Bth Amt Ex 2007d10 Se

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Master's Degree Thesis Master's Degree Thesis ISRN: BTH-AMT-EX--2007/D-10--SE ISRN: BTH-AMT-EX--2007/D-10--SE

Supervisor:

Supervisor: Sharon Sharon Kao-Walter, Kao-Walter, Ph.D. Ph.D. Mech. Mech. Eng.Eng.

Department of Mechanical Engineering Department of Mechanical Engineering

Blekinge Institute of Technology Blekinge Institute of Technology

Karlskrona, Sweden Karlskrona, Sweden 2007 2007

Harikishan Mandalapu

Sandeep Karanamsetty

Parameter Analysis of Creep

Models of PP/CaCo3

Nanocomposites

0 0 55000 0 1100000 0 1155000 0 2200000 0 2255000 0 33000000 0 0 0.02 0.02 0.04 0.04 0.06 0.06 0.08 0.08 0.1 0.1 .. Time(Seconds) Time(Seconds) S S t t r r a a i i n n Experimental Experimental Theoritical Theoritical

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The Parameter analysis of creep

models of PP/CaCo3

nanocomposites

Harikishan Mandalapu

Sandeep Karanamsetty

Department of Mechanical

Department of Mechanical EngineeringEngineering Blekinge Institute of Technology Blekinge Institute of Technology

Karlskrona, Sweden Karlskrona, Sweden

2007 2007

Thesis submitted for completion of Master of Science in Mechanical Thesis submitted for completion of Master of Science in Mechanical Engineering with emphasis on Structural Mechanics at the Department of Engineering with emphasis on Structural Mechanics at the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden.

Sweden. Abstract: Abstract:

The present report is about the parameter analysis of creep models o The present report is about the parameter analysis of creep models o Nanocomp

Nanocomposites osites (Isotactic (Isotactic polypropylpolypropylene ene and and CaCo3).The CaCo3).The parametricparametric analysis of the nanocomposites under creep was carried out, and the analysis of the nanocomposites under creep was carried out, and the parameters

parameters related related to to creep creep model model are are determineddetermined by by comparing comparing to to thethe experimental results. The influence of these parameters on the creep was experimental results. The influence of these parameters on the creep was studied. Using commercially available software ABAQUS, Finite studied. Using commercially available software ABAQUS, Finite Element Calculations were done for elastic and creep conditions. The Element Calculations were done for elastic and creep conditions. The results obtained from theoretical analysis were verified with the results obtained from theoretical analysis were verified with the Experimental Results. Also Abaqus results are compared with the Experimental Results. Also Abaqus results are compared with the Experimental results. Experimental results were obtained from the Experimental results. Experimental results were obtained from the experiments conducted by the Department of Chemistry, Huazhong experiments conducted by the Department of Chemistry, Huazhong University of Science and Technology, China and by Department o University of Science and Technology, China and by Department o Advanced Materials and Technology, College of Engineering, Peking Advanced Materials and Technology, College of Engineering, Peking

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Acknowledgements

This thesis work is carried out at the Department of Mechanical This thesis work is carried out at the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, under Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, under the supervision of

the supervision of Dr.Sharon Kao-Walter.Dr.Sharon Kao-Walter.

We consider it is our duty to acknowledge the people without whose We consider it is our duty to acknowledge the people without whose assistance this work could not have been undertaken at all. We record our assistance this work could not have been undertaken at all. We record our deep sense of gratitude to Dr.Sharon Kao-Walter, Department of deep sense of gratitude to Dr.Sharon Kao-Walter, Department of Mechanical Engineering, Blekinge Institute of Technology, Sweden, for Mechanical Engineering, Blekinge Institute of Technology, Sweden, for herher invaluable guidance and kind cooperation extended by her throughout the invaluable guidance and kind cooperation extended by her throughout the course of this work.

course of this work.

Our thanks go to the almighty God for giving us the opportunity to be able Our thanks go to the almighty God for giving us the opportunity to be able to complete this project.

to complete this project.

We wish to express our sincere appreciation to Tech. Etienne Mfoumou for We wish to express our sincere appreciation to Tech. Etienne Mfoumou for his help regarding ABAQUS analysis.

his help regarding ABAQUS analysis.

We also thank all our faculty members and our classmates for their We also thank all our faculty members and our classmates for their encouragement, discussion, comments and many innovative ideas in encouragement, discussion, comments and many innovative ideas in carrying out this work.

carrying out this work.

Finally, we would like to dedicate this work to our parents in India for their Finally, we would like to dedicate this work to our parents in India for their moral support and inspiration.

moral support and inspiration.

Karlskrona, April 2007 Karlskrona, April 2007 Harikishan Mandalapu, Harikishan Mandalapu, Sandeep Karanamsetty. Sandeep Karanamsetty.

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1 Notations

E

E Young’s Young’s Modulus Modulus [Mpa][Mpa]

σ

σ Stress Stress [Mpa][Mpa]

ε

ε StrainStrain

cc ε

ε Creep Creep StrainStrain

cc ε

ε

&&

Strain Strain raterate

ν

ν Poisson’s Poisson’s ratioratio

D

D Constitutive Constitutive matrixmatrix t

t Time Time [Seconds][Seconds]

R

R Universal Universal gas gas constant constant [Cal/(mol)(K)][Cal/(mol)(K)] A

A Material ConstantMaterial Constant n

n Material ConstantMaterial Constant m

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Abbreviations

PP

PP _Isotactic_{Isotactic polypropy}_{polypropylene}_lene PN

PN _{poly-oxyethylene}_{poly-oxyethylene} HDPE

HDPE _{High density}_{High density polyethylene}_polyethylene CaCo

CaCo₃₃ _{Calcium Carbonate}_{Calcium Carbonate} LVDT

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2 Background

Nanocomp

Nanocomposites osites refer refer to to materials materials consisting consisting of of at at least least two two phases phases withwith one dispersed in another that is

one dispersed in another that is called matrix and forms a three-dimensionalcalled matrix and forms a three-dimensional network. Nanocomposites have been studied extensively mainly for network. Nanocomposites have been studied extensively mainly for improved physical properties.

improved physical properties.

