PRELIMINARY REPORT

Jordan Suls and Zach Nelson

Instructor

N.M. Ghoniem

5/30/16

Multi-physics Design and

Thermal Analysis of an IC

Engine Piston and Cylinder

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Contents 1. Abstract ................................................................................................................................. 3

2. List of Symbols ....................................................................................................................... 4

3. Table of Figures ...................................................................................................................... 5

4. Introduction and Background ................................................................................................... 6

5. Definition and Conceptual Design ............................................................................................ 8

6. Preliminary Design ................................................................................................................ 10

6.1 Hand (Matlab) Calculations of Thermal and Fluid Loads .................................................... 10

6.2 Two-Dimensional Thermoelastic Analysis ........................................................................ 12

6.3 Axisymmetric Segregated Fluid Flow and Heat Transfer .................................................... 12

7. Detailed Design .................................................................................................................... 16

7.1 Three-dimensional Solid Modeling ................................................................................... 16

7.2 Analytical Heat Release Model ........................................................................................ 17

7.3 Multi-physics Design Approach ....................................................................................... 19

7.4 Fluid Flow and Heat Transfer Simulation: Non-Isothermal Flow ......................................... 19

7.5 Thermomechanical Stress Analysis .................................................................................. 20

8. Failure Analysis .................................................................................................................... 22

8.1 Material Data and Design Limits ...................................................................................... 22

8.2 Design Code Allowable Properties ................................................................................... 23

8.3 Strength Safety Factors ................................................................................................... 24

8.4 Creep-Fatigue Safety Factors ........................................................................................... 24

8.5 Fracture Failure Assessment ............................................................................................ 29

9. Final Design Optimization ..................................................................................................... 31

10. Conclusions and Recommendations ..................................................................................... 33

11. References ......................................................................................................................... 34

12. Appendices (Matlab Codes and Material Property Data) ........................................................ 36

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1. Abstract A two-dimensional, axisymmetric model was produced and analyzed with the use of Finite Element Method (FEM) for an IC engine piston and cylinder. With the use of Matlab and the assumed normal and off-normal operational conditions, preliminary thermal load calculations were produced. Using these parameters, an initial 2-D model was generated using SolidWorks and the necessary materials were selected. An aluminum alloy (AlSi) was considered to be applicable for this design based on the operation requirements. With COMSOL, a segregated 2-D axisymmetric fluid flow and heat transfer model was produced for a transient combustion cycle to determine the temperature distribution in the piston and cylinder. Then, an analytical heat release model was used to determine the pressure and temperature of the gas as a function of the crank angle, along with the heat transfer coefficient. A coupled fluid flow and heat transfer analysis revealed the maximum temperature of 670 K along the piston head and a maximum stress state of 80 MPa in the pin holes during the ignition part of the cycle. A fatigue-creep damage accumulation model was used to determine cycles to failure. The fatigue model resulted in 2.7 x108 cycles to failure and the creep showed a 33.3 hour lifetime in its max stress state. This resulted in a total number of cycles to failure of 3.17x107 cycles, below the targeted lifetime for the piston. The fracture assessment used Paris law to calculate cycles to failure separately, which resulted in 2.32x108 cycles. A design optimization was done to find a different material that had a higher time to rupture than AlSi. The new material selected was Aluminum 6061-T6, which increased the number of cycles to failure of fracture to 6.065x109 cycles, rupture time to 307 hours at high stress state, and fatigue life to 1.57x1013. This changed the creep fatigue lifetime to 3.976x108 cycles, which is higher than the lifetime requirement of 3.13x108 cycles and results in a safety factor of 1.26 for creep-fatigue.

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2. List of Symbols Symbol Description U Fluid Velocity (m/s) h Convective Heat Transfer Coefficient (W/m^2 K) T Temperature (K) q Heat Flux (W/m^2) α Thermal Expansion (1/K) κ Thermal Conductivity (W/mK) ρ Density (kg/m^3) ν Thermal Diffusivity (m^2/s) µ Dynamic Viscosity (kg/ms) k Ratio of Specific Heat Cp Specific Heat at Constant Pressure (kJ/kgK) Cv Specific Heat at Constant Volume (kJ/kgK) b Bore Cylindrical Length (m) s Stroke length (m) a, m Heat Transfer Constants in Combustion Engine d Diameter of Piston (m) h Piston Height (m) A Cross-Sectional Area of Piston (m^2) Nu Nusselt Number K1c Fracture Toughness (MPA-m0.5) Re Reynolds Number R Gas Constant (J/mol K) θ Crank angle (Degrees) σ Stress (MPa) Q Activation Energy, heat release tr Time to rupture (s) ε strain A,n Creep constants Cp, n Paris Law Constants A Crack length (m) N Number of cycles d damage n’, K’ Ramberg-Osgood Model Constants f Mass fraction of fuel burned V Volume (m^3) C1 Fracture constant P Applied Pressure (MPa) r Radius (m)

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3. Table of Figures Figure 1. Basic geometry of the reciprocating internal combustion engine………………………7 Figure 2. The four-stroke operating cycle………………………………………………………...8 Figure 3. Piston part and design sketch in Solidworks……………………………………………9 Figure 4. Cylinder part and design sketch in Solidworks…………………………………………9 Figure 5. Pin, connecting rod and crankshaft in Solidworks……………………………………...9 Figure 6. Pressure-Volume Diagram of Cycles………………………………………………….11 Figure 7. Three-Dimensional deformed state with Von Mises Stress Distribution……………...13 Figure 8. Two-Dimension undeformed state with Von Mises Stress Distribution……………...13 Figure 9. Temperature distribution along the piston and cylinder……………………………….14 Figure 10. Velocity Distribution (Left) and Pressure Gradient (Right) of the combusted gas…..15 Figure 11. Thermal Gradient of expanded gas…………………………………………………...15 Figure 12. Exploded and cross-sectional assembly views……………………………………….16 Figure 13. Detailed drawing of assembly’s cross-sectional view………………………………..16 Figure 14. Burn fraction………………………………………………………………………….17 Figure 15. Burn Rate……………………………………………………………………………..17 Figure 16. Pressure vs. Crank Angle…………………………………………………………….18 Figure 17. Temperature vs. Crank Angle………………………………………………………..19 Figure 18. 3-D Cylinder, combusted gas and piston mesh………………………………………20 Figure 19. 3-D Temperature distribution of combusted gas and cylinder……………………….20 Figure 20. 2-D Temperature distribution of combusted gas and cylinder……………………….20 Figure 21. Resulting Mesh of Piston…………………………………………………………….21 Figure 22. Von Mises Stress Distribution (Pa)…………………………………………………..22 Figure 23. Strain Distribution (mm/mm)………………………………………………………...22 Figure 24. Temperature Distribution (K)………………………………………………………...22 Figure 25. Piston guideline dimensions………………………………………………………….23 Figure 26. Strain Amplitude vs. Number Reversal Cycles to failure……………………………26 Figure 27. Strain Amplitude vs. Number of Reversals to failure………………………………..26 Figure 28. Alternating Stress-Strain Relationship……………………………………………….27 Figure 29. Cylinder Pressure on Piston…………………………………………………………..27 Figure 30: Finding Mode I Fracture Toughness Constant……………………………………….29 Figure 31. Final Stress Distribution……………………………………………………………...33 Figure 32. Temperature Distribution…………………………………………………………….33

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4. Introduction and Background

