parameter analysis of creep models of pp/caco3 nanocomposites831640/fulltext01.pdf · parameters...

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

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

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden

2007

Harikishan Mandalapu

Sandeep Karanamsetty

Parameter Analysis of Creep Models of PP/CaCo3

Nanocomposites

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

Time(Seconds)

Stra

in

ExperimentalTheoritical

The Parameter analysis of creep models of PP/CaCo3

nanocomposites

Harikishan Mandalapu Sandeep Karanamsetty

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden 2007

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

The present report is about the parameter analysis of creep models ofNanocomposites (Isotactic polypropylene and CaCo3).The parametricanalysis of the nanocomposites under creep was carried out, and theparameters related to creep model are determined by comparing to theexperimental results. The influence of these parameters on the creep wasstudied. Using commercially available software ABAQUS, FiniteElement Calculations were done for elastic and creep conditions. Theresults obtained from theoretical analysis were verified with theExperimental Results. Also Abaqus results are compared with theExperimental results. Experimental results were obtained from theexperiments conducted by the Department of Chemistry, HuazhongUniversity of Science and Technology, China and by Department ofAdvanced Materials and Technology, College of Engineering, PekingUniversity, Beijing, China. Keywords: Creep strain, PP/CaCO3, Nanocomposites, FEM.

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Acknowledgements This thesis work is carried out at the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, under the supervision of Dr.Sharon Kao-Walter. 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 deep sense of gratitude to Dr.Sharon Kao-Walter, Department of Mechanical Engineering, Blekinge Institute of Technology, Sweden, for her invaluable guidance and kind cooperation extended by her throughout the course of this work. Our thanks go to the almighty God for giving us the opportunity to be able to complete this project. We wish to express our sincere appreciation to Tech. Etienne Mfoumou for his help regarding ABAQUS analysis. We also thank all our faculty members and our classmates for their encouragement, discussion, comments and many innovative ideas in carrying out this work. Finally, we would like to dedicate this work to our parents in India for their moral support and inspiration. Karlskrona, April 2007 Harikishan Mandalapu, Sandeep Karanamsetty.

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Contents 1 Notations 5 2 Background 7 3 Introduction to creep 8 3.1 Primary Creep 11 3.2 Secondary Creep 12 3.3 Tertiary Creep 12 3.4 Creep under variable loading 12 4 Experimental Work 14 4.1 Introduction 14 4.2 Tensile test for creep measurement 14 4.3 Experimental Results and parameter Analysis Work 16 4.4 Curve fitting with three methods 20 4.4.1 Method 1 20 4.4.2 Method 2 27 4.4.3 Method 3 31 4.5 Discussion and conclusion of parameter analysis 38 5 ABAQUS Model 39 6 Modeling and Simulation 45 6.1 ABAQUS/CAE Model 45 7 Conclusions and Further Work 49 8 References 50

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Appendices Appendix A 52 A1 Finite Element Method 52 A2 Modeling, Simulation and Results 54 A2.1 Pre-processing 54 A2.2 Simulation 57 A2.3 Post-processing 58 A2.4 Simulation Results 60 Appendix B ABAQUS/CAE Input files 61

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1 Notations E Young’s Modulus [Mpa] σ Stress [Mpa] ε Strain

cε Creep Strain

cε& Strain rate ν Poisson’s ratio D Constitutive matrix t Time [Seconds] R Universal gas constant [Cal/(mol)(K)] A Material Constant n Material Constant m Material Constant

6

Abbreviations PP Isotactic polypropylene PN poly-oxyethylene HDPE High density polyethylene CaCo3 Calcium Carbonate LVDT Linear Variable Displacement Transducer

7

2 Background Nanocomposites refer to materials consisting of at least two phases with one dispersed in another that is called matrix and forms a three-dimensional network. Nanocomposites have been studied extensively mainly for improved physical properties. Isotactic polypropylene (PP) is used as a most common plastic for the manufacturing of automotive parts, home appliances, for construction process, etc [3].But this PP is notch sensitive and brittle under severe conditions of deformation, such as at low temperatures or high impact rates, which makes limited its wider range of usage for the manufacturing processes. Blending PP with rigid inorganic particles such as CaCO3 is the best way to improve the stiffness and toughness of the PP.Dispersion quality of CaCO3 particles played a crucial role in toughening efficiency. The nanocomposites composed of isotactic PP and CaCo3 nanoparticles were fabricated by melt extrusion [3]. A nonionic modifier, poly-oxyethylene (PN), was added to the PP/CaCo3 mixture by dry mixing before melt extrusion. The dispersion of CaCo3 particles was greatly improved by this PN modifier [2]. Isotactic polypropylene (PP) and calcium carbonate CaCo3 nanocomposites mixture have various applications in automotive, construction and in other fields due to their high impact strengths and toughness. Different kinds of nanocomposites were tested to study the creep mechanism of components. An experimental setup was made and tensile creep tests were carried out on the specimen with different composite compositions each time. The aim of our thesis is to analyze the better creep model by studying the parameters. The material parameters are determined by comparing to a creep model from the experimental data and are verified along with the values estimated from theoretical formula. The analysis is done considering three methods with a defined creep model. We try to match the experimental results for the creep behavior with the results obtained from the theoretical formula. ABAQUS and Matlab were used to perform the 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 in Engineering materials and structures manifested by the accumulation of plastic deformation over prolonged time periods under steady or variable loading conditions [11].

