Creep. Creep behavior of viscoelastic polymeric materials

Transcription

1 B1 Version: 2.2_EN Date: 15. March BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS FACULTY OF MECHANICAL ENGINEERING DEPARTMENT OF POLYMER ENGINEERING Creep Creep behavior of viscoelastic polymeric materials CHECK THE VALIDITY OF NOTE ON WEBPAGE OF THE DEPARTMENT!

2 Version: 2.1_EN Date: 15. April LOCATION OF THE PRACTICE THIS PRACTICE WILL BE HELD IN BUILDING T, GROUND FLOOR LAB CONTENT 1. The aim of the practice Theoretical background Creep and stress relaxation Deformation components of polymers The creeping Determining of the creeping curve Determining the model parameters from creeping curve Stress relaxation Practice description and tasks Equipments and machines Report of the lab practice Creep 2/13

3 Version: 2.1_EN Date: 15. April THE AIM OF THE PRACTICE The aim of the laboratory exercise is to study the creeping behavior and stress relaxation of thermoplastic polymers and to get acquainted with the most common model which describes this phenomenon. During the laboratory practice the students will determine the parameters of the socalled Burgers-model and get knowledge about the long term habit of polymers. 2. THEORETICAL BACKGROUND Polymer products and parts used in everyday life are subject to loading and respond to that with continuous strain, for example the plastic shopping bag s handle which elongate under load of the inside (carried) products. This deformation can lead to failure within hours while the load does not change. This phenomenon can be observed in case of polymer products in general and is called creeping. In some engineering constructions this constant dimension change can cause problems, hence in case of intensive loading, that is why dimensioning needs to be done for maximum deformation instead of stress peak CREEP AND STRESS RELAXATION The phenomenon of creeping is that the polymer material responds to static stress loading with deformation change, and then, although stress is constant, strain and deformation increases in time. When inspecting very long times (decades/century) every engineering material is creeping. The general creep test is starting with instantaneous load stimulus, after that the load is constant. The materials response to this load is instantaneous deformation followed by further continuously increasing deformation (fig 3.a and fig 3.e). Besides gathering information on the response of the material the aim is to decompose the curve hence determining the deformation components at a given load level. Stress relaxation is a phenomenon similar to creeping, when instead of a constant loading the strain of the material is constant. In cases of stress relaxation the test starts with instantaneous elongation, and then it is held constant. During the test the initial high stress in the polymer material decreases as a function of time, i.e. the material relaxes (fig 6.) DEFORMATION COMPONENTS OF POLYMERS Viscoelastic polymers show different behavior compared to metals. At metals an exact elastic limit can be defined. Below this limit metals show ideally elastic behavior, above it metals have plastic-elastic deformation. At polymers there is no such limit and permanent residual deformation (irreversible) is formed even at moderate loads. Another difference is that polymers have no linear relation between stress and deformation and the behavior of polymers depend strongly on the temperature, the applied level and duration of stress. This is why it is hard to describe the relations between these parameters in the case of polymers. For the most common and accepted modelling method, the reversible and irreversible deformation components are divided into three different classes. In this approximation the total Creep 3/13

4 Version: 2.1_EN Date: 15. April deformation (ε) contains an instantaneous elastic deformation (εe), a time-dependent delayed elastic deformation or relaxation (εd) and a time-dependent viscous (residual) deformation, i.e. permanent set (εp). (t) e + d(t) + p(t) Stress-deformation curves of deformation components are schematically depicted in Fig. 1. during loading and load removal. Figure 1. Deformation components of polymers (a) The elastic deformation is instantaneous and can be described by the change of the atomic distances and the distortion of the valence angles between fixed chemical bonds. This phenomenon characterizes purely elastic deformation therefore it is mechanically and thermodynamically reversible. The simplest model for this component is the ideal spring which follows Hook s law. This model have linear characteristic which means another simplification compared to Figure 1. (b) The delayed elastic deformation is described by the orientation or straightening of polymer chains as they can be present in different conformations. This procedure is built up retarded and after removing the load the restoration of the structure is retarded too. The loading and load removal curves are not the same, hysteresis can be observed. The area of hysteresis-loop dissipated as heat and it is related to deformation work coefficient. This component mechanically reversible but thermodynamically it is irreversible. This characteristic of the material can described with a combination of a linear dash pot and an ideal spring. If we connect these models in parallel we get the Kelvin-Voigt model. (c) The residual deformation or permanent set indicates displacement with relative relocation of the polymer chains (the centers of the gravity of the molecules are moving relative to each other). The work of deformation in this case is fully dissipated as heat. That means this component is irreversible both mechanically and thermodynamically. Creep 4/13

