MPa) sf (MPa) eps b q
0.00035 7 27 0.013 0.63 0.56
0.004 8 29.3 0.010 0.63 0.51
0.027 9 31.7 0.009 0.70 0.47
0.20 10 35.4 0.008 0.68 0.70
2.1 11 39.5 0.0075 0.70 0.76
29 12 42.5 0.006 0.75 0.59
91 13 46.5 0.005 0.75 0.54
G. Dean, B. Read / Polymer Testing 20 (2001) 677–683
It can be seen that the comparison between the function and experimental data is very satisfactory except at the highest rate of 91 s⫺1. The departure here may arise from experimental error in the data which were determined using the falling weight impact method. Alternatively, it may indicate the onset of a breakdown of Eq. (4a) at this rate. With any predictive method, it is necessary to establish the limitations in the functions especially when used for extrapolation. The validity of Eq. (4a) over a wider range of strain rate can be explored through the use of measurements at lower temperatures as discussed below. Evidence that Eqs. (3) and (4) represent an over simplification of yielding kinetics has been reported for some polymers [2,3] and attributed to a contribution to the yield stress from a secondary (lower temperature) relaxation process. Assuming that this process gives an additive contribution to the yield stress, it follows that the sf vs log e˙ p plot (Fig. 3) will show a change to a steeper gradient at a strain rate dependent on the kinetics of the secondary relaxation. This change of gradient occurs at lower strain rates if the temperature is lowered, as shown in Fig. 5. In this figure, the maximum stress sy at each strain rate is identified from tensile tests at different absolute temperatures T and is plotted as the ratio sy/T against log strain rate. By normalising with respect to the absolute temperature T, the analysis shows that the gradients of both regions (with and without contributions from the secondary relaxation respectively) are
Fig. 4. Comparisons of tensile data with calculated values using Eq. (7) and the parameters in Table 2.
the data and yield values for sfo and soo of 38 MPa and 10.6 MPa respectively. The parameters b and c are 3.6 MPa and 1.1 MPa respectively. The variation of the parameter eps can also be adequately described by a linear dependence on log e˙ p as also shown in Fig. 3. Thus taking eps⫽eso⫺d log e˙ p
(5)
values for eso of 0.0077 and for d of 0.0013 have been derived. In general it will be possible to simplify Eq. (4) with negligible loss of accuracy. It can be seen from the parameters derived from Fig. 3 that Eq. (4) can be written sf⫽sfo(1⫹a log e˙ p)
681
(6)
so⫽soo(1⫹a log e˙ p) where a is approximately 0.090. Eq. (2) then takes the form s⫽(sfo⫺(sfo⫺soo)exp⫺(ep/eps)b)(1⫺qep)(1⫹a log e˙ p) (7) It is worth noting that if eps is assumed constant, independent of strain rate, then the strain and strain-rate dependencies of s in Eq. (7) are separable. This can make implementation of materials properties as a userdefined hardening function easier in certain FE packages. Fig. 4 shows comparisons of Eq. (7) with the experimental data using values for the parameters given in Table 2.
Fig. 5. Plots of sy/T against log strain rate for the propylene– ethylene copolymer at different temperatures. Here, sv refers to the peak value of stress on a stress/strain curve.
Table 2 Values for the parameters in Eqs. (7) and (5) used to model strain and strain-rate dependent yield in the propylene copolymer sfo (MPa)
soo (MPa)
a
q
b
eso
d
38
10.6
0.090
0.6
0.69
0.007
0.0013
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the same at each temperature. The transition to a steeper gradient is then easier to identify, and it is possible to estimate where this will occur at temperatures around and above ambient.
