Polymer Materials Sciencept.bme.hu/~vas/PhD_Polymer Materials Science... · 2017-02-16 · Behavior...
Transcript of Polymer Materials Sciencept.bme.hu/~vas/PhD_Polymer Materials Science... · 2017-02-16 · Behavior...
1
1
Polymer Materials ScienceBMEGEPT9107, 2+0+0, 3 Credits
Lecturer: Prof. Dr. László Mihály Vas
Budapest University of Technology and EconomicsDepartment of Polymer Engineering
2017.02.16.
5. Phenomenological Modeling of the Mechanical Behavior of Polymers
2
Polymer Materials ScienceBooks, textbooks, lecture notes, guides
� G. Bodor: Structural investigation of polymers. Akadémai Kiadó, Budapest; Ellis Horwood, Chichester, 1991.
� I.M. Ward, D.W. Hadley: An introduction to the mechanical properties of solid polymers. J. Wiley & Sons, Chichester – New York, 1993.
� T.A. Osswald, G. Menges: Materials Science of polymers for engineers. Hanser Pub., New York, 1996.
� L.M. Vas: Lecture notes, ppt slides, http://pt.bme.hu/~vas
� G. Strobl: The Physics of Polymers. Concepts of Understanding theirStructures and Behaviour. Springer Verlag, Berlin. 1996.
2017.02.16.
2
2017.02.16. 3
Content of Polymer Materials ScienceRecapitulation
� Polymer materials, typical material classes, molecular and
morphological structure of polymers, polymer blends and alloys
� Testing methods of polymer structures
� Mechanical behavior of polymer materials
� Behavior of polymers under changing temperature, humidity and
other environmental factors
� Phenomenological modeling of the mechanical behaviors of
solid polymers
� Strength and fracture-mechanical properties of polymers
� Statistical-mechanical modeling of polymers
4
Classification of PolymersRecapitulation
� Classification respect of structure
• Linear polymers (linear, chain molecular structure)- Semi-crystalline polymers (e.g. PE, PP, PA, PAN, PET)
- Amorphous polymers (PVC, PS, PMMA, PC)
• Crosslinked polymers (network structure – amorphous polymers.) - Elastomers (weakly cross-linked, e.g. rubbers: NR, BR, PUR)
-Duromers/Thermosets (strongly cross-linked; resins: e.g. UP, EP, VE)
� Classification in respect of thermal and mechanical behavior
• Thermoplastics (they can be molten reversibly ⇒ linear polymers; e.g. PE, PP, PA, PET, PVC, PS, PMMA, PC)
• Non-thermoplastics- Linear polymers (Kevlar, PAN, cellulose, chitin, protein)- Crosslinked polymers (elastomers, duromers/thermosets)- Semi-crosslinked & semi-crystalline polymers (wool fiber, PEX)
2017.02.16.
3
5
Mechanical properties Recapitulation
� Micro- and macro-deformation components
Microdeformation components Macrodeformation components
• Energy elastic (εU) - reversible
• Entropy elastic (εS) - reversible
• Energy dissipating (εD) - irreversible
→→→→
→→→→
→→→→
• Elastic (εe) (Mech: reversible)
(Tdyn: reversible)
• Delayed elastic (εd) (Mech: reversible)
(Tdyn: irreversible)
• Remaining (εr) (Mech: irrev.)
(Tdyn: irreversible)
2017.02.16.
6
Mechanical properties Recapitulation
� General scheme of mechanical tests
A – sample, material-operator: Y(t)=A[X](t)
Stimulus Response
2017.02.16.