A piston is a component of a reciprocating internal combustion (IC) engine. It is the moving component which is contained inside a cylinder and achieves gas-tight by piston rings. In the IC engine, it functions to transfer force from expanding gas in the cylinder to the crankshaft via a connecting rods. As an integral part in an engine, it is necessary to determine the piston temperature distribution in order to control the thermal stresses and deformations within the acceptable levels [1]. Pistons in an IC engine are usually made of aluminum alloy which has a thermal expansion coefficient of 80% higher than that of the cylinders, which are made of cast iron. Therefore, the analysis of pistons thermal behavior is crucial when designing a safe and efficient IC engine. Thermal analysis is a branch of material science where the properties of materials are studies as they change with temperature. Finite Element Analysis (FEA) method are commonly used for thermal analysis. Due to the complicated working environment for piston, there are many methods proposed to determine the optimal design. In this project, the piston is simulated at 3000 RPM with a vehicle speed at 90 Km/h and Aluminum Silicon alloy (AlSi) is used as preliminary material, which has material properties of [2]:

Table 1 Material Properties of Piston

Material AlSi

Thermal conductivity [W/m °C] 155

Thermal expansion 10-6 [1/°C] 21

Density [kg/m3] 2700

Specific heat [J/kg °C] 960

Poisson's ratio 0.3

Young's modulus 90 The alloy chosen was a common alloy used for high performance engines, AlSi12CuMgNi. The composition of the alloy can be found in Table 2.

Table 2: Chemical Composition of AlSi Alloy Element Si Cu Mg Ni Fe Mn Zn Ti Al wt % 11-13 0.8-1.3 0.8-1.3 1.3 0.7 0.3 0.3 0.2 Rest

4.1 Engine Operating Cycles

This project is focused on reciprocating engines, where the piston moves back and forth in a cylinder and transmits power through a connecting rod and crank mechanism to the drive shaft as shown in Fig 1.

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Figure 1 Basic geometry of the reciprocating internal combustion engine

The steady rotation of the crank produces a cyclical piston motion. The piston comes to rest at the top-center (TC) crank position and bottom-center (BC) crank position when the cylinder volume is a minimum or maximum respectively. The minimum cylinder volume is called the clearance volume VC. The volume swept out by piston, the difference between the maximum or total volume VT and the clearance volume, is called the displaced or swept volume VD. The majority of reciprocating engines operate on what is known as the four-stoke cycle. Each cylinder requires four strokes of its piston – two revolutions of the crankshaft – to complete the sequence of events which produces one power stroke. This four-stroke cycle comprises [3]:

Figure 2 The four-stroke operating cycle

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1. An intake stroke, which starts with the piston at TC and ends with the piston at BC, which

draws fresh mixture into the cylinder.

2. A compression stroke, when both valves are closed and the mixture inside the cylinder is

compressed to a small fraction of its initial volume. Toward the end of the compression

stroke, combustion in initiated and the cylinder pressure rises more rapidly.

3. A power stroke, or expansion stroke, which starts with the piston at TC and ends at BC as

the high temperature, high pressure gases push the piston down and force the crank to rotate.

As the piston approaches BC the exhaust valve opens to initiate the exhaust process and drop

the cylinder pressure to close to the exhaust pressure.

4. An exhaust stroke, where the remaining burned gases exit the cylinder: first because the

cylinder pressure may be substantially higher than the exhaust pressure; then as they are

swept out by the piston as it moves toward TC.

In this project our focus is the heat flux generated towards the piston during the power stroke process. The mathematical model of simulation is established initially, and the FEA is carries out by using COMSOL. The simulation is done with the following assumptions [2]:

- the effect of piston motion on the heat transfer is neglected,

- the rings and skirt are fully engulfed in oil and there are no cavitations,

- the conductive heat transfer in the oil film was neglected.

Base on the FEA analysis, the temperature and stress distributions of pistons are thus evaluated, which provide references for future piston redesign.

5. Definition and Conceptual Design Our IC engine design consults the dimensions for Lamborghini V12 6.0 Liter engine. The

design of our piston starts with the revolving of a basic sketch as shown in the Fig 3, which has a diameter of 90 mm and overall length of 62mm. Holes are opened transversely through piston for the pin which connects the piston with connecting rod after.

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Figure 3 Piston part and design sketch in Solidworks

In our cylinder design as shown in Fig 4, based on assumptions we set inner diameter to be 90 mm and overall length to be 130 mm, inside which gives a stroke length of 80 mm and fluid thickness of 5 mm (distance between top of the piston to cylinder head).

Figure 4 Cylinder part and design sketch in Solidworks

In addition, pins, connecting rods and crankshaft are modeled in Solidworks to provide a realistic motion constrain in case of necessary.

Figure 5 Pin, connecting rod and crankshaft in Solidworks

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Based on the design dimensions, our modeled V12 engine reaches a displacement of 372.7 cc which is exactly 6.0 Liter.

6. Preliminary Design For initial design considerations, a simple 2-D axisymmetric model was used (as

shown in Fig. 3 for preliminary analyses. The piston was essentially treated as a solid cylinder such that a plane strain condition could be implemented (εzz=0). To ascertain the possible material choices for the piston and cylinder, it was first necessary to perform some hand calculations to estimate the loading environment each component would be subjected to. With the use of Matlab, the following derivations were performed.

6.1 Hand (Matlab) Calculations of Thermal and Fluid Loads

Heat Flux during the Combustion Process: Conversion from gallons per mile to liters per kilometer:

5.88%&''()*

100-.'/*∗ 2.35

'.3/4* ∗ -.'/*

%&''()* ∗ 5-=13.82'.3/4*

1005-

Based on the assumed highway speed of 90 km/hr and the associated fuel efficiency at that speed of 13.82 liter/km, the resulting heat transfer was found to be:

13.82'.3/4

1005-∗29.889

1'.3/4∗905-

1ℎ4∗

1ℎ4

3600*= 1029859/*

With a piston radius of 45 mm and a piston cylinder height of 50 mm, the heat flux was determined to be 5.023 MW/m2.

=.*3()>?4@&A/B4/& = C4D + 2C4ℎ = C . 045 D + 2C . 045 . 061 = .02361-D

G =H/&3I4&)*@/4

B=102985

. 02361= 4.309

MW

-D

Heat Transfer Coefficient: The estimated heat transfer coefficient is found from Nusselt and Reynolds number, where a and m are coefficients related to the combustion process [4]:

L? = & M/ N,ℎP

5= &

QP

R

N

, ℎ =. 06

S-T

10.4

. 09-

8-*

. 09-

100U − 6-D

*

.WX

= 5419.3S

-DT

Plugging in values used from our model of the piston and cylinder for the bore length, and assuming a gas thermal conductivity of 0.06 W/mK, a diffusivity of 100E-6 m^2/s, and a gas velocity which is equal to the piston velocity (U = Upiston = 2*RPM*s/60 = 2*3000 rev/min*0.08 m/60 = 8m/s), we find a heat transfer coefficient of 5419.3 W/m^2K.

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Temperature between Gas and Piston: From the heat flux and heat transfer coefficient, we can find the temperature change between the gas and the piston. Assuming an initial temperature of the piston to be 300 C (573.15 K), the gas temperature is found to be approximately 1368.27 K:

G = ℎ∆I, ∆I =4.309U6

S

-D

5419.3S

-DT

= 795.12T, I%&* = I[.*3() +∆I = 573.15 + 795.12 = 1368.27T

See Matlab calculation in the Appendix. Pressure of Gas upon Ignition: To estimate the pressure of the air when it is ignited, the combustion process is modeled until it is ignited. A simple diagram of this cycle is shown below:

Figure 6: Pressure-Volume Diagram of Cycles

The first cycle (1-2) is during the intake stage, where the air coming in is at a constant pressure and temperature when the chamber is expanding. The second cycle (2-3) is an adiabatic process of compression, where the intake valve is closed and the gas is compressed, reducing the volume to its level at 1 and increasing the temperature and pressure of the gas. The final cycle shown (3-4) is the ignition, where the volume is kept constant and the gas is ignited, raising the temperature and pressure. To start, the gas is assumed to be at room ambient conditions (P1=101.325 kPa, T1= 300 K) and the volume is the piston at full displacement (V1=π*R2*l = π*0.0452*0.005=1.9x10-5 m3). The first cycle is assumed a constant pressure and temperature (P2=P1, T2=T1) as air comes in through a valve as the piston expands. The volume of the gas when the cylinder is fully retracted is 5.289x10-4 m3 (V2= π*0.0452*.085=5.289x10-4 m3). The second cycle (2-3) is an adiabatic compression of the gas until it is fully compressed. The piston is again at full displacement (V3=V1), and the pressure and temperature at full compression is shown below. Gamma is assumed to be 7/5 (diatomic gas).

