Multi-mode revisited

Multi-mode revisited Testing the application of shift factors S.J.M Hellenbrand 515217 MT 7.29 Coaches: Ir. L.C.A. van Breemen Dr. Ir. L.E. Govaert 2-7- 7

Contents Contents 1 Introduction 2 I Polymers 5 1.1 Intrinsic deformation behaviour. 5 1.2 Physical ageing and mechanical rejuvenation... 6 2 Deformation behaviour 7 2.1 Linear viscoelastic deformation 7 2.2 Plastic deformation 8 2.3 Non-linear viscoelastic deformation..9 3 Experimental 11 2.1 Material and processing conditions... 11 2.2 Thermal treatment.. 11 2.3 Mechanical testing. 11 4 Results 12 4.1 Tensile tests 12 4.2 Creep tests.. 14 5 Conclusion 17 Bibliography 18 Appendices 19 A Tensile tests 19 B Spectrum and shifting 21 C Creep tests.. 23 1

Introduction To analyze and optimize the mechanical performance of polymer materials and products is a time-consuming process. A good solution for this problem is applying numerical techniques. These are cost-efficient and consume less time. But before doing so, an accurate constitutive model description of the materials intrinsic stress-strain behaviour is needed. Figure 1: Global representation of intrinsic stress-strain curve for a polymer with pre- and post-yield part. I : single-mode approach of the actual curve II : actual strain-stress curve; multi-mode approach of the actual curve The stress-strain model of a polymer can be divided in two parts (see figure 1): the pre- and the post-yield. For the past several years, a single Maxwell element and a nonlinear spring put in series (see figure 2(a)) were used to describe the stress-strain curve (see figure 1 line I). This is called single-mode due to the single Maxwell element within the model. With the current implementation, the use of multiple Maxwell elements is possible. These multiple Maxwell elements are put in series with a nonlinear spring (see figure 2(b)), this results in a more accurate approach of the actual stress-strain behaviour (see figure 1 line II). Due to the increased number of Maxwell elements this is referred to as: multi-mode. The amount of needed modes is limited and can be calculated with an optimisation routine. Line II in figure 1 resembles the simulated curve of the materials intrinsic behaviour. 2

(a) (b) Figure 2: (a) Single-mode: a single Maxwell element in series with a nonlinear spring. (b) Multi-mode : multiple Maxwell elements in series with a nonlinear spring. With a single-mode approach, the obtained pre-yield behaviour is linear elastic. Therefore it isn t an accurate approach, because energy, the difference between the actual and the single-mode curve, is not taken into account (see figure 1). This is problematic when describing post-mortem behaviour. Figure 3: Two methods of indentation [3] For indentation, as an example, the description of the post-mortem behaviour is very important. If a sphere/pyramid shaped indenter (see figure 3) is pressed into a material, the surface of indention is measured and the load divided by the area of the impression upon the material gives the hardness. Predicting the post-mortem behaviour of the material accordingly to a single-mode approach, the simulated impression (see figure 4(a)) will be smaller, as compared to the multi-mode approach. This results in an incorrect prediction of the hardness of the material. (a) (b) Figure 4: (a) Force-impression diagram for indentation, material response according to single-mode (I) and multi-mode (II) approach (figure 1). (b) Post-mortem cross-section of an impression with a sphere shaped indenter. 3

With a more accurate description of the intrinsic stress-strain curve, this will not occur, resulting in a more realistic result. The objective of this project will be to determine whether or not it is possible to create a unique set of intrinsic material parameters for describing a polymer s intrinsic deformation. This unique dataset will be implemented into a numerical program and the results will be compared with results obtained from mechanical testing. 4

Chapter 1 Polymers 1.1 Intrinsic deformation behaviour When a polymer homogeneously deforms, it shows a specific behaviour. This is referred to as the intrinsic deformation behaviour, a general representation can be given in a true stress-strain plot (see figure 1.1). Figure 1.1: schematic representation of the deformation behaviour of a polymer material When following the intrinsic deformation curve from figure 1.1, first along comes a linear viscoelastic range. This is a time-dependent response which is fully reversible when deformations are small. When increasing the deformation the material becomes nonlinear viscoelastic. If the material s deformation increases further the yield point is reached, the deformation becomes irreversible. Stress-induced plastic flows set in which leads to a structural evolution, thus reducing the material s resistance to plastic flow. This is called strain-softening. During strain-softening the strength of the material reduces. With further deformation of the material, strain-hardening takes place. The orientation of the molecules leads to an increase of stress. 5