Isotactic polypropylene (PP) is used as a most common plastic for the Isotactic polypropylene (PP) is used as a most common plastic for the manufacturing of automotive parts, home appliances, for construction manufacturing of automotive parts, home appliances, for construction process,

process, etc etc [3].But [3].But this this PP PP is is notch notch sensitive sensitive and and brittle brittle under under severesevere conditions of deformation, such as at

conditions of deformation, such as at low temperatures or high impact rates,low temperatures or high impact rates, which makes limited its wider range of usage for the manufacturing which makes limited its wider range of usage for the manufacturing processes. Blending PP with

processes. Blending PP with rigid inorganic particles rigid inorganic particles such as such as CaCO3 is CaCO3 is thethe best

best way way to to improve improve the the stiffness stiffness and and toughness toughness of of the the PP.DispersionPP.Dispersion quality of CaCO3 particles played a crucial role in toughening efficiency. quality of CaCO3 particles played a crucial role in toughening efficiency. The nanocomposites composed of isotactic PP and CaCo

The nanocomposites composed of isotactic PP and CaCo33 nanoparticlesnanoparticles

were fabricated by melt extrusion [3]. A nonionic modifier, were fabricated by melt extrusion [3]. A nonionic modifier, poly-oxyethylene (PN), was added to the PP/CaCo

oxyethylene (PN), was added to the PP/CaCo33 mixture by dry mixing mixture by dry mixing

before

before melt melt extrusion. extrusion. The The dispersion dispersion of of CaCoCaCo33 particles was greatly particles was greatly

improved by this PN modifier [2]. Isotactic polypropylene (PP) and improved by this PN modifier [2]. Isotactic polypropylene (PP) and calcium carbonate CaCo

calcium carbonate CaCo33 nanocomposites mixture have various nanocomposites mixture have various

applications in automotive, construction and in other fields due to their high applications in automotive, construction and in other fields due to their high impact strengths and toughness. Different kinds of nanocomposites were impact strengths and toughness. Different kinds of nanocomposites were tested to study the creep mechanism of components. An experimental setup tested to study the creep mechanism of components. An experimental setup was made and tensile creep tests were carried out on the specimen with was made and tensile creep tests were carried out on the specimen with different composite compositions each time.

different composite compositions each time.

The aim of our thesis is to analyze the better creep model by studying the The aim of our thesis is to analyze the better creep model by studying the parameters.

parameters. The The material material parameters parameters are are determined determined by by comparing comparing to to aa creep model from the experimental data and are verified along with the creep model from the experimental data and are verified along with the values estimated from theoretical formula. The analysis is

values estimated from theoretical formula. The analysis is done consideringdone considering three methods with a defined creep model. We try to match the three methods with a defined creep model. We try to match the experimental results for the creep behavior with the results obtained from experimental results for the creep behavior with the results obtained from the theoretical formula. ABAQUS and Matlab were used to perform the the theoretical formula. ABAQUS and Matlab were used to perform the necessary finite element analysis and mathematical calculations. An necessary finite element analysis and mathematical calculations. An overview of creep behavior is observed from the results.

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3 Introduction to Creep

Creep takes place i

Creep takes place in Engineering materials and structures manifested by then Engineering materials and structures manifested by the accumulation of plastic deformation over prolonged time periods under accumulation of plastic deformation over prolonged time periods under steady or variable

steady or variable loading conditions [11].loading conditions [11].

At elevated temperatures and at constant stress or load many materials At elevated temperatures and at constant stress or load many materials continue to deform at slow rate. This behavior is also called as

continue to deform at slow rate. This behavior is also called as creep.creep. InIn other words high temperature progressive deformation of a material at other words high temperature progressive deformation of a material at constant stress is also called as

constant stress is also called as creepcreep..

Creep deformation does not happen suddenly.

Creep deformation does not happen suddenly. Creep Creep is the term used to is the term used to describe the tendency of a material to move or to deform permanently to describe the tendency of a material to move or to deform permanently to relieve stresses.

relieve stresses.

There are different stages of creep. Creep can be subdivided into three There are different stages of creep. Creep can be subdivided into three categories primary, steady state creep and t

categories primary, steady state creep and tertiary.ertiary.

The following figure1 illustrates the different stages of creep in a simple The following figure1 illustrates the different stages of creep in a simple way,

way,

Figure1:

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The long-term behavior of modern structures, whose final static The long-term behavior of modern structures, whose final static configuration is frequently the result of a complex sequence of phases of configuration is frequently the result of a complex sequence of phases of loading and restraint conditions, are influenced largely by creep. Creep loading and restraint conditions, are influenced largely by creep. Creep substantially modifies the initial stress and strain patterns, increasing the substantially modifies the initial stress and strain patterns, increasing the deformations induced by sustained loads, Relaxing the stresses due to deformations induced by sustained loads, Relaxing the stresses due to sustained imposed deformations, (artificially introduced, e.g. by jacking, or sustained imposed deformations, (artificially introduced, e.g. by jacking, or due to natural causes like shrinkage or settlements) activating the delayed due to natural causes like shrinkage or settlements) activating the delayed additional restraints. A special emphasis is given to the compact additional restraints. A special emphasis is given to the compact formulations derived directly from the fundamental theorems of the theory formulations derived directly from the fundamental theorems of the theory of linear viscoelasticity for nano

of linear viscoelasticity for nano materials.materials.

Creep tests are carried out on a specimen loaded [9], e.g., in tension or Creep tests are carried out on a specimen loaded [9], e.g., in tension or compression, usually at constant load, inside a furnace which is maintained compression, usually at constant load, inside a furnace which is maintained at a constant temperature T.The extension of the specimen is measured as a at a constant temperature T.The extension of the specimen is measured as a function of time. A typical

function of time. A typical creep curvecreep curve for metals, polymers, and ceramicsfor metals, polymers, and ceramics is represented in Fig2.

is represented in Fig2.

The response of the specimen loaded by

The response of the specimen loaded by σ σ ₀₀at timeat time t t

=

00can be divided intocan be divided into

elastic

elastic andand plastic plastic part as part as ), ), ,, (( )) (( // ₀₀ 0 0 0 0 σ σ E E T T ε ε p p σ σ T T ε ε

=

+

(3.1)(3.1) Where

Where E E ((T T ))the modulus of Elasticity. The creep strain in Fig2 is can thenthe modulus of Elasticity. The creep strain in Fig2 is can then be expressed

be expressed according toaccording to k k cc ε ε t t ε ε α α t t ε ε 0 0 )) ((

−

=

_(3.2)_(3.2)

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Figure2: Stages in creep [10] Figure2: Stages in creep [10] Where

Where k k < 1 in the < 1 in the primary, k primary, k = 1 in the = 1 in the secondary,secondary, andand k >1k >1 in the in the tertiarytertiary creep stage. These terms

creep stage. These terms correspond to a decreasing,constant,and increasingcorrespond to a decreasing,constant,and increasing strainrate,respectively, and were introduced by ANDRADE(1910).These strainrate,respectively, and were introduced by ANDRADE(1910).These three creep stages are often called

three creep stages are often called transient creep, steady creep,transient creep, steady creep, and and accelerating creep;

accelerating creep; respectively. respectively.

