Polymer Materials Sciencept.bme.hu/~vas/PhD_Polymer Materials Science... · 2017-02-16 · Behavior of Polymers 2 Polymer Materials Science ... G. Strobl: The Physics of Polymers

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Polymer Materials ScienceBMEGEPT9107, 2+0+0, 3 Credits

Lecturer: Prof. Dr. László Mihály Vas

Budapest University of Technology and EconomicsDepartment of Polymer Engineering

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5. Phenomenological Modeling of the Mechanical Behavior of Polymers

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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.

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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

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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)

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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)

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Mechanical properties Recapitulation

� General scheme of mechanical tests

A – sample, material-operator: Y(t)=A[X](t)

Stimulus Response

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Step function Ramp function Sinusoidal function

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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)

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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)

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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,…)

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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

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� Qualitative modeling – Model-elements 1.

Spring Viscous element

Inertial element St. Venant element

Engineering stress:

Strain:

Load:

Uniaxial tensile load

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� 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

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Tensile Shear

Tensile Shear

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Phenomenological modeling 4.� Models of deformation components

Def. components Model Motion law

Elastic Spring

Remaining Viscose element

Delayed elastic

Kelvin-Voigt element

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� Deformation components – delayed elastic deformation

Kelvin-Voigt element

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� Qualitative modeling – Creep (ATP and WCE)

ATP WCE

LDPE

Burgers model Stuart model

ATP WCE

Creep compliance,:

Burgers response to creep stimulus:

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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

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� 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

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εεεεe

εεεεd

εεεεr

εεεεe

εεεεe

εεεεr

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� 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

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MEASUREMENT:

Model-response:

εεεεe εεεεe

εεεεe

εεεεr

εεεεr

εεεεd εε εε r

εε εε d

εεεεd

εεεεr

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� Qualitative modeling – Stress relaxation of WCE

Standard-Solid model

WCE

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MEASUREMENT:

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� Qualitative modeling – 5-parameter model

Burgers model:

E∞=0

Standard-Solid model:

E2=∞ and/or η2=∞

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Union of Burgers and Standard-Solid models

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� 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

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GM

MODEL POLYMER

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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

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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.)

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� 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

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� Boltzmann’s superposition principle (BS) – Basic LVE

equations in time domain

Response (Y) to arbitrary stimulus (X):

(solution with Laplace-transform)

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� Dynamic qualitative modeling – Kelvin-Voigt model

Complex complianceComplex

Hooke’s law

Loss factor

Complex

stimulus

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� Dynamic qualitative modeling – Maxwell model

Complex modulus

Loss factor

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� 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)

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Phenomenologi-

cal modeling

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� 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

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� 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)

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Time

domain

Spectrum

domainFrequency

domain

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� 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

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� 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)

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� 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)

↔

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� LE relationships for anisotropic material

Orthotropic

(9 indep.

const.)

Monotropic(transversally

isotropic)

(5 indep.

const.)

Tensile modulus of 2D orthotropic

material in direction α

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� 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).

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Arrhenius-type diagrams

Retting, W.: Hanser-Verlag, 1992.

Temperature dependent peak values of relaxation spectra of PVC and PP at ambient

temperature

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Relaxation spectrum of PVC at ambient temperature Relaxation spectrum of PP at ambient temperature

Main

max. Cryst.

max.

Main max.

Secondary max.

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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.

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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

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� 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

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� 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.)

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Polymer

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� 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.

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Some methods for describing the NLVE

behavior

• Semi-empirical, heuristic solutions• Application of nonlinear model-elements•Modification of the Boltzmann equations

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� 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

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� Semi-empirical solutions – Nonlinear creep description of POM (the parameters depend on the time and/or the load)

ηo(σ), EK(σ), τ(t)

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Creep

Recovering

Loading level

Own weight

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� NLVE modeling – Semi-empirical solution

Describing stress relaxation with hyperbolic power function:

Describing stress relaxation with general exp. function by Kohlrausch:

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� 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:

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� 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.):

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Polymer Materials Sciencept.bme.hu/~vas/PhD_Polymer Materials Science... · 2017-02-16 · Behavior of Polymers 2 Polymer Materials Science ... G. Strobl: The Physics of Polymers