1.2 Physical ageing and mechanical rejuvenation. With gaining stress and temperature the intermolecular interactions become less, resulting in a higher molecular mobility. This can be translated in material deformations which occur on an accelerated timescale. This can be explained in terms of the presence of a non-equilibrium thermodynamic state. When a polymer melt cools down it follows the equilibrium melt line and the non-equilibrium glass line (see figure 1.2(b)). The mobility of molecules decreases when following this trajectory. After cooled down the polymer starts to age to reach the equilibrium glass. With ageing the mobility of the molecules is further reduced resulting in an increase of the yield stress (see figure 1.2(a)). (a) Figure 1.2: (a) schematic representation of the influence of aging time t e on the yieldstress and post-yield behaviour. (b) schematic illustration of physical aging: volume as a function of temperature. (b) 6

Chapter 2 Deformation behaviour The deformation behaviour of a polymer is based on the inter- and intra-molecular interactions. Each of these interactions represent a specific molecular motion. The intermolecular interactions give rise to a spectrum of relaxation times. With increase of stress and/or temperature this relaxation mechanism is accelerated. For describing the intrinsic deformation accurately, all these contributions of the various relaxation mechanisms in the material, this includes time dependence, stress and temperature dependence, should be included in the model. Also included in an accurate model should be a proper description of large strain plasticity: strain hardening. This because of a rubber-like stress contribution observed at large deformation. The strain hardening is a result of intra-molecular interactions and independent from stress and temperature influences. Even the entire prior history is relevant, not only the momentary values of temperature and stress. Temperature and stress during processing and service life should also be taken into account. 2.1 Linear viscoelastic deformation. Linear viscoelastic behaviour is often described with the use of a Boltzmann single integral representation in its relaxation form: t σ ( t) = E( t t ) ɺ ε( t ) dt (2.1) The relaxation modulus E(t), the viscoelastic function, contains the information concerning time-dependent material behaviour. This function can be visualised by a mechanical model with a sufficient number of Maxwell elements and is graphically represented in figure (2.1). Accordingly to figure (2.1), the equation for the a generalized linear Maxwell element results in equation (2.2), where τ i = ηi Ei. 7

Figure 2.1: schematic representation of the generalized Maxwell model n t E( t) = Ei exp (2.2) i= 1 τ i 2.2 Plastic deformation A fluid-like approach is the most common way to describe the yield of a polymer. Hereby the material is thought to be a strongly nonlinear fluid with very high relaxation times. According to Tobolsky and Eyring [4], this behaviour can be described with the use of a single nonlinear Maxwell element with one, temperature and stress-activated, relaxation time. In figure (2.2) a schematic representation of a nonlinear Maxwell element is given. The dashpot represents the stress-dependant viscosity, η( σ ), which determines the plastic flow. The spring represents the initial elastic response that is described by the modulus E. Figure 2.2: schematic representation of a nonlinear Maxwell model The analytical expression for stress-dependent viscosity: σ σ η( σ ) = = * ɺ ε ( σ ) U σv ɺ ε exp sinh RT kt (2.3) Equation (2.3) can be written as: ( σ σ ) ( σ σ ) η( σ ) = η (2.4) sinh 8

With the introduction of a stress-dependent shift function a, defined as: Reduces equation (2.4) further to: a σ ( σ σ ) ( σ σ ) ( σ ) = (2.5) sinh η( σ ) = η ( σ ) (2.6) a σ 2.3 Non-linear viscoelastic deformation Between the linear viscoelastic range and the yield-point lays the nonlinear viscoelastic part (see figure 1.1). To accurate describe the material s behaviour in this range of the deformation, a model should be used that contains the characteristics of the bounding regions of both adjacent ranges. When for simplicity the stress dependence, a σ, is taken the same for each relaxation time. This results in a relation in which all relaxation times are the same function of temperature. A mechanical model is shown in figure (2.1) This results in a generalized nonlinear Maxwell model: n t E( t) = Ei exp i= 1 τ i ( σ ) (2.7) where τ i ( σ )( = τ a i σ ( σ )) and E i refer to the i th number of elements. Maxwell element and n is the At different levels of stress, the model should accurately describe the material s behaviour with respect to the boundary conditions. For stresses belowσ this implies that the shift function a σ equals one, all relaxation times become linear. At moderate stresses, the typical time and stress-dependence observed in the nonlinear viscoelastic region of the intrinsic deformation curve is obtained. While high stresses result in yield, this due to all relaxation times decreasing exponentially. The stress activation of a molecular process is governed by the total applied stress, according to the Eyring theory. This makes the total stress dependent shift function a σ the same for all relaxation times. Which implies that time-stress superposition is similar to time-temperature superposition. This due to all relaxation times being the same function of temperature. As a result of this, a stress-reduced time ψ can now be defined: t dt ψ = (2.8) a σ [ σ ( t )] 9