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=2\2] = =3\3], =3 = 101.325U3 ∗5.28U − 4

WX

1.9U − 5WX

= 5.1748=&

=3\3

I3==2\2

I2, I3 =

5.1748=& ∗ 1.9U − 5-^ ∗ 300T

101.325U3=& ∗ 5.28U − 4-^= 901T

Once the gas is ignited (cycle 3-4), the gas temperature found above (1368.27K=T4) can be applied. Assuming a constant volume, the pressure applied can be calculated.

=3\3

I3==4\4

I4, =4 =

=3I4

I3= 5.1748=& ∗

1368.27K

901K= 7.868=&

See Matlab calculation in Appendix.

6.2 Two-Dimensional Thermoelastic Analysis

The required equations to perform a 2-D thermoelastic analysis are as follows: Equilibrium Equation:

ab,b − c?a + da = 0 In this case, for stationary components such as the cylinder, the acceleration term (?a) is zero and removed from the equation. Stress-Strain Relations:

ab = Aabefgef − hab I − Ii = jgee + 2kgab − hlabm where,

h =nU

1 − 2o&)pm = I − Ii

Strain-Displacement Relations:

gqr =1

2?a,b + ?b,a

With these 3 sets of equations, the displacements, strain and stress can all be found as function of the applied body, surface and thermal loads applied. Furthermore, these equations can be simplified for the case of an isotropic, elastic case, such as this on. This leads to the following equation: Navier’s Equation:

k∇D?a + j + k ?e,ea − hm,a + da − c?a = 0

6.3 Axisymmetric Segregated Fluid Flow and Heat Transfer

For this type of simulation, a weakly coupled COMSOL Multi-physics FEA was used to model the heat transfer process between the combustion gas and the combustion chamber components. For this type of analysis, the temperature distribution on the piston and cylinder are

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determined based on the pressure and heat flux that are present during the combustion process. Then the structural simulation is modeled using the previous determined thermal loads and strains to calculate the stresses in the components. This analysis is considered to be “Sequential Coupling”. For this process, COMSOL solves the control equations of fluids and structures at each time step and exchanges the calculation data to realize the coupling solution. The analysis treats each physical field as relatively independent, which eases the computational requirements for the solution of complex processes. This primary drawback of this analysis is that the effects of friction-driven heat transfer between the piston and cylinder are ignored. Although this assumption can alter the accuracy of the results, the heat generation from friction is essentially negligible when compared to that of the combustion process. Fluid Flow During the combustion process, the ignition of the fuel causes the rapid gas expansion which results in the piston moving one stroke length. To model the combustion process, the gas is given a flow velocity of 8 m/s acting uniformly on the top surface of the piston. The pressure in the chamber, which was calculated to be 7.5 MPa, was used in a coupled COMSOL Finite Element Analysis to model the fluid flow and structural response. In COMSOL, Fluid-Structural multi-physics allows for the velocity and pressure of the compressible gas to determine the primary stresses in the Piston and Cylinder. The results of the analysis are shown below in Figures 7 and 8.

Figure 7 and 8 Three-Dimensional deformed state and Two-Dimension undeformed state with Von Mises Stress

Distribution

The maximum stress found was approximately 35 MPa and was located at the sharp edges of the piston rings and top land. For the cylinder, the maximum stress level was found at the contact points between the piston and cylinder. The value of the Von Mises stress was 25 MPa. For this

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analysis, the pressure and velocity were considered to be uniform across the top of the piston and along the cylinder walls. Heat Transfer for Fluids and Solids

The heat transfer simulation of the combusted gas was split into two analyses, one during the initiation of the third stroke cycle and one at the end once the gas has fully expanded. Both models used a 2-D axisymmetric piston and cylinder with a turbulent gas flow. The Reynold’s number was calculated using the following equations:

M/ =?p

o= 7200 > 4000

where u is the gas velocity, v is the gas diffusivity, and d is the cylinder diameter. With this information, the following results were found for a fuel mixture of 14.7:1 ratio of air to gasoline, a piston composed of AlSi and a cast iron cylinder. Governing Equations: Turbulent Flow

∇ ∙ cv = 0

c v ∙ ∇w = ∇ ∙ −[x + k + ky ∇v + v∇ −2

3k + ky ∇ ∙ v x + z

Heat Transfer c{|v ∙ ∇I + ∇ ∙ } = ~

} = −5∇I Results: At the beginning of the combustion cycle, the heat flux generated from the gas combustion produces a maximum temperature in the gas of 2200 K. From convective heat transfer, this leads to a maximum temperature on the piston surface of 691 K.

Figure 9 Temperature distribution along the piston and cylinder

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Next, a model was created for the end of the third stroke cycle, once the gas has fully expanded. The underlying equations used were the same. For this scenario, the pressure and velocity distribution for the expansion of the fuel was analyzed and shown below in Figure 10.

Figure 10 Velocity Distribution (Left) and Pressure Gradient (Right) of the combusted gas

The resulting temperature gradient throughout the expanded gas is shown below in Figure 11. It can be seen that the cylinder and piston allow for the thermal diffusion of the high temperatures gas.

Figure 11 Thermal Gradient of expanded gas

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These preliminary results give an initial expectation of the thermal conditions the piston must be able to withstand. Based on the thermal properties of the material selected, AlSi alloy will be a good candidate for the desired application.

7. Detailed Design

7.1 Three-dimensional Solid Modeling

The major components to be used in FEA is shown in below:

Figure 12 Exploded and cross-sectional assembly views

A more detailed drawing of assembly’s cross-sectional view:

Figure 13 Detailed drawing of assembly’s cross-sectional view

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7.2 Analytical Heat Release Model

Wiebe Finite Heat Release Model With the known heat input of the gas during combustion, a Wiebe function can be used to determine the heat release as a function of the crankshaft angle. The Wiebe function, shown below, gives the mass fraction of fuel burned as a function of the crankshaft angle [3].

@ = 1 − exp −&m − mi

∆m

Ç

The Wiebe function gives the fraction of heat added (f) as a function of the crank angle, (θ), the crank angle where combustion begins (θ0), the combustion duration (Δθ), and constants (a and n), which are generally 5 and 3 for four-stroke engines [3]. Generally, the burn duration is 60 degrees. The burn fraction and rate plots as function of the crank angle are given in Figures 14 and 15.

Figures 14 and 15: Burn fraction and rate plots.

The heat release as a function of the crank angle is:

É~

Ém= ~aÇ

p@

pm

This parameter is used to ultimately find the change in pressure for the crankshaft angle. The equation used is:

p=

pm=5 − 1

\

É~

Ém−5=

\

p\

pm

where the volume of the cylinder is known and the change of volume is related to change of the position of the piston within in the cylinder. Using the derivations described by Ferguson et al., the gas pressure in the cylinder was modeled analytically [5]. The results are shown in Figure 16.