3. The application of materials models 3.1. Limitations of elastic–plastic models For an elastic–plastic model which assumes that plastic deformation is described by a von Mises yield criterion, the only data requirements are elastic properties and a hardening curve represented by a plot of yield stress against plastic strain. If rate dependent behaviour is to be modelled then curves spanning a range of strain rate are needed, as shown in Fig. 2. However, in order to establish how valid the von Mises criterion is for a particular material, it is necessary to compare hardening functions obtained under different stress states. In this way, the sensitivity of yielding to the hydrostatic component of stress can be investigated. Tensile data are generally used for this purpose as data acquisition is relatively straightforward, together with results from shear or uniaxial compression tests. Results are analysed using a yield criterion with sensitivity to the hydrostatic component of stress [4,5]. The simplest of these is sT ⫽
冑3(l+1) 2l
J1/2 2D ⫹
(l−1) J 2l 1
(8)
where J1/2 2D is the 2nd invariant of the deviatoric stress tensor and is a measure of the effective shear stress in the material. J1 is another stress invariant and is the hydrostatic stress component. The parameter sT is a tensile yield stress and will depend upon plastic strain ep. The function sT(ep) is the plastic strain hardening function and is shown at different strain rates by the data in Fig. 2. The parameter l is a measure of the sensitivity of yielding to the hydrostatic stress. Its magnitude can be determined by analysis of the tests under different stress states and lies typically in the range 1.1–1.2. The yield behaviour of many materials cannot be readily modelled using Eq. (8). Such materials are characteristically multiphase polymers that have been specially formulated by blending or copolymerisation to improve toughness. A common example of such a material is acrylonitrile butadiene styrene (ABS). Results of tests carried out on a typical grade of this material under uniaxial tension, uniaxial compression and shear are shown in Fig. 6. If the results of the shear and compression tests are analysed using Eq. (8), then a value for l=1.2 is obtained. However, if tension and compression data or tension and shear data are compared, then the derived values for l are significantly higher and must vary with plastic strain as a consequence of the different shape of
Fig. 6. Experimental stress/strain curves for an acrylonitrile– butadiene–styrene (ABS) polymer obtained from tests under uniaxial compression, uniaxial tension and shear.
the tensile curve. Clearly plastic deformation under a tensile stress is different from that under compression or shear. The plastic deformation mechanism that determines behaviour under tension is associated with the nucleation of multiple crazes or cavities in the polymer. Since this requires a dilatational component of stress or strain, the process is not observed in compression or shear. In the case of ABS, the rubber phase acts as centres for nucleation of cavities which serve to produce enhanced yielding around the rubber particles leading to a reduction in the tensile yield stresses relative to equivalent stresses in shear and compression. This behaviour is revealed by measurements of Poisson’s ratio with strain as shown in Fig. 7. Initially, Poisson’s ratio is observed to rise slightly with strain consistent with a plastic deformation mechanism that is predominantly shear in character. At a strain of around 0.02, the Poisson’s ratio decreases as cavities start to nucleate, and a deformation mechanism is initiated that involves significant dilatational strain. This cavity nucleation process
Fig. 7. Comparison of tensile and shear test data for the ABS polymer showing the onset of dilatational plastic deformation in the tensile test at a strain of about 0.02.
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parisons of the measured tensile behaviour with calculations based on the new model. 4. Conclusions
Fig. 8. Comparison of tensile stress/strain data for the ABS polymer with the curve predicted usng the new model with hardening data from a shear test.