Step function Ramp function Sinusoidal function
4
7
Modeling mechanical properties
� ‘Black box’ modeling of mechanical behaviors
A – sample, material-operator, A: X→YA=A[X](t)
M – model, model-operator, M: X→YM=M[X](t)
2/16/2017
Measured response
Model response
Objective of modeling: Creating model M so that deviation
YM-YA is minimum in a given sense and in time interval [0,t]
or at least for ε>0 the next inequality stands:
Stimulus,
excitation
X and Y are
mechanical
quantities
(deformation
and load
properties)
8
Modeling mechanical properties
� Phenomenological modeling• Linear elastic (LE) material models (for metals, or for certain polymers in case of small deformation);
• Linear viscoelastic (LVE) material models (for polymers when the deformation is relatively small);
• Nonlinear elastic (NLE) material models (for metals and polymers in case of large deformation and monotonic increasing or decreasing load);
• Nonlinear elastoplastic (NLEP) material models (for metals in case of large deformation and arbitrary loading mode)
• Nonlinear viscoelastic (NLVE) material models (for polymers in case of large deformation and arbitrary loading mode).
� Structural-mechanical modeling• Statistical polymer-network model of elastomers;
• Statistical fiber-bundle-cells model of strongly oriented linear polymers;
• Other models of compounds/mixtures/composites (layer models, homogenization,…)
2017.02.16.
5
9
Phenomenological modeling 1.
METHODS OF THE LINEAR VISCOELASTIC THEORY
� Qualitative modeling – Formal description of responses• Mechanical model-elements and basic models
• Analogous mechanical model-elements
• Models of the deformation-components
• Qualitative models of creep
• Qualitative models of stress relaxation
� Quantitative modeling – Description with given error• Quantitative models of stress relaxation, relaxation spectrum
• Quantitative models of creep, retardation spectrum
� Boltzmann’s superposition principle – basic LVE equations
� Modeling in frequency domain
� Relation-graph of LVE material characteristics
2017.02.16.
10
Phenomenological modeling 2.
� Qualitative modeling – Model-elements 1.
Spring Viscous element
Inertial element St. Venant element
Engineering stress:
Strain:
Load:
Uniaxial tensile load
2017.02.16.
6
11
Phenomenological modeling 3.
� Mechanical analogous model elements
Spring
Viscose element
Hooke’s law:
E – elastic modulus
Newton’s law:
η – dynamic viscosity factor
σ=F/Ao - stress, ε=∆l/lo- strain
2017.02.16.
Tensile Shear
Tensile Shear
12
Phenomenological modeling 4.� Models of deformation components
Def. components Model Motion law
Elastic Spring
Remaining Viscose element
Delayed elastic
Kelvin-Voigt element
2017.02.16.
7
13
Phenomenological modeling 6.
� Deformation components – delayed elastic deformation
Kelvin-Voigt element
2017.02.16.
14
Phenomenological modeling 7.
� Qualitative modeling – Creep (ATP and WCE)
ATP WCE
LDPE
Burgers model Stuart model
ATP WCE
Creep compliance,:
Burgers response to creep stimulus:
2017.02.16.
MEASUREMENTS:
MODELING:
ATP = Amorphous thermoplastics
WCE = Weakly crosslinked elastomers = R= Rubbers
Burgers model � Stuart model when η1→∞
εεεεe εε εε e
εεεεr
εεεεd εε εε d
εε εε e
εε εε e
εε εε dεε εε rεε εε d
8
15
Phenomenological modeling 8.
� Qualitative modeling
– creep of ATP
Constructing the
model response –summing up the component
deformations (εe, ε
d, ε
r)
point by point in the time
domain:
ATP
2017.02.16.
εεεεe
εεεεd
εεεεr
εεεεe
εεεεe
εεεεr
16
Phenomenological modeling 10.
� Qualitative modeling – Stress relaxation of ATP
Burgers model:
It describes the
whole relaxation
process of ATP
in formally
correct way ATP
ATP
Maxwell model:For the loaded state
only
Relaxation modulus:
εr εm
2017.02.16.
MEASUREMENT:
Model-response:
εεεεe εεεεe
εεεεe
εεεεr
εεεεr
εεεεd εε εε r
εε εε d
εεεεd
εεεεr
9
17
Phenomenological modeling 11.
� Qualitative modeling – Stress relaxation of WCE
Standard-Solid model
WCE
2017.02.16.
MEASUREMENT:
18
Phenomenological modeling 12.