Hereby is the relaxation time for a stress well belowσ and a σ the ratio of the relaxation time at a stressσ. When all parts of the model are put together in one formula for describing the entire range of deformation it leads to the following stress-strain relation: with t σ ( t) = E ( ψ ψ ) ɺ ε ( t ) dt (2.9) t dt ψ = and a σ [ σ ( t )] ψ = t a σ dt [ σ ( t )] 1

Chapter 3 Experimental 3.1 Material The material used for injecting moulding was a commercial grade of polycarbonate supplied by General Electric Plastics Europe. This material is more commonly known as LEXAN11R. The tensile bars used for the experiment were tensile bars, produced accordingly to ASTM D638 type IV, with a melt temperature from 28 C and a mold temperature from 3 / 4 C. This has been done with an Arburg Allrouder 532 5-15. 3.2 Thermal treatments: The annealing of samples was preformed at two different annealing temperatures (T a = 1 o C and T a = 12 o C) with both the same annealing time (t a =143 hours). 3.3 Mechanical Testing: Stress relaxation, tensile and creep experiment were preformed on a MTS Elastomer Test System 831, equipped with an Instron strain gage extensometer (catalogue No: 262-62), and a thermostatically controlled oven. The measure length of the extensometer was 25mm and the range was ± 2.5mm. All tests were preformed at a controlled temperature of 23 o C. The tensile tests and relaxation tests have been done for samples with three different ages: as received, 143hours at1 o C and143hours at12 o C. The creep tests have been done with the as received tensile bars. 2 3 4 The tensile tests were done at three different strain rates: ɺ ε = 1, ɺ ε = 1 andɺ ε = 1. Creep test were done at eight different force levels. The relaxation tests were done with a continuous applied strain of 1%, 2% and 3%. 11

Chapter 4 Results 4.1 Tensile tests Tensile test were done with as received Lexan11R at three different strain rates. In figure 4.1 the results are plotted in a strain- stress diagram. These are the average values of nine experiments at specific strain rate. Figure 4.1: results tensile test Lexan11R as The maximum driving stress of the tensile experiments are plotted in a strain rate versus stress plot (see figure 4.2(a)). Figure 4.2(b) shows the same data but now with error lines representing 2 times the standard deviation. (a) (b) Figure 4.2: (a) maximum true stress for Lexan11R, (b) maximum true stress with error lines 12

With the measured data, the spectrum of the material can be retrieved. And by applying a shift factor it can be shown if shifting is possible. The yieldpoint is calculated by, with: τ 2 1 ( A ) S ( ) 3τ 3 3 σ y = ln 2 3ɺ ε + A + σ r λy λ (4.1) 3 + µ 3 + µ 3 + µ S ( ) A = S A t Because the rate constant, εɺ, and strain hardening, σ r, are the same for all experiment, these can be neglected when obtaining the shift factor. This implies that only S A is responsible for different material behaviour. Now formula 4.1 can be simplified to, σ = y 3τ S 3 + µ A (4.2) Shifting occurs between two different situations, taken this into account in equation (4.2) this results in the shift factor: 3τ σ y σ ( ) 2 y = S 1 A S 2 A (4.3) 1 3 + µ The shift factor: S = ( S S ) (4.4) A A2 A1 3 + µ S A = ( σ y σ ) 2 y (4.5) 1 3τ For polycarbonate τ =.7 MPa and µ =.8 are used (page 9, table 5.2, [1]). 3 + µ The right part of equation (4.5) is a constant and can be replaced with C 1 =. So 3τ equation (4.5) results in; S = σ σ C (4.6) ( ) A y2 y1 1 The spectra of three differently aged samples are plotted in figure 4.3(a) and 4.3(b). In figure 4.3(b) shifting is applied on A1 and A12 with Ar as the reference situation. All the curves follow the Ar-curve. (Ar = as received, A1 = annealed at 1 o C for 143 hours, A12 = annealed at 12 o C for 143 hours). Shifting is proven to be possible. 13

