Modeling the creep damage of P91 steel using peridynamics_IMECE2019 - - PDF document

Modeling the creep damage of P91 steel using peridynamics_IMECE2019 presentation

Presentation · November 2019

15

3 authors, including: Some of the authors of this publication are also working on these related projects: Study of damping properties of composite polymers View project Coupling of peridynamics with finite element methods: Problems and Solutions View project Shank Kulkarni Pacific Northwest National Laboratory

Xiaonan Wang ANSYS

All content following this page was uploaded by Shank Kulkarni on 15 November 2019.

Modeling the creep damage of P91 steel using peridynamics

ASME 2019 International Mechanical Engineering Congress and Exposition

Shank S. Kulkarni †,# Alireza Tabarraei † Xiaonan Wang †,*

November 11-14, 2019 Salt Lake City, UT, USA

†Department of Mechanical engineering and engineering science,

University of North Carolina at Charlotte.

#Pacific Northwest National Laboratory, Richland, Washington. *ANSYS corp., Canonsburg, Pennsylvania.

Presentation outline

• 1. Introduction/ Motivation
• 2. Peridynamic theory
• 3. Creep formulation
• 4. Numerical implementation
• 5. Results and discussion
• 6. Conclusion

2

Introduction to Creep

■ Time-dependent deformation under certain applied load. ■ More severe at high temperatures. ■ Plays an important role in deciding life of components such as: pipes in power plants, turbine blades, heat exchangers etc. ■ Three basic stages:

– Primary – Secondary – Tertiary

3

Primary creep Secondary creep Tertiary creep time (t) strain (ɛ) tf ɛf fracture

} Elastic strain

𝜖2ε 𝜖t2 < 0 𝜖2ε 𝜖t2 > 0 𝜖2ε 𝜖t2 = 0

Failure due to creep of turbine blade [1]

[1] M. S. Haque, An improved Sin-hyperbolic constitutive model for creep deformation and damage. University of Texas, 2015

Liu- Morakami creep damage model

■ Fairly accurate for secondary as well as tertiary creep. ■ Damage parameter: 𝐸

4

No damage

0 < 𝐸 < 1

Failure Material constant

ሶ 𝜁𝑗𝑘

𝑑 = 3

2 𝐷𝑛𝜏𝑓𝑟

𝑜2 𝑇𝑗𝑘

𝜏𝑓𝑟 𝑓𝑦𝑞 2(𝑜2 + 1) 𝜌 1 + ൗ 3 𝑜2 𝜏1 𝜏𝑓𝑟 𝐸 ൗ

3 2

𝑇𝑗𝑘 = Deviatoric stress 𝜏𝑓𝑟 = von Mises stress 𝜏1 = max principle stress

𝛽 is multi-axiality parameter, for uniaxial condition 𝛽 = 0

𝑒𝐸 𝑒𝑢 = 𝐸𝑛 1 − 𝑓−𝑟2 𝑟2 𝜏𝑠

𝑞𝑓𝑟2𝐸

Material constants (for tertiary creep)

𝜏𝑠 = 𝛽𝜏1 + (1 − 𝛽)𝜏𝑓𝑟

Need of a new method

■ Simulating crack propagation is an extremely challenging task. ■ Many issues

– Crack always propagate along element boundaries – Remeshing

■ Strong numerical method was needed which can predict crack growth accurately with ease.

5

Some other methods:

■ Extended FEM

– Introduced by Belytsckho in 1999 – Crack propagation is modeled using enrichment functions – Challenging in simulating large number of cracks together

■ Meshless Methods

– Approximation is built at nodes only – Examples: Smooth particle hydrodynamics Element free Galerkin Reproducing kernel particle – Requires higher order integration schemes

6

[1] [2]

[1] Belytschko T, Gracie R, Ventura G. A review of extended/generalized finite element methods for material

• modeling. Modelling and Simulation in Materials Science and Engineering. 2009 Apr 2;17(4):043001.

[2] Chen JS, Hillman M, Chi SW. Meshfree methods: progress made after 20 years. Journal of Engineering

• Mechanics. 2017 Jan 23;143(4):04017001.