The results (3.1) and (3.2) from the creep test justify a classification of The results (3.1) and (3.2) from the creep test justify a classification of material behavior in

material behavior in threethree disciplines: elasticity, plasticity, and creepdisciplines: elasticity, plasticity, and creep mechanics.

mechanics.

Due to a proposal of HAUPT (2000) one can also distinguish four theories Due to a proposal of HAUPT (2000) one can also distinguish four theories of material behavior as follows:

of material behavior as follows:

••

_{The theory of}_{The theory of elasticity}_{elasticity is concerned with the}_{is concerned with the rate-independent}_{rate-independent}

behavior

behavior without hysteresis.without hysteresis.

••

_{The theory of}_{The theory of plasticity}_plasticity _{specifies the}_{specifies the rate-independent}_{rate-independent behavior}_{behavior with}_with

hysteresis. hysteresis.

••

_{The theory of}_{The theory of viscoelasticity}_{viscoelasticity} _{describes the}_{describes the rate-dependent}_{rate-dependent behavior}_behavior

without equilibrium hysteresis. without equilibrium hysteresis.

••

_{The theory of}_{The theory of viscoplasticity}_{viscoplasticity} _{is devoted to the rate-dependent}_{is devoted to the} _{rate-dependent}

behavior

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The creep behavior exists in two of the above listed categories, namely in The creep behavior exists in two of the above listed categories, namely in the theories of

the theories of viscoelasticityviscoelasticity andand viscoplasticity.viscoplasticity.

3.1 Primary Creep

Primary creep, Stag

Primary creep, Stage I, is a period of decreasing cre I, is a period of decreasing creep rate [9]. eep rate [9]. PrimaryPrimary creep is a period

creep is a period of primarof primarily transient creep. ily transient creep. During this During this periodperiod deformation takes place and the resistance to creep increases until stage II deformation takes place and the resistance to creep increases until stage II For constant-temperature creep behavior,

For constant-temperature creep behavior,ε ε cc is thus given by is thus given by

)) ,, (( t t f f cc σ σ ε ε

=

(3.3) (3.3) Several mathematical forms exist to represent the function

Several mathematical forms exist to represent the function f f ((σ σ ,,t t ););one ofone of

these is the Norton-Baily creep law: these is the Norton-Baily creep law:

,, m m n n cc A Aσ σ t t ε ε

=

(3.4)(3.4)

Where the parameters

Where the parameters A A,,nn,,mm depend on the material depend on the material and temperature. Theyand temperature. They can be determined in a uniaxial creep test.

can be determined in a uniaxial creep test. If the stress

If the stress σ σ in (3.4) is assumed to be constant the creep ratein (3.4) is assumed to be constant the creep rate

cc d d

≈

ε ε

&&

isis given by given by .. 1 1 − −

=

nn mm cc Am Amσ σ t t ε ε

&&

(3.5)(3.5)

Inserting the time

Inserting the time t t from (3.4) into (3.5), we arrive at the relation from (3.4) into (3.5), we arrive at the relation

( ( )) m m m m cc m m n n m m cc mAmA 1 1 1 1 −−

=

σ σ ε ε ε ε

&&

(3.6)(3.6)

Which characterizes the

Which characterizes the strain-hardening-theory,strain-hardening-theory, i.e., this strain rate i.e., this strain rate equation (3.6) includes

equation (3.6) includes stressstress and and strainstrain as variables. In contrast to (3.6) theas variables. In contrast to (3.6) the strain rate equation (3.5) contains

strain rate equation (3.5) contains stressstress andand timetime as variables and isas variables and is therefore called the

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3.2 Secondary Creep

Secondary creep, Stage II, is a period of roughly constant creep rate [9]. Secondary creep, Stage II, is a period of roughly constant creep rate [9]. Stage II

Stage II is referred to is referred to as steady as steady state creep. state creep. Creep deformations of theCreep deformations of the secondary stage are large and of similar character to pure plastic secondary stage are large and of similar character to pure plastic deformations.

deformations.

3.3 Tertiary Creep

Tertiary creep, Stage III, occurs when there is a reduction in cross sectional Tertiary creep, Stage III, occurs when there is a reduction in cross sectional area due to necking or effective reduction in area due to internal void area due to necking or effective reduction in area due to internal void formation [9]. The tertiary creep phase is accompanied by the formation of formation [9]. The tertiary creep phase is accompanied by the formation of microscopic cracks on the grain boundaries, so that damage-accumulation microscopic cracks on the grain boundaries, so that damage-accumulation occurs. In some cases voids are caused by a given stress history and, occurs. In some cases voids are caused by a given stress history and, therefore, they are distributed

therefore, they are distributed anisotropically among the grain boundaries.anisotropically among the grain boundaries.

3.4 Creep under Variable Loading

Norton-Bialy

Norton-Bialys’ s’ creep law creep law as as expressed by expressed by the equation the equation (3.3) has (3.3) has been usedbeen used extensively in analyzing creep problems [4]. The simplicity of this law extensively in analyzing creep problems [4]. The simplicity of this law helps in arriving at analytical solutions with acceptable accuracy for creep helps in arriving at analytical solutions with acceptable accuracy for creep problems

problems involving involving steady steady loadings. loadings. For For situations situations when when the the appliedapplied stresses vary with time, either continuously or according to step changes, stresses vary with time, either continuously or according to step changes, the use of Norton-Bialys’ law becomes inaccurate since the phases of the use of Norton-Bialys’ law becomes inaccurate since the phases of primary creep cannot be

primary creep cannot be neglected at neglected at every load change. every load change. To overcome this,To overcome this, the idea that equation (3.3) expresses the creep rate as a function of stress the idea that equation (3.3) expresses the creep rate as a function of stress

σ

σ and current time t, and current time t,

i.e., )

i.e., ε ε

&&

cc

=

f f ((σ σ ,,t t ) has has been been replaced replaced by by considering considering ε ε

&&

cc

=

f f ((σ σ ,,ε ε cc)).The.The

derivation of such functions is as

derivation of such functions is as follows:follows:

The primary and secondary creep strains are expressed [4] by Equation as The primary and secondary creep strains are expressed [4] by Equation as follows: follows: m m n n cc t t A Aσ σ ε ε

=

(3.7)(3.7) Where n >> 1 and m

Where n >> 1 and m

≤

11.The time derivative gives.The time derivative gives

− −

=

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This derivation results in the

This derivation results in the known Time-Hardeknown Time-Hardening rule, where creepning rule, where creep strain rate is expressed as a function of the stress

strain rate is expressed as a function of the stress σ σ and time t. and time t.