At elevated temperatures and at constant stress or load many materials continue to deform at slow rate. This behavior is also called as creep. In other words high temperature progressive deformation of a material at constant stress is also called as creep.

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

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

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

Figure1: Different stages of creep [11].

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The long-term behavior of modern structures, whose final static configuration is frequently the result of a complex sequence of phases of loading and restraint conditions, are influenced largely by creep. Creep substantially modifies the initial stress and strain patterns, increasing the deformations induced by sustained loads, Relaxing the stresses due to sustained imposed deformations, (artificially introduced, e.g. by jacking, or due to natural causes like shrinkage or settlements) activating the delayed additional restraints. A special emphasis is given to the compact formulations derived directly from the fundamental theorems of the theory of linear viscoelasticity for nano materials. 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 at a constant temperature T.The extension of the specimen is measured as a function of time. A typical creep curve for metals, polymers, and ceramics is represented in Fig2. The response of the specimen loaded by 0σ at time 0=t can be divided into elastic and plastic part as

),,()(/ 000 TTE p σεσε += (3.1) Where )(TE the modulus of Elasticity. The creep strain in Fig2 is can then be expressed according to

kc tt αεεε 0)( −= (3.2)

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Figure2: Stages in creep [10] Where k < 1 in the primary, k = 1 in the secondary, and k >1 in the tertiary creep stage. These terms correspond to a decreasing,constant,and increasing strainrate,respectively, and were introduced by ANDRADE(1910).These three creep stages are often called transient creep, steady creep, and accelerating creep; respectively. The results (3.1) and (3.2) from the creep test justify a classification of material behavior in three disciplines: elasticity, plasticity, and creep mechanics. Due to a proposal of HAUPT (2000) one can also distinguish four theories of material behavior as follows:

• The theory of elasticity is concerned with the rate-independent behavior without hysteresis.

• The theory of plasticity specifies the rate-independent behavior with hysteresis.

• The theory of viscoelasticity describes the rate-dependent behavior without equilibrium hysteresis.

• The theory of viscoplasticity is devoted to the rate-dependent behavior with equilibrium hysteresis.

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The creep behavior exists in two of the above listed categories, namely in the theories of viscoelasticity and viscoplasticity. 3.1 Primary Creep Primary creep, Stage I, is a period of decreasing creep rate [9]. Primary creep is a period of primarily transient creep. During this period deformation takes place and the resistance to creep increases until stage II For constant-temperature creep behavior, cε is thus given by

),( tfc σε = (3.3)

Several mathematical forms exist to represent the function );,( tf σ one of these is the Norton-Baily creep law:

,mnc tAσε = (3.4)

Where the parameters mnA ,, depend on the material and temperature. They can be determined in a uniaxial creep test.

If the stress σ in (3.4) is assumed to be constant the creep rate cd ε&≈ is given by

.1−= mn

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

( )

mm

cm

nm

c mA11 −

= εσε& (3.6)

Which characterizes the strain-hardening-theory, i.e., this strain rate equation (3.6) includes stress and strain as variables. In contrast to (3.6) the strain rate equation (3.5) contains stress and time as variables and is therefore called the time-hardening-law.

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3.2 Secondary Creep Secondary creep, Stage II, is a period of roughly constant creep rate [9]. Stage II is referred to as steady state creep. Creep deformations of the secondary stage are large and of similar character to pure plastic deformations. 3.3 Tertiary Creep 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 formation [9]. The tertiary creep phase is accompanied by the formation of microscopic cracks on the grain boundaries, so that damage-accumulation occurs. In some cases voids are caused by a given stress history and, therefore, they are distributed anisotropically among the grain boundaries. 3.4 Creep under Variable Loading Norton-Bialys’ creep law as expressed by the equation (3.3) has been used extensively in analyzing creep problems [4]. The simplicity of this law helps in arriving at analytical solutions with acceptable accuracy for creep problems involving steady loadings. For situations when the applied stresses vary with time, either continuously or according to step changes, the use of Norton-Bialys’ law becomes inaccurate since the phases of primary creep cannot be neglected at every load change. To overcome this, the idea that equation (3.3) expresses the creep rate as a function of stress σ and current time t, i.e., ),( tfc σε =& has been replaced by considering ),( cc f εσε =& .The derivation of such functions is as follows: The primary and secondary creep strains are expressed [4] by Equation as follows:

mnc tAσε = (3.7) Where n >> 1 and m 1≤ .The time derivative gives

1−= mnc tAmσε& (3.8)

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This derivation results in the known Time-Hardening rule, where creep strain rate is expressed as a function of the stress σ and time t. Another formulation known as “strain-hardening” may be derived from the above equation (3.8) eliminating time t, as given by Equation (3.7), namely

m

n

c

At

1

⎟⎟⎠

⎞⎜⎜⎝

⎛=

σε (3.9)

Substitution into Equation (3.8) yields

mm

n

cn

AAm

/1

)(−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

σεσε&

or

mmcmnmc mA /)1(//1 )()( −= εσε& (3.10)

Equation (3.10) expresses the creep rate cε& as a function of the stress σ and the current creep strain cε .Experiments indicate that the strain-hardening formulation is to be favored over time-hardening formulation. However, nothing the large scatter in creep data, the use of the simple law of time hardening becomes justifiable in deriving analytical solutions. Evidently, and strain hardening offers no difficulty in seeking numerical solutions. Both formulations as given above are applicable only for situations where no stress reversals occur, a situation where modified rules have to be used .Also both formulations do not account well for the 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 CaCo3 nanoparticles [2] are used to reinforce thermoplastic polymers which have wide applications in many areas. The addition of these nanocomposites to the polymers increases their toughness and stiffness. The major drawback of these polymers is creep which occurs at stresses below the yield stress of the polymer materials. Nanocomposites with combination of surface modifiers such as poly-oxyethylene (PN) are good for obtaining uniform dispersion in the polymer matrix and have better mechanical behavior [3] than the original polymer matrix materials. The present work is to analyze the creep behavior of these composites with different PN content. 4.2 Tensile test for Creep Measurement The experimental setup used for the testing the creep behavior [2] is shown in the Figure (4) below.