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Figure 16: Pressure vs. Crank Angle

Woschni Heat Transfer Coefficient One method for estimating the heat transfer coefficient is using the assumed correlation that:

L? = 0.035M/N Woschni used the assumption that k is proportional to T0.75, µ is proportional to T0.62 and p=ρRT. The final correlation can be expressed as [6]:

ℎ = 3.26PÑi.D[i.ÖIÑi.XXÜi.Ö where b is the bore and w is the averaged gas velocity within the cylinder. The average gas velocity for each stroke cycle is given as:

Ü =

6.18Q|aáàâÇ@(4ä)3&5/&)pUãℎ&?*3

2.28Q|aáàâÇ@(4{(-[4/**.()

2.28Q|aáàâÇ + 0.00324Ii\∆=i

\i=i@(4{(-P?*3.()

Using the results of the Non-Isothermal Flow analysis, the temperature and pressure of the gas at the piston head can be used to final the heat transfer coefficient at each stroke cycle. The corresponding values are given in Table 3.

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Figure 17: Temperature vs. Crank Angle

Table 3: Woschni Heat Transfer Coefficient Stroke Pressure Gas Temp Heat Transfer Coefficient

Intake 101 kPa 293K 150 (W/m2-K) Compression 4.1 MPa 510K 600 (W/m2-K) Combustion 7.86 MPa 2100K 2300 (W/m2-K) Exhaust 101 kPa 700K 200 (W/m2-K)

Using the heat transfer coefficients from the Woschni correlation and the heat release from the Wiebe model, the temperature of the gas at the corresponding crank angle was modeled. The analytic model gives a gas temperature of 2100K during the combustion cycle and 700K at the end of the expansion process. 7.3 Multi-physics Design Approach

The 3-D model of the piston and cylinder were imported from SolidWorks into COMSOL. First, a coupled Turbulent Fluid flow and Heat Transfer analysis was done for the fuel during the combustion and expansion process. A Non-Isothermal Fluid flow was used to model the fluid velocity and temperature distribution that will be present on the piston during this portion of the stroke cycle. This allows for the temperature to be variable in the fluid which led to a temperature gradient across the surface of the piston head. The results from this analysis were then utilized in a Thermomechanical analysis to determine the stresses generated by the fuel pressure and the convective heat flux during the fuel combustion. The four stroke steps were analyzed independently. The intake was considered to be stress free, the compression only dealt with the stresses from the pressurized gas, the combustion involved a thermomechanical analysis for the pressure and temperature, and the exhaust considered the remaining thermal stresses from the elevated fuel temperature. With this process, the critical stress area of the piston could be determined and analyzed later for creep, fatigue and fracture. 7.4 Fluid Flow and Heat Transfer Simulation: Non-Isothermal Flow

The Non-Isothermal Flow simulation couples heat transfer in the fluid with the turbulent fluid flow of the gas in the cylinder as it combusts and expands. The fluid properties are dependent on the temperature of the fluid. Also, it is necessary to show how the temperature of the fluid

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changes as it expands and receives convective cooling from the cylinder walls. The cylinder has water jackets which contain water coolant and cool the gas. The heat energy generated during the combustion of the fuel was used as an applied load along with the heat flux in the cylinder water jacket from the room temperature coolant. The final gas temperature of the fluid that is present on the piston head at the end of the combustion cycle is calculated. The results show that the fluid is 780K at the piston interface.

Figure 18: 3-D Cylinder, combusted gas and piston mesh

Figures 19 and 20: 3-D and 2-D Temperature distribution of combusted gas and cylinder

7.5 Thermomechanical Stress Analysis

With the results from the Non-Isothermal Flow model, the temperature gradient present on the piston was used along with the pressure from the combusted gas. To achieve a more accurate result, the convective heat transfer coefficient of the gas was calculated using the Woschni heat transfer model. Piston Mesh

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A 3-D mesh was generated using the COMSOL preprocessor and triangular elements. The resulting mesh is shown below in Figure 21.

Figure 21: Resulting Mesh of Piston

Applied Loads, Boundary Conditions and Multi-physics

A heat flux was placed on the head of the piston, which was a function of the calculated heat transfer coefficient and gas temperature. The pin hole where the piston is connected to the crankshaft was used as a fixed constraint. The Thermal Stress model in COMSOL was used to combine the Heat Transfer module with a Solid Mechanics module using Thermal Expansion and Temperature Coupling Multiphysics to compute the total stress in the piston. Results: Combustion

The Von Mises Stress distribution is shown in Figure 22, where the maximum stress was found to be 80 MPa on the circular pin holes. The temperature distribution within the piston, from Figure 24, gives a maximum temperature at the piston head of 670K and a temperature at the pin hole to be 500K.

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Figure 22: Von Mises Stress Distribution (Pa) Figure 23: Strain Distribution (mm/mm)

Figure 24: Temperature Distribution (K)

8. Failure Analysis The critical aspects of failure analysis for this application are creep, fatigue and fracture due to the cyclic thermal and mechanical loading on the piston. This study determined analytically the number of cycles to failure for the piston using the COMSOL FEA results. 8.1 Material Data and Design Limit

The relevant material properties for this fatigue analysis are displayed in Table 4. These values are based on previous experimental data for an AlSi alloy at various temperatures.

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Table 4: AlSi Alloy Fatigue Properties Temp (°C) Cyclic

Strength Coefficient K’ (N/mm2)

Cyclic Strain Hardening Exponent n’

Fatigue Strength Coefficient σ’f (MPa)

Fatigue Strength Exponent b

Fatigue Ductility Coefficient ε’f

Fatigue Ductility Exponent b

150 370 0.11 194 -0.054 0.013 -0.49 250 241 0.11 118 -0.054 0.0347 -0.49 350 104 0.11 44.7 -0.054 0.13 -0.49

Lifetime Requirements for the Piston Based on the US Department of Transportation’s data, the average driver travels about 13,500 miles per year. Using a lifetime of 10 years, the piston should be able to withstand 135,000 miles. The conversion of the number of miles to the number of cycles is given below.

LfaåçàaNç =é.*3&)A/{&4é4.o/)

I.4/{.A?-@/4/)A/ãI4&)*-.**.()M&3.(

2

For the Lamborghini, the front tire size is 255/35 R19, which corresponds to a 26 inch diameter. The Lamborghini Aventador has a 5th gear ratio of 1.18 and a final drive axle ratio of 5.06. Note that a single load cycle corresponds to two engine cycles. The final calculations are:

LfaåçàaNç = 135,000-.'/*63360.)

-.'/

14/o('?3.()

26 ∗ C.)Aℎ

1.18 ∗ 5.06

2= 3.13 ∗ 10ÖAèA'/*

8.2 Design Code Allowable Properties

Geometry

The design codes will be referenced off of Mahle GmbH “Piston Design Guidelines” for a 4 stroke cylinder engine. To start, a list of geometries acceptable for pistons is listed below, along with our dimensions [7]. Figure 25 shows referenced dimensions and Table 5 shows the guideline dimension ranges vs. the current modes

Figure 25: Piston guideline dimensions

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Table 5: Piston Guideline Dimensions vs. Current Dimensions [7] Current Piston Geometry (mm) Guideline Dimensions

Diameter [mm] 90 65-105 Total Height [GL/D] .677 0.6-0.7

Compression Height [KH/D] .34 0.3-0.45 Pin Diameter [BO/D] .24 0.2-0.26

Top Land Height [mm] 5 2-8 Second Ring Land Height [St/D] .05 0.040-0.055

Groove Height for First Piston Height [mm] 2 1.0-1.75 Skirt Length [SL/D] 0.5 0.4-0.5

Pin Boss Spacing [NA/D] .22 0.20-0.35 Crown Thickness [s/D] .1 0.06-0.1

The current piston follows the guidelines for the piston groove height. The piston groove height was changed to 1.5 mm to meet these guidelines.