Another formulation known as “strain-hardening” may be

Another formulation known as “strain-hardening” may be derived from thederived from the above equation (3.8) eliminating time t,

above equation (3.8) eliminating time t, as given by Equation (3.7), namelyas given by Equation (3.7), namely

m m n n cc A A t t 1 1

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

σ σ ε ε (3.9) (3.9)

Substitution into Equation (3.8) yields Substitution into Equation (3.8) yields

m m m m n n cc n n A A Am Am // 1 1 )) (( − −

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

σ σ ε ε σ σ ε ε

&&

or or m m m m cc m m n n m m cc m m A A11// (( )) // (( ))(( −−11))//

=

σ σ ε ε ε ε

&&

(3.10)(3.10)

Equation (3.10) expresses the creep rate

Equation (3.10) expresses the creep rate ε ε

&&

cc as a function of the stress as a function of the stress σ σ

and the current creep strain

and the current creep strainε ε cc.Experiments indicate that the strain-.Experiments indicate that the

strain-hardening formulation is to be favored over time-strain-hardening formulation. hardening formulation is to be favored over time-hardening formulation. However, nothing the large scatter in creep data, the use of the simple law However, nothing the large scatter in creep data, the use of the simple law of time hardening becomes justifiable in deriving analytical solutions. of time hardening becomes justifiable in deriving analytical solutions. Evidently, and strain hardening offers no difficulty in seeking numerical Evidently, and strain hardening offers no difficulty in seeking numerical solutions. Both formulations as given above are applicable only for solutions. Both formulations as given above are applicable only for situations where no stress reversals occur, a situation where modified rules situations where no stress reversals occur, a situation where modified rules have to be used .Also both formulations do not account well for the have to be used .Also both formulations do not account well for the important phenomenon of creep recovery due to unloading or variable important phenomenon of creep recovery due to unloading or variable cyclic loading.

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4 Experimental Work

4.1 Introduction

The nanocomposites composed of isotactic PP and CaCo

The nanocomposites composed of isotactic PP and CaCo33 nanoparticles [2] nanoparticles [2]

are used to reinforce thermoplastic polymers which have wide applications are used to reinforce thermoplastic polymers which have wide applications in many areas. The addition of these nanocomposites to the polymers in many areas. The addition of these nanocomposites to the polymers increases their toughness and stiffness. The major drawback of these increases their toughness and stiffness. The major drawback of these polymers

polymers is is creep creep which which occurs occurs at at stresses stresses below below the the yield yield stress stress of of thethe polymer materials.

polymer materials. NanocomNanocomposites with posites with combination of combination of surface modifierssurface modifiers such as poly-oxyethylene (PN) are good for obtaining uniform dispersion in such as poly-oxyethylene (PN) are good for obtaining uniform dispersion in the polymer matrix and have better mechanical behavior [3] than the the polymer matrix and have better mechanical behavior [3] than the original polymer matrix materials.

original polymer matrix materials.

The present work is to analyze the creep behavior of these composites with The present work is to analyze the creep behavior of these composites with different PN content.

different PN content.

4.2 Tensile test for Creep Measurement

The experimental setup used for the testing the creep behavior [2] is shown The experimental setup used for the testing the creep behavior [2] is shown in the Figure (4) below.

in the Figure (4) below.

Figure3: Schematic presentation of Test setup used for Experimental Figure3: Schematic presentation of Test setup used for Experimental

analysis [2] analysis [2] CPU CPU Weight Weight Carrier Carrier 50mm 50mm Specimen Specimen LVDT

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Specimen

Specimen PP PP CaCoCaCo33 PN PN Young’sYoung’s

Modulus Modulus Poisson’s Poisson’s ratio ratio Material Material Constant Constant Code Code (Wt (Wt %) %) (Wt (Wt %) %) (Wt(Wt %) %) (Gpa) (Gpa) ν ν nn PP PP 100 100 0 0 0 0 1.21 1.21 0.34 0.34 10.2810.28 PPC-0.75 PPC-0.75 84.25 84.25 15 15 0.75 0.75 1.55 1.55 0.36 0.36 8.718.71 PPC-1.5 PPC-1.5 83.5 83.5 15 15 1.5 1.5 1.25 1.25 0.34 0.34 11.7611.76 PPC-2.25 PPC-2.25 82.75 82.75 15 15 2.25 2.25 1.31 1.31 0.32 0.32 12.2012.20 Table1:

Table1: Combinations of composites taken for experimentsCombinations of composites taken for experiments

The densities of PP and CaCO3 are 0.96 gm/cm

The densities of PP and CaCO3 are 0.96 gm/cm33 and 2.55 gm/cm and 2.55 gm/cm33 respectively.

respectively.

The device used to carry out the tensile creep test [2] consists of a LVDT The device used to carry out the tensile creep test [2] consists of a LVDT with precession of 0.02mm, control box, computer, weights and carrier. with precession of 0.02mm, control box, computer, weights and carrier. The tests were done at four different stresses 12.33MPa, 17.33MPa, The tests were done at four different stresses 12.33MPa, 17.33MPa, 20.67MPa, 24MPa respectively. The tests were carried

20.67MPa, 24MPa respectively. The tests were carried out in the out in the laboratorylaboratory at a controlled temperature of 22

at a controlled temperature of 2200C with variation ofC with variation of

±

2200C. The slightC. The slight change in the temperatures is negligible on the tensile properties of the change in the temperatures is negligible on the tensile properties of the material. The dimensions of the

material. The dimensions of the specimen tested were 50x30x10mmspecimen tested were 50x30x10mm33..

Generally, the whole creep process is divided in two three phases like Generally, the whole creep process is divided in two three phases like primary, secondar

primary, secondary steady state and teritiary.Though the creep rate iy steady state and teritiary.Though the creep rate is rathers rather high in the primary stage than in the secondary steady stage, the creep high in the primary stage than in the secondary steady stage, the creep strain is not important compared to the total deformation because the rate strain is not important compared to the total deformation because the rate slows down continuously and the duration is limited.

slows down continuously and the duration is limited.

In this work, we are only interested in the second stage which occupies In this work, we are only interested in the second stage which occupies longer duration and the creep rate remains constant. So, the steady stage longer duration and the creep rate remains constant. So, the steady stage influences the dimensional stability of the structure. In the tertiary stage influences the dimensional stability of the structure. In the tertiary stage there is increase in the creep rate which causes final failure in short time. there is increase in the creep rate which causes final failure in short time. So, the present work concentrates on the first two stages of creep to study So, the present work concentrates on the first two stages of creep to study the effect of creep deformation and creep rate of the steady stage. The the effect of creep deformation and creep rate of the steady stage. The tensile test is carried for duration of four hours for loads 12.33MPa, tensile test is carried for duration of four hours for loads 12.33MPa, 17.33MPa, and 20.67MPa respectively. But for load 24MPa the failure 17.33MPa, and 20.67MPa respectively. But for load 24MPa the failure occurred within one hour.