Figure3: Schematic presentation of Test setup used for Experimental analysis [2]

CPUWeight

Carrier

50mm

Specimen

LVDT Control box

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Specimen PP CaCo3 PN Young’s Modulus

Poisson’s ratio

Material Constant

Code (Wt %) (Wt %) (Wt %)

(Gpa) ν n

PP 100 0 0 1.21 0.34 10.28 PPC-0.75 84.25 15 0.75 1.55 0.36 8.71 PPC-1.5 83.5 15 1.5 1.25 0.34 11.76 PPC-2.25 82.75 15 2.25 1.31 0.32 12.20

Table1: Combinations of composites taken for experiments The densities of PP and CaCO3 are 0.96 gm/cm3 and 2.55 gm/cm3

respectively. 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. The tests were done at four different stresses 12.33MPa, 17.33MPa, 20.67MPa, 24MPa respectively. The tests were carried out in the laboratory at a controlled temperature of 220C with variation of 02± C. The slight change in the temperatures is negligible on the tensile properties of the material. The dimensions of the specimen tested were 50x30x10mm 3 . Generally, the whole creep process is divided in two three phases like primary, secondary steady state and teritiary.Though the creep rate is rather 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 slows down continuously and the duration is limited. 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 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. 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 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 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 nanocomposites with different compositions of PP, CaCo3, PN.The curves are as shown below [2].

0 2000 4000 6000 8000 10000 12000 14000 160000.008

0.01

0.012

0.014

0.016

0.018

0.02

Time(S)

Stra

in

Strain Vs Time under 12.33Mpa Experimental

PP0PPC-0.75PPC-1.75PPC-2.25

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

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0 5000 10000 150000.01

0.015

0.02

0.025

0.03

0.035

Time(S)

Stra

inStrain Vs Time under 17.33Mpa Experimental

PPPPC0.75PPC1.5PPC2.25


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Time(S)

Stra

in

Strain Vs Time under 20.67Mpa Experimental

PP0PPC-0.75PPC-1.75PPC-2.25


18

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time

Stra

in

Strain Vs Time Under 24Mpa Experimental


Figure7: Strain versus time under 24MPa [2]

The creep rate is calculated from the values obtained from the experimental data. We have, Total strain = elastic strain + creep strain

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

1−= mnc tAmσε& (4.2) Now the creep rate for different composites is plotted for the various stresses. They are shown in the fig.8, 9 below. Also Logarithmic creep vs. the logarithmic stress values are also plotted for the verification purpose.

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12 14 16 18 20 22 240

0.5

1

1.5

2

2.5

3x 10-5

Stress(Mpa)

Stra

in ra

te


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

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4-7

-6.5

-6

-5.5

-5

-4.5

Log(Stress)

Log(

Cre

ep-ra

te)

PP

PPC0.75

PPC1.5

PPC2.25

Figure 9: Creep rate of steady stage versus stress of tested materials

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4.4 Curve fitting with three different methods The analysis was carried out considering three different methods. In method 1 we assume the creep model to have the formula of

mnc tAσε = 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. We tried to estimate the creep strain rate and the total strain with the calculated parameters. The results obtained are compared with the experimental results. Analysis of the above comparison is done to study the creep behavior in this particular method. In method 2 we have considered the same creep model similar to method 1. 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 parameters in this method and also analysis was carried out similar to method 1.From the comparison of the experimental and analytical results conclusions were drawn. The method 3 was carried out assuming the same creep model similar to the above two methods. Here we have assumed that the material parameters ‘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 method 2. 4.4.1 Method 1 We know the creep formula as,

mnc tAσε = (4.3) Since the secondary creep rate is considered in the experiment, and we know cε is almost linearly dependent on time and when the exponent ‘m=1’ in the above expression. The values of ‘A’ can be calculated for four different stresses by comparing 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 ‘A’ assuming the other parameter ‘m’=1.The values of ‘A’ are estimated from the above logarithmic Creep rate Vs. logarithmic stress graphs.