(a) Figure 4.3: (a) Linear relaxation modulus from all material, as received and aged, with the same strain rate, before shifting. (b) Linear relaxation modulus from all material with shifting. (b) Here shifting is done accordingly to time-temperature superposition. In the next paragraph, shifting is done accordingly to time-stress superposition. 4.2 Creep tests By creep test the master curve is obtained. In figure 4.4(a) all experiment are plotted. To obtain the true linear compliance master curve in figure 4.4(b) the result of the creep tests need to be shifted horizontally along the logarithmic time axis with respect to the 15MPa reference curve. (a) Figure 4.4: (a) Creep (τ eq ) compliance, starting at 15 Mpa with increasing steps of 5 Mpa till 4 Mpa. (b) Master curve of creep (τ eq ) compliance after shifting at a reference stress of 15 Mpa. (b) Shifting is done according to: a ( ) c a ( ) 15 σ = σ σ (4.7) 14

with c as a constant that accounts for the shift of 15MPa master curve with respect to the linear range, with ( ) a σ σ as in equation (2.5), c = a 1 σ (15) (4.8) The Eyring viscosity parameter (equation 2.5) which describes the nonlinearity in the stress response is determined by two parameters, the zero-shear viscosity η and the non-linearity parameterτ. By fitting a plot of the viscosity as a function of the equivalent stress, these material constants can be determined. To accomplish this, viscosity data needs to be extracted from creep tests at different stress levels. Figure 4.5: Logarithm of shift factors with respect to 15 MPa, obtained from creep tests as function of equivalent stress. The plateau-creep rate is determined at an imposed constant creep loadσ where the e creep rate reaches a constant value and therefore defines the viscosityη = σ / ɺ ε. e In terms of the Maxwell model, the plateau-viscosity η pl is equal to the sum of the e e viscosities ηi = ηi a σ of all the Maxwell modes separately: pl pl σ η η a( τ ) = = e e pl pl i eq ɺ ε pl i (4.6) Figure 4.6: Sherby-Dorn plot for the determination of the plateau-creep rate at 4 MPa 15

This plateau-creep rate was obtained from a Sherby-Dorn plot, a graph of the creep rate versus creep strain. Only at the highest creep load of 4 MPa, a plateau-creep was observed due to the limited experimental time window. From a Sherby-Dorn (see figure 4.6) the value of 4 1 the plateu-creep can be estimated: ɺ ε pl = 2 1 s e 4 This leads to a plateau-creep viscosity of: η = 2 1 Mpa. The ratio of the shift factors at these levels equals the ratio of the plateau-creep viscosities at two different stress levelsτ 1 eq andτ 2 eq accordingly to equation 4.6 and can be written as: pl e 1 1 pl ( eq ) aσ ( eq ) e 2 2 pl ( eq ) aσ ( eq ) η τ τ = (4.7) η τ τ This implies that the value of the creep viscosity at 4 MPa may be used to convert the plot of shift factors (figure 4.5) to a graph of viscosity as a function of equivalent stress (figure 4.7). In contrast to figure 4.5, a logarithmic stress axis is used to show more clearly the linear region characterized by the zero-shear viscosity η Figure 4.7: Zero-shear viscosity η, the solid line is a fit using a single Eyring function The solid line is a fit using a single Eyring function. In figure 4.7 is shown that this solid line and the data point obtained from the experiment follow the same path. It can be said that creep is determined by the same Eyring function (equation (2.5)) as used in shifting between materials with different age (see 4.1 Tensile tests). 16

Chapter 5 Conclusion For the past several years, single mode approach was used to describe the intrinsic deformation behaviour of a material. For predicting post-yield behaviour it was accurate enough, but not in the pre-yield range. Due to energy which is not taken into account at single mode approach, the material has a different behaviour after deformation. For indentation this implies, that material simulated with a single mode approach will result in an incorrect prediction of for instance, the hardness of the material. Therefore the single mode approach is extended to a multi mode approach. The intrinsic deformation behaviour of a material can be described more accurately, this due to multiple parallel Maxwell elements. To use this model, a unique set of material parameters for describing a polymer s intrinsic deformation can be obtained by the use of tensile- and/or creep tests. It has been proven that the difference in age can be described with a unique set of relaxation times and modili, simply by shifting with the state parameter S A. 17