is responsible for the observed reduction in tensile yield stresses. 3.2. A new model for toughened plastics An alternative elastic–plastic model is currently being developed based on the Gurson model [6] for porous plasticity in metals. The Gurson model describes enhanced yielding caused by the nucleation of cavities around rigid inclusions in the material. These are nucleated by the debonding of the inclusions and the matrix at a critical level of plastic strain. This model is being modified to take into consideration three important features of toughened plastics. 1. A modification of the yield criterion to include a sensitivity to the hydrostatic component of stress in the absence of cavities. 2. A modification of the hardening function of the material so that it changes progressively from the function for a toughened polymer (without cavities) to the function appropriate to the matrix polymer as cavities are nucleated and the toughening phase becomes ineffective. 3. A change in the criterion for cavity nucleation so that the critical factor is the volumetric strain. In this way, cavities will not be formed under compressive or shear stress states. The new model is being developed and evaluated through an analysis of results from tests under different states of stress. Fig. 8 shows some early predictions using the model to analyse data from the ABS polymer shown in Fig. 6. The hardening function used in the model calculations is obtained from the stress/strain curve measured in a shear test. Fig. 8 then shows com-
Computer methods based on finite element analysis are able to calculate the performance of plastics components under impact. For these calculations, tensile stress/strain data are needed that have been determined over a wide range of strain rate up to, and ideally, beyond 100 s⫺1. There are problems with the measurement of stress/strain curves at strain rates above around 1 s⫺1. At rates above this, errors in measurement occur associated with the generation of transient forces and with difficulties in the measurement of strain. The data required by a finite element analysis are tensile stress as a function of plastic strain and are derived from measured stress/strain curves. Mathematical functions have been proposed here that accurately model stress/plastic strain curves for a propylene–ethylene copolymer and the dependence of these curves on strain rate. Using this model, stress/strain curves at high strain rates can be calculated by extrapolation of measurements made at low and moderate strain rates where measurement accuracy is adequate. The validity of the extrapolation can be verified by selected measurements at a lower temperature. A new materials model has been developed for plastics that acquire high toughness through the nucleation of cavities under dilatational stress. The model is able to predict tensile behaviour from a knowledge of behaviour under shear. Further work is needed to make additional refinements to the model and assess its applicability to a wider range of types of polymer. Acknowledgements This work was carried out with funding from the DTI as part of their programme on the Characterisation of the Performance of Materials. References [1] H. Eyring, J. Chem. Phys. 4 (1936) 283. [2] J.A. Roetling, Polymer 6 (1965) 311. [3] C. Bauwens-Crowet, J.-C. Bauwens, G. Homes, J. Polym. Sci. A2 7 (1969) 735. [4] S.S. Sternstein, L. Ongchin, Am. Chem. Soc. Polym. Prepr. 10 (1969) 117. [5] R. Raghava, R.M. Caddell, G.S. Yeh, J. Mater. Sci. 8 (1973) 225. [6] A.L. Gurson, J. Eng. Mater. Technol., Trans ASME 99 (1977) 2.
Property Modelling
Modelling the behaviour of plastics for design under impact G. Dean, B. Read Centre for Materials Measurement and Technology, National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK Received 9 October 2000; accepted 19 December 2000
Abstract Computer methods based on finite element analysis are able to predict the performance of plastics under impact loading. Although the accuracy of results depends on the model used to describe the deformation behaviour of the polymer, whichever model is used, the analysis requires stress/strain data over a wide range of strain rate. These data are most conveniently measured in tension, but procedures are currently not available for determining results at high strain rates. ISO standards for tensile property measurement are applicable for strain rates up to around 0.1 s⫺1. To simulate behaviour under impact, data are required at rates that are 3 or 4 orders of magnitude higher than this. For accurate data acquisition at these higher rates, attention needs to be paid to apparatus design in order to minimise contributions from transient forces arising from resonances and the propagation of shock waves in the apparatus. In addition, procedures and extensometers are not routinely available for determining strains at the higher rates of deformation. This paper illustrates the acquisition of data over a wide range of strain rates through a combination of measurements at low and moderate strain rates with extrapolation of these data to higher rates. In order to maximise accuracy at moderate strain rates, suitable designs need to be selected for the transducers, the test specimen geometry and the test assembly. Extrapolation is achieved by the use of mathematical functions to model the stress/strain curves and their rate dependence. Reference is also made to the development of a new materials model for describing the deformation behaviour of toughened plastics at large strains. 2001 NPL. Published by Elsevier Science Ltd. All rights reserved. Keywords: Measurement; Plastics; Modelling; Impact; Materials; FEA
1. Computer aided design Most modern grades of plastics are very tough materials that are capable of sustaining large strain levels prior to failure, making them an attractive option for many applications. Such applications include housings for domestic and electronic appliances, which may have to sustain accidental impact without showing signs of damage, and interior components of motor vehicles, which may be impacted by occupants in a crash and must minimise damage to the human body part involved. In these applications and others, it is important that the performance and the safe operating limits of the component under impact are known. Substantial savings can be made in both time and cost if this can be done through
competent design rather than the moulding and testing of prototypes. Computer methods are available based on finite element analysis that enable predictions to be made of performance under realistic loading situations. There are two factors that limit the wider use of these computer methods. Firstly, a lack of recognised and routine test methods for measuring some of the data required by the analysis, and secondly, limited knowledge of the validity for plastics of the materials models employed by the computer analysis, and thus uncertainty in the accuracy of derived results. This paper discusses the experimental issues associated with data measurement, and a procedure for determining properties by modelling and extrapolation is proposed. The limitations of elastic–plastic materials
0142-9418/01/$ - see front matter 2001 NPL. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 0 1 ) 0 0 0 0 3 - 4
678
G. Dean, B. Read / Polymer Testing 20 (2001) 677–683
models for describing non-linear deformation in plastics are also discussed and the basis for a new model is explained.