� Qualitative modeling – 5-parameter model
Burgers model:
E∞=0
Standard-Solid model:
E2=∞ and/or η2=∞
2017.02.16.
Union of Burgers and Standard-Solid models
10
19
Phenomenological modeling 14.
� Quantitative modeling – Stress relaxation
Generalized
Standard-Solid
model
Generalized
Maxwell
model (GM)
Response of Standard-
Solid model :
Only one kind of
relaxation time (τ)
Response of polymer
sample:
Several kinds of
relaxation time (ϑ→τ)
τi=ηi/Ei; i=1,…,n
Solution: Modeling the
measured process of several
kinds of τ relaxation times by generalized Maxwell model
2/16/2017
GM
MODEL POLYMER
20
Phenomenological modeling 15.
� Quantitative modeling – Stress relaxation
Generalized Maxwell model (a)
its stress relaxation (b)
and the discrete relaxation
spectrum (c)
(Ei,τi)n – discrete-, H(lnτ) – continuous relaxation spectrum
E(t) – relaxation modulus
2017.02.16.
11
21
Phenomenological modeling 16.
� Quantitative modeling – Stress relaxation
Continuous relaxation spectrum (CRS), H(lnττττ)
Effect of temperature (1: 25oC, 2: 40oC, 3: 50oC,
4: 60oC) in case of LDPE
Relation of CRS to the structural elements of filled,
crosslinked polymer
(Urzsumcev-Makszimov: MK, Bp. 1982) • Range of spectrum becomes wider and wider with inreasing the
molecular mass of polymer. (Javorszkij B.M.-Detlaf A.A.: Fizikai zsebkönyv. Műszaki K. Bp. 1974.)
2017.02.16.
22
Phenomenological modeling 17.
� Quantitative modeling – Creep (ATP and WCE)
• Generalized Kelvin-Voigt model (GKV) (a), its creep (b)
and the discrete retardation
spectrum (c)
• Generalized Stuart (a) and Burgers (b) models
ATPWCE
L(lnτ) –continuous
retardation
spectrum
J(t) – Creep compliance2/16/2017
GKVGKV
12
23
Phenomenological modeling 18.
� Boltzmann’s superposition principle (BS) – Basic LVE
equations in time domain
Response (Y) to arbitrary stimulus (X):
(solution with Laplace-transform)
2017.02.16.
24
Phenomenological modeling 19.
� Dynamic qualitative modeling – Kelvin-Voigt model
Complex complianceComplex
Hooke’s law
Loss factor
Complex
stimulus
2017.02.16.
13
25
Phenomenological modeling 20.
� Dynamic qualitative modeling – Maxwell model
Complex modulus
Loss factor
2017.02.16.
26
Phenomenological modeling 21.
� Dynamic quantitative modeling – LVE complex elastic modulus and complex compliance
Generalized Maxwell model
based complex modulus (X=ε):
Generalized Kelvin-Voigt model
based complex modulus (X=σ):
Basic LVE equations in frequency domain:
Relation of E* and E(t), as well as J* and J(t) (BS)
2017.02.16.
14
27
Phenomenologi-
cal modeling
22.
� Summary of
LVE functions for
characterizing
polymer materials
Retting, W.: Hanser-Verlag, 1992.2017.02.16.
Stress relaxation Bending vibration Tensile test - LVE
These formulae provide: Time dependent modulus E(t)
Relaxation spectrum H(t)
Experimental determination of the time dependent modulus and the relaxation spectrum based
on stress relaxation, bending vibration, and tensile tests (acc. to [15])
Dehnung = StrainSpannung = StressZeit = TimeResonanzkurve = Resonance curveFrequenz = FrequencyKraft = ForceRelaxationsmodul = Relaxation modulus
tg α ist nur eine Funktion of t = tan αis the function of t onlyDämpfung = Damping/Loss factorSpeichermodul = Storage modulusVerlustmodul = Loss modulusTangentenmodul = Tangent modulus
28
Phenomenological modeling 23.