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From fig.14 it can be found that there is a huge deviation in the results between experimental and theoretical when we use Average ‘A’ value. One 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 applying suitable numerical methods. The new values are substituted once again. The calculations are performed by Matlab. The value of ‘A’ is obtained by iterative calculations. The results from theoretical formula are verified with the experimental results. Results are plotted in Matlab as shown below, The results are tabulated as shown below: PP PPC-0.75 PPC-1.5 PPC-2.25 n 10.28 8.71 11.76 12.20 m 1 1 1 1 Average ‘A’

3.338 1910−×

19.746 1810−× 6.993 2110−×

30.72 2210−×

Approached ‘A’

0.8011 1910−×

0.938 1710−×

1.1785 2110−×

4.362 2210−×

Table2: Results from method 1

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0 5 10 15 20 250

1

2

3

4

5

6x 10

-5

Stress

Cre

ep-s

train

-rate

For PP

ApproachedExperimentalAverage

Figure10: Creep strain rate Vs Stress for PP

0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

-5

Stress

Cre

ep-s

train

-rate

For PPC0.75


Figure11: Creep strain rate Vs Stress for PPC0.75

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0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2x 10

-4

Stress

Cre

ep-s

train

-rate

For PPC1.5


Figure12: Creep strain rate Vs Stress for PPC1.5

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2x 10

-4

Stress

Cre

ep-s

train

-rate

For PPC2.25 Before Iterative Approximation of A


Figure13: Creep strain rate Vs stress for PPC2.25

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From the results above it can be understood that the parameter ‘m’ = 1 does not give desired experimental results, and we proceed to method 2. We have the total strain given by

tAtdt

d no

c

o )(σεεεε +=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

(4.4) Now strain vs. time results are plotted as shown in Fig.14-18,

0 2000 4000 6000 8000 10000 12000 14000 160000.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

Time(Seconds)

Stra

in

CASE1:Strain Vs Time for PP under 12.33Mpa


Figure14: Strain Vs time for PP under 12.33Mpa

25

0 2000 4000 6000 8000 10000 12000 14000 160000.01

0.015

0.02

0.025

0.03

0.035

Time(Seconds)

Stra

inCASE1:Strain Vs Time for PP under 17.33Mpa



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0.02

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0.028

Time(Seconds)

Stra

in

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


Figure16: Strain Vs time for PPC0.75 under 17.33Mpa

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0.015

0.02

0.025

0.03

0.035

Time(Seconds)

Stra

in




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in




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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 assumptions made, and also the creep model we assumed in our case may not be appropriate. So we proceed to method 2. 4.4.2 Method 2 In this method we assume that the creep stain rate cε& defined by the creep strain rate expressed as a function of the stress σ and time t (i.e.) 1−= mnc tAmσε& .The parameter ‘m’ remaining constant during the creep stage, with varying ‘A’ Since the secondary creep rate has much significance in the design fields, we consider secondary creep here. From the Norton-Bialy’s creep laws:

mnc tAσε = (4.5) Now equation (4) expresses the creep rate as a function of stress σ and current time t, i.e., ),( tfc σε =& has been replaced by considering ),( cc f εσε =& .The derivation of such functions is as follows. The time derivative gives

1−= mnc tAmσε& (4.6) This derivation results in the known Time-Hardening rule, where creep strain rate is expressed as a function of the stress σ and time t. 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 tried at four different stresses. We arrive at four different ‘A’ values. Similar approach as the method 1 is done here for ‘A’ value. The results are plotted for experimental vs. theoretical.

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The results are tabulated as shown below,

PP PPC-0.75 PPC-1.5 PPC-2.25 n 10.28 8.71 11.76 12.20 m 0.82 0.82 0.82 0.82

Average ‘A’

3.2193 1910−×

2.0577 1710−× 5.0453 2010−×

30.0123 2210−×

Approached ‘A’

1.033 1810−×

8.9016 1710−×

2.563 2010−×

3.052× 2110 −

Table3: Results from method 2

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10-5

Stress(Mpa)

Cre

ep-s

train

-rate

CASE2:For PP With average A and Constant m values

ApproachedExperimental

Figure19: Creep strain rate Vs Stress for method 2

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

29

0 2000 4000 6000 8000 10000 12000 14000 160000.01

0.015

0.02

0.025

0.03

0.035

Time(Seconds)

Stra

in




0 5000 10000 150000.01

0.012

0.014

0.016

0.018

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0.024

0.026

0.028

Time(Seconds)

Stra

in




30

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0.015

0.02

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0.03

0.035

Time(Seconds)

Stra

in




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0.02

0.025

0.03

0.035

Time(Seconds)

Stra

in




31

The total strain in this method is given by

mno

co tA )(σεεεε +=+= (4.7)

The Relative error in % for method 2 is as shown below,

PP PPC0.75 PPC1.5 PPC2.25

12.33(Mpa) 23.45 27.45 27.10 25.70 17.33(Mpa) 22.36 24.90 26.70 25.50 20.67(Mpa) 20.44 22.96 26.97 26.34

24(Mpa) 15.34 13.89 14.67 15.56 Table4: Relative error in % of results when compared to experimental

results. From the above results it can be understood that the values of parameter ‘m’ < 1and the corresponding ‘A’ values does not give desired experimental results, and therefore method 3 was introduced. 4.4.3 Method 3 Here we assume that the material parameters ‘A’ and ‘m’ vary at each stress for the different materials, and we proceed to calculate the parameters from the creep equation.

1−= mnc tAmσε& (4.8)

We try for different time t, for example 2000, 4000 sec respectively at different stresses. And by performing necessary calculations we get the 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 experimental values.