2. Determination of data required for impact analysis 2.1. Materials models and data Non-linear stress/strain behaviour of plastics is generally associated with an enhanced stress–relaxation, or creep arising from an increase in molecular mobility induced by elevated stresses. Much of the strain is recoverable at levels below the yield strain. Satisfactory models of non-linear viscoelasticity have yet to be developed, and most FE packages consider material non-linearity in terms of elastic–plastic models that were developed for metals and adapted for use with plastics and other materials. With elastic–plastic models, calculations of stress and strain distributions at low strains (ⱕ0.01) are based on the theory of linear elasticity using, for isotropic materials, values of the Young’s modulus E and elastic Poisson’s ratio ne. The onset of non-linearity in a stress– strain curve is then ascribed to plastic deformation and occurs at a stress level regarded as the first yield stress. The subsequent increase in yield stress with strain is associated with the effects of strain hardening. The hardening behaviour is required by the computer analysis and is most commonly defined by a plot of tensile yield stress against the tensile plastic strain. This is derived from a measured tensile stress/strain curve by subtracting the elastic component of strain at each stress level. Since plastics are viscoelastic materials, the hardening curves are dependent upon the strain rate associated with the applied load. 2.2. Experimental methods Much of the material property data required by a finite element analysis of the performance of a component under impact loading can be derived from a series of tensile stress/strain curves measured over a range of strain rates. For many applications, the strain rate should reach or exceed 100 s⫺1. Standard test methods for measuring tensile behaviour are confined to strain rates below 0.1 s⫺1. Using servo-hydraulic test machines, smaller specimens and light weight extensometers, reliable results can generally be obtained to strain rates around 1 s⫺1. Above this value experimental difficulties, especially with strain measurement, can significantly reduce the accuracy of results. The development of high-speed extensometers that operate at higher strain rates has been reported, although these devices are not used routinely for data acquisition.
A simpler approach to determining strain in such tests is to derive values from cross-head movement or the change in grip separation. An apparent or average strain is obtained from the ratio of the change in grip separation to the length of the specimen between the grips prior to load application. [Note, if the measurement of cross-head movement is used then a correction will, in general, be necessary arising from the compliance of the test assembly.] Since the test specimen is usually waisted in the gauge region, this average strain will differ from actual strain in the gauge region of the specimen, where the corresponding stress is calculated. It is possible to estimate the actual strain values from the average strain using a correlation of these quantities determined from lower speed tests, where actual strain can be measured with extensometers. The correlation obtained will be specific to a particular test machine, the specimen size and geometry and the strain level in the specimen when this is above the limit for linear behaviour. There will also be a small dependence upon material properties and hence test speed. There remains some concern over the use of this approach for determining strain in materials where the strain is not uniform in the gauge section of the specimen beyond the peak in the stress/strain curve. Materials that show a visible neck region are an extreme example of this. In this way, it is generally possible to obtain valid tensile data up to test speeds of around 1 m/s which will correspond to strain rates of typically 10 s⫺1. At test speeds above this, additional problems arise. Owing to the finite time for the hydraulic ram to accelerate to the set speed, the test will not be performed under constant strain rate. In general, this is not a major problem since in the initial part of the test where the behaviour is linear, the strain rate is less than in the latter part where plastic deformation is dominant. Hence, elastic and plastic behaviour should be analysed using different values of strain rate. A more serious problem arises with the rapid accelerations associated with load application that give rise to transient forces. Such forces are recorded by the force transducer and superimposed onto the measured force on the specimen. These transient forces arise from excitation of a resonance of the force transducer and from multiple reflections of the acoustic pulse generated at one end of the specimen by the sudden load application. The influence of these forces on the accuracy of measurements can be minimised by using components of low mass in the test assembly and a high stiffness force transducer and by filtering or smoothing the force signal. 2.3. Data acquisition by materials modelling The measurement of data characterising rate-dependent behaviour suitable for impact analyses is laborious