� LVE functions for material characterization – Relation graph
Approximate numerical relationships (DMA software):
Schwarzl, Ninomiya-Ferry: E(t)↔E*(ω), J(t)↔J*(ω)Hopkins-Hamming: E(t)↔J(t)
2017.02.16.
Time
domain
Spectrum
domainFrequency
domain
15
29
Phenomenological modeling 24.
� Relationship between the relaxation modulus and the creep
compliance
� Relationships between the relaxation and retardation spectra
(Ferry J.D.: Viscoelastic properties of polymers. J. Wiley, New York, 1961.)
Linear polymer: Ee=E∞=0, η>0
Crosslinked polymer: Ee=E∞>0, η=∞
Basis equations:
(relaxation and creep)
Steady state flow
viscosity
2017.02.16.
30
Phenomenological modeling 25.
� Relationship between the relaxation modulus and the
molecular mass distribution
• Relationship between the characteristic relaxation time (τ) and the molecular mass in case of linear polymers (k, b are constants)
• Relationship between the relaxation modulus and the probability density
function of the molecular mass ϕ(m):
(Urzsumcev-Makszimov: MK 1982)
2017.02.16.
16
31
Phenomenological modeling 26.
� Generalization of LVE relationships for multiaxial load
and anisotropic material
Linear elastic (LE) material behavior
– (Hooke’s law)
• Uniaxial tensile load and pure shear
• Multiaxial load
Cijkl – 4th order tensor
E – elasticity matrix (6x6)
Linear viscoelastic (LVE) material
behavior
• Uniaxial tensile load and pure shear
• Multiaxial load
E(t) – relaxation modulus matrix (6x6)
↔
2017.02.16.
32
Phenomenological modeling 27.
� LE relationships for anisotropic material
Orthotropic
(9 indep.
const.)
Monotropic(transversally
isotropic)
(5 indep.
const.)
Tensile modulus of 2D orthotropic
material in direction α
2017.02.16.
17
33
Phenomenological modeling 28.
� Quantitative modeling – Effect of temperature 1.
The viscosity as well as the relaxation time (constant) depend on the temperature
(T) according to the so called Arrhenius type relation:
Using the Arrhenius type relations and recording the variation of e.g. the relaxation
spectrum H(lnτ) as a function of 1/T reciprocial temperature (Arrhenius-variable) gives the so called Arrhenius-type diagram that is suitable for illustrating simply the
temperature dependent structural-mechanical behavior of the polymer materials.
This illustration can be based on the WLF equation. Extending the similar effects
principle and the WLF equation for other environmental (moisture content, pressure), and
loading parameters relationships similar to those above can be obtained, moreover this
gives possibility studying the addition of different effects (see the long term behavior).
2017.02.16.
34
Phenomenological modeling 29.
� Quantitative modeling – Effect of temperature 2.
Arrhenius-type diagrams
Retting, W.: Hanser-Verlag, 1992.
Temperature dependent peak values of relaxation spectra of PVC and PP at ambient
temperature
2017.02.16.
Relaxation spectrum of PVC at ambient temperature Relaxation spectrum of PP at ambient temperature
Main
max. Cryst.
max.
Main max.
Secondary max.
18
35
Phenomenological modeling 30.
� Quantitative modeling – Effect of temperature 3.
Utilizing the temperature-time superposition and the shifting factor aT the effect of
temperature (T) can be taken into account in the LVE equations.
Increasing T the values t and τ decrease but the spectrum area remains constant:
Urzsumcev-Makszimov: MK 19822017.02.16.
36
Phenomenological modeling 31.
Characterizing LVE behaviors:
• The relaxation modulus and the creep compliance do not depend on the levels of εo/γo (indiagram: γo< γc) and σo/τo loads as stimuli;• The time constants of the up- and unloading parts of the creep- or relaxation curves are identical;
• To pure sinusoidal stimulus the response is pure sinusoidal that is there are no harmonics;
• The complex elastic modulus and the complex compliance do not depend on the εo or σo stimulus-
amplitudes;
• The isochrones (see Chapter 3) are linear;
• The responses to stimuli of different types can be calculated from one another (e.g. the tensile test curve,
the relaxation and creep curves can be determined from one
another)
� Limits of LVE behavior
2017.02.16.