32

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

Material PP PPC0.75 Stress(Mpa) A m A m

12.33 3.883 1610−× 0.52 6.97 1410−× 0.35

17.33 2.58× 1710 − 0.52 1.38 1410−× 0.31

20.67 2.79 1710−× 0.41 7.35 1510−× 0.27

24 4.30 1810−× 0.57 3.54 1610−× 0.60

Table5: Results from method 3 with assumed model

Material PPC1.5 PPC2.25 Stress(Mpa) A m A m

12.33 5.83 1710−× 0.30 3.51 1710−× 0.27

17.33 6.77× 1810 − 0.22 1.93 1810−× 0.23

20.67 2.05 1910−× 0.43 1.8762010−×

0.60

24 6.70 2110−× 0.81 1.2612110−×

0.90

Table6: Results from method 3 with assumed model

33

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

-5

Stress(Mpa)

Cre

ep-s

train

-rate

CASE3:For PP With average A and m values

ApproachedExperimental

Figure24: Creep strain rate Vs Stress for method 3

The total strain is given by,

mno

co tA )(σεεεε +=+= (4.9)

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

34

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Time(Seconds)

Stra

in




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Time(Seconds)

Stra

in

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



35

0 5000 10000 150000.01

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0.035

Time(Seconds)

Stra

inCASE3:Strain Vs Time for PPC1.5 under 17.33Mpa



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0.015

0.02

0.025

0.03

0.035

0.04

Time(Seconds)

Stra

in



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

36

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 above. Plots between A and stress values and also between m and stress values are as shown

12 14 16 18 20 22 2410

14

1015

1016

1017

1018

1019

1020

1021

1022

Stress(Mpa)

A

A Vs.Sigma

*PPvPPC0.75.PPC1.5+PPC2.25

Figure29: Plot of A Vs Stress for method 3

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12 14 16 18 20 22 240.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stress(Mpa)

m

m Vs.Sigma

* PPv PPC0.75. PPC1.5+ PPC2.25

Figure30: m Vs Stress plot for method 3

Relative error in % for case3 is as shown below,

PP PPC0.75 PPC1.5 PPC2.25

12.33(Mpa) 6.06 2.58 7.10 8.76 17.33(Mpa) 6.79 2.90 6.70 7.66 20.67(Mpa) 6.80 2.96 6.97 6.77

24(Mpa) 2.79 1.89 2.67 2.56 Table7: Relative error in % of results when compared to experimental

results for method 3 4.4.1.1 The results show that: 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 shows that material constant ‘A’ decreases with the increase in stress. Nanocomposites PP and PP0.75 have similar behavior from the figure.29. 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 the material model

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4.5 Discussion and Conclusion of parameter analysis From the above three different methods we observed that the method 3 results were appropriate when compared to the experimental results. So we conclude that the parameters ‘A’ and ‘m’ vary with each material at specific stress value. This shows that creep behavior is dependent on the material parameters. ‘A’, ‘n’ and ‘m’ are constants for the specific temperature conditions and stresses. The other material constant ‘n’ also influences the creep behavior. For our convenience we assumed the parameter to be constant for each material at different stresses. If the parameter ‘n’ was not assumed to be constant then the study of the creep behavior will become difficult to analyze. The creep equation will have three varying material parameters which vary at each stress. We conclude that ‘A’ and ‘m’ values influence the creep behavior which is clear from the plots between the experimental and theoretical results. Also there exists unique ‘A’ and ‘m’ values for each material under different stresses for a definite creep model defined.

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5 ABAQUS Model The creep model is described in different way in the ABAQUS. We now try to estimate the values of ‘ ’ and ‘ ’ for the model. The ABAQUS model is described as

= (5.1)

(5.2) Where is creep strain rate Total strain is,

(5.3) Where, is ordinary elastic strain. The values of ‘ ’ and ‘ ’ calculated for the ABAQUS model are shown in the table.

Material PP PP0.75 Stress(Mpa)

12.33 20.12 1710−× -0.48 2.44 1410−× -0.65

17.33 1.406× 1710 − -0.48 4.218 1510−× -0.69

20.67 1.14 1710−× -0.59 1.98 1510−× -0.73

24 2.45 1810−× -0.43 2.12 1610−× -0.40

Table8: Results for the ABAQUS model from method 3

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Material PPC1.5 PPC2.25 Stress(Mpa)

12.33 1.74 1710−× -0.70 9.53 1810−× -0.73

17.33 1.50 1810−× -0.78 4.439 1910−× -0.77

20.67 8.81 2010−× -0.57 1.126 2010−× -0.40

24 5.44 2110−× -0.19 1.13 2110−× -0.10

Table 9: Results for the ABAQUS model from method 3

Also results from the ABAQUS model are calculated using these ‘ ’ and ’ ’ values, and plotted along with the experimental results for the selected materials at different stresses. By applying the parameters from Table.7 and Table.8 in ABAQUS creep calculation, the strain and strain rate were obtained for different materials. The results were shown in fig.31 to 38 together with the experimental curves. The good agreement can be observed from the figures. The procedure to run the ABAQUS and same results were described in Appendix A.

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0 2000 4000 6000 8000 10000 12000 14000 160000.01

0.012

0.014

0.016

0.018

0.02

0.022

Time(Seconds)

Stra

inStrain Vs Time for PP under 12.33Mpa

ExperimentalABAQUS


0 2000 4000 6000 8000 10000 12000 14000 160000.01

0.015

0.02

0.025

0.03

0.035

0.04

Time(Seconds)

Stra

in

PP under 17.33Mpa

ExperimentalABAQUS


42

0 2000 4000 6000 8000 10000 12000 14000 160000.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

Time(Seconds)

Stra

in

Strain Vs Time for PPC0.75 under 12.33Mpa

ExperimentalABAQUS


0 2000 4000 6000 8000 10000 12000 14000 160000.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028

Time(Seconds)

Stra

in

PPC0.75 under 17.33Mpa

ExperimentalABAQUS


43

0 2000 4000 6000 8000 10000 12000 14000 160000.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

