The Mechanical Behaviour of Poly(Vinyl Butyral) at Different Strain Magnitudes and Strain Rates

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1 View metadata, citation and similar papers at core.ac.uk brought to you by CORE 2 Journal of Materials Science manuscript No. provided by Spiral - Imperial College Digital Repository 3 (will be inserted by the editor) 4 5 6 7 8 9 The mechanical behaviour of poly(vinyl butyral) at different 10 11 12 strain magnitudes and strain rates 13 14 15 P. A. Hooper · B. R. K. Blackman · J. P. Dear 16 17 18 19 20 Received: date / Accepted: date 21 22 23 24 Abstract The mechanical behaviour of poly(vinyl butyral) (PVB) at small (< 0.1%) and 25 26 large strains (> 200%) is investigated experimentally over a range of strain rates in order 27 28 to provide data for the development and validation of constitutive models. The small-strain 29 30 response is investigated using dynamic mechanical analysis at frequencies from 1Hz to 31 32 100Hz and temperatures from −80◦Cto70◦C. It is found that a generalized Maxwell model 33 34 adequately describes the material behaviour in the small-strain regime. The large-strain re- 35 36 sponse is investigated using a high-speed servo hydraulic test machine at strain-rates from 37 38 0.2s−1 to 400s−1. It is found that the PVB response is characterised by a time-dependent 39 40 steep initial rise in stress followed by a hyperelastic type response until failure. No current 41 42 material model completely captures this time-dependent behaviour at large-strains. 43 44 45 Keywords PVB · Poly(vinyl butyral) · Laminated glass · DMA · Large strain · High 46 47 strain-rate · Viscoelasticity 48 49 P. A. Hooper · B.R.K. Blackman · J. P. Dear 50 51 Department of Mechanical Engineering, Imperial College London, Exhibition Road, London, SW7 2AZ, UK 52 53 Tel.: +4420 7594 7128 54 E-mail: [email protected] 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 2 5 6 1 Introduction 7 8 9 Polyvinyl butyral (PVB) is primarily used as an interlayer material in the manufacture of 10 11 laminated glass. Its mechanical response is highly non-linear, time dependent and it is capa- 12 13 ble of undergoing extensions to several times its initial length and recovering without signif- 14 15 icant permanent deformation. The behaviour of PVB at two distinct strain magnitudes are of 16 17 practical interest. Firstly, the small-strain behaviour plays an important role in determining 18 19 the bending behaviour of uncracked laminated glass panes. Secondly, the large-strain be- 20 21 haviour is of interest in cracked laminated glass where the PVB acts as bridge between glass 22 23 fragments and can undergo large tensile extensions. The mechanical response at different 24 25 rates of strain is also of significant importance due to applications ranging from quasi-static 26 27 loading through to impact, ballistic and blast loading regimes. 28 29 30 Bennison et al. [1] and van Duser et al. [2] have investigated the small-strain behaviour 31 32 of a Dupont Butacite interlayer in the development of a finite element model to predict the 33 34 behaviour of laminated glass plates subject to wind pressure loading. They included vis- 35 36 coelastic effects by using a generalized Maxwell series to account for the time-dependent 37 38 shear modulus of the interlayer. Terms in the Maxwell model were determined experimen- 39 40 tally using dynamic mechanical analysis. Variation in shear modulus at different tempera- 41 42 tures was also taken into account by using the Williams-Landel-Ferry (WLF) equation [3] 43 44 to shift the time dependent shear modulus curve to a different temperature. The mechanical 45 46 behaviour of PVB at large-strains has not been widely reported in the literature. 47 48 In this paper the behaviour of a single grade of PVB is investigated experimentally over 49 50 a wide range of strain magnitudes and strain rates. The behaviour over these ranges needs 51 52 to be fully understood before physically based models of laminated glass (particularly after 53 54 the glass plies fracture) and other composites containing PVB can be developed. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 3 5 6 2 Methods 7 8 9 The mechanical behaviour of PVB was investigated using two experimental methods. The 10 11 small-strain behaviour was investigated using dynamic mechanical analysis (DMA) at fre- 12 ◦ ◦ 13 quencies from 1Hz to 100Hz and temperatures from −80 Cto70C. The large-strain be- 14 15 haviour was investigated using a servo hydraulic tensile test machine at displacement rates 16 17 between 0.01 m/s and 15 m/s. The interlayer material tested was Saflex PVB produced by 18 19 Solutia Inc. with product number RB-41. 20 21 22 23 2.1 Small-strain behaviour 24 25 26 The small-strain viscoelastic behaviour of PVB is of interest when considering the response 27 28 of a laminated glass pane to loading before the glass plies fracture. Under these conditions 29 30 the strain in the PVB is limited by the failure strain of the glass plies (typically 0.1%). The 31 32 small-strain viscoelastic response has been investigated using DMA. The following sections 33 34 cover some background information on viscoelasticity, the DMA experimental technique 35 36 and analysis methods used. 37 38 39 Linear viscoelasticity 40 41 42 The viscoelastic properties of a material can be investigated by subjecting a sample to an 43 44 oscillatory load. When a viscoelastic material is subjected to an oscillating load the strain 45 46 ε lags behind the applied stress σ due to the viscous component of the material response. 47 48 Figure 1a shows an applied sinusoidal stress and the resulting out-of-phase strain response 49 50 with a phase angle δ.Itisusefultodefine two moduli which correspond to the elastic 51 52 and viscous components of stress. Figure 1b shows the stress signal decomposed into a 53 54 component in phase with the strain and a component out-of-phase with the strain. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 4 5 6 Tδ/2π Strain 7 Stress 8 9 10

11 , σ 12 13 14 15 T 16 17 Time 18 19 (a) Lag in stress. 20 21 22 Strain Stress in-phase 23 Stress out-of-phase 24 25