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and prone to errors unless suitable precautions are taken. An alternative approach to obtaining plastic strain hardening behaviour at high strain rates is to use mathematical models of stress/strain curves measured at low and intermediate strain rates to extrapolate behaviour to higher rates where experimental difficulties arise. The advantages of this approach are: 1. Data acquisition is achieved with the use of minimal testing effort and expensive apparatus. The testing procedures required for data extrapolation are relatively straightforward. 2. Extensive materials property data can be readily calculated from a mathematical function together with values for the material parameters in this function that have been derived from selected tests. This is a convenient and efficient way of accessing materials properties in a package that allows a user to define the expression for rate-dependent hardening. 3. If valid functions are available for modelling ratedependent behaviour, then materials properties at high strain rates derived by extrapolation will be more accurate than measured values. This may be particularly relevant for those materials in which the strain distribution in the gauge length of the specimen is not uniform after the peak load has been reached. Measured stresses and strains are then average values which can depart significantly from true stress and strain in regions of strain localisation. To address this latter problem, true stress and strain measurements can be made using a video extensometer and multiple, short gauge regions along the specimen. By measuring lateral strain as well as longitudinal strain, in these discrete regions in the tensile specimen, values for Poisson’s ratio and thus true stresses can be calculated. Video extensometers can only be used at low or moderate strain rates but by extrapolating precise measurements at these rates to higher rates, it should be possible to achieve higher accuracy in true stress and strain values than is possible by measurement of average values. The realisation of these objectives depends critically on the validity of the model functions used to describe strain and strain-rate dependence. Some suitable functions are proposed below and their application illustrated using extensive property data measured on a propylene– ethylene copolymer containing 8% ethylene. 2.4. Modelling plastic strain hardening Fig. 1 shows true stress/true strain curves for the propylene copolymer measured in tension at strain rates ranging from 3×10−4 s⫺1 to 90 s⫺1. These strain rates refer to strain levels of 0.03 and above where plastic
Fig. 1. Tensile stress–strain curves for a propylene–ethylene copolymer measured at different test speeds. The values for the plastic strain rate for each test are shown with each curve.
deformation is dominant. At strain rates between 0.0003 s⫺1 and 0.03 s⫺1 measurements were obtained using the ISO multipurpose test specimen and cross-head speeds of 1, 10 and 100 mm/min. Strain measurements at low values (ⱕ0.01) were made using contacting extensometers to measure both the longitudinal and transverse strains. At higher strains, a contacting lateral extensometer was used for transverse strain measurements, and a video extensometer was used to obtain longitudinal strains. From these measurements, Poisson’s ratio was derived over the strain range and used to calculate values for the true tensile stress. No significant variation of Poisson’s ratio with strain rate was discernible from these tests. Data at strain rates between 0.2 and 30 were obtained using tensile specimens having the same geometry as the standard multipurpose specimen but scaled to half the size. A servohydraulic test machine was used to achieve cross-head speeds in the range 10 to 1000 mm/s. Contacting extensometers were used to measure longitudinal strain up to a strain of about 0.1 in tests carried out at speeds of 10 and 100 mm/s. At higher strains, and in tests at the higher speed of 1 m/s, longitudinal strains were derived from measurements of cross-head movement. Data at the highest strain rate were obtained from falling weight impact tests at a speed at contact of 4 m/s. Lateral strain measurements were not made in tests at strain rates above 0.03 s⫺1. Since tests at lower rates showed that any variation of Poisson’s ratio with strain rate was small, true stresses in the higher rate tests were calculated using Poisson’s ratio data obtained at the lower rates. Values for tensile modulus E were obtained at each strain rate from measurements below a strain level of 0.005. These were used to derive curves of stress s against plastic strain ep based on the assumption that the
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parameters b and q are essentially rate independent for this material and can be assumed to take average values of 0.69 and 0.6 respectively without any loss of predictive accuracy. The parameter eps decreases with strain rate and reflects a shift in the strain at which the peak in stress occurs with increasing strain rate as observed in Fig. 2. This dependence can be either modelled as shown below or assumed constant for convenience, if the associated loss of precision is acceptable. 2.5. Modelling rate-dependent behaviour
Fig. 2. True tensile stress/true tensile plastic strain curves derived from the data in the figure.