8.3 Strength Safety Factors

The ultimate strength of AlSi is 154 MPa at 200°C. For the given maximum stress of 80 MPa, the strength safety factor n = 1.92. This value is too low to be considered acceptable for use in a high performance V12 engine.

8.4 Creep-Fatigue Safety Factors Creep-Fatigue Analysis The main cause of failure in the piston will to be a result of thermomechanical fatigue. The cyclic loading induced in the cylinder during the four stroke process will progressively cause damage accumulation. The total damage will be a function of both fatigue and creep. A lifetime assessment will be used to determine the number of cycles the piston can perform before the damage causes crack growth that leads to failure. Based on the results of the Finite Element Analysis, the stress-strain distributions in the piston during each step of the stroke (Intake, Compression, Power and Exhaust stroke) is known and can be used to calculate the mean and alternating stress. For this application, the high cycle fatigue is desired since the local cyclic stress is sufficiently smaller than the ultimate stress and the piston needs to be able to withstand a large number cycles to be effectively utilized in an automobile. Fatigue Analysis Under the uniaxial loading conditions, a Ramsberg-Osgood model can be used to model the stress-strain relations. The elastic-plastic behavior is given as:

g = gç + g| =`

U+

`

T

êÇ

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Where n is the strain hardening coefficient and K is the strength coefficient. Here the strain is a function of the elastic strain and the plastic strain. For the case of cyclic loading, the cyclic stress-strain relationship can be represented as [8]:

gë =ë

U+

ë

T′

ê

Çì The alternating stress, ë, is the difference between the maximum and minimum stress divided by two, as shown below:

ë =`Nëî − `NaÇ

2

Generally, S-N curves are used to relate the alternating stress (S) to the number of loading cycles to failure (N). The Basquin Equation states that the alternating stress is related to the number of load reversals (2Nf) by the following equation:

Ugë,çfëáàaï = ë = `ñå 2Låó

The material parameters, fatigue strength coefficient (`ñå) and fatigue strength exponent (b), are found from previous experimental research. This relationship is for high cycle fatigue, where elastic deformation is more significant than plastic deformations. Therefore the lifetime is a result of the material strength. For low cycle fatigue, plastic deformation is the dominant factor so the material’s ductility determines the lifetime of the material. The low cycle fatigue relationships are given as:

gë,|fëáàaï = gñå 2Låï

Figure 26 shows the correlation between the low and high cycle fatigue equations, where 2Nt is the point where the elastic and plastic strain are equal. If Nf < Nt, then low cycle fatigue is dominant. To combine the two equation, the total strain amplitude is found as the addition of the elastic and plastic strain amplitude [8]:

gë = gë,çfëáàaï + gë,|fëáàaï =`ñå

U 2Lå

ó+ gñå 2Lå

ï

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Figure 26: Strain Amplitude vs. Number Reversal Cycles to failure

The fatigue strength coefficient is temperature dependent and must be selected based on the operating temperature of the piston. The piston temperature is considered to be independent of the operating state. The stroke state changes at a high frequency so the temperature is assumed to be constant over the entire stroke and corresponds to the maximum temperature which occurs during combustion. The temperature variation is only transient during the initial start-up of the engine but will be essentially steady-state after a couple of cycles. Using the material fatigue properties, a S-N curve was made in Matlab to show the low cycle, high cycle and combined fatigue analytic models. The transition between low cycle and high cycle fatigue occurs at where the two are equal at the same strain amplitude value (2Nt). This corresponds to a value of 103 cycles (see Figure 27).

Figure 27: Strain Amplitude vs. Number of Reversals to failure

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The strain amplitude relationship to the alternating stress was plotted in Matlab using the Ramsberg-Osgood Model developed. The effects of strain hardening are included to give the elastic-plastic constitutive relations. Therefore, for the critical stress amplitude found in the COMSOL FEA, the corresponding strain amplitude and cycles to failure was found (see Figure 28).

Figure 28: Alternating Stress-Strain Relationship

For the alternating stress of 40 MPa, the strain amplitude was found to be 0.00045 mm/mm. This corresponds to a value of 2Nf equal to 2.7x108. Creep Analysis In order to model the creep damage on the piston over time, A model from Robinson [8] was used to accumulate the damage. The model is shown below, where t is the amount of time of exposure, and tf is the time until rupture.

3a

3åa= 1

a

To do this, the cycle will be modeled as high exposure and low exposure. The high exposure happens from the end of compression until after ignition and the low exposure occurs during the other parts of the cycle. The diagram in Figure 30 models the piston cylinder pressure as a function of the crank angle [10].

Figure 29: Cylinder Pressure on Piston

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Taking this model, we can approximate the pressure as a step function from approximately 0-100 degrees as high exposure, where the large loads are mechanical, and 100-720 degrees as low exposure, where the large loads are thermal. To determine the creep effect on the piston, the strain rate of the material from creep is determined. This is found from Norton’s Equation, shown below. The strain rate is assumed to be relatively constant for the majority of the fatigue cycle since it is in secondary stage.

This can be used along with the initial thermal and elastic strains to approximate the time until rupture.

gò + gy + gï = g Jin and Jong experimentally determined the strain rate at a stress of 40 MPa and a temperature of 300 C for AlSi alloys and found an approximate strain rate of 3E-5 [10]. Using an activation energy found from Dandrea-Lakes of 150 KJ/mol [10], and an estimated n value of 5.2 from Jin and Jong, the A constant can be found:

B = 3 ∗ 10ÑX/( 40 X.D ∗ /ã[êXi∗êiö

Ö.^êõ∗ ^iiúDW^.êX = 6.6 Then, the equivalent stress can be applied to this equation from our analysis, and the strain rate can be found using our equivalent temperature of 300 C.

g = 23.15X.D ∗ 6.6 ∗ exp−150 ∗ 10^

8.314 ∗ 300 + 273.15= 1.76 ∗ 10Ñù

Finally, time to rupture can be found by taking the relationship developed from Jin and Jong [12]:

3ûD

3ûê=

ê

D

Çü††úê

3ûD = 400 ∗ 60 ∗30

23.15

ù.D

= 119715* Estimating the cycles to failure:

Lå =3°a¢°

3û,°a¢°+3fâ£

3û,fâ£= 1

100

720∗ * ∗ 4 ∗

Lå

2Q|aáàâÇ= 3û

10

72∗1

50∗ Lå = 119715

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Lå,ï§çç| = 4.31 ∗ 10W Cumulative Damage For the case of creep-fatigue, the total damage can be found by combining the Palmgren-Miner (fatigue life) and Robinson rule (creep life). The resulting equation is as follows:

pa =

a

La

Låa+

3a

3åaa

= 1

a

Here, Ni is the number of cycles at the given alternating stress, Nfi is the number of cycles to failure at the given alternating stress, ti is the time spent at the stress-temperature combination, and tfi is the creep failure life.

The number of cycles to failure from creep and fatigue found from previous sections are used in this calculation to find the n value, which is the same under both conditions. A ratio of 10/72 of high stress and an estimated 50 cycles/ second is used in the creep cycles to relate the time at rupture to cycles to failure for creep. The cycles until failure in the AlSi is found to be 3.71x107 cycles. The MATLAB code shows the calculation in the Appendix.