26 , 27 σ 28 29 30 31 32 Time 33 34 (b) Stress components. 35 36 Fig. 1: Sinusoidal stress and strain for a linear viscoelastic material. 37 38 39 40 The amplitude of the stress in-phase with the strain divided by the strain amplitude 41 42 is referred to as the storage modulus and is denoted by E in tension and G in shear. The 43 44 amplitude of the stress out-of-phase with the strain divided by the strain amplitude is referred 45 46 to as the loss modulus and is denoted by E in tension and G in shear. The ratio of the 47 48 moduli is equal to the tangent of the phase angle 49 50 51 52 53 E G 54 = = tanδ (1) 55 E G 56 57 58 59 60 61 62 63 64 65 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Fig. 2: DMA tensile fixture. 27 28 29 30 For tanδ = 0 the stress and strain are in-phase (the loss modulus is equal to zero) and the 31 32 material behaviour is purely elastic. For tanδ = 1 the viscous component of stress is as large 33 ◦ 34 as the elastic component and the stress and strain are 45 out-of-phase. At a phase angle of 35 36 90◦ the storage modulus is equal to zero and the material behaviour is purely viscous. 37 38 39 40 41 Dynamic mechanical analysis 42 43 44 In dynamic mechanical analysis (DMA) a sinusoidal strain is applied to a sample and the 45 46 resulting stress signal is measured in order to calculate storage modulus, loss modulus and 47 48 tanδ. This is usually performed over ranges of frequencies and temperatures to characterise 49 50 a material at different time scales. A TA Instruments Q800 DMA machine was used to 51 52 test the small-strain viscoelastic behaviour of PVB in extension. Figure 2 shows the tensile 53 54 fixture used to apply oscillatory strains to the sample. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 6 5 6 In this fixture the top of the sample is held stationary between grips and the bottom of 7 8 the sample is connected to a drive shaft which oscillates up and down. The initial length of 9 . . 10 the sample l0 was 16mm, the width w was 4 45mm and the thickness was 1 52mm. A small 11 12 preload was applied to the sample to ensure that it did not buckle when oscillated. The stress 13 14 in the sample was then calculated from the cross sectional area and the applied force. The 15 16 strain was found from the original length of the sample and the displacement of the drive 17 18 shaft. The PVB was tested with a strain amplitude of 0.1% at three frequencies; 1Hz, 10Hz 19 20 and 100Hz. 21 22 The test fixture shown in Fig. 2 was enclosed inside a chamber so that the temperature 23 24 could be controlled and varied. A liquid nitrogen cooling stage was used to reduce the tem- 25 26 perature in the chamber to below ambient and an internal heating element was used to raise 27 28 the temperature above ambient. A thermocouple placed close to the sample was used to 29 30 provide feedback for the temperature controller. The PVB was tested under isothermal con- 31 32 ◦ ◦ ◦ ditions at 5 C temperature increments from −80 Cto70 C. At each temperature increment, 33 34 data was collected at the three test frequencies. 35 36 37 38 39 40 41 Time-temperature superposition 42 43 44 The Time-Temperature Superposition (TTS) principle states that the moduli (and other vis- 45 46 coelastic functions) measured over a range of frequencies at one temperature can be super- 47 48 posed with the moduli measured over a range of frequencies at a different temperature. That 49 50 is to say there is an equivalence between time and temperature for the material response. 51 52 TTS can be used to expand the tested frequency range by performing tests at different tem- 53 54 peratures and reducing the results to a single reference temperature T . 55 0 56 57 58 59 60 61 62 63 64 65 1 2 3 4 7 5 6 The reduction of data to a single temperature is performed by shifting the frequency 7 8 values collected at one temperature by a factor, aT , until the curve overlaps with the curve 9 10 at the reference temperature. The shift factor is equal to 11 12 13 ω 14 = 0 aT ω (2) 15 T 16 17 where ω0 is the frequency that gives a particular response at the reference temperature 18 19 and ωT is the frequency that gives the same response at another temperature. Calculating 20 21 the shift factors for each test temperature allows a master curve to be produced where all the 22 23 data have been reduced to a single reference temperature. This technique was applied to the 24 25 data acquired using DMA to expand the frequency range beyond the machine capabilities 26 27 which were between 0.01Hz and 200Hz. The shift factors of the data were determined by 28 29 manually shifting the data to a reference temperature of 20◦C using the Rheology Advantage 30 31 Data Analysis Software provided by TA Instruments. 32 33 The Williams-Landel-Ferry (WLF) equation [3] is an empirical relation between tem- 34 35 perature and shift factors of the form 36 37 38 39 C1 (T − T0) logaT = − (3) 40 C2 + T − T0 41 42 where C1 and C2 are constants. The manually determined shift factors were fitted to 43 44 the WLF equation using nonlinear regression to find C1 and C2. This was done for shift 45 46 factors between 0◦C and 70◦C. Below 0◦C the PVB was found to be in the glassy regime 47 48 and the shift factors were no longer described by the WLF equation. The shift factors were 49 50 recalculated using the WLF equation to provide a refined master curve. Once a master curve 51 52 has been generated the WLF equation can be used to extrapolate it to different reference 53 54 temperatures. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 8 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Fig. 3: Generalized Maxwell viscoelastic material model. 35 36 37 Generalized Maxwell model 38 39 40 The small-strain viscoelastic behaviour of many polymers can be described using the gen- 41 42 eralized Maxwell model [4]. The model is shown in Fig. 3 and consists of a series of elastic 43 44 spring and viscous damper pairs in parallel. Under rapid loading the damping elements do 45 46 not have time to deform and the stiffness is defined by the summation of the spring elements. 47 48 For infinitely slow loading the stresses in the damping elements are zero and the stiffness is 49 50 defined by the single undamped spring. This model is commonly used to define viscoelastic 51 52 behaviour in finite element programs such as Abaqus. 53 54 The Generalized Maxwell model can be defined mathematically by 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 9 5 6 7 n − /τ ( )= + t i 8 G t G∞ ∑ Gie (4) 9 i=1 10 where G(t) is the shear relaxation modulus and is a function of time t, G∞ is the long 11 12 term shear modulus and Gi and τi are the shear modulus and relaxation times associated 13 14 with each spring and damper pair. The instantaneous shear modulus G0 is defined as 15 16 17 n 18 G0 = G∞ + ∑ Gi (5) 19 i=1 20 21 The frequency data collected using DMA and TTS were converted to relaxation data 22 23 using the approximation [4] 24 25 26 G(t)= G(ω) − 0.40G(0.40ω)+0.014G(10ω) (6) 27 ω=1/t 28 29 where ω is the frequency in radians per second. The measured tensile storage and loss 30 31 functions E(ω) and E(ω) were converted to G(ω) and G(ω) using the relation E = 3G 32 33 (assuming an incompressible material) [4]. Evaluation of the loss modulus at the required 34 35 frequencies was performed using linear interpolation. Nonlinear regression was used to fit 36 37 the generalized Maxwell model to the shear relaxation modulus data and determine the 38 39 values of G∞, G and τ . 40 i i 41 42 43 2.2 Large strain tensile tests 44 45 46 The large strain behaviour of PVB is of interest when considering the response of cracked 47 48 laminated glass to loads such as those from blast and impact. In these situations the PVB acts 49 50 as a bridging ligament between glass fragments and can experience large extensions before 51 52 failing. A servo-hydraulic tensile test machine was used to test PVB samples in tension at 53 54 displacement rates between 0.01 m/s and 15 m/s. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 10 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Fig. 4: PVB tensile test sample dimensions. 