Bibliography [1] Klompen, E. (25). Mechanical properties of solid polymers: constitutive modeling of long and short term behaviour. University Press Facilties, Eindhoven. [2] Tervoort, T.A. (1995). A multi-mode approach to finite, three-dimensional, nonlinear viscoelastic behavior of polymer glasses. Eindhoven. [3] http://bruce.cs.cf.ac.uk/bruce/lvm/methods%211-119/la%2method%2117/la_117.jpg (22/6/27) [4] Tobolsky, A. and Eyring, H. (1943). Mechanical properties of polymeric materials. J. Chem. Phys.,4,283-295. 18

Appendices A Tensile tests Lexan 11 R as received Lexan 11 R at strain rate 1e( 2) driving stress [MPa] 7 6 5 4 3 2 1 strain rate = 1e( 2) strain rate = 1e( 3) strain rate = 1e( 4) driving stress [MPa] 7 6 5 4 3 2 1 Ar A1 A12.2.4.6.8 true strain [ ].2.4.6.8 true strain [ ] Lexan 11 R 1c 143h Lexan 11 R at strain rate 1e( 3) driving stress [MPa] 7 6 5 4 3 2 1 strain rate = 1e( 2) strain rate = 1e( 3) strain rate = 1e( 4) driving stress [MPa] 7 6 5 4 3 2 1 Ar A1 A12.2.4.6.8 true strain [ ].2.4.6.8 true strain [ ] Lexan 11 R 12c 143h Lexan 11 R at strain rate 1e( 4) driving stress [MPa] 7 6 5 4 3 2 1 strain rate = 1e( 2) strain rate = 1e( 3) strain rate = 1e( 4) driving stress [MPa] 7 6 5 4 3 2 1 Ar A1 A12.2.4.6.8 true strain [ ].2.4.6.8 true strain [ ] 19

8 LEXAN 11 R with error marge True stress [MPa] 7 A12 A1 Ar 6 1 4 1 3 1 2 strain rate [s 1 ] 2

B Spectum and shifting 1 4 Lexan 11 R as received Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 strain rate = 1e( 2) strain rate = 1e( 3) strain rate = 1e( 4) 1 1 4 1 8 1 12 1 16 1 2 1 24 Time [s] 1 4 Lexan 11 R 1c 143h Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 strain rate = 1e( 2) strain rate = 1e( 3) strain rate = 1e( 4) 1 1 4 1 8 1 12 1 16 1 2 1 24 Time [s] 1 4 Lexan 11 R 12c 143h Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 strain rate = 1e( 2) strain rate = 1e( 3) strain rate = 1e( 4) 1 1 4 1 8 1 12 1 16 1 2 1 24 1 28 Time [s] 21

1 4 Lexan 11 R at strain rate 1e( 2) 1 4 LEXAN 11 R shifted at strain rate 1( 2) Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 Ar A1 A12 Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 Ar A1 A12 1 1 4 1 8 1 12 1 16 1 2 1 24 1 28 Time [s] 1 2 1 1 4 1 8 1 12 1 16 1 2 1 24 Time [s] 1 4 Lexan 11 R at strain rate 1e( 3) 1 4 LEXAN 11 R shifted at strain rate 1( 3) Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 Ar A1 A12 Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 Ar A1 A12 1 1 4 1 8 1 12 1 16 1 2 1 24 1 28 Time [s] 1 2 1 1 4 1 8 1 12 1 16 1 2 1 24 Time [s] 1 4 Lexan 11 R at strain rate 1e( 4) 1 4 LEXAN 11 R shifted at strain rate 1( 4) Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 Ar A1 A12 Relaxation modulus [MPa] 1 3 1 2 1 1 1 1 1 Ar A1 A12 1 1 4 1 8 1 12 1 16 1 2 1 24 1 28 Time [s] 1 2 1 1 4 1 8 1 12 1 16 1 2 1 24 Time [s] 22

C Creep tests 4.5 Creep(τ eq ) complicance 4 Compliance [GPa 1] 3.5 3 2.5 2 1.5 1 1 1 1 2 1 3 1 4 1 5 1 6 Time [s] 4.5 Master curve of creep(τ eq ) complicance 4 Cpmpiance [Gpa 1] 3.5 3 2.5 2 1.5 1 1 2 1 4 1 6 1 8 1 1 1 12 1 14 Time [s] 23

5 Logarithm shift factor log(a 15 ) [ ] 5 1 15 2 1 2 3 4 τ eq [MPa] 1 3 Sherby Dorn Strain rate [s 1 ] 1 4.16.18.2.22.24 Strain [ ] 5 Zero shear viscosity η 4 η e log(η e ) [MPa s] 3 2 1 1 1 1 τ eq [MPa] 24