total strain e is the sum of elastic and plastic components. Thus ep⫽e⫺
s E
(1)
The resulting curves are shown in Fig. 2 and are the ratedependent hardening data required by a finite element analysis. Each of these curves can be accurately modelled by a function of the form s⫽[so⫹(sf⫺so)(1⫺exp⫺(ep/eps)b)](1⫺qep)
(2)
With this curve, stress values range between so (the first yield stress) corresponding to zero plastic strain and a value of around sf (the initial flow stress) corresponding approximately to the plateau stress. The parameter eps represents some mean strain within the strain range over which the stress rises rapidly between so and sf, and the parameter b influences the width of that range. The term containing the parameter q describes a small decrease of stress with plastic strain beyond the strain at which the peak in stress occurs. Values for these parameters giving the best fit of Eq. (2) to each of the curves in Fig. 2 are shown in Table 1. It is evident from Table 1 that the main strain rate dependence occurs in the parameters so and sf. The
The variation of the yield stress of polymers with strain rate is generally described using the Eyring equation [1]. In its most simple form, this can be written sy⫽A⫹B log e˙ p
(3)
where sy is usually the peak stress in a tensile test carried out at the strain rate e˙ p and A and B are temperaturedependent materials parameters. The rate-dependence of the parameters so and sf in Table 1 have accordingly been modelled using the expressions sf⫽sfo⫹b log e˙ p
(4a)
and so⫽soo⫹c log e˙ p
(4b)
Plots of sf and so taken from Table 1 against log e˙ p are shown in Fig. 3. Eq. (4) gives a satisfactory fit to
Fig. 3. Plots of the parameters sf, so and eps shown in Table 1 against log plastic strain rate.
Table 1 Values for the parameters in Eq. (2) giving the fits shown to the data in Fig. 2 e˙ p (s⫺1) so (MPa) sf (MPa) eps b q
0.00035 7 27 0.013 0.63 0.56
0.004 8 29.3 0.010 0.63 0.51
0.027 9 31.7 0.009 0.70 0.47
0.20 10 35.4 0.008 0.68 0.70
2.1 11 39.5 0.0075 0.70 0.76
29 12 42.5 0.006 0.75 0.59
91 13 46.5 0.005 0.75 0.54
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It can be seen that the comparison between the function and experimental data is very satisfactory except at the highest rate of 91 s⫺1. The departure here may arise from experimental error in the data which were determined using the falling weight impact method. Alternatively, it may indicate the onset of a breakdown of Eq. (4a) at this rate. With any predictive method, it is necessary to establish the limitations in the functions especially when used for extrapolation. The validity of Eq. (4a) over a wider range of strain rate can be explored through the use of measurements at lower temperatures as discussed below. Evidence that Eqs. (3) and (4) represent an over simplification of yielding kinetics has been reported for some polymers [2,3] and attributed to a contribution to the yield stress from a secondary (lower temperature) relaxation process. Assuming that this process gives an additive contribution to the yield stress, it follows that the sf vs log e˙ p plot (Fig. 3) will show a change to a steeper gradient at a strain rate dependent on the kinetics of the secondary relaxation. This change of gradient occurs at lower strain rates if the temperature is lowered, as shown in Fig. 5. In this figure, the maximum stress sy at each strain rate is identified from tensile tests at different absolute temperatures T and is plotted as the ratio sy/T against log strain rate. By normalising with respect to the absolute temperature T, the analysis shows that the gradients of both regions (with and without contributions from the secondary relaxation respectively) are
Fig. 4. Comparisons of tensile data with calculated values using Eq. (7) and the parameters in Table 2.