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4.3 Experimental Results and Parameter Analysis

Curves are plotted for creep strain versus time for different stresses for the Curves are plotted for creep strain versus time for different stresses for the nanocomposites with different compositions of PP, CaCo

nanocomposites with different compositions of PP, CaCo33, PN.The curves, PN.The curves

are as shown below [2]. are as shown below [2].

0 0 2200000 0 4400000 0 6600000 0 8800000 0 101000000 0 112200000 0 114400000 0 1166000000 0.008 0.008 0.01 0.01 0.012 0.012 0.014 0.014 0.016 0.016 0.018 0.018 0.02 0.02

Time(S)

S S t t r r a a i i n n

Strain Vs Time under 12.33Mpa Experimental

PP0 PP0 PPC-0.75 PPC-0.75 PPC-1.75 PPC-1.75 PPC-2.25 PPC-2.25

Figure 4: Strain versus time under 12.33MPa [2] Figure 4: Strain versus time under 12.33MPa [2]

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0 0 5500000 0 110000000 0 1155000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035

Time(S)

Strain Vs Time under 17.33Mpa Experimental

PP PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25 PPC2.25 Figure 5:

Figure 5: Strain versus time under 17.33MPa [2]Strain versus time under 17.33MPa [2]

0 0 2200000 0 4400000 0 6600000 0 8800000 0 101000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09 0.09 0.1 0.1 Time(S) Time(S) S S t t r r a a i i n n

Strain Vs Time under 20.67Mpa Experimental Strain Vs Time under 20.67Mpa Experimental

PP PP 00 PPC-0.75 PPC-0.75 PPC-1.75 PPC-1.75 PPC-2.25 PPC-2.25 Figure 6:

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0 0 55000 0 1100000 0 1515000 0 2200000 0 2255000 0 3300000 0 3355000 0 4400000 0 44550000 0 0 0.02 0.02 0.04 0.04 0.06 0.06 0.08 0.08 0.1 0.1 0.12 0.12 0.14 0.14

Time

Strain Vs Time Under 24Mpa Experimental

PP PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25 PPC2.25

Figure7: Strain versus time under 24MPa [2] Figure7: Strain versus time under 24MPa [2]

The creep rate is calculated from the values obtained from the experimental The creep rate is calculated from the values obtained from the experimental data.

data.

We have, We have,

Total strain = elastic strain +

Total strain = elastic strain + creep straincreep strain

(4.1) (4.1) The creep rate is defined by the formula,

The creep rate is defined by the formula, 1 1 − −

=

nn mm cc t t Am Amσ σ ε ε

&&

_(4.2)_(4.2) Now

Now the the creep creep rate rate for for different different composites composites is is plotted plotted for for the the variousvarious stresses. They are shown in the fig.8, 9 below. Also Logarithmic creep vs. stresses. They are shown in the fig.8, 9 below. Also Logarithmic creep vs. the logarithmic stress values are

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1 12 2 114 4 116 6 118 8 220 0 222 2 2244 0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3x 10x 10 -5 -5 Stress(Mpa) Stress(Mpa) S S t t r r a a i i n n r r a a t t e e PP PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25 PPC2.25

Figure 8: strain rate of steady stage versus stress of tested materials Figure 8: strain rate of steady stage versus stress of tested materials

1 1 11..005 5 11..1 1 11..115 5 11..2 2 11..225 5 11..3 3 11..335 5 11..44 -7 -7 -6.5 -6.5 -6 -6 -5.5 -5.5 -5 -5 -4.5 -4.5 Log(Stress) Log(Stress) L L o o g g ( ( C C r r e e e e p p - - r r a a t t e e ) ) PP PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25 PPC2.25

Figure 9: Creep rate of

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4.4 Curve fitting with three different methods

The analysis was carried out

The analysis was carried out considering three different methods.considering three different methods.

In method 1 we assume the creep model to have the formula of In method 1 we assume the creep model to have the formula of

m m n n cc t t A Aσ σ ε ε

=

We proceed to determine the material parameters for the creep model from We proceed to determine the material parameters for the creep model from the experimental data. Also we assume the parameter m to be equal to one. the experimental data. Also we assume the parameter m to be equal to one. We tried to estimate the creep strain rate and the total strain with the We tried to estimate the creep strain rate and the total strain with the calculated parameters. The results obtained are compared with the calculated parameters. The results obtained are compared with the experimental results. Analysis of the above comparison is done

experimental results. Analysis of the above comparison is done to study theto study the creep behavior in this particular method.

creep behavior in this particular method. In method 2 we have considered the

In method 2 we have considered the same creep model similar to same creep model similar to method 1.method 1. Here we have assumed the parameter ‘m’ to be constant for each material Here we have assumed the parameter ‘m’ to be constant for each material independent of stress and with varying ‘A’. We also determine the material independent of stress and with varying ‘A’. We also determine the material parameters

parameters in in this this method method and and also also analysis analysis was was carried carried out out similar similar toto method 1.From the comparison of the experimental and analytical results method 1.From the comparison of the experimental and analytical results conclusions were drawn.

conclusions were drawn.

The method 3 was carried out assuming the same creep model similar to t The method 3 was carried out assuming the same creep model similar to thehe above two methods. Here we have assumed that the material parameters above two methods. Here we have assumed that the material parameters ‘A’ and ‘m’ vary at each stress for different materials. Analysis was done ‘A’ and ‘m’ vary at each stress for different materials. Analysis was done for the creep model using the same approach similar to method 1 and for the creep model using the same approach similar to method 1 and method 2.

method 2.

4.4.1 Method 1 4.4.1 Method 1

We know the creep formula as, We know the creep formula as,

=

_(4.3)_(4.3)

Since the secondary creep rate is considered in the experiment, and we Since the secondary creep rate is considered in the experiment, and we know

know ε ε cc is almost linearly dependent on time and when the exponent is almost linearly dependent on time and when the exponent

‘m=1’

‘m=1’ in the in the above expression.above expression. The values of ‘A’ can

The values of ‘A’ can be calculated for four different stresses by comparingbe calculated for four different stresses by comparing to the experimental results and the average value is considered, and the to the experimental results and the average value is considered, and the same is substituted in the above formula. We try to estimate the parameter same is substituted in the above formula. We try to estimate the parameter ‘A’

‘A’ assuming the other parameter assuming the other parameter ‘m’=1‘m’=1.The values of ‘A’ are estimated.The values of ‘A’ are estimated from the above logarithmic Creep rate