5 Version: 2.1_EN Date: 15. April Name of deformation component Notation Model Representation, parameters elastic deformation e Ideal spring (Hook s law) residual deformation or time-dependent viscous deformation p Linear dash-pot (Newton s law of viscosity) time-dependent viscoelastic deformation or relaxation d Kelvin-Voigt model (spring and dash-pot in parallel) 2.3. THE CREEPING Based on earlier presented modeling elements the well-known 4-parameter also known as Burgers-model can be constructed. This model contains the least elements which qualitatively describes the creep behavior of thermoplastic polymers (Fig. 2.). It made of a spring, a dash pot and a Kelvin-Voigt model connected in series. Figure 2. 4 parameter or Burgers-model In case of σ = σ 0 = constant load the superposition of deformation components is true, that means Burgers-model s elongation function can be created from the single models response, they can be added together like in Figure 3. This is why parameters of Burgers-model can be determined from the measured creep curve (in indirect way). Creep 5/13

6 Version: 1.2_EN Date: 17. March a, The creeping stimulus is an instantaneous stress, where: σ = σ 0 = constant b, Response of an ideal spring which follows Hooke s law: σ = E ε ε e (t) = σ 0 E 1 c, Response of an ideal dash pot which follows Newton s law of viscosity: σ = η dε dt ε p(t) = σ 0 η 1 t d, Response of the Kelvin-Voigt model (spring and dash pot in parallel): ε d = σ 0 (1 e E 2 E 2 η 2 t ) e, Response of the Burgers-model after superposition of b), c) and d) equation: ε(t) = σ 0 + σ 0 t + σ 0 (1 e E 2 E 1 η 1 E 2 η 2 t ) Knowing the previous equations long time deformations can be predicted from a relatively short measurement. From a 2-3 minutes long examination the deformation can be estimated for days. In everyday practice these creeping tests are carried out for days or weeks to help predicting the deformation of polymer products during their lifecycle. Figure 3. Stimulus of creep, and responses of single models and sum of these model elements Creep 6/13

7 Version: 1_EN Date: 1. April DETERMINING OF THE CREEPING CURVE After the Burgers-model has chosen to describe the behavior of real polymer, the parameters of this model can be determined, which parameters are the elastic modulus of the springs (E1 and E2 [MPa]) and characteristic viscosity of dash pots ( 1 and 2 [Pas]). The parameters are determined from a real creep curve. At the creep test a standard dogbone (also known as dumbbell) specimen is fixed between the clamps of a load machine. The initial gripping length is l0. The load machine starts the test by applying load (stress) on the specimen and continuously registers the displacement (Δl) in µm. It can be done the same ways as at tensile tests. The test is performed until a specific, preselected time (T). The start of test is t=0 and the end of test is t=t time. The given Δl(t) deformation curve is converted to ε(t) elongation for further calculations with the formula below: ε(t) = Δl(t) l 0 (1) 2.5. DETERMINING THE MODEL PARAMETERS FROM CREEPING CURVE The normalized creeping curve (Fig. 4.) determined this way can be used to determine the parameters for the examined material. Figure 4. Evaluation of creeping curve From the instantaneous elongation at t=0 time (ε0), the value of E1 (parameter of the spring) can be determined: ε(t = 0) = ε 0 = σ 0 E 1 E 1 = σ 0 ε 0 [MPa] (2) where σ 0 = F A At t=t moment the following simplification can be introduced: Creep 7/13

8 Version: 1_EN Date: 1. April τ 2 = η 2 E 2 [s] (3) where τ 2 is a time-dimensioned value. Choosing a long measuring time (T), when T>>τ 2 the exponential term in 4 parameter model s equation will be the following: Caused by this statement it can be concluded at t=t: e T/τ 2 0 (4) ε(t) σ 0 E 1 + σ 0 η 1 T + σ 0 E 2 (5) Further simplification is that the tangent line to ε(t) curve at t=t is nearly the same as the asymptote of ε(t) curve at t time. Construct a tangent line to the ε(t) curve at t=t time (end of test), and draw a parallel line with it in the (0, ε0) point. In this point construct a parallel line with the x axis, too. These lines divides the ε(t) curve to the three deformation components (εe(t), εd(t), εp(t)) as you can see at Fig. 4. This decomposed curve is almost the same as the ideal Burgers-model s curve (Fig. 3.). After this geometric construction we can measure εp(t=t) and εd (t=t), from this section and hence E2 and η2 can calculated as follows: ε d (t = T) = σ 0 E 2 E 2 σ 0 ε d (t=t) ε p (t = T) = σ 0 T η η 1 σ 0 T (7) 1 ε p (t=t) For the missing η2 parameter the 2 time constant must have determined. It can be determined from the intersection of the tangent line drafted to ε(t) curve at t=0 time and the asymptote at t= time. This intersection point projected to axis x gives the 2 constant. This method is very inaccurate, because it is difficult to construct a correct tangent at the initial part of the curve and that is why the following method has to be used: Calculate εrel(t) value at t= 2 time: Replaced τ2 by equation (3): (6) ε d (t = τ 2 ) = σ 0 (1 e E2 η2 τ2 ) (8) E 2 ε d (t = τ 2 ) = σ 0 E 2 (1 e 1 ) (9) The value of (1 e 1 ) expression is 0,63. With use of equation (5): ε d (t = τ 2 ) 0,63 ε d (t = T) Creep 8/13