19 20 21 22 PVB tensile specimens 23 24 25 Tensile test specimens were cut from sheet PVB of thickness 0.76mm (Solutia product 26 27 number RB-41) using a die. The sample geometry used was that of specimen Type 2 outlined 28 29 in BS ISO 37:2005 [5] and is shown in Fig. 4. The specimens were cut in perpendicular di- 30 31 rections to check for anisotropy in the mechanical response. Anisotropic behaviour could 32 33 not be distinguished from between-sample variability so the material was assumed to be 34 35 isotropic. The 20mm central test section length was marked with thin black lines using a 36 37 permanent marker pen prior to testing to enable the strain to be monitored during the test 38 39 using a high-speed camera. Digital callipers were used to measure the thickness and width 40 41 of the test section at three locations to an accuracy of 0.03mm. 42 43 44 45 Test equipment 46 47 48 An Instron VHS high strain-rate test machine was used to load the PVB test samples in 49 50 tension. The test samples were connected to the test machine through lightweight titanium 51 52 alloy grips. All connecting components between the sample and load cells were kept as light 53 54 as possible to minimise transient forces under acceleration. A piezoelectric load-cell (Model 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 11 5 6 222B manufactured by PCB Piezotronics Inc) was used to measure load at displacement 7 8 rates above 0.1 m/s. At slower rates it was found that the discharge time constant of the 9 10 load-cell and amplifier circuit was too short with respect to the rate at which the force was 11 12 changing, resulting in drift in the recorded signal. To mitigate this problem a strain gauge 13 14 load-cell was used at displacement rates of 0.1 m/s and slower since strain gauges do not 15 16 suffer from the loss of charge as piezoelectric load cells do. 17 18 The top of the sample was connected to the test machine actuator through a lost-motion 19 20 device as shown in Fig. 5. The device gives the actuator enough travel to accelerate to the 21 22 desired test velocity before loading the sample. Vibrations occurring at the onset of loading 23 24 were damped out with a thin rubber washer. A support was provided to prevent the PVB 25 26 samples from collapsing under the weight of the loss motion device and to ensure they were 27 28 taught prior to loading. The test temperature was the ambient room temperature which was 29 30 25 ± 3◦C. 31 32 The force from both load cells and the actuator displacement were recorded by an Imatek 33 34 C2008 data acquisition system. The position of the actuator was tracked internally by the 35 36 test machine using a linear variable differential transformer (LVDT). A high-speed camera 37 38 was used to record extension and strain. An automated lighting system that switched on just 39 40 prior to loading was used to minimise heating of the sample. Both the high-speed camera 41 42 and the data acquisition hardware were triggered by the actuator displacement, just prior to 43 44 the end of travel of the lost-motion device, to ensure synchronous recording. 45 46 47 2.2.1 Optical strain measurement 48 49 50 Images of the sample deformation captured by high-speed camera were post-processed to 51 52 calculate strain in the sample. A software program was developed using the OpenCV com- 53 54 puter vision library to track the position of lines on the samples over time [6]. The Lucas- 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 12 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Fig. 5: PVB tension test setup with actuator cutaway showing lost-motion device. 46 47 48 Kanade (LK) method of feature tracking was used to track the motion of selected features 49 50 between images [7]. The four edges created by the two lines marked on the test specimens 51 52 were tracked as shown by the green dots in Fig. 6. The test section length in each frame 53 54 was determined from the average of the distance between the two outer markers and the two 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 13 5 6 7 8 9 10 11 12 13 14 15 16 17 Fig. 6: Tracking of test section length on a PVB tensile test sample. 18 19 20 inner markers before the specimen was loaded. The engineering strain was then calculated 21 22 using the original length, determined from the frame just prior to the start of loading and the 23 24 difference in length in all subsequent frames. No attempt was made to calibrate and correct 25 26 for lens distortion, which was found to be less than a pixel over the specimen gauge length. 27 28 An image of a steel rule positioned in place of a test specimen was analysed and no deviation 29 30 in the scale length at the image centre and at the image edge could be found. 31 32 33 Data analysis 34 35 36 A typical engineering stress-strain response for a PVB tensile test is shown in Fig. 7. The 37 38 strain in the sample was calculated from tracking markers on the sample in images captured 39 40 by the high-speed camera. The images were captured at known intervals from the time the 41 42 camera was triggered, enabling the calculated strains to be easily synchronised with the 43 44 force-time data. However, the force was captured at a higher frequency than the images 45 46 used to calculate strain. It was therefore necessary to interpolate the strain data to the times 47 48 at which the force data was sampled. The stress at failure, σ , and the strain to failure, ε , 49 f f 50 were calculated at the time of maximum force. Two moduli were also calculated, the small- 51 52 strain modulus, E , and the large strain modulus E corresponding to the gradient of the 53 0 20% 54 stress-strain curve at 20% strain. The stress at the point where the two moduli lines intersect, 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 14 5 6 40 7 σ 8 f 9 10 11 30 12 13 14 Ε 0 Ε 15 20 20% 16 17 Eng. Stress (MPa) 18 σ 19 ov 20 10 21 22 23 24 0 ε 25 0 0.5 1 1.5 2 f 2.5 26 Eng. Strain 27 28 29 Fig. 7: Typical stress-strain curve from a PVB tensile test. 30 31 32 asshowninFig.7,isdefined as the overstress σov and is a measure of the increase in stress 33 34 due to viscous effects. 35 36 37 When considering large deformations it is useful to define new measures of stress and 38 39 strain in addition to their engineering definitions. For most materials a large tensile extension 40 41 is accompanied by a reduction in cross-sectional area, resulting in an increase in stress. The 42 σ 43 true stress t takes this reduction in area into account by using the current cross-sectional 44 45 area in its definition rather than the original cross-section area. It is often related to the 46 σ 47 engineering stress by assuming that the deformation occurs under a condition of constant 48 49 volume and uniform strain (that is no necking). This results in the relation 50 51 52 53 54 σ = σ (1 + ε) (7) 55 t 56 57 58 59 60 61 62 63 64 65 1 2 3 4 15 5 6 where ε is the engineering strain. This assumption is usually valid for rubbery materials 7 8 since Poisson’s ratio is approximately 0.5 [8,9]. The extension ratio, or stretch, λ is defined 9 10 as the ratio of current length l to initial length l0 and is related to the engineering strain by 11 12 13 l Δl 14 λ = = 1 + = 1 + ε (8) l l 15 0 0 16 These definitions of true stress and stretch are commonly used in hyperelastic constitu- 17 18 tive models. 19 20 21 22 3Results 23 24 25 3.1 Small-strain behaviour 26 27 28 The tensile storage and loss modulus of PVB measured using DMA are presented as a func- 29 30 tion of temperature in Fig. 8. At temperatures below approximately 5◦C the PVB was found 31 32 to be in the glassy region and the storage modulus was in the order of 1GPa. Between 5◦C 33 34 and 10◦C the modulus began to drop rapidly in magnitude as the PVB entered its transition 35 36 region. At temperatures above approximately 40◦C the PVB was in the rubbery region and 37 38 the storage modulus dropped to the order of 1MPa. The effect of frequency can be seen 39 40 as a shift of the curve along the temperature axis. An increase in frequency increased the 41 42 temperature at which the transition region occurred. 43 44 The loss modulus in the glassy region was relatively constant and was in the order of 45 46 100MPa. This was approximately one tenth of the storage modulus and corresponds to a 47 48 phase angle of 6◦ between the stress and strain. In the transition zone the loss modulus 49 50 was of the same order of magnitude as the storage modulus and in some cases exceeded 51 52 it, corresponding to a phase angle of approximately 45◦ between the stress and strain. The 53 54 viscoelastic behaviour of PVB is therefore much more pronounced in the transition region. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 16 5 6 This is of importance because the likely operating temperature range of the material coin- 7 8 cides with its transition region. 9 10 Figure 9 shows the storage and loss moduli that have been reduced to a reference tem- 11 12 perature of 20◦C using TTS and the WLF equation. It shows that the material enters the 13 14 glassy region at frequencies of approximately 104 rad/s and is in the rubbery region at fre- 15 16 quencies below 1 rad/s. An approximation between strain rate and angular frequency can be 17 18 made by representing the sinusoidal loading as a triangular wave form. In this case the strain 19 20 rate ε˙ and angular frequency ω are related by 21 22 23 24 ε ε˙ = 0 ω 25 π (9) 26 27 28 where ε0 is the strain amplitude (ε0 = 0.001 here). The PVB therefore shows glassy 29 30 behaviour for strain rates above 3.2s−1 and rubbery behaviour at strain rates below 3.2 × 31 32 10−4 s−1 for a temperature of 20◦C. 33 34 The shift factors and WLF equation fit used to reduce the data to a reference temper- 35 36 ature of 20◦C are shown in Figure 10. The WLF equation shows good agreement with the 37 38 manually determined shift factors. The WLF constants were determined to be C1 = 8.87 and 39 ◦ 40 C2 = 55.5 C using nonlinear regression. The WLF equation can be used to shift the reduced 41 42 data to a new reference temperature. At a reference temperature of 25◦C the data would be 43 44 shifted approximately one decade higher on the frequency scale. The strain rate required 45 46 to produce glassy behaviour would increase at this higher reference temperature. Similarly 47 48 a reference temperature of 15◦C would correspond to a shift of approximately one decade 49 50 lower on the frequency scale, reducing the strain rate required to produce glassy behaviour. 51 52 The shear relaxation modulus calculated using Equation 6 is plotted as a function of 53 54 time in Fig. 11. The generalized Maxwell fit determined using nonlinear regression is also 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 17 5 6 10000 7 GlassyTransition Rubbery 8 9 10 1000 11 12 13 100 14 15