19
37
Phenomenological modeling 32.
� Relation of relaxation-(a) and tensile test (b) curves of a LVE material
Relaxation curve is strictly monotonic ⇒ LVE tensile test curve strictly monotonic
decreasing and convex from below increasing and concave from below
2017.02.16.
38
Phenomenological modeling 33.
� LE, LVE, and NLVE ranges on the tensile test curve of
a polymer
LE = Linear elasticLVE = Linear viscoelasticNLVE = Nonlinear viscoelastic
LVE range:
Where no irreversible
structural change and
deformation occur (here the
model parameters are
constants). E.g.:
PVC: <0,5%
PE: <0,1%
PC: <1%
(Ehrenstein’s book: page 104.)
2/16/2017
Polymer
20
39
Phenomenological modeling 34.
� Properties of NLVE behavior – Because of irreversible
structural changes at larger load the mechanical behavior of materials changes
hence the parameters, that are constants at smaller load, change as well.
Characterizing NLVE behaviors :
• The relaxation modulus and the creep compliance do depend on the
levels of εoand σ
oloads as stimuli;
• The time constants of the up- and unloading parts of the creep- or relaxation curves are not identical;
• To pure sinusoidal stimulus the response is periodic, but not pure
sinusoidal that is there are harmonics;
• The complex elastic modulus and the complex compliance do depend
on the εoor σ
ostimulus-amplitudes;
• The isochrones (see Chapter 3) are nonlinear;
• The responses to stimuli of different types cannot be calculated from
one another.
2/16/2017
40
Phenomenological modeling 35.
Some methods for describing the NLVE
behavior
• Semi-empirical, heuristic solutions• Application of nonlinear model-elements•Modification of the Boltzmann equations
2017.02.16.
21
41
Phenomenological modeling 36.
� NLVE modeling – Semi-empirical solutions
Describing creep with a power equation of Nutting-type:
Describing creep with an equation by Kauzmann, Eyring, and Nielsen:
K(t) – time dependent creep compliance
σc – a kind of critical stress
K, α, n>0
2017.02.16.
42
Phenomenological modeling 37.
� Semi-empirical solutions – Nonlinear creep description of POM (the parameters depend on the time and/or the load)
ηo(σ), EK(σ), τ(t)
2017.02.16.
Creep
Recovering
Loading level
Own weight
22
43
Phenomenological modeling 38.
� NLVE modeling – Semi-empirical solution
Describing stress relaxation with hyperbolic power function:
Describing stress relaxation with general exp. function by Kohlrausch:
2017.02.16.
44
Phenomenological modeling 39.
� NLVE modeling – Nonlinear model-elements
• St. Venant elements (model of ideally plastic body)
• Model of Coulomb friction (direction dependent but constant resistance)
• Application of nonlinear spring:
• Application of nonlinear viscous element:
� The viscosity is deformation rate dependent (Oswald - de Waele, Bingham, Carreau-
type liquids)
� The viscosity is deformation dependent : Kovács-type direction-dependent viscous
element – Viscosity η increases when the piston goes upward and decreases at
moving downward.
� Pfefferle-type nonlinear viscous element for describing the creep – E.g. in the Kelvin-
Voigt model exchanging the Newton-type element for a Pfefferle element a the
solution obtained at creep stimulus is as follows:
2017.02.16.
23
45
Phenomenological modeling 40.
� NLVE modeling – Modifications of the Boltzmann equation, or in case of a stimulus containing an initial jump the following so called Boltzmann-
Volterra equation (core function K(t) is the derivative of the normalized relaxation modulus):
� Boltzmann-Persoz principle:
� Boltzmann-Frese principle (Assumed: the material properties are the same for tensile or compression load hence only the deformations of odd exponents remain in the
integral series.):
2017.02.16.