And hence… Peridynamics!

■ A non-local continuum mechanics formulation developed in 2000 by Silling [1]. ■ Main purpose is to model complex fracture patterns and crack growth. ■ Peridynamic theory is valid in the presence of discontinuities in displacement field.

7

Vs. 𝜍 ሷ 𝒗 = න

𝐼

𝑔 𝒗 − 𝒗, 𝒚′ − 𝒚, 𝑢 𝑒𝑊

𝑦′ + 𝑐 𝒚, 𝑢

𝜍 ሷ 𝒗 = 𝛼 ∙ 𝜏(𝒚, 𝑢) + 𝑐 𝒚, 𝑢

Theoretical background

■ Any point x interacts directly with other points within a distance δ called ‘Horizon’. ■ The material within distance δ is called family of x : ■ Peridynamic equation of motion: ■ Where, force function f is:

8

𝜍 𝜖2𝒗 𝒚, 𝑢 𝜖𝑢2 = න

𝐼

𝑔 𝒗 𝒚′, 𝑢 − 𝒗 𝒚, 𝑢 , 𝒚′ − 𝒚, 𝑢 𝑒𝑊

𝑦′ + 𝑐 𝒚, 𝑢

𝐼 = 𝒚′ ∈ ℜ: 𝒚′ − 𝒚 < 𝜀

Peridynamic domain before and after deformation.

stretch Relative position Relative displacement 𝜽 = 𝒗 𝒚′, 𝑢 − 𝒗 𝒚, 𝑢 𝑡 = 𝝄 + 𝜽 − 𝝄 𝝄 𝝄 = 𝒚′ − 𝒚 𝑔 𝝄, 𝜽 = 𝝄 + 𝜽 𝝄 + 𝜽 𝑑0𝑡 Micro- modulus function 𝑑 𝝄 = ቐ 2𝐹 𝜀2𝐵 , 𝝄 ≤ 𝜀 0, 𝝄 > 𝜀

State based PD

■ Developed in order to overcome following limitations on bond based PD

– Limitation on Poisson ratio – Difficulty of transforming material model from classical mechanics to PD

9

X' X fxx'

X' X fxx'

ρ ሷ u 𝐲, t = න

H

𝐔 𝐲, t 𝐲′ − 𝐲 − 𝐔 𝐲′, t x − 𝐲′ dVξ + 𝐜 𝐲, t 𝜍 ሷ u 𝐲, t = න

𝐼

𝑔 𝒗 𝒚′, 𝑢 − 𝒗 𝒚, 𝑢 , 𝒚′ − 𝒚, 𝑢 𝑒𝑊

𝑦′ + 𝑐 𝒚, 𝑢

Force vector state Equivalent to second

• rder tensor

State based PD

𝐔 at each 𝐲 is a function of deformation of all bonds connected at 𝐲

Bond based PD

Dynamic to static conversion

■ PD is dynamic formulation by nature. ■ Specific attention has to be paid in order to solve static problems. ■ Some methods are:

– Stability estimation – Energy minimization method – Dynamic relaxation method – Adaptive dynamic relaxation method

10

𝜍 𝑦 ሷ 𝑣 𝑦, 𝑢 = න

𝐼

𝑔

𝑞 𝑗𝑜𝑢 𝑦, 𝑢 + 𝑐(𝑦, 𝑢)

𝜍 𝑦 ሷ 𝑣 𝑦, 𝑢 + Λ 𝜍 𝑦 ሶ 𝑣 𝑦, 𝑢 = න

𝐼

𝑔

𝑞 𝑗𝑜𝑢 𝑦, 𝑢 + 𝑐(𝑦, 𝑢)

Numerical implementation

■ Discretization of domain ■ Apply boundary conditions ■ Carry out static analysis ■ For every time increment: – Calculate 𝜏𝑓𝑟, 𝜏1 and 𝜏𝑠 – Using Liu-Morakami calculate increment in creep and damage at every point – Update E – Evaluate force vector – Carry out static analysis again ■ Update time till D reaches 1

11

Discretization (Mesh) Apply BC Calculate stress and 𝜏𝑓𝑟, 𝜏1 and 𝜏𝑠 Calculate 𝜁𝑗𝑘

𝑑 and D

End Start Static analysis Loop over time (t = ∆t, …,n∆t) Static analysis with updated values Evaluate force vector and apply force Update E’ = E(1-D)

Flow chart showing creep damage model algorithm

12

 Evaluate stress at each gauss point.  Evaluate increment in creep strain in damage at each gauss point.  Calculate force required to achieve the incremented creep strain by:  Apply this force at each node.  Repeat the process for all elements.  Solve equilibrium equation again.