)

L@ï§çç|+

)

L@åëàa¢•ç= 1, ) =

1

1L@ï§çç|

+1

L@åëàa¢•ç

8.5 Fracture Failure Assessment

To model the fracture of the piston, the COMSOL model shows the maximum stress occurring in the hole where the pin goes through the piston. A compressive stress from ignition and a reaction force from the pin will result in an axial stress in tension on the hole of the piston. The pin causes tension on the hole axially and the compressive pressure on the top of the piston will cause a hoop stress in tension. The member can then be assumed to be an infinite plate under axial tension (lambda=0) emanating from a circular hole. Figure 30 below from MAE296B Lecture 16 slides shown below are used to find a C1 value [13].

Figure 30: Finding Mode I Fracture Toughness Constant

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The infinite stress in must be found to find K1. The stress can be found assuming the piston acts as a thick pressure vessel with an external pressure applied of 7.5 MPa. A derivation of the axial stress shown with Lame’s Equation with only an external pressure applied leads to the equation below [14]:

¶ = −ß®§®

©

§®©Ñ§™

© = 7.5 ∗õX©

õX©Ñ^W©= 23.158=&

Assuming the radius of the hole will be much larger than the crack length (a/(R+a)=0), C1 will be initially at 3.365. Finally, with a crack size at minimum detectable length of 1 micrometer, a KI value can be found:

T´ = {´` C& = 3.365 ∗ 23.15 ∗ C ∗ 1 ∗ 10Ñù = .1388=& - The local stress can be assumed as well, with stress concentration factor of 3 assumed (transverse circular hole in tube in tension) [15]:

`Nëî = ë = Tà ÇâN

p

é=74

90= .82,

24

é=22

90= .24, {1 = 3, {2 = .94, {3 = 6.12, T3 = 3.825,

` = 3.825 ∗ 23.15 = 88.548=&

To determine how the material will fail (yielding or brittle fracture) maximum crack length values are calculated for each situation. Brittle fracture uses the KIc value, which is 25 MPa-m1/2 [16].

&Nëî =

T´ï¶{1

D

C=

2523.15 ∗ 2.1825

D

C= .078-

Since this number is very high, yielding is more likely. To find this value, the ratio of the max stress to the yield stress is equivalent to the initial area of the cross section to the final area of the cross section at which it will fail, which is determined by the crack length:

¨

`Nëî=

Bi

Båëaf•§ç

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31

&Nëî =Bi

3 ∗ è./'p4&3.(− Céa − 2 ∗ 0.022 = 0.0734-

Yielding will be the failure as the crack grows. Finally, the C1 value will change a large amount as the crack grows (a/R+a)= {0,0.75} , so an average value will be taken to determine the cycles until failure. The C1 value at amax =1, so the average=3.365+1/2=2.1825. The Paris Law integral can now be evaluated to get the cycles until failure. Assume a Cp value of 1x10-11 and an n value of 2.87 [17] and the alternating stress is 23.15 MPa.

p&

pL= {| ∆T

Ç = {| {1∆` C&Ç,

p&

&ÇD

= {|({ê∆`)ÇC

ÇD pL

Æå

ê

ëNëî

ëNaÇ

2

2 − 2.87∗ &

DÑD.ÖWD

. 0734

1U − 6= 2.1825D.ÖW ∗ 2.4 ∗ 10Ñêi ∗ 23.15U6D.ÖW ∗ C

D.ÖWD ∗ L@

Lå = 2.32 ∗ 10ÖAèA'/*

9. Final Design Optimization Through the design process, creep failure was the limiting factor. To optimize the piston design, a material change was made that focused on finding a material that a higher time to rupture than AlSi. Since aluminum alloys are recommended for high performance pistons, an aluminum alloy of 6061-T6 was selected. This is a common aluminum alloy and had a much higher rupture time than the AlSi, along with other more beneficial properties. Table 6 below shows these properties of Aluminum 6061-T6 compared to AlSi.

Table 6: 6061-T6 Vs. AlSi Properties Property AlSi Aluminum 6061-T6 [ASM] Thermal conductivity [W/m °C] 155 167 Thermal expansion 10−6 [1/°C] 21 25.2 Density [kg/m3] 2700 2700 Specific heat [J/kg °C] 960 896 Poisson's ratio 0.3 0.33 Young's modulus [GPa] 90 68.9 Yield Strength [MPa] 145 276 Fracture Toughness [MPa m1.2] 25 29

Strength Safety Factor

The ultimate strength of Aluminum 6061-T6 is 159 MPa at 200°C. For the given maximum stress of 75 MPa, the strength safety factor n = 2.12.

Creep

From design optimization change to 6061 Aluminum, the rupture time is tested at 4 ksi (27.6 MPa) at 550 and 600 F (287.78 and 315.56), getting average time to rupture values of 758 and 131.5 hrs [19]. Extrapolating between these values assuming a 1/ln(tr) relationship (from

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Norton’s equation above) an estimated time to rupture of 307 hours is calculated. The following cycles to failure is then 3.976x108 cycles. Fracture

From the design optimization, the material was changed to 6061-T6 aluminum alloy, which has paris law constants Cp of 3.7086 × 10−12 and an n value of 4.1908 [20]. Using the equations from above, This results in an amax value of 0.1049 m from brittle fracture, and 6.065x109 cycles to failure (Nf), higher than the calculated cycles to failure from the AlSi Alloy. See the supporting MATLAB code in appendix for both materials.

Damage Accumulation and Final Safety Factor

The damage accumulation resulted in 3.976x108 cycles to failure from creep-fatigue. This led to a final safety factor of 1.27.

) =&A3?&'AèA'/*3(@&.'?4/

é/*.4/pØ.@/3.-/=3.976 ∗ 10Ö

3.13 ∗ 10Ö= 1.27

Table 7: Al 6061-T6 Fatigue Properties Temp (°C) Cyclic

Strength Coefficient K’ (N/mm2)

Cyclic Strain Hardening Exponent n’

Fatigue Strength Coefficient σ’f (MPa)

Fatigue Strength Exponent b

Fatigue Ductility Coefficient ε’f

Fatigue Ductility Exponent b

20 424 0.089 383 -0.053 0.207 -0.628 100 401 0.089 365 -0.053 0.209 -0.628 200 320 0.089 202 -0.053 0.343 -0.628

Figures 31 and 32: Final Stress Distribution and Temperature Distribution

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10. Conclusions and Recommendations Based on the results of the segregated fluid flow and heat transfer Finite Element Analyses,

the models give a decent preliminary expectation of what the loading environment for the piston and cylinder will be. To continue this analysis, the thermoelastic stresses from a coupled fluid-thermal-structural analysis will be performed to get a more accurate determination of the stresses during one loading cycle. Then, a failure assessment will be implemented to model the effects of fatigue, fracture and creep. To further optimize this design, other materials will be considered to prevent against the effects of viscoplasticity. The utilization of Functionally Graded Materials (FGMs) or coatings to give better thermal performance will have to be analyzed to assess their efficacy. The main failure mode was due to creep, and aluminum alloys are proven to be a reliable material in piston design due their thermal properties, lightweight, and creep-fatigue properties. From this, a piston was designed to withstand the desired lifetime of cycles with a safety factor of 1.27 after design optimization.