Time(Seconds)

Stra

inStrain Vs Time for PPC1.5 under 12.33Mpa

ExperimentalABAQUS


0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

Time(Seconds)

Stra

in

Strain Vs Time for PPC1.5 under 24Mpa


Figure36: Strain Vs time for PPC1.5 under 24Mpa

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0 2000 4000 6000 8000 10000 12000 14000 160000.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

Time(Seconds)

Stra

in

PPC2.25 under 12.33Mpa

ExperimentalABAQUS


0 500 1000 1500 2000 25000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time(Seconds)

Stra

in

Strain Vs Time for PPC2.25 under 24Mpa

ExperimentalABAQUS

Figure38: Strain Vs time for PPC2.25 under 24Mpa

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6 Modeling and Simulation 6.1 ABAQUS/CAE The ABAQUS software [15] suite has an unsurpassed reputation for technology, quality, and reliability and provides a powerful and complete solution for both routine and sophisticated linear and nonlinear engineering problems. ABAQUS delivers a unified FEA environment that is a compelling alternative to implementations involving multiple products and vendors. With ABAQUS/CAE we can quickly and efficiently create, edit, monitor, diagnose, and visualize advanced ABAQUS analyses. ABAQUS/CAE integrates modeling, analysis, job management, and results visualization in a consistent, easy-to-use environment that is simple to learn for new users yet highly productive for experienced users. Familiar concepts such as feature-based, parametric modeling make ABAQUS/CAE a modern and effective pre- and postprocessor for engineering specialists. The finite element analysis of the creep model has been done in ABAQUS. The assumptions made in our modeling are as follows 1. The material is assumed to be isotropic. 2. The material is assumed to be homogeneous. The step by step procedure of the analysis of the creep model in ABAQUS is shown in the Appendix A. It includes creation of part, assigning of various material properties, creation of the section and the assembly, steps, applying of loads and boundary conditions, meshing of the model. Finally, creation of job and submitting the job for the analysis. After submitting the job for analysis the ABAQUS software performs the creep analysis of the material model. The results can be interpreted as shown below.

46

The creep formation in the material can be seen for different step times in the following figures,

FOR STEP TIME = 125 SECONDS

Figure39: Starting stage of Creep of PPC2.25 under 24Mpa.

From the figure39 the propagation of the creep at step time 125seconds can be observed. Since the object is constrained at one end the values of the creep strain vary according to the coloured regions.

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Figure40: CE 22 for PPC2.25 under 24Mpa at certain step time.

From the fig.40 the propagation of creep at step time 960 seconds can be observed. The center region has the maximum creep strain and the region tends to expand with the increase in step time.

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Figure41: Creep formation at certain step time From the Fig.39 to 41 the propagation of creep for step time 2500 seconds can be observed. From the Fig.39 we can notice the creep phenomena with the dark region formed in the center. Also from the Fig.39 to 41 we can conclude that the increase in step time results in the increase in the creep strain up to a certain time period.

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7 Conclusions and further work ’A’ and ’m’ are constants and they are functions of stress. From the above analysis we conclude that the material constants ‘A’ and ‘m’ influence the creep strain for the nanomaterials studied. Erroneous values of ‘A’ results in the huge deviation in the creep strain. This is evident from the plots between the experimental and the theoretical values. So, approximation of the ‘A’ value from the available range of values is done in order to reduce the deviations in the results. Similarly suitable ‘m’ values were calculated for the above three cases. Plots of creep strain Vs strain at different applied stresses shows that all the materials are in the first two creep stages for the maximum duration of the experiment. A better creep model is required for describing nanocomposites. Since creep varies with material model, each material can be described using different creep model. The results obtained from the analysis were not agreeing the experimental results for certain loads in the first two cases for strain values. The possible reason could be the assumptions made during the evaluation step. Our assumption for the material constant m = 1 in our case may not be appropriate, because the possible range of ‘m’ values are from 0 to 1.The approached values of ‘A’ gave better results than the approximated ‘A’ values. There exist a unique ‘A’ and ‘m’ values for each material at different stress conditions. This is evident from the case 3. Still, better results can be achieved by employing better approximation methods. Creep Analysis is carried out in Abaqus.The results are satisfactory. From the ABAQUS results the creep strain CE22 is shown in the figures. The propagation of creep is properly described in the results. We can observe the creep phenomena with the varying step time. Better results can be achieved by employing advanced approximation techniques, and by considering more creep models. Hopefully, research can be carried out on these outlined setbacks for rectifying the same.

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8 References 1. Computational Engineering Textbook by Goran Broman,

2. Reference to Creep behavior of polypropylene/CaCo3 nanocomposites with nonionic modifier by Jinlong Ahang,Shu-Lin Bai,Centre for Advanced Composite Materials, Department of Advanced Materials and technology, College of Engineering, Peking University, Beijing 100871,China

3. Cao, Guozhong. Nanostructures and Nanomaterials. Singapore: World Scientific Publishing Company, Incorporated, 2004. 4. Engineering Solid Mechanics (Fundamentals and Applications) by Abdel- Rahman Ragab, Salah Eldin Bayoumi. 3. Papalambros, P. Y., (2000), Extending the Optimization Paradigm in

Engineering Design, Department of Mechanical Engineering, University of Michigan, USA.

4. McKay, M. D., Beckman, R. J. and Conover, W.J., (1979), A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, Vol. 21, No. 2, May 1979.