FEM vs PD

 Evaluate stress at each node.  Evaluate increment in creep strain in damage at each node.  Calculate force required to achieve the incremented creep strain by:  Apply this force at each neighbor node.  Repeat the process for all neighbors then repeat for all nodes.  Solve equilibrium equation again.

(Δ𝑔𝑑)𝑢= න

Ω

𝑪𝑈𝑫Damage

𝑓𝑞

Δ𝜁𝑑dΩ (Δ𝑔𝑑)𝑢= 𝜕𝑫Damage

𝑓𝑞

Δ𝜁𝑑𝑳−1𝜊

Influence function and damage

■ Influence function: ■ Bond breaking in peridynamics:

– Due to elongation beyond certain limit.

■ Bond breaking in peridynamics creep model:

– Due to elongation beyond certain limit. – Damage parameter of point x reached critical value. – Damage parameter of point x’ reached critical value.

13

𝜍 ሷ u 𝐲, t = න

𝐼

ω( ξ ) 𝑔 𝒗 𝒚′, 𝑢 − 𝒗 𝒚, 𝑢 , 𝒚′ − 𝒚, 𝑢 𝑒𝑊

𝑦′ + 𝑐 𝒚, 𝑢

x x’ ξ

D ≥ 1 D′ ≥ 1 ω( ξ )

Influence function ൯ ෝ ω ξ , 𝐫, 𝐫′ = 𝜕𝜊( ξ )ωd(D, D′ Damage parameters at 𝒚 and 𝒚′

ωd D, D′ = 0 if {D = 1 or D′ = 1}

Influence function

■ Influence function is chosen as:

14

𝜕𝜊 ξ = 𝑓−|𝜊|2

𝑚2

[1] Kilic, B. and Madenci, E., 2010. An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theoretical and Applied Fracture Mechanics, 53(3), pp.194-204.

[1]

Selection of damping coefficient

■ Material parameters:

– Young’s Modulus: 148 GPa – Poison's ratio: 0.3

■ Aim:

– To check accuracy of code – To select optimum damping coefficient

15

Parametric study in order to chose optimum value

• f damping coefficient Λ.

Comparison of results from PD and FEM Problem geometry

Material properties

16

Material : P91 steel at 650° C

E (GPa) 132 𝝋 0.3 𝑫𝒏 1.092 × 10−20 𝒐𝟑 8.462 𝑬𝒏 2.952 × 10−16 𝒒 6.789 𝒓𝟑 3.2 𝜷

• 0. 215

[1] Hyde, T. H., M. Saber, and W. Sun. "Testing and modelling of creep crack growth in compact tension specimens from a P91 weld at 650 C." Engineering Fracture Mechanics 77.15 (2010): 2946-2957.

[1]

Creep parameters Elastic parameters

Examples: Uniaxial tension (plane strain)

17

■ Dimension: 10 mm × 100 mm ■ Mesh size: 0.5 mm ■ Applied load: 70, 82, 87, 93, 100 MPa

Displacement of point A with respect to pseudo time

Results

18

Variation of creep strain versus time

Examples: 3D Uniaxial tension

19

■ Dimension: 10 mm × 10 mm × 100 mm ■ Mesh size: 0.5 mm ■ Applied load: 240 MPa, 260 MPa, 280 MPa and 300 MPa

Geometry of 3D uniaxial tension specimen Comparison of creep strain using PD model and experiment for different loading values.