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11. References [1] A. S. Mendes, Structural Analysis of the Aluminum Cylinder Head for a High-Speed Diesel Engine, SAE Paper 2007-01-2562 [2] Calbureanu, M., Malciu, R., Tutunea, D., Ionescu, A., & Lungu, M. (2013). Finite element modeling of a spark ignition engine piston head. Recent Advances in Fluid Mechanics and Heat & Mass Transfer, Recent Advances in Mechanical Engineering Series, (3), 61-64. [3] Heywood, J. B. (1988). Internal combustion engine fundamentals (Vol. 930). New York: Mcgraw-hill. [4] Kirkpatrick, K. Piston Heat Transfer. Colorado State University Internal Combustion Engines Web Page. [5] Ferguson, C. R. (1986). Internal combustion engines, applied thermosciences. New York: Wiley. [6] Woschni, 1967, "A universally applicable equation for the instantaneous heat transfer coefficient in the internal combustion engine", SAE Paper 670931 [7] Mahle. Piston Design Guidelines. Springer Fachmedien Wiesbaden GmbH 2012. [8] Shaha, S., Czerwinski, F., Kasprzak, W., Friedman, J., & Chen, D. (2015). Monotonic and cyclic deformation behavior of the Al–Si–Cu–Mg cast alloy with micro-additions of Ti, V and Zr. International Journal of Fatigue, 70, 383-394. doi:10.1016/j.ijfatigue.2014.08.001 [9] Robinson, E.L. (1952) Effect of temperature variation on the long-time rupture strength of steels. Trans. ASME, 74, 777 –781. [10] P S Shenoy and A Fatemi. Dynamic analysis of loads and stresses in connecting rods. Department of Mechanical, Industrial, and Manufacturing Engineering, The University of Toledo, Toledo, Ohio, USA. The manuscript was received on 25 June 2005 and was accepted after revision for publication on 6 February 2006. JMES105 # IMechE 2006 Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science [11] Song Hak Jin*, Jo Un Jong. Assessment of Thermal Cycling Creep Lifetime of Al-Si Alloys.

Department of Mechanics of Materials, Kim Chaek University of Technology, Pyongyang, Korea. International Journal of Mechanics and Applications 2015, 5(2): 41-47 DOI:

10.5923/j.mechanics.20150502.03 [12] Jay Christian Dandrea · Roderic Lakes. Creep and creep recovery of cast aluminum alloys. Received: 29 July 2008 / Accepted: 9 July 2009 / Published 28 July 2009 Mech Time- Depend Mater (2009) 13: 303–315

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[13] Ghoniem, Nasr. MAE 296 B Failure in Mechanical Design-II: High Temperature Components Lecture 16: Fracture Mechanics-II. UCLA [14] Thick Walled Cylinders. University of Washington. http://courses.washington.edu/me354a/Thick%20Walled%20Cylinders.pdf [15] Roark. Chapter 6 Stress Concentrations. Rensselaer Hartford. http://www.ewp.rpi.edu/hartford/~ernesto/Su2012/EP/MaterialsforStudents/Aiello/Roark- Ch06.pdf [16] Aluminum 413.0-F Die Casting Alloy. Matweb [17] Alberto Carpinteri, Marco Paggi. Are The Paris’ Law Parameters Dependent on each Other? Dipartimento di Ingegneria Strutturale e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24 – 10129 Torino. [18] Aluminum 6061-T6. ASM Aerospace Specification Metals Inc. Matweb [19] Kaufman, JG. Parametric Analyses of High-temperature Data for Aluminum Alloys. December 2008. [20] Ehsan Hedayati, Mohammad Vahedi. Using Extended Finite Element Method for Computation of the Stress Intensity Factor, Crack Growth Simulation and Predicting Fatigue Crack Growth in a Slant-Cracked Plate of 6061-T651 Aluminum. World Journal of Mechanics. Vol.4 No.1(2014) Article ID:42192,7 pages

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12. Appendices (Matlab Codes and Material Property Data) Appendix A: Matlab Heat Transfer Estimation: % Piston Heat Transfer % Heat Transfer Coefficient k=.06; %gas thermal conductivity (W/mK) b=.09; %bore cylindrical length mu=100*10^-6; %gas thermal diffusivity RPM=3000; %rotation speed (rev/min) stroke=.08; %stroke length (m) Upiston=2*RPM*stroke/60; %velocity of piston (m/s) a=10.4; %constants for internal combustion engine (ICE) m=3/4; %constants for internal combustion engine (ICE) h=a*k/b*(Upiston*b/mu)^m %heat transfer coefficient (W/m^2K) Re=Upiston*b/mu %reynolds number % Heat Flux eff=13.82/100; %fuel efficiency of car at 90 km/s (liters/km) gas_energy=29.8*10^6; %energy of gasoline (J/liter) car_velocity=90; %velocity of car (km/s) q=eff*gas_energy*car_velocity*1/3600; %rate of heat transfer (J/s) Area=pi()*.045^2+2*pi()*.045*.062; %effective surface area of piston (m^2) flux=q/Area %heat flux (W/m^2) delta_T=flux/h %Change in temperature (K)

Appendix B: Matlab Pressure at Ignition Estimation % Code to find Pressure at Ignition T1=300; % Initial Temperature V1=.005*pi()*.045^2; % Initial Volume P1=101.325*10^3;% Initial Pressure V2=.085*pi()*.045^2; % Volume at full expansion P2=P1; % Pressure and temperature assumed constant (intake) T2=T1; V3=V1; % Volume is at full compression again gamma=7/5; % diatomic gas constant P3=P2*V2^gamma/V3^gamma; % Pressure at full compression T3=P3*V3*T2/(P2*V2); % temperature at full compression T4=1368.27; % Temperature after ignition V4=V3; % Volume is constant after ignition P4=T4*P3/T3 % Pressure at ignition

Appendix C: Creep-Fatigue Damage Accumulation

%Damage Accumulation nf=2.7*10^8;% fatigue cycles to failure tf=119715; % creep time to rupture cycles_s=50; %amount of cycles per second high_stress_ratio=10/72; % ration during cycle when high stress is ocurring nf_creep=tf*cycles_s/high_stress_ratio; %calculate Nf for creep n_tot=1/(1/nf_creep+1/nf); %find cycles to failure Appendix D: Fracture and Paris Law

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%AlSi fracture C1=2.1825; %average C1 value for fracture p=7.5*10^6; %pressure applied Kf=3.825; %stress concentration factor Do=.09; %outer diameter t=.008;% thickness Di=Do-2*t; %inner diameter d=.022; %hole diameter sigma_max=23.15; %Axial Stress max sigma_min=0;%Axial Stress min sigma_yield=145;% yield strength a=1*10^(-6);% estimated min crack length K1_max=C1*sigma_max*sqrt(pi()*a); %max stress intensity factor value K1_min=C1*sigma_min*sqrt(pi()*a); %min stress intensity factor value C=1*10^-11; %paris law constant n=2.87; %paris law constant frac_tough=25*10^6; %fracture toughness amax1=(frac_tough/(C1*sigma_max*10^6))^2/pi(); %max crack length from fracture Ao=t*(pi()*Di-(2*d+a)); %initial area sigma_local=sigma_max*Kf;% max stress at hole yield_ratio=sigma_yield/sigma_local; %safety factor amax2=-(Ao/(yield_ratio*t)-pi()*Di)-2*d;%max crack length from yield, Nf=2/(2-n)*(amax1^((2-n)/2)-a^((2-n)/2))/(C1^n*C*sigma_max^n*pi()^(n/2)); %cycles to failure %6061 fracture C1=2.1825; %average C1 value for fracture p=7.5*10^6; %pressure applied Kf=3.825; %stress concentration factor Do=.09; %outer diameter t=.008; % thickness Di=Do-2*t; %inner diameter d=.022; %hole diameter sigma_max=23.15; %Axial Stress max sigma_min=0;%Axial Stress min sigma_yield=276;% yield strength a=1*10^(-6);% estimated min crack length K1_max=C1*sigma_max*sqrt(pi()*a); %max stress intensity factor value K1_min=C1*sigma_min*sqrt(pi()*a); %min stress intensity factor value C=3.7086*10^-12; %paris law constant n=4.1908; %paris law constant frac_tough=29*10^6; %fracture toughness amax1=(frac_tough/(C1*sigma_max*10^6))^2/pi(); %max crack length from fracture Ao=t*(pi()*Di-(2*d+a)); %initial area sigma_local=sigma_max*Kf;% max stress at hole yield_ratio=sigma_yield/sigma_local; %safety factor amax2=-(Ao/(yield_ratio*t)-pi()*Di)-2*d;%max crack length from yield, Nf=2/(2-n)*(amax1^((2-n)/2)-a^((2-n)/2))/(C1^n*C*sigma_max^n*pi()^(n/2)); %cycles to failure