5. TheodorBalderes,"Finite elementmethod”,inAccessScienceby McGraw- Hill.

6. http://websok.libris.kb.se/websearch/search?SEARCH_ONR=10133679

7. Introduction to Finite Elements in Engineering by Tirupathi R.Chandrupatla and Ashok D.Belegundu.

8. Introduction to the Finite Element Method by Niels Ottosen and Hans Petersson

9. Creep Mechanics by J.Betten 2nd Edition

10. Materialsengineer.com/CA-Creep-Stress-Rupture.htm Copyright © 1999 Metallurgical Consultants

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11. Ritchie, R.O. "Mechanical Behavior of Materials Lecture Notes,"

University of California, Berkeley. Rhoads, Fall 1993.

12. en.wikipedia.org/wiki/Creep_(deformation)

13. Hertzberg, R.W. "Deformation and Fracture Mechanics of Engineering Materials." 3rd ed., New York. Wiley & Sons, 1989

14. Article relating to creep analysis research group, POLITECNICO DI TORINO, Italy.

15. ABAQUS Standard manual.

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Appendix A A1 Finite Element Method All the physical phenomena encountered in engineering mechanics [7] are modeled by differential equations, and it is very complicated to solve these equations by normal analytical methods. The finite element method is the one approach by which differential equations can be solved in an approximate manner as shown in Fig.42. Model Approximation

Figure42: Steps in Engineering Mechanics Analysis [7] The differential equation or equations, which describe the physical problem considered, are assumed to hold over a certain region. This region may be one-, two-, or three-dimensional. It is a characteristic feature of the finite element method that instead of seeking approximations that hold directly over the entire region, the region is divided into smaller parts, so called finite elements, and the approximation is then carried out over each element. For instance, even though the variable varies in a highly in nonlinear manner over the entire region, it may be a fair approximation to assume that the variable varies in a linear fashion over each element. The collection of all elements is called a finite element mesh. When the type of approximation which is to be applied over each element has been selected, the corresponding behavior of each element can then be determined. This can be performed because the approximation made over each element is fairly simple. Having determined the behavior of all

Differential Equation

Physical Phenomenon

Finite Element Equations

53

elements, these elements are then patched together, using some specific rules, to form the entire region, which eventually enables us to obtain an approximate solution for the behavior of the entire body. The finite element (FE) method can be applied to obtain approximate solutions for arbitrary differential equations. As the FE method is a numerical means of solving general differential equations, it can be applied to various physical phenomena.Inorder to emphasize this aspect, we shall be concerned here with FE formulation of such diverse problems as heat conduction, torsion of elastic shafts, diffusion, ground water flow, and the elastic behavior of one-, two- and three-dimensional bodies, including beam and plate analysis. As previously mentioned, it is a characteristic feature of the FE method that the region,ie.,the body, is divided into smaller parts,i.e.the elements, for which a rather simple approximation is adopted. This approximation is usually a polynomial. The approximation over each element means that an approximation is adopted for how the variable changes over the element. This approximation is, infact, some kind of interpolation over the element, where it is assumed that the variable is known at certain points in the element. These points are called nodal points and they are often located at the boundary of each element. The precise manner in which the variable changes between its values at the nodal points is expressed by the specific approximation, which may be linear, quadratic, cubic, etc. The Finite Element Method is a numerical approach which results in the establishment of systems of equations often involving thousands of unknowns. FEM is the numerical analysis technique for obtaining approximate solutions to many types of engineering problems. The need for numerical methods arises from the fact that for most engineering problems analytical solutions does not exist. While the governing equations and boundary conditions can usually be written for these problems, and difficulties introduced by either irregular geometry or other discontinuities render the problems intractable analytically. To obtain a solution, the engineer must make simplifying assumptions reducing the problem to one that can be solved, or a numerical procedure must be used. In an analytic solution, the unknown quantity is given by a

54

mathematical function valid at an infinite number of locations in the region under study, while numerical methods provide approximate values of the unknown quantity only at discrete points in the region. In the finite element method, the region of interest is divided into numerous connected sub regions or elements within which approximating functions (usually polynomials) are used to represent the unknown quantity. A2 Modeling, Simulation and Results in ABAQUS The ABAQUS procedure consists of the following processes: A2.1 Pre-Processing In the Pre-Processing part the model of the specimen is created for the analysis. There are different steps in this process, they are as follows

• Part In this module it has been taken a three-dimensional, solid part with extrusion property to create the required element for analysis. The sketcher size was taken as 200.A beam with length 30mm, height 50mm and width 10mm has been created as shown in the figure.

55

Figure43: Model of nanocomposite PP

• Property In this module it has been defined the properties of the material like poisons ratio, Young’s Modulus, etc; The properties of the materials used are shown in the following table [2]

Specimen Young’s Modulus Poisson’s ratio Material Constant

Code (Gpa) ν n PP 1.21 0.34 10.28 PPC-0.75 1.55 0.36 8.71 PPC-1.5 1.25 0.34 11.76 PPC-2.25 1.31 0.32 12.20

Also this is the module where it has been defined the material behaviors like elastic and creep with the required properties and data.

56

• Assembly

In this module an independent mesh on instance for the analysis has been created.

• Step In this module the analysis procedure was defined and the visco procedure has been taken and the step time was taken as from 4000 to 16000 Seconds depending on the stresses applied.

• Load In this module the boundary conditions and the load conditions required for our analysis has been considered. The beam was fixed at one end and the load has been applied at the other end as shown in the figure below.