16 ’ (MPa) E 17 10 18 19 20 21 1 22 1 Hz 23 10 Hz 100 Hz 24 0.1 25 -80 -60 -40 -20 0 20 40 60 80 26 Temperature (°C) 27 28 (a) Storage modulus. 29 30 10000 31 GlassyTransition Rubbery 32 33 34 1000 35 36 37 38 100 39 40 ’’ (MPa) E 41 10 42 43 44 45 1 46 47 1 Hz 10 Hz 48 100 Hz 0.1 49 -80 -60 -40 -20 0 20 40 60 80 50 Temperature (°C) 51 52 (b) Loss modulus. 53 54 55 Fig. 8: Tensile storage and loss moduli vs temperature for PVB. 56 57 58 59 60 61 62 63 64 65 1 2 3 4 18 5 6 103 7 Rubbery Transition Glassy 8 9

10 2 11 10 12 13 14 15 101 ’’ (MPa)

16 E ’, 17 E 18 19 20 100 21 22 23 Storage E’ 24 Loss E’’ 10-1 25 10-4 10-2 100 102 104 106 108 26 Frequency (rad/s) 27 28 ◦ 29 Fig. 9: Tensile storage and loss moduli vs frequency reduced to a temperature of 20 C. 30 31 32 shown. It was found that six Maxwell elements were enough to describe the shear relaxation 33 34 modulus over this time scale. Table 1 gives the calculated generalized Maxwell material 35 36 constants. These parameters can be used in ﬁnite element models to include viscoelastic 37 38 effects at small strains. 39 40 41 42 43 3.2 High strain-rate tensile tests 44 45 46 A typical time history for a high strain-rate tensile test on PVB is shown in Fig. 12. The 47 48 ﬁgure shows engineering stress, strain and strain rate for a sample tested at 1 m/s. The stress 49 50 trace shows a steep initial rise in stress after slack in the lost motion device has been taken 51 52 up. After the stress reached around 10MPa the rate at which the stress was increasing slowed 53 54 by a factor of approximately 10. The stress continued to increase until failure at a stress of 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 19 5 6 5 Shift factors 7 WLF fit 8 4 C =8.87 C1 9 2=55.5 10 3 11 12 2 13 1

14 ) T 15 a 0

16 log( 17 -1 18 19 -2 20 21 -3 22 23 -4 24 -5 25 0 10 20 30 40 50 60 70 80 26 Temperature (°C) 27 28 29 Fig. 10: Manually determined shift factors and the WLF equation ﬁtusedtoreducedatato 30 ◦ 31 a temperature of 20 C. 32 33 34 Table 1: Generalized Maxwell model terms. 35 36 G / τ 37 i i G0 i (s) 38 . × −5 39 1 0.49016 2 45 10 40 2 0.40844 2.21 × 10−3 41 42 3 0.08522 4.98 × 10−2 43 . × −1 44 4 0.01389 6 24 10 45 5 0.00159 2.49 × 101 46 47 6 0.00200 1.00 × 103 48 * 49 Instantaneous shear modulus G0 = 178MPa, long-term shear modulus 50 51 G∞ = 0.125MPa 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 20 5 6 103 Experiment 7 Generalized Maxwell fit 8 9

10 2 11 10 12 13 14 15 101 ) (MPa) t

16 ( 17 G 18 19 20 100 21 22 23 24 10-1 25 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 26 Time (s) 27 28 ◦ 29 Fig. 11: Shear relaxation curve at a reference temperature of 20 C. 30 31 32 33MPa and a strain of 220%. The strain rate during the test began at 27s−1 and slowed at an 33 34 approximately linear rate to 20s−1 at the time of failure. Small oscillations in the strain rate 35 36 can be seen at a time of 0.07s, resulting in small oscillations in the recorded stress. These 37 38 were a result of small oscillations in the actuator velocity. 39 40 The drop in gradient at 10MPa is not a sign that the material has yielded. Almost all of 41 42 the extension was recoverable, with samples returning to within 2.5% of the original gauge 43 44 length after the test. The initial steep rise in stress is a result of the viscous component of the 45 46 material response and depends on the applied strain rate. 47 48 Figure 13 shows engineering stress-strain curves for PVB samples tested at strain rates 49 50 between 0.2s−1 and 400s−1. Curves at strain rates of 8s−1 and 200s−1 are not presented 51 52 for clarity. The plot shows that the stress-strain response of PVB has a strong strain rate 53 54 − dependence. At a strain rate of 0.2s 1 the material behaves similarly to a hyperelastic ma- 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 21 5 6 40 3 Stress 7 Strain 8 35 Strain rate 9 2.5