Appendix E: Gas Heat Release and Temperature

function [] = WoschniHeatTransfer()

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% Gas cycle heat release code with Woschni heat transfer clear(); thetas = -20; % start of heat release (deg) thetad = 40; % duration of heat release (deg) r =10; % compression ratio gamma = 1.4; % gas const Q = 27.22; % dimensionless total heat release beta = 2.22; % dimensionless volume a = 5; % weibe parameter a n = 3; % weibe exponent n omega =314.1; % engine speed rad/s c = 0; % mass loss coeff s = 0.1; % stroke (m) b = 0.1; % bore (m) T_bdc = 300; % temp at bdc (K) tw = 1.2; % dimensionless cylinder wall temp P_bdc = 100; % pressure at bdc (kPa) Up = s*omega/pi; % mean piston speed (m/s) step=1; % crankangle interval for calculation/plot NN=360/step; % number of data points theta = -180; % initial crankangle thetae = theta + step; % final crankangle in step % initialize results data structure save.theta=zeros(NN,1); save.vol=zeros(NN,1); % volume save.press=zeros(NN,1); % pressure save.work=zeros(NN,1); % work save.heatloss=zeros(NN,1); % heat loss save.mass=zeros(NN,1); % mass left save.htcoeff=zeros(NN,1); % heat transfer coeff save.heatflux=zeros(NN,1); % heat flux (W/m^2) fy=zeros(4,1); % vector for calulated pressure, work, heat and mass loss fy(1) = 1; % initial pressure (P/P_bdc) fy(4) = 1; % initial mass (-) %for loop for pressure and work calculation for i=1:NN, [fy, vol, ht,hflux] = integrate_ht(theta,thetae,fy); % print values % fprintf('%7.1f %7.2f %7.2f %7.2f \n', theta,vol,fy(1),fy(2),fy(3)); % reset to next interval theta = thetae; thetae = theta+step; save.theta(i)=theta; % put results in output vectors save.vol(i)=vol; save.press(i)=fy(1); save.work(i)=fy(2); save.heatloss(i)=fy(3); save.mass(i)=fy(4); save.htcoeff(i)=ht; save.hflux(i)=hflux;

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end % end of pressure and work for loop [pmax, id_max] = max(save.press(:,1)); % find max pressure thmax=save.theta(id_max); % and crank angle ptdc=save.press(NN/2)/pmax; w=save.work(NN,1); % w is cumulative work massloss =1- save.mass(NN,1); eta=w/Q; % thermal efficiency imep = eta*Q*(r/(r -1)); %imep/P1V1 eta_rat = eta/(1-r^(1-gamma)); % output overall results fprintf(' Weibe Heat Release with Heat and Mass Loss \n'); fprintf(' Theta_start = %5.2f \n', thetas); fprintf(' Theta_dur = %5.2f \n', thetad); fprintf(' P_max/P1 = %5.2f \n', pmax); fprintf(' Theta @P_max = %7.1f \n',thmax); fprintf(' Net Work/P1V1 = %7.2f \n', w); fprintf(' Heat Loss/P1V1 = %7.2f \n', save.heatloss(NN,1)); fprintf(' Mass Loss/m = %7.3f \n',massloss ); fprintf(' Efficiency = %5.2f \n', eta); fprintf(' Eff./Eff. Otto = %5.2f \n', eta_rat); fprintf(' Imep/P1 = %5.2f \n', imep); %plot results figure(); plot(save.theta,save.work,'-',save.theta,save.heatloss,'--','linewidth',2 ) set(gca, 'Xlim',[-180 180],'fontsize', 18,'linewidth',1.5); hleg1=legend('Work', 'Heat Loss','Location','NorthWest'); set(hleg1,'Box', 'off') xlabel('Crank Angle \theta (deg)','fontsize', 18) ylabel('Cumulative Work and Heat Loss','fontsize', 18) print -deps2 WoschiWorkHeat plot(save.theta,save.press,'-','linewidth',2 ) set(gca, 'fontsize', 18,'linewidth',1.5,'Xlim', [-180 180]); xlabel('Crank Angle (deg)','fontsize', 18) ylabel('Pressure (bar)','fontsize', 18) print -deps2 WoschniP figure(); plot(save.theta,save.htcoeff,'-','linewidth',2 ) set(gca, 'fontsize', 18,'linewidth',1.5,'Xlim', [-180 180]); %legend(' Work', ' Heat Loss','Location','NorthWest') xlabel('Crank Angle \theta (deg)','fontsize', 18) ylabel('Heat transfer coefficient h (W/m^2-K)','fontsize', 18) print -deps2 Woschnihtcoeff figure(); plot(save.theta,save.hflux,'-','linewidth',2 ) set(gca, 'fontsize', 18,'linewidth',1.5,'Xlim', [-180 180]); %legend(' Work', ' Heat Loss','Location','NorthWest') xlabel('Crank Angle \theta (deg)','fontsize', 18) ylabel('Heat flux q{"} (MW/m^2)','fontsize', 18) print -deps2 Woschniheatflux

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function[fy,vol,ht, hflux] = integrate_ht(theta,thetae,fy) % ode23 integration of the pressure differential equation % from theta to thetae with current values of fy as initial conditions [tt, yy] = ode23(@rates, [theta thetae], fy); % put last element of yy into fy vector for j=1:4 fy(j) = yy(length(tt),j); end % pressure differential equation function [yprime] = rates(theta,fy) vol=(1.+ (r -1)/2.*(1-cosd(theta)))/r; dvol=(r - 1)/2.*sind(theta)/r*pi/180.; %dvol/dtheta dx=0.; if(theta>thetas) % heat release >0 dum1=(theta -thetas)/thetad; x=1-exp(-(a*dum1^n)); dx=(1-x)*a*n*dum1^(n-1)/thetad; %dx/dthetha end P=P_bdc*fy(1); %P(theta) (kPa) T=T_bdc*fy(1)*vol; % T(theta) (K) term4=T_bdc*(r-1)*(fy(1)-vol^(-gamma))/r; % comb. vel. increase U=2.28*Up + 0.00324*term4; % Woschni vel (m/s) ht = 3.26 *P^(0.8)*U^(0.8)*b^(-0.2)*T^(-0.55); %Woschni ht coeff hflux=ht*T_bdc*(fy(1)*vol/fy(4) - tw)/10^6; %heat flux MW/m^2 h = ht*T_bdc*4/(1000*P_bdc*omega*beta*b); %dimensionless ht coeff term1= -gamma*fy(1)*dvol/vol; term3= h*(1. + beta*vol)*(fy(1)*vol/fy(4) - tw)*pi/180.; term2= (gamma-1)/vol*(Q*dx - term3); yprime(1,1)= term1 + term2 - gamma*c/omega*fy(1)*pi/180; yprime(2,1)= fy(1)*dvol; yprime(3,1)= term3; yprime(4,1)= -c*fy(4)/omega*pi/180; end %end of function rates end % end of function integrate_ht end % end of function HeatReleaseHeatTransfer