the data and yield values for sfo and soo of 38 MPa and 10.6 MPa respectively. The parameters b and c are 3.6 MPa and 1.1 MPa respectively. The variation of the parameter eps can also be adequately described by a linear dependence on log e˙ p as also shown in Fig. 3. Thus taking eps⫽eso⫺d log e˙ p
(5)
values for eso of 0.0077 and for d of 0.0013 have been derived. In general it will be possible to simplify Eq. (4) with negligible loss of accuracy. It can be seen from the parameters derived from Fig. 3 that Eq. (4) can be written sf⫽sfo(1⫹a log e˙ p)
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(6)
so⫽soo(1⫹a log e˙ p) where a is approximately 0.090. Eq. (2) then takes the form s⫽(sfo⫺(sfo⫺soo)exp⫺(ep/eps)b)(1⫺qep)(1⫹a log e˙ p) (7) It is worth noting that if eps is assumed constant, independent of strain rate, then the strain and strain-rate dependencies of s in Eq. (7) are separable. This can make implementation of materials properties as a userdefined hardening function easier in certain FE packages. Fig. 4 shows comparisons of Eq. (7) with the experimental data using values for the parameters given in Table 2.
Fig. 5. Plots of sy/T against log strain rate for the propylene– ethylene copolymer at different temperatures. Here, sv refers to the peak value of stress on a stress/strain curve.
Table 2 Values for the parameters in Eqs. (7) and (5) used to model strain and strain-rate dependent yield in the propylene copolymer sfo (MPa)
soo (MPa)
a
q
b
eso
d
38
10.6
0.090
0.6
0.69
0.007
0.0013
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the same at each temperature. The transition to a steeper gradient is then easier to identify, and it is possible to estimate where this will occur at temperatures around and above ambient.
3. The application of materials models 3.1. Limitations of elastic–plastic models For an elastic–plastic model which assumes that plastic deformation is described by a von Mises yield criterion, the only data requirements are elastic properties and a hardening curve represented by a plot of yield stress against plastic strain. If rate dependent behaviour is to be modelled then curves spanning a range of strain rate are needed, as shown in Fig. 2. However, in order to establish how valid the von Mises criterion is for a particular material, it is necessary to compare hardening functions obtained under different stress states. In this way, the sensitivity of yielding to the hydrostatic component of stress can be investigated. Tensile data are generally used for this purpose as data acquisition is relatively straightforward, together with results from shear or uniaxial compression tests. Results are analysed using a yield criterion with sensitivity to the hydrostatic component of stress [4,5]. The simplest of these is sT ⫽
冑3(l+1) 2l
J1/2 2D ⫹
(l−1) J 2l 1
(8)
where J1/2 2D is the 2nd invariant of the deviatoric stress tensor and is a measure of the effective shear stress in the material. J1 is another stress invariant and is the hydrostatic stress component. The parameter sT is a tensile yield stress and will depend upon plastic strain ep. The function sT(ep) is the plastic strain hardening function and is shown at different strain rates by the data in Fig. 2. The parameter l is a measure of the sensitivity of yielding to the hydrostatic stress. Its magnitude can be determined by analysis of the tests under different stress states and lies typically in the range 1.1–1.2. The yield behaviour of many materials cannot be readily modelled using Eq. (8). Such materials are characteristically multiphase polymers that have been specially formulated by blending or copolymerisation to improve toughness. A common example of such a material is acrylonitrile butadiene styrene (ABS). Results of tests carried out on a typical grade of this material under uniaxial tension, uniaxial compression and shear are shown in Fig. 6. If the results of the shear and compression tests are analysed using Eq. (8), then a value for l=1.2 is obtained. However, if tension and compression data or tension and shear data are compared, then the derived values for l are significantly higher and must vary with plastic strain as a consequence of the different shape of
Fig. 6. Experimental stress/strain curves for an acrylonitrile– butadiene–styrene (ABS) polymer obtained from tests under uniaxial compression, uniaxial tension and shear.