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From fig.14 it can be found that there is a huge deviation in the results From fig.14 it can be found that there is a huge deviation in the results between experimen

between experimental and theoretical when we use Average ‘A’ value. Onetal and theoretical when we use Average ‘A’ value. One of the possible reasons could be the error in the calculation of ‘A’ value of the possible reasons could be the error in the calculation of ‘A’ value from the experimental data. Now we tried to approach the ‘A ‘values by from the experimental data. Now we tried to approach the ‘A ‘values by applying suitable numerical methods. The new values are substituted once applying suitable numerical methods. The new values are substituted once again. The calculations are performed by Matlab. The value of ‘A’ is again. The calculations are performed by Matlab. The value of ‘A’ is obtained by iterative calculations. The results from theoretical formula are obtained by iterative calculations. The results from theoretical formula are verified with the experimental results. Results are plotted in Matlab as verified with the experimental results. Results are plotted in Matlab as shown below,

shown below,

The results are tabulated as shown below: The results are tabulated as shown below:

PP PP PPC-0.75 PPC-0.75 PPC-1.5 PPC-1.5 PPC-2.25PPC-2.25 n 10.28 n 10.28 8.71 8.71 11.76 11.76 12.2012.20 m m 1 1 1 1 1 1 11 Average Average ‘A’ ‘A’ 3.338 3.338

×

1010−−1919 19.746 19.746

×

1010−−1818 6.993 6.993

×

1010−−2121 30.7230.72

×

1010−−2222 Approached Approached ‘A’ ‘A’ 0.8011 0.8011

×

1010−−1919 0.9380.938

×

1010−−1717 1.17851.1785

×

1010−−2121 4.3624.362

×

1010−−2222

Table2: Results from method 1 Table2: Results from method 1

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0 0 55 1100 1155 2200 2255 0 0 1 1 2 2 3 3 4 4 5 5 6 6x x 1010 -5 -5 Stress Stress C C r r e e e e p p - - s s t t r r a a i i n n r r - - a a t t e e For PP For PP Approached Approached Experimental Experimental Av Averageerage

Figure10: Creep strain rate Vs Stress for PP Figure10: Creep strain rate Vs Stress for PP

0 0 55 1100 1155 2200 2255 0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 x 10x 10 -5 -5 Stress Stress C C r r e e e e p p - - s s t t r r a a i i n n r r - - a a t t e e For PPC0.75 For PPC0.75 Approached Approached Experimental Experimental Av Averageerage

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0 0 55 1100 1155 2200 2255 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 x x 1010 -4 -4 Stress Stress C C r r e e e e p p - - s s t t r r a a i i n n r r - - a a t t e e For PPC1.5 For PPC1.5 Approac Approac hedhed

Experimental Experimental Average Average

Figure12: Creep strain rate Vs Stress for PPC1.5 Figure12: Creep strain rate Vs Stress for PPC1.5

0 0 55 1100 1155 2200 2255 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 x x 1010 -4 -4 Stress Stress C C r r e e e e p p - - s s t t r r a a i i n n r r - - a a t t e e For PP

For PP C2.25 BC2.25 B efefore Iteraore Iterativtive Ae A pproximation of Approximation of A

Ap

Ap proacproac hedhed Experimental Experimental Average Average

Figure13:

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From the results above it can be

From the results above it can be understoounderstood that the parameter ‘m’ = 1 d that the parameter ‘m’ = 1 doesdoes not give desired experimental results, and we

not give desired experimental results, and we proceed to method 2.proceed to method 2. We have the total strain given by

We have the total strain given by

t t A A t t dt dt d d _n_n o o cc o o ε ε ((σ σ )) ε ε ε ε ε ε

⎟⎟⎟⎟

=

+

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+

=

(4.4) (4.4) Now strain v

Now strain vs. time results are plotted as shs. time results are plotted as shown in Fig.14own in Fig.14-18,-18,

0 0 2200000 0 4400000 0 6600000 0 8800000 0 110000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.011 0.011 0.012 0.012 0.013 0.013 0.014 0.014 0.015 0.015 0.016 0.016 0.017 0.017 0.018 0.018 0.019 0.019 0.02 0.02 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

CASE1:Strain Vs Time for PP under 12.33Mpa CASE1:Strain Vs Time for PP under 12.33Mpa

Experimental Experimental Theoritical Theoritical

Figure14:

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0 0 2200000 0 4400000 0 6600000 0 8800000 0 110000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

Figure15:

Figure15: Strain Vs time for PP under 17.33MpaStrain Vs time for PP under 17.33Mpa

0 0 2200000 0 4400000 0 6600000 0 8800000 0 101000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.012 0.012 0.014 0.014 0.016 0.016 0.018 0.018 0.02 0.02 0.022 0.022 0.024 0.024 0.026 0.026 0.028 0.028 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

CASE1:Strain Vs Time for PPC0.75 under 17.33Mpa CASE1:Strain Vs Time for PPC0.75 under 17.33Mpa

Figure16:

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0 0 2200000 0 4400000 0 6600000 0 8800000 0 101000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 Time(Seconds) Time(Seconds) S S t t a r r a i i n n

Figure17: Strain Vs time for PPC01.5 under 17.33Mpa Figure17: Strain Vs time for PPC01.5 under 17.33Mpa

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From the fig.14 to 18 above we can see the strain vs. time is linear which is From the fig.14 to 18 above we can see the strain vs. time is linear which is not correct according to the experimental results. This may be due to the not correct according to the experimental results. This may be due to the assumptions made, and also the creep model we assumed in our case may assumptions made, and also the creep model we assumed in our case may not be appropriate. So we

not be appropriate. So we proceed to method 2.proceed to method 2. 4.4.2 Method 2

4.4.2 Method 2

In this method we assume that the creep stain rate

In this method we assume that the creep stain rate ε ε

&&

cc defined defined by by thethe

creep strain rate expressed as a function of the stress

creep strain rate expressed as a function of the stress σ σ and time t and time t

(i.e.)

(i.e.)ε ε

&&

cc

=

Am Amσ σ nnt t mm−−11 .The parameter ‘m’ remaining constant during the .The parameter ‘m’ remaining constant during the

creep stage, with varying ‘A’ creep stage, with varying ‘A’

Since the secondary creep rate has much significance in the design fields, Since the secondary creep rate has much significance in the design fields, we consider secondary creep here. From the Norton-Bialy’s creep laws: we consider secondary creep here. From the Norton-Bialy’s creep laws:

=

_(4.5)_(4.5) Now

Now equation (4) expresses the creep rate as a function of stressequation (4) expresses the creep rate as a function of stress σ σ and and

current time t, current time t,

i.e., )

i.e., ε ε

&&

cc

=

f f ((σ σ ,,t t ) has has been been replaced replaced by by considering considering ε ε

&&

cc

=

f f ((σ σ ,,ε ε cc)).The.The

derivation of such functions is as

derivation of such functions is as follows.follows. The time derivative gives

The time derivative gives 1 1 − −

=

&&

_(4.6)_(4.6)

This derivation results in the known Time-Hardening rule, where creep This derivation results in the known Time-Hardening rule, where creep strain rate is expressed as a function of the stress

strain rate is expressed as a function of the stress σ σ and time t. and time t.