10 ) 11 -1 30 12 2 13 25 14 15 20 1.5 16 17 Eng. Strain 15 18 1 19 10 20 Eng. Stress (MPa), Strain rate (s 21 0.5 22 5 23 24 0 0 25 20 40 60 80 100 120 140 160 26 Time (ms) 27 28 m/ 29 Fig. 12: Engineering stress, strain and strain rate time histories for a test at 1 s. 30 31 32 terial, showing large non-linear deformation up to the point of failure. At higher strain rates 33 34 the curves show an initial gradient that increases with strain rate. Above strains of 10% the 35 36 curves all have similar shapes except for the stress offset resulting from the steep initial 37 38 rise in stress. This suggests that the small-strain behaviour is greatly affected by the vis- 39 − 40 cous response where as the large-strain behaviour is not. At a strain rate of 400s 1 small 41 42 oscillations can be seen in the trace after the steep initial rise in stress. The period of these 43 44 oscillations was approximately 290 μs and matched the natural period of the load-cell and 45 46 grips as determined from oscillations after sample break (see Fig. 15). 47 48 49 Small-strain modulus 50 51 52 The small-strain modulus, E , was calculated from the initial steep gradient section of each 53 0 54 stress-strain curve and is shown as a function of strain rate in Fig. 14. The small-strain 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 22 5 6 40 7 8 35 9 10 11 30 12 13 25 14 -1 400 s 15 20 16 -1 17 60 s

Eng. Stress (MPa) 15 18 -1 19 20 s 20 10 -1 21 2 s 22 5 -1 23 0.2 s 24 0 25 0 0.5 1 1.5 2 2.5 26 Eng. Strain 27 28 29 Fig. 13: Summary of engineering stress vs engineering strain at different strain rates. 30 31 32 . −1 −1 33 modulus varies between 20MPa and 1GPa over strain rates between 0 2s and 400s . 34 . −1 −1 35 The modulus values observed between 0 2s and 60s are approximately proportional to 36 37 the square root of strain rate 38 39 40 41 42 / 43 ε˙ 1 2 = 44 E0 E0,0 (10) ε˙0 45 46 47 48 where E , is the small-strain modulus at reference strain rate ε˙ . The small-strain ref- 49 0 0 0 50 erence modulus was determined using nonlinear regression and was found to be E , = 51 0 0 52 51±3MPa at a reference strain rate, ε ,of1s−1. The measured small-strain modulus values 53 0 54 − remain within 20% of this empirical ﬁt except at strain rates of 200s 1 and above. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 23 5 6 1000 Ε 7 Ε 0 20% 8 Empirical fit 9 10 11 12 100 13 14 15 16

17 Modulus (MPa) 18 10 19 20 21 22 23 1 24 0.1 1 10 100 1000 25 Strain rate (s-1) 26 27 28 29 Fig. 14: Log-log plot of tensile modulus at different strains as a function of strain rate and 30 31 empirical fit of Equation 10. 32 33 34 35 Limitations at high-rates 36 37 38 At high strain rates the variance in the measured modulus values increases. To accurately 39 40 measure modulus, the natural period of the load-cell and connecting components must be 41 42 short when compared to the rise time of the applied force. If this is not the case then the 43 44 dynamic response of the load-cell will be too slow and the measured force will not track 45 46 the applied force. Figure 15 shows the natural oscillations of the load-cell and connecting 47 48 components after a load is suddenly removed. With this setup the natural period of the load- 49 50 cell was observed to be approximately 290 μs. The rise time of the applied force is a function 51 52 of strain rate and the modulus. At at strain rate of 400s−1 the rise time to reach the observed 53 54 overstress of 20MPa would be 50 μs, assuming that the modulus can be calculated using 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 24 5 6 150 7 8 9 100 10 11 12 50 13 14 15 0 16 17 Force (N) 18 -50 19 290 us 20 21 -100 22 23 24 -150 25 6 7 8 9 10 26 Time (ms) 27 28 29 Fig. 15: Unforced vibrations of the load-cell after a load is suddenly removed. 30 31 32 Equation 10. This predicted rise time is approximately one sixth of the natural period of the 33 34 load-cell and the measured force no longer tracks the applied force, leading to inaccurate 35 36 modulus values. At a strain rate of 60s−1 the rise time is approximately twice the natural 37 38 period of the load-cell and is the practical limit for determining modulus. 39 40 Other factors also hinder the measurement of modulus in the small-strain region at high 41 42 strain rates. Stress waves take a finite amount of time to propagate along the sample length 43 44 and the sample can only be assumed to be in a uniform state of stress if they can be neglected. 45 46 To be neglected the waves must decay quickly when compared to the rise time of the applied 47 48 loading. The wave speed in the sample can be estimated using the relation 49 50 51 52 53 = E0 54 c ρ (11) 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 25 5 6 where ρ is the density of PVB. Applicability of this equation is based on the assumption that 7 8 at high strain rates the material is effectively linear elastic for small strains. Using a modulus 9 − 10 of 1GPa for a test at a strain rate of 400s 1 and a density of 1100 kg/m3, the wave speed in 11 12 the specimen is in the order of 1000 m/s. For a 40mm long specimen (distance between 13 14 grips) the stress wave at the start of loading would take 40 μs to propagate down the sample. 15 − 16 This propagation time is almost equal to the estimated rise time at a strain rate of 400s 1. 17 18 Typically the stress wave will propagate up and down the sample several times, depending 19 20 on damping characteristics, before it has decayed to a level where dynamic equilibrium can 21 22 be assumed. The stress and strain in the sample will therefore not be uniform in the small- 23 24 strain region for the higher strain rate tests and the calculated modulus may be inaccurate. At 25 − 26 a strain rate of 60s 1 the wave speed can be calculated as approximately 600 m/s if Equation 27 28 10 is used to determine modulus. At this speed the wave would travel the length of the 29 30 sample seven times during the rise time, giving the wave enough time to decay so that 31 32 dynamic equilibrium can be assumed before the end of the small-strain region. 33 34 The high-speed camera frame rate also limits the measurable modulus at high strain 35 36 rates. At the highest rate of 400s−1 the frame rate used was 15kHz, giving strain values 37 38 every 66 μs. If the stress rises from 0MPa to 20MPa in the inter-frame time, the maximum 39 40 modulus that can be determined is approximately 800MPa for a strain rate of 400s−1.To 41 42 capture the modulus in the small-strain region, the frame interval should be at most half the 43 44 rise time of the input force. 45 46 47 Modulus at 20% strain 48 49 50 The modulus calculated at 20% strain, E , is also shown in Fig. 14. The calculated val- 51 20% 52 ues vary between about 7MPa and 10MPa over the range of tested strain rates. The peak 53 54 − modulus was observed at a strain rate of 8s 1 and the values decrease at a similar rate for 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 26 5 6 both higher and lower strain rates. The variance in calculated modulus also increases at high 7 8 strain rates for the same reasons discussed in the small-strain modulus section. 9 10 11 12 Failure stress and strain 13 14 15 Stress and strain at failure are presented in Fig. 16 as a function of strain rate. The failure 16 17 stress shows a small increase as the strain rate increases. At 0.2s−1 the mean failure stress 18 19 was 30MPa and at 400s−1 the mean failure stress was found to have increased to 38MPa. 20 21 The increasing failure stress can be approximated by the expression 22 23 24 25 ε˙ σf = mσ log + σf,0 (12) 26 ε˙0 27 28 σ 29 where mσ is the increase in failure stress per decade of strain rate and f,0 is the failure 30 ε˙ 31 stress at a reference strain rate 0. These constants were determined using nonlinear regres- 32 = . ± . σ = . ± . 33 sion and were found to be mσ 2 58 0 3MPa per decade and f,0 30 6 0 6MPafora 34 ε −1 35 reference strain rate, 0,of1s . 36 37 Failure strain values varied between 200% and 225%. The failure strain was found to 38 −1 −1 39 decrease slightly over the tested strain rate range of 0.2s to 400s . An expression similar 40 41 to that for the failure stress can be used to approximate failure strain 42 43 44 45 ε˙ εf = mε log + εf,0 (13) 46 ε˙0 47 48 where mε is the change in failure strain per decade of strain rate and ε , is the fail- 49 f 0 50 ure strain at a reference strain rate ε˙ . These constants were determined using nonlinear 51 0 52 regression and where found to be mε = −0.05 ± 0.02 per decade and ε , = 2.2 ± 0.04 for a 53 f 0 54 − reference strain rate of 1s 1. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 27 5 6 60 2.5 7 8 9 50 10 2 11 12 40 13 1.5 14 15 30 16 Strain 17 1 Eng. Stress (MPa) 18 20 19 20 0.5 21 10 Failure stress 22 Failure strain Empirical fit stress 23 Empirical fit strain 0 0 24 0.1 1 10 100 1000 25 Strain rate (s-1) 26 27 28 29 Fig. 16: Semi-log plot of failure stress and strain vs strain rate and empirical fits of Equations 30 31 12 and 13. 32 33 34 Overstress 35 36 37 The overstress, calculated at the intersection between E0 and E20%, is shown as a function of 38 39 strain rate in Fig. 17. The overstress can be thought of as the contribution of viscosity in the 40 41 small-strain region. At at a strain rate of 0.2s−1 the observed overstress was approximately 42 43 1.3MPa. At the highest strain rate of 400s−1 the overstress increased to 19MPa. This order 44 45 of magnitude increase in overstress is much greater than the increase in failure stress, indi- 46 47 cating that the effect of strain rate is more pronounced at small strains than at large strains. 48 49 An empirical cube root expression of the form 50 51 52 53 1/ ε˙ 3 54 σ = σ ov η ε (14) 55 ˙0 56 57 58 59 60 61 62 63 64 65 1 2 3 4 28 5 6 Experimental data 7 Empirical fit 8 9 10 11 12 13 10 14 15 16