the tensile curve. Clearly plastic deformation under a tensile stress is different from that under compression or shear. The plastic deformation mechanism that determines behaviour under tension is associated with the nucleation of multiple crazes or cavities in the polymer. Since this requires a dilatational component of stress or strain, the process is not observed in compression or shear. In the case of ABS, the rubber phase acts as centres for nucleation of cavities which serve to produce enhanced yielding around the rubber particles leading to a reduction in the tensile yield stresses relative to equivalent stresses in shear and compression. This behaviour is revealed by measurements of Poisson’s ratio with strain as shown in Fig. 7. Initially, Poisson’s ratio is observed to rise slightly with strain consistent with a plastic deformation mechanism that is predominantly shear in character. At a strain of around 0.02, the Poisson’s ratio decreases as cavities start to nucleate, and a deformation mechanism is initiated that involves significant dilatational strain. This cavity nucleation process
Fig. 7. Comparison of tensile and shear test data for the ABS polymer showing the onset of dilatational plastic deformation in the tensile test at a strain of about 0.02.
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parisons of the measured tensile behaviour with calculations based on the new model. 4. Conclusions
Fig. 8. Comparison of tensile stress/strain data for the ABS polymer with the curve predicted usng the new model with hardening data from a shear test.
is responsible for the observed reduction in tensile yield stresses. 3.2. A new model for toughened plastics An alternative elastic–plastic model is currently being developed based on the Gurson model [6] for porous plasticity in metals. The Gurson model describes enhanced yielding caused by the nucleation of cavities around rigid inclusions in the material. These are nucleated by the debonding of the inclusions and the matrix at a critical level of plastic strain. This model is being modified to take into consideration three important features of toughened plastics. 1. A modification of the yield criterion to include a sensitivity to the hydrostatic component of stress in the absence of cavities. 2. A modification of the hardening function of the material so that it changes progressively from the function for a toughened polymer (without cavities) to the function appropriate to the matrix polymer as cavities are nucleated and the toughening phase becomes ineffective. 3. A change in the criterion for cavity nucleation so that the critical factor is the volumetric strain. In this way, cavities will not be formed under compressive or shear stress states. The new model is being developed and evaluated through an analysis of results from tests under different states of stress. Fig. 8 shows some early predictions using the model to analyse data from the ABS polymer shown in Fig. 6. The hardening function used in the model calculations is obtained from the stress/strain curve measured in a shear test. Fig. 8 then shows com-
Computer methods based on finite element analysis are able to calculate the performance of plastics components under impact. For these calculations, tensile stress/strain data are needed that have been determined over a wide range of strain rate up to, and ideally, beyond 100 s⫺1. There are problems with the measurement of stress/strain curves at strain rates above around 1 s⫺1. At rates above this, errors in measurement occur associated with the generation of transient forces and with difficulties in the measurement of strain. The data required by a finite element analysis are tensile stress as a function of plastic strain and are derived from measured stress/strain curves. Mathematical functions have been proposed here that accurately model stress/plastic strain curves for a propylene–ethylene copolymer and the dependence of these curves on strain rate. Using this model, stress/strain curves at high strain rates can be calculated by extrapolation of measurements made at low and moderate strain rates where measurement accuracy is adequate. The validity of the extrapolation can be verified by selected measurements at a lower temperature. A new materials model has been developed for plastics that acquire high toughness through the nucleation of cavities under dilatational stress. The model is able to predict tensile behaviour from a knowledge of behaviour under shear. Further work is needed to make additional refinements to the model and assess its applicability to a wider range of types of polymer. Acknowledgements This work was carried out with funding from the DTI as part of their programme on the Characterisation of the Performance of Materials. References [1] H. Eyring, J. Chem. Phys. 4 (1936) 283. [2] J.A. Roetling, Polymer 6 (1965) 311. [3] C. Bauwens-Crowet, J.-C. Bauwens, G. Homes, J. Polym. Sci. A2 7 (1969) 735. [4] S.S. Sternstein, L. Ongchin, Am. Chem. Soc. Polym. Prepr. 10 (1969) 117. [5] R. Raghava, R.M. Caddell, G.S. Yeh, J. Mater. Sci. 8 (1973) 225. [6] A.L. Gurson, J. Eng. Mater. Technol., Trans ASME 99 (1977) 2.