We solve them to find out the value for ‘m’. The values of ‘A’ are We solve them to find out the value for ‘m’. The values of ‘A’ are calculated by substituting the values of ‘m’ in above equations. The same is calculated by substituting the values of ‘m’ in above equations. The same is tried at four different stresses. We arrive at four different ‘A’ values. tried at four different stresses. We arrive at four different ‘A’ values. Similar approach as the method 1 is

Similar approach as the method 1 is done here for ‘A’ value. The results aredone here for ‘A’ value. The results are plotted for exp

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The results are tabulated as

The results are tabulated as shown below,shown below,

PP PP PPC-0.75 PPC-0.75 PPC-1.5 PPC-1.5 PPC-2.25PPC-2.25 n n 10.28 10.28 8.71 8.71 11.76 11.76 12.2012.20 m 0.82 m 0.82 0.82 0.82 0.82 0.82 0.820.82 Average Average ‘A’ ‘A’ _3.2193_3.2193 1919 10 10−−

×

2.05772.0577 17 17 10 10−−

×

5.0453 5.0453

×

1010−−2020 30.012330.0123

×

1010−−2222 Approached Approached ‘A’ ‘A’ _1.033_1.033 1818 10 10−−

×

_8.9016_8.9016 1717 10 10−−

×

_2.563_2.563 2020 10 10−−

×

_3.052_3.052

×

2121 10 10−− Table3:

Table3: Results from method 2 Results from method 2

0 0 55 1100 1155 2200 2255 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.4 1.4 1.6 1.6 1.8 1.8 2 2 x 10x 10 -5 -5 Stress(Mpa) Stress(Mpa) C C r r e e e e p p - s s t t r r a a i i n n - - r r a a t t e e

CASE2:For PP With average A and Constant m values CASE2:For PP With average A and Constant m values

App

Approroacheachedd Experimental Experimental

Figure19:

Figure19: Creep strain rate Vs Stress for method 2Creep strain rate Vs Stress for method 2

The results were drawn in the same figures together with the Experimental The results were drawn in the same figures together with the Experimental results from Fig.20 to 23 for

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Figure20: Strain Vs time for PP under 17.33Mpa Figure20: Strain Vs time for PP under 17.33Mpa

0 0 5500000 0 110000000 0 1155000000 0.01 0.01 0.012 0.012 0.014 0.014 0.016 0.016 0.018 0.018 0.02 0.02 0.022 0.022 0.024 0.024 0.026 0.026 0.028 0.028 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

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0 0 5500000 0 110000000 0 1155000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

Figure22:

Figure22: Strain Vs time for PPC1.5 under 17.33MpaStrain Vs time for PPC1.5 under 17.33Mpa

0 0 2200000 0 4400000 0 6600000 0 8800000 0 101000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 Time(Seconds) Time(Seconds) S S t t r r a a i i n n CASE2:St

CASE2:St rain Vs Time for PPC2.25 under 17rain Vs Time for PPC2.25 under 17.33Mpa.33Mpa

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The total strain in this

The total strain in this method is given bymethod is given by

m m n n o o cc o o ε ε ε ε A A((σ σ )) t t ε ε ε ε

=

+

=

+

(4.7) (4.7)

The Relative error in % for method 2 is

The Relative error in % for method 2 is as shown below,as shown below, PP PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25PPC2.25 12.33(Mpa) 12.33(Mpa) 23.45 23.45 27.45 27.45 27.10 27.10 25.7025.70 17.33(Mpa) 17.33(Mpa) 22.36 22.36 24.90 24.90 26.70 26.70 25.5025.50 20.67(Mpa) 20.67(Mpa) 20.44 20.44 22.96 22.96 26.97 26.97 26.3426.34 24(Mpa) 24(Mpa) 15.34 15.34 13.89 13.89 14.67 14.67 15.5615.56 Table4:

Table4: Relative error in % of results when compared to experimental Relative error in % of results when compared to experimental results.

results.

From the above results it can be understood that the values of parameter From the above results it can be understood that the values of parameter ‘m’ < 1and the corresponding ‘A’ values does not give desired ‘m’ < 1and the corresponding ‘A’ values does not give desired experimental results, and therefore method 3

experimental results, and therefore method 3 was introduced.was introduced. 4.4.3 Method 3

4.4.3 Method 3

Here we assume that the material parameters ‘A’ and ‘m’ vary at each Here we assume that the material parameters ‘A’ and ‘m’ vary at each stress for the different materials, and we proceed to calculate the stress for the different materials, and we proceed to calculate the parameters fro

parameters from the creep equationm the creep equation.. 1 1 − −

=

&&

_(4.8)_(4.8)

We try for different time t, for example 2000, 4000 sec respectively at We try for different time t, for example 2000, 4000 sec respectively at different stresses. And by performing necessary calculations we get the different stresses. And by performing necessary calculations we get the values of the parameters can be obtained. Now we plot the curves for time values of the parameters can be obtained. Now we plot the curves for time vs. strain using the obtained parameters, and they are verified with the vs. strain using the obtained parameters, and they are verified with the experimental values.

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The values of A and m for each material at different stress values are The values of A and m for each material at different stress values are shown in the table

shown in the table below,below,

Material Material PP PP PPC0.75PPC0.75 Stress(Mpa) Stress(Mpa) A A m m A A mm 12.33 12.33 _3.883_3.883 1616 10 10−−

×

0.520.52 _6.97_6.97 1414 10 10−−

×

0.350.35 17.33 17.33 _2.58_2.58

_×

1717 10 10−− 0.520.52 1.381.38

×

1010−−1414 0.310.31 20.67 20.67 _2.79_2.79 1717 10 10−−

×

0.410.41 _7.35_7.35 1515 10 10−−

×

0.270.27 24 24 _4.30_4.30 1818 10 10−−

×

0.570.57 _3.54_3.54 1616 10 10−−

×

0.600.60 Table5:

Table5: Results from method 3 with assumed model Results from method 3 with assumed model

Material Material PPC1.5 PPC1.5 PPC2.25PPC2.25 Stress(Mpa) Stress(Mpa) A A m m A A mm 12.33 12.33 _5.83_5.83 1717 10 10−−

×

0.300.30 _3.51_3.51 1717 10 10−−

×

0.270.27 17.33 17.33 _6.77_6.77

_×

1818 10 10−− 0.220.22 1.931.93

×

1010−−1818 0.230.23 20.67 20.67 _2.05_2.05 1919 10 10−−

×

0.43 0.43 1.8761.876 20 20 10 10−−

×

0.60 0.60 24 24 _6.70_6.70 2121 10 10−−

×

0.81 0.81 1.2611.261 21 21 10 10−−

×

0.90 0.90 Table6:

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0 0 55 1100 1155 2200 2255 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.4 1.4 1.6 1.6 1.8 1.8 2 2 x 10x 10 -5 -5 Stress(Mpa) Stress(Mpa) C C r r e e e e p p - - s s t t r r a a i i n n r r - - a a t t e e

CASE3:For PP With average A and m values CASE3:For PP With average A and m values

Appro

Approachedached Experimental Experimental

Figure24: Creep strain rate Vs Stress for method 3 Figure24: Creep strain rate Vs Stress for method 3 The total strain is given by,

The total strain is given by,

m m n n o o cc o o ε ε ε ε A A((σ σ )) t t ε ε ε ε

=

+

=

+

(4.9) (4.9)

The results were drawn in

The results were drawn in the same figures together with the the same figures together with the ExperimentalExperimental results, as shown from Fig.25 to 28 for the different materials.