17 Overstress (MPa) 18 19 20 21 22 23 1 24 0.1 1 10 100 1000 25 Strain rate (s-1) 26 27 28 29 Fig. 17: Log-log plot of overstress as a function of strain rate and empirical fit of Equation 30 31 14. 32 33 34 was fitted to the data using non-linear regression where ση = 2.58 ± 0.04MPa and is 35 −1 36 the overstress at a reference strain rate ε˙0 of 1s . This empirical fit is shown on Fig. 17. 37 38 The observed data remains within 20% of this empirical fit except for one value recorded at 39 − − 40 0.2s 1 and both values recorded at a strain rate of 8s 1. 41 42 43 True stress 44 45 46 At high levels of strain the engineering stress is not an accurate measure of stress in the ma- 47 48 terial due to changes in cross-sectional area. Many hyperelastic constitutive material models 49 50 describe the material response in terms of true stress and the stretch ratio. As no necking 51 52 was observed, the true stress can be found from engineering stress by assuming that there 53 54 is no change in volume of the deformed material. Under the condition of constant volume 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 29 5 6 140

7 -1 -1

8 60 s 9 120 400 s -1 20 s 10 -1 -1 11 100 2 s 0.2 s 12 13 14 80 15 16 60 17 18 True Stress (MPa) 19 40 20 21 22 20 23 24 0 25 1 1.5 2 2.5 3 3.5 26 Stretch ratio, λ 27 28 29 Fig. 18: Summary of true stress vs stretch ratio at different strain rates. 30 31 32 an increase in length of stretch ratio λ will result in a reduction in cross sectional area of 33 34 1/λ . The true stress, σt, is therefore equal to the engineering stress multiplied by the stretch 35 − 36 ratio . Figure 18 shows true-stress vs stretch ratio curves at strain rates between 0.2s 1 and 37 38 400s−1. 39 40 41 42 4 Discussion 43 44 45 Small-strain behaviour and temperature effects 46 47 48 The small-strain behaviour of PVB is of primary interest in the bending response of lami- 49 50 nated glass. In an uncracked laminate the maximum strain in the PVB interlayer is limited 51 52 to below the failure strain of the glass plies. The strain to failure of the glass plies is ap- 53 54 proximately 0.1% if a failure strength of 80MPa and an elastic modulus of 72GPa [10] are 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 30 5 6 assumed. However, the PVB interlayer is not likely to experience strain of this magnitude in 7 8 the uncracked response phase because the predominant mode of deformation is in bending. 9 10 Under such conditions the PVB interlayer lies on the neutral axis where its main function 11 12 is to transfer horizontal shear force and maintain the separation distance between the glass 13 14 plies. 15 16 The amount of horizontal shear force that is transferred will depend on the shear modu- 17 18 lus and therefore strain rate and temperature. At high deformation rates, or low temperatures, 19 20 the PVB will be able transfer a significant amount of shear force between the glass plies and 21 22 the laminated pane will have a bending stiffness of similar magnitude to an equivalent mono- 23 24 lithic pane of the same overall thickness. At slow deformation rates, or high temperatures, 25 26 the PVB will not transfer a significant amount of shear stress and its only function will be to 27 28 separate the glass plies. Under these conditions the laminate will behave as two independent 29 30 glass layers with twice the stiffness of one ply on its own. These two extremes were first 31 32 investigated by Norville et al. [11]. The generalized Maxwell viscoelastic material model 33 34 calibrated here allows the effect of time and temperature to be taken into account when 35 36 considering the shear transfer across the PVB interlayer. 37 38 Figure 19 compares the data acquired in this paper for a Solutia Saflex PVB interlayer 39 40 against data published by van Duser et al. [2] on a DuPont Butacite interlayer. The two 41 42 curves share a similar shape and the moduli in the glassy and rubbery region are of the 43 44 same orders of magnitude. However, the Butacite curve is shifted by almost two decades 45 46 towards longer time scales. Comparing the storage modulus vs temperature curves of the two 47 48 materials shows that Butacite’s glass transition temperature T is approximately 5◦Cto10◦C 49 g 50 higher than that of Saflex. Referring to the shift factors in Fig. 10 shows that a difference 51 52 of 5◦Cto10◦C corresponds to a shift of one to two decades in time and therefore accounts 53 54 for the observed shift. Manufacturers vary the quantity of plasticiser added to the PVB resin 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 31 5 6 103 Saflex 7 Butacite 8 9