Figure44: Boundary conditions and Load for material

57

• Mesh

In this module the finite element mesh for the element for the analysis has been generated. Global seeds have been assigned for the instance. And meshing was done for the region. The meshed element used for the analysis is as shown in the following figure.

Figure45: Meshing of the model A2.2 Simulation

• Job In this module the job has been created and submitted for analysis.

58

A2.3 Post-Processing

• Visualization This module is used to read the output database that ABAQUS/CAE generated during the analysis and to view the results of the analysis. All the results required are generated by this module using the options needed. The results are generated from the field output. The creep strain CE 22 values are determined. The results for the creep analysis are shown in the following figures,

Figure46: S,Mises for PPC2.25 under 24Mpa

59

Figure47: Tensors and Vectors for the Model PPC2.25 under 24Mpa

Figure48: Undeformed shape of PPC2.25 under 24Mpa

60

Figure49: Deformed shape for PPC2.25 under24Mpa A2.4 Simulation Results

Figure50: Creep Strain Vs Time under 24Mpa from ABAQUS For PPC2.25 For a particular element selected.

61

Appendix B ABAQUS/CAE Input files Input file for PP under 12.33Mpa *Heading PP0 ** Job name: PP0 Model name: Model-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=PP0 *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=PP0-1, part=PP0 *Node 1, 15., 25., 10. 2, 15., 20., 10. 3, 15., 15., 10. 4, 15., 10., 10. 5, 15., 5., 10. 6, 15., 0., 10. 7, 15., -5., 10. 8, 15., -10., 10. 9, 15., -15., 10. 10, 15., -20., 10. 11, 15., -25., 10. 12, 15., 25., 5. 13, 15., 20., 5. 14, 15., 15., 5. 15, 15., 10., 5. 16, 15., 5., 5. 17, 15., 0., 5.

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18, 15., -5., 5. 19, 15., -10., 5. 20, 15., -15., 5. 21, 15., -20., 5. ...so on 213, -15., 10., 5. 214, -15., 5., 5. 215, -15., 0., 5. 216, -15., -5., 5. 217, -15., -10., 5. 218, -15., -15., 5. 219, -15., -20., 5. 220, -15., -25., 5. 221, -15., 25., 0. 222, -15., 20., 0. 223, -15., 15., 0. 224, -15., 10., 0. 225, -15., 5., 0. 226, -15., 0., 0. 227, -15., -5., 0. 228, -15., -10., 0. 229, -15., -15., 0. 230, -15., -20., 0. 231, -15., -25., 0. *Element, type=C3D8I 1, 34, 35, 46, 45, 1, 2, 13, 12 2, 35, 36, 47, 46, 2, 3, 14, 13 3, 36, 37, 48, 47, 3, 4, 15, 14 4, 37, 38, 49, 48, 4, 5, 16, 15 5, 38, 39, 50, 49, 5, 6, 17, 16 6, 39, 40 .............so on..... 108, 206, 207, 218, 217, 173, 174, 185, 184 109, 207, 208, 219, 218, 174, 175, 186, 185 110, 208, 209, 220, 219, 175, 176, 187, 186 111, 210, 211, 222, 221, 177, 178, 189, 188 112, 211, 212, 223, 222, 178, 179, 190, 189 113, 212, 213, 224, 223, 179, 180, 191, 190

63

114, 213, 214, 225, 224, 180, 181, 192, 191 115, 214, 215, 226, 225, 181, 182, 193, 192 116, 215, 216, 227, 226, 182, 183, 194, 193 117, 216, 217, 228, 227, 183, 184, 195, 194 118, 217, 218, 229, 228, 184, 185, 196, 195 119, 218, 219, 230, 229, 185, 186, 197, 196 120, 219, 220, 231, 230, 186, 187, 198, 197 *Nset, nset=_PickedSet2, internal, generate 1, 231, 1 *Elset, elset=_PickedSet2, internal, generate 1, 120, 1 ** Region: (PP0:Picked) *Elset, elset=_PickedSet2, internal, generate 1, 120, 1 ** Section: PP0 *Solid Section, elset=_PickedSet2, material=PP0 1., *End Instance ** *Nset, nset=_PickedSet4, internal, instance=PP0-1, generate 1, 221, 11 *Elset, elset=_PickedSet4, internal, instance=PP0-1, generate 1, 111, 10 *Elset, elset=__PickedSurf5_S4, internal, instance=PP0-1, generate 10, 120, 10 *Surface, type=ELEMENT, name=_PickedSurf5, internal __PickedSurf5_S4, S4 *End Assembly ** ** MATERIALS ** *Material, name=PP0 *Creep, law=TIME 2.158e-16, 10.28, -0.48 *Elastic 1210., 0.34 ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Displacement/Rotation

64

*Boundary _PickedSet4, 1, 1 _PickedSet4, 2, 2 _PickedSet4, 3, 3 ** ---------------------------------------------------------------- ** ** STEP: CREEP ** *Step, name=CREEP, nlgeom=YES CREEPTEST *Visco, cetol=0.01 1600., 16000., 0.16, 16000. ** ** LOADS ** ** Name: Load-1 Type: Pressure *Dsload _PickedSurf5, P, -12.33 ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Department of Mechanical Engineering, Master’s Degree Programme Blekinge Institute of Technology, Campus Gräsvik SE-371 79 Karlskrona, SWEDEN

Telephone: Fax: E-mail:

+46 455-38 55 10 +46 455-38 55 07 [email protected]

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