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0 0 2200000 0 4400000 0 6600000 0 8800000 0 110000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 0.04 0.04 Time(Seconds) Time(Seconds) S S t t r r a a i i n n CASE3:St

CASE3:St rain Vs rain Vs TTime for PP ime for PP undeunder 17.33Mpar 17.33Mpa

Figure25: Strain Vs time for PP under 17.33Mpa Figure25: Strain Vs time for PP under 17.33Mpa

0 0 5500000 0 110000000 0 1155000000 0.01 0.01 0.012 0.012 0.014 0.014 0.016 0.016 0.018 0.018 0.02 0.02 0.022 0.022 0.024 0.024 0.026 0.026 0.028 0.028 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

CREEP3:Strain Vs Time for PPC0.75 under 17.33Mpa CREEP3:Strain Vs Time for PPC0.75 under 17.33Mpa

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0 0 5500000 0 110000000 0 1155000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 Time(Seconds) Time(Seconds) S S t t r r a a i i n n

0 0 2200000 0 4400000 0 6600000 0 8800000 0 110000000 0 112200000 0 114400000 0 1166000000 0.01 0.01 0.015 0.015 0.02 0.02 0.025 0.025 0.03 0.03 0.035 0.035 0.04 0.04 Time(Seconds) Time(Seconds) S S t t r r a a i i n n CASE3:Strain V

CASE3:Strain V s s Time foTime for PPC2.25 under 17.33Mpar PPC2.25 under 17.33Mpa

Figure28: Strain Vs time for PP2.25 under 17.33Mpa Figure28: Strain Vs time for PP2.25 under 17.33Mpa

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From the calculations it is evident that there exists one set of ‘A’ and ‘m’ From the calculations it is evident that there exists one set of ‘A’ and ‘m’ values for each material at different stresses. This is clear from the plots values for each material at different stresses. This is clear from the plots above.

above.

Plots between A and stress values and also between m and stress values are Plots between A and stress values and also between m and stress values are as shown as shown 1 122 1144 1166 1188 2200 2222 2244 10 101414 10 101515 10 101616 10 101717 10 101818 10 101919 10 102020 10 102121 10 102222 Stress(Mpa) Stress(Mpa) A A A

A VsVs.S.Sigmaigma

*PP *PP vPPC0.75 vPPC0.75 .PPC1.5 .PPC1.5 +PPC2.25 +PPC2.25

Figure29: Plot of A Vs Stress for method 3 Figure29: Plot of A Vs Stress for method 3

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1 122 1144 1166 1188 2200 2222 2244 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 Stress(Mpa) Stress(Mpa) m m m Vs.Sigma m Vs.Sigma * P * PPP v PPC0.75 v PPC0.75 . PPC1.5 . PPC1.5 + PP + PP C2.2C2.255 Figure30:

Figure30: m Vs m Vs Stress plot Stress plot for method for method 33 Relative error in % for case3 is

Relative error in % for case3 is as shown below,as shown below, PP PP PPC0.75 PPC0.75 PPC1.5 PPC1.5 PPC2.25PPC2.25 12.33(Mpa) 12.33(Mpa) 6.06 6.06 2.58 2.58 7.10 7.10 8.768.76 17.33(Mpa) 17.33(Mpa) 6.79 6.79 2.90 2.90 6.70 6.70 7.667.66 20.67(Mpa) 20.67(Mpa) 6.80 6.80 2.96 2.96 6.97 6.97 6.776.77 24(Mpa) 24(Mpa) 2.79 2.79 1.89 1.89 2.67 2.67 2.562.56 Table7:

Table7: Relative error in % of results when compared to experimental Relative error in % of results when compared to experimental results for method 3

results for method 3 4.4.1.1 The results show that:

4.4.1.1 The results show that:

From the figure.29 it is evident that for the nanocomposites PPC1.5 and From the figure.29 it is evident that for the nanocomposites PPC1.5 and PPC2.25 the material constant Vs stress graph behavior is similar, also it PPC2.25 the material constant Vs stress graph behavior is similar, also it shows that material constant ‘A’ decreases with the increase in stress. shows that material constant ‘A’ decreases with the increase in stress. Nanocomp

Nanocomposites osites PP PP and and PP0.75 PP0.75 have have similar similar behavior behavior from from the the figure.29.figure.29. For nanocomposites PP there is a small increase in the ‘A' value with For nanocomposites PP there is a small increase in the ‘A' value with increase in stress at a particular instant, this may be due to the variation in increase in stress at a particular instant, this may be due to the variation in the material model

Bth Amt Ex 2007d10 Se

Harikishan Mandalapu

Harikishan Mandalapu

Sandeep Karanamsetty

Sandeep Karanamsetty

Parameter Analysis of Creep

Parameter Analysis of Creep

Models of PP/CaCo3

Models of PP/CaCo3

Nanocomposites

Nanocomposites

The Parameter analysis of creep

The Parameter analysis of creep

models of PP/CaCo3

models of PP/CaCo3

nanocomposites

nanocomposites

Harikishan Mandalapu

Harikishan Mandalapu

Sandeep Karanamsetty

Sandeep Karanamsetty

Acknowledgements

Acknowledgements

Contents

Contents

1 Notations

1 Notations

&&

Abbreviations

Abbreviations

2 Background

2 Background

3 Introduction to Creep

3 Introduction to Creep

=

=

=

=

+

+

−

−

=

=

••

••

••

••

3.1 Primary Creep

3.1 Primary Creep

=

=

=

=

≈

≈

&&

=

=

&&

=

=

&&

3.2 Secondary Creep

3.2 Secondary Creep

3.3 Tertiary Creep

3.3 Tertiary Creep

3.4 Creep under Variable Loading

3.4 Creep under Variable Loading

&&

=

=

&&

=

=

=

=

≤

≤

=