10 2 11 10 12 13 14 15 101 ) (MPa) t

16 ( 17 G 18 19 20 100 21 22 23 24 10-1 25 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 26 Time (s) 27 28 ◦ 29 Fig. 19: Comparison of shear relaxation curves between Butacite and Saflex PVB at 20 C. 30 31 32 to control the Tg of the material for different commercial markets. This is likely to be the 33 34 cause of the difference observed here. The variation of Tg between interlayer manufactures 35 36 needs to be considered when analysing the response of laminated glass because the transition 37 38 region lies within the expected operating temperature range. A higher Tg will lead to a stiffer 39 40 interlayer which may be of benefit in an uncracked laminated pane. However, it may be 41 42 detrimental in a cracked pane of laminated glass as it may lead to brittle behaviour and 43 44 premature failure of the interlayer at high strain-rates. 45 46 The small-strain modulus E0 can be compared qualitatively with the tensile storage mod- 47 48 ulus E using the relation between frequency and strain rate given in Equation 9. The small- 49 50 strain modulus varied between 28MPa and 300MPa over an equivalent angular frequency 51 52 range of 0.32 rad/s to 32 rad/s. The frequency range to produce the same tensile storage modu- 53 54 lus values was between approximately 10 rad/s and 1000 rad/s. Although these values are over 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 32 5 6 10 times larger than the equivalent angular frequencies, the gradient of the storage modulus 7 8 curve in the transition region matches that of the empirical fit used to describe the small- 9 10 strain modulus. The tensile storage modulus E was reduced to a reference temperature of 11 ◦ ◦ 12 20 C whereas the small-strain modulus was measured at a temperature of 25 ± 3 C. The 13 14 shift in frequency is of the same direction and magnitude as though the storage modulus 15 ◦ 16 were measured at a temperature of 25 C. 17 18 19 Low strain rate large-strain behaviour 20 21 −1 22 At a strain rate 0.2s the small-strain modulus E0 approaches the same value as the mod- 23 24 ulus at 20% strain, E20%. The stress-strain curve at this strain rate is of similar form to that 25 26 of a hyperelastic material. Common constitutive models for hyperelastic behaviour include 27 28 the Gent [12], Arruda-Boyce [13] and van der Waals models [14]. These models have been 29 30 used by Muralidhar et al. [15] to consider the bridging behaviour in cracked laminates. Fig- 31 32 ure 20 compares a non-linear regression fit of the three models against the experimental data 33 34 at 0.2s−1. All three models show good agreement with the experimental data at this strain 35 36 rate. The Gent and Arruda-Boyce curves are almost indistinguishable over this range. Whilst 37 38 these models show good agreement at this strain rate they do not take into account the time 39 40 dependent component of the material response. They are unable to capture the high initial 41 42 modulus observed at higher strain rates. Even at a strain rate of 0.2s−1 hysteresis and stress 43 44 relaxation effects will be ignored by these models. Modelling the large-strain behaviour of 45 46 PVB with a hyperelastic model is therefore of limited value. 47 48 49 Time dependent large-strain behaviour 50 51 52 The nonlinear viscoelastic behaviour of some elastomers and polymers can be captured by 53 54 assuming that the stress can be represented by a separable equation of the form 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 33 5 6 140 Experiment 7 Gent 8 Arruda-Boyce 9 120 van der Waals 10 11 100 12 13 14 80 15 16 60 17 18 True Stress (MPa) 19 40 20 21 22 20 23 24 0 25 1 1.5 2 2.5 3 3.5 26 Stretch ratio, λ 27 28 29 Fig. 20: Comparison of hyperelastic material models with an experiment at a strain rate of 30 . −1 31 0 2s . 32 33 34 35 36 σ(λ,t)= f (λ)g(t) (15) 37 38 39 where f (λ) and g(t) are stretch and time dependent functions respectively. The function 40 41 f (λ) is usually represented by a hyperelastic material model and g(t) is defined using a 42 43 Prony series of the form 44 45 46 N 47 − /τ ( )= + t i 48 g t g∞ ∑ gie (16) i=1 49 50 where g∞ and g are dimensionless constants and τ are relaxation times. This form 51 i i 52 of representing nonlinear viscoelastic behaviour is commonly employed in finite element 53 54 packages such as Abaqus [14]. For Equation 15 to be applicable to a material the stretch 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 34 5 6 100 7 8 9 10 11 Eng. Strain 12 1.72 400 s-1 13 1.46 1.23 14 60 s-1 1.02 15 10 16 20 s-1 0.83 17 0.65 18 Eng. Stress (MPa) 0.50 19 2 s-1 0.35 20 0.22 21 0.11 22 0.2 s-1 23 24 1 25 0.0001 0.001 0.01 0.1 1 10 100 26 Time (s) 27 28 29 Fig. 21: Log-log plot of stress vs time for different levels of strain. 30 31 32 33 and time dependent components of the material response need to be separable. Smith [16] 34 35 devised a simple method for determining whether this is the case for materials loaded at 36 37 constant strain rates. He suggested that if stress vs time curves are plotted on logarithmic 38 39 axes for different strain rate tests, lines of constant strain would be parallel if the material 40 41 can be represented by Equation 15. 42 43 44 Figure 21 shows lines of constant strain overlaid on stress vs time curves for five samples 45 46 tested at strain rates from 0.2s−1 to 400s−1. It can clearly be seen that the lines of constant 47 48 strain are not parallel at engineering strains below 1.2 and therefore Equation 15 cannot be 49 50 used to represent the nonlinear viscoelastic behaviour of PVB over this range. At low strains 51 52 the constant strain line has a steeper gradient than that at high strains. This indicates that 53 54 effect of time is more pronounced at low strains than at high strains. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 35 5 6 Recent constitutive models proposed by Hoo Fatt and Ouyang [17,18] have attempted 7 8 to describe the large-strain behaviour of Styrene Butadiene Rubber (SBR) at high strain 9 10 rates. SBR, like PVB, shows an initially high modulus at small strains for high strain rates 11 12 and a lower modulus at large strains. Their model consists of a low stiffness hyperelastic 13 14 spring in parallel with a high initial stiffness hyperelastic spring and a nonlinear damper. At 15 16 low strain rates the behaviour tends towards that of the low stiffness hyperelastic spring. At 17 18 high strain rate the behaviour tends towards that of the combined stiffness of both springs. 19 20 The assumed hyperelastic function was empirical and was chosen to provide a smooth fit 21 22 to the observed SBR data. The viscosity function was also an empirical relation that was 23 24 dependent on the strain and strain rate, rather than being based on the underlying structure 25 26 of the material. It was found that the hyperelastic function was capable of describing PVB 27 28 stress-strain response at low strain rates. However, the model was unable to capture the high 29 30 initial modulus seen in the high strain rate tests accurately. Calibration of the model was also 31 32 complex, requiring four constants to describe the behaviour of both hyperelastic springs and 33 34 seven constants to describe the damping function. 35 36 37 38 39 40 41 42 43 44 The lack of a physically based constitutive model that completely describes the be- 45 46 haviour of PVB at all strains and strain rates means that it is not currently possible to model 47 48 the behaviour of cracked laminated glass on the level of individual cracks. This does not 49 50 mean that an empirical model of the global behaviour of cracked laminated glass cannot be 51 52 developed. One such approach in the modelling of blast loaded laminated glass is described 53 54 by Hooper et al. [19]. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 36 5 6 5 Conclusions 7 8 9 10 This paper describes an experimental investigation into the behaviour of PVB at different 11 12 strain rates and strain magnitudes. The small-strain viscoelastic behaviour has been mea- 13 14 sured using DMA. It was found that the PVB tested had a glassy tensile modulus of the 15 16 order of 1GPa and a rubbery modulus of the order of 1MPa. The PVB showed a transition 17 18 between the two moduli over a temperature range of 5◦Cto40◦C. A generalized Maxwell 19 20 model was found to provide an accurate description of the shear relaxation modulus at time 21 22 scales ranging from 10−8 sto103 s for small strains. 23 24 25 Differences in glass transition temperature in PVB produced by different manufacturers 26 27 were identified. DuPont Butacite had a glass transition temperature approximately 5◦Cto 28 29 10◦C higher than that of Saflex PVB produced by Solutia. A temperature shift of ±5◦C about 30 31 a reference temperature of 20◦C was found to be equivalent to a decade shift in frequency. 32 33 A low operating temperature would therefore result in a stiffer interlayer, possibly leading 34 35 to brittle behaviour at high strain rates. 36 37 38 The large-strain tensile behaviour of PVB was investigated using a high-speed servo- 39 40 hydraulic testing machine. It was found that the engineering stress at failure varied from 41 42 30MPa at a strain rate of 0.2s−1 to 38MPa at 400s−1. These correspond to a true stress 43 44 at failure of 95MPa and 120MPa respectively due a reduction in cross sectional area. The 45 46 strain to failure was found to vary between 225% and 200%, showing a slight reduction over 47 48 the same strain rate range. A small-strain modulus was also determined from the large-strain 49 50 tensile tests. Moduli between the DMA and large-strain tests were qualitatively similar, tend- 51 52 ing towards a tensile modulus of 1GPa at short time scales. Differences in test temperature 53 54 made drawing a direct quantitative comparison difficult. 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 37 5 6 The overstress in the large-strain tensile test was found to vary between 1.3MPa and