Material : 316 steel at 600° C

Examples: Bending

20

Geometry of plate Results (a) creep strain and (b) Damage parameter

Material : 316 steel at 600° C ■ Dimension: 10 mm × 40 mm ■ Mesh size: 0.5 mm ■ Applied load: 100 N

Conclusions

■ Peridynamic formulation to predict the creep in metals using stress based damage model was presented. ■ Liu-Murakami creep damage model was selected due to its accuracy along with less computational cost. ■ Static solution was obtained using dynamic relaxation method. ■ Different creep problem were solved using new model and results are in good agreement with finite element solutions as well as experiments.

21

Future challenges

■ Simulation of crack propagation due to creep. ■ Skin effect near crack tip and newly formed crack surface is still a challenge. ■ Formulation can be extended to finite deformations. ■ Effect of plasticity can be studied.

22

References

23

1. Shank S. Kulkarni, Alireza Tabarraei. "An analytical study of wave propagation in a peridynamic bar with nonuniform discretization." Engineering Fracture Mechanics, Vol: 190, pp: 347-366, 2018.

• 2. Shank S. Kulkarni, Alireza Tabarraei. "A stochastic analysis of the damping property of filled elastomers." Macromolecular Theory and

Design, vol 28 (2), 1800062, 2019. 3. Xiaonan Wang, Shank S. Kulkarni, Alireza Tabarraei.”Concurrent coupling of peridynamics and classical elasticity for elastodynamic problems.” Computer Methods in Applied Mechanics and Engineering, Vol:344, pp: 251-275, 2019.

• 4. Shank S. Kulkarni, Xiaonan Wang, Alireza Tabarraei.”State based peridynamic formulation for metal creep using classical creep

damage models.” (Manuscript under preparation).

• 5. Shank S. Kulkarni, Alireza Tabarraei. "Study of damping properties of polymer matrix composites through wave propagation." ASME 2017

International Mechanical Engineering Congress and Exposition, Tampa Florida, USA, Nov-2017, pp: V001T03A018.

• 6. Shank S. Kulkarni, Alireza Tabarraei, Xiaonan Wang. “Study of Spurious wave reflection at the interface of Peridynamics and Finite element

region.” ASME 2018 International Mechanical Engineering Congress and Exposition, Pittsburgh PA, USA, Nov-2018, pp: V009T12A054.

• 7. Shank S. Kulkarni, Alireza Tabarraei, Pratik Ghag. “A Finite element approach for study of wave attenuation characteristics of Epoxy polymer

composite.” ASME 2018 International Mechanical Engineering Congress and Exposition, Pittsburgh PA, USA, Nov-2018, pp: V009T12A042.

• 8. Shank S. Kulkarni, Alireza Tabarraei, Pratik Ghag. "Effect of properties of interphase layer on damping properties of polymer composites using

sensitivity analysis." ASME 2019 International Mechanical Engineering Congress and Exposition, Salt Lake City, Utah, USA, 2019. 9. Xiaonan Wang, Shank S. Kulkarni, Alireza Tabarraei. “Seamless coupling of peridynamics and finite element method in commercial software of finite element to solve elasto-dynamic problems.” ASME 2019 International Mechanical Engineering Congress and Exposition, Salt Lake City, Utah, USA, 2019.

• 10. Shank S. Kulkarni, Alireza Tabarraei. “A nonlinear visco-hyper elastic constitutive model for modeling behavior of polyurea at large deformations.”

ASME 2019 International Mechanical Engineering Congress and Exposition, Salt Lake City, Utah, USA, 2019.

• 11. Shank S. Kulkarni, Xiaonan Wang, Alireza Tabarraei. "Modeling the creep damage of P91 steel using peridynamics." ASME 2019

International Mechanical Engineering Congress and Exposition, Salt Lake City, Utah, USA, 2019.

• 12. Shank S. Kulkarni, Alireza Tabarraei, Satyam Shukla. “Study of the effect of carbon nano-tube waviness and volume fraction on the damping

property of a polymer composite.” ASME 2019 International Mechanical Engineering Congress and Exposition, Salt Lake City, Utah, USA, 2019.

A Big Thank You!

Any Questions?

You can find me at: Shank.Kulkarni@pnnl.gov skulka17@uncc.edu Ph: 770-380-7100