9 Version: 1_EN Date: 1. April From this deduction now the τ2 can be described by more accurate ways. First step is to measure 0,63 εrel(t=t) distance to the y axis at (t=0) time from ε0, then draw a parallel in this point with ε(t) curve s asymptote (t=t time). In order to get τ2 project the section of this line and creeping curve to axis x. Finally η2 can be described from equation 3: η 2 = E 2 τ 2 (10) By knowing the parameters of Burgers-model, it is possible to predict deformation at another arbitrary σ* load or T* time with some degree of precision STRESS RELAXATION In case of linear structured polymers, if we use instantaneous deformation stimulus (ε0), the material s response is a decreasing stress compared to starting value and tends to zero. The initial totally elastic deformation starts to transform into residual and time-dependent viscoelastic deformations. If load applied through enough time all the ε0 deformation transforms to residual deformation (permanent set) and stress ceases totally. Using the Maxwell-model (Fig 5.) is a possible way to model this phenomenon. This model can be described with a spring and a dash pot connected in series. Figure 5. Maxwell-model Due to the instantaneous deformation the spring responses with elongation, then the deformation is taken over by the dash pot. In parallel with this process the stress decrease and dissolves (Fig 6.). a) b) Figure 6. Stress relaxation s stimulus (a), response to this load (b) Creep 9/13

10 Version: 1_EN Date: 1. April From the response curve σ(t) at relaxation test, in case of ε0 constant load, at a given (chosen) time the relaxation modulus can calculated: E r (t) = σ(t) ε 0 (11) The relaxation modulus is a time-dependent parameter. If we want to compare two different materials, those must be compered at the same period of time PRACTICE DESCRIPTION AND TASKS The aim of laboratory practice is to study the creeping curve of polymers, determine the parameters of the Burgers-model and examination of stress relaxation. Determining a creeping curve of a polymer specimen. Make the fitting which was previously shown in the 2.5 paragraph Determination of the parameters from the creeping curve Calculate the expectable deformation in T* time at * stress Determination of the relaxation modulus of a polymer product at T=120 s 1.1. EQUIPMENTS AND MACHINES ZWICK Z005 UNIVERSAL TENSILE TESTER (FIGURE 6.) Maximal force: 5 kn Precision: ± 2 μm Speed range: 0, mm/min Figure 6. Zwick Z005 Universal Tensile Tester Creep 10/13

11 Version: 1_EN Date: 1. April REPORT OF THE LAB PRACTICE Name: Results: Neptun: Date: Instructors s signature: Instructor s name: 1. Laboratory exercise 2. Raw data: Determining a creeping curve of a polymer specimen. Make the fitting which was previously shown in the 2.5 paragraph Determination of the parameters from the creeping curve Calculate the expectable deformation in T* time at * stress Determination of the relaxation modulus of a polymer product at T=120 s Material of the specimen: Load, F = [N] Cross-section, A (axb) = [mm 2 ] Stress, 0 = [MPa] Starting length of specimen, l0 = [mm] 3. Measured and calculated results l (T)= [mm] (T)= T= [s] - - le (T)= [mm] e (T)= - E1 = [MPa] - lp (T)= [mm] p (T)= - - 1= [Pas] ld (T)= [mm] d (T)= - E2= [MPa] τ2= [s] - 2= [Pas] (T*)= Creep 11/13

12 Version: 1_EN Date: 1. April Stress relaxation measuring: Material of specimen: Cross-section A (axb) = [mm 2 ] Starting length of specimen, l0 = [mm] Elongation = [mm] Force at T=0 = [N] Stress at T=0 = [MPa] Force at T=120 s = [N] Stress at T=120 s = [MPa] Relaxation modulus at 120 s = [MPa] Creep 12/13