7 − − 8 19MPa over a strain rates range of 0.2s 1 to 400s 1. After the initial sharp increase in 9 10 stress at small strains the curves at all strain rates showed the same general form, similar 11 12 to that of a hyperelastic material. The influence of time effects on the stress were found 13 14 to be larger at small strains. No current model completely captures the large-strain time- 15 16 dependent behaviour of PVB, making it difficult to develop a physically based model of the 17 18 behaviour of cracked laminated glass on the level of individual cracks. However, this does 19 20 not mean that an empirical model of the global behaviour of cracked laminated glass cannot 21 22 be developed. 23 24 25 Acknowledgements The authors acknowledge the Engineering and Physical Sciences Research Council 26 27 (EPSRC) and Arup for financially supporting Dr Paul Hooper. We acknowledge the many constructive dis- 28 cussions with David C. Smith, Ryan A. M. Sukhram and David Hadden of Arup. 29 30 31 32 33 References 34 35 36 1. S.J. Bennison, A. Jagota, C.A. Smith, Journal of the American Ceramic Society 82, 1761 (1999) 37 38 2. A. van Duser, A. Jagota, S.J. Bennison, Journal of Engineering Mechanics 125(4), 435 (1999). DOI 39 10.1061/(ASCE)0733-9399(1999)125:4(435) 40 41 3. M.L. Williams, R.F. Landel, J.D. Ferry, Journal of the American Chemical Society 77(14), 3701 (1955). 42 43 DOI 10.1021/ja01619a008 44 4. J.D. Ferry, Viscoelastic properties of polymers, Thrid edn. (John Wiley & Sons, 1980) 45 46 5. BS ISO 37:2005, Rubber, vulcanized or thermoplastic. Determination of tensile stress-strain properties. 47 48 (British Standards Institute, London, 2005) 49 6. G. Bradski, A. Kaehler, Learning OpenCV (O’Reilly Media, 2008) 50 51 7. B.D. Lucas, T. Kanade, in Proceedings of the 1981 DARPA Imaging Understanding Workshop (1981), 52 53 pp. 121–130 54 8. L.R.G. Treloar, The physics of rubber elasticity (Oxford University Press, London, 1975) 55 56 57 58 59 60 61 62 63 64 65 1 2 3 4 38 5 6 9. P.K. Freakley, A.R. Payne, Theory and practice of engineering with rubber (Applied Science Publishers, 7 1978) 8 9 10. D. Cormie, G. Mays, P. Smith (eds.), Blast effects on buildings, Second edn. (Thomas Telford, 2009) 10 11 11. H.S. Norville, K.W. King, J.L. Swofford, Journal of Structural Engineering 124(1), 46 (1998). DOI 12 10.1061/(ASCE)0733-9399(1998)124:1(46) 13 14 12. A.N. Gent, Rubber chemistry and technology 69, 59 (1996) 15 16 13. E.M. Arruda, M.C. Boyce, Journal of the Mechanics and Physics of Solids 41, 389 (1993) 17 14. Dassault Systmes Simulia Corp., Providence, RI, 2009, Abaqus theory manual v6.9 18 19 15. S. Muralidhar, A. Jagota, S.J. Bennison, S. Saigal, Acta Materialia 48, 4577 (2000) 20 16. T.L. Smith, Transactions of the Society of Rheology 6, 61 (1962) 21 22 17. M.S.H. Fatt, X. Ouyang, International Journal of Solids and Structures 44, 6491 (2007) 23 24 18. M.S.H. Fatt, X. Ouyang, Mechanics of Materials 40, 1 (2008) 25 19. P.A. Hooper, R.A.M. Sukhram, B.R.K. Blackman, J.P. Dear, International Journal of Solids and Struc- 26 27 tures Accepted (2011) 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65