See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/337290782 Modeling the creep damage of P91 steel using peridynamics_IMECE2019 presentation Presentation · November 2019 DOI: 10.13140/RG.2.2.30012.80002 CITATIONS READS 0 15 3 authors , including: Shank Kulkarni Xiaonan Wang Pacific Northwest National Laboratory ANSYS 50 PUBLICATIONS 128 CITATIONS 18 PUBLICATIONS 189 CITATIONS SEE PROFILE SEE PROFILE 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 All content following this page was uploaded by Shank Kulkarni on 15 November 2019. The user has requested enhancement of the downloaded file.
November 11-14, 2019 Salt Lake City, UT, USA ASME 2019 International Mechanical Engineering Congress and Exposition Modeling the creep damage of P91 steel using peridynamics Shank S. Kulkarni †,# Alireza Tabarraei † Xiaonan Wang †, * † 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 Failure due to creep of turbine blade [1] life of components such as: pipes in power plants, turbine blades, heat exchangers etc. fracture ɛ f 𝜖 2 ε 𝜖t 2 > 0 𝜖 2 ε 𝜖 2 ε 𝜖t 2 < 0 𝜖t 2 = 0 ■ Three basic stages: strain ( ɛ ) Tertiary creep Primary creep – Primary Secondary creep – Secondary } Elastic strain – Tertiary t f time (t) 3 [1] M. S. Haque, An improved Sin-hyperbolic constitutive model for creep deformation and damage. University of Texas, 2015
ሶ Liu- Morakami creep damage model ■ Fairly accurate for secondary as well as tertiary creep. Material constants (for tertiary creep) ■ Damage parameter: 𝐸 1 − 𝑓 −𝑟 2 𝑒𝐸 𝑞 𝑓 𝑟 2 𝐸 𝑒𝑢 = 𝐸 𝑛 𝜏 𝑠 𝑟 2 0 < 𝐸 < 1 No damage Failure 𝜏 𝑠 = 𝛽𝜏 1 + (1 − 𝛽)𝜏 𝑓𝑟 Material constant 𝑑 = 3 𝑜 2 𝑇 𝑗𝑘 2(𝑜 2 + 1) 𝜏 1 𝛽 is multi-axiality parameter, 3 2 2 𝐷 𝑛 𝜏 𝑓𝑟 𝐸 ൗ 𝜁 𝑗𝑘 𝑓𝑦𝑞 𝜏 𝑓𝑟 𝜏 𝑓𝑟 3 𝑜 2 for uniaxial condition 𝛽 = 0 𝜌 1 + ൗ 𝑇 𝑗𝑘 = Deviatoric stress 𝜏 𝑓𝑟 = von Mises stress 𝜏 1 = max principle stress 4
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 [1] ■ Meshless Methods – Approximation is built at nodes only – Examples: Smooth particle hydrodynamics Element free Galerkin Reproducing kernel particle – Requires higher order integration schemes [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. 6
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. 𝑔 𝒗 − 𝒗, 𝒚 ′ − 𝒚, 𝑢 𝑒𝑊 Vs. 𝜍 ሷ 𝒗 = න 𝑦 ′ + 𝑐 𝒚, 𝑢 𝜍 ሷ 𝒗 = 𝛼 ∙ 𝜏(𝒚, 𝑢) + 𝑐 𝒚, 𝑢 𝐼 7
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 : 𝐼 = 𝒚 ′ ∈ ℜ: 𝒚 ′ − 𝒚 < 𝜀 ■ Peridynamic equation of motion: Peridynamic domain before and after deformation. 𝜍 𝜖 2 𝒗 𝒚, 𝑢 𝑔 𝒗 𝒚 ′ , 𝑢 − 𝒗 𝒚, 𝑢 , 𝒚 ′ − 𝒚, 𝑢 𝑒𝑊 = න 𝑦 ′ + 𝑐 𝒚, 𝑢 𝜖𝑢 2 𝐼 ■ Where, force function f is: 𝑔 𝝄, 𝜽 = 𝝄 + 𝜽 𝝄 + 𝜽 𝑑 0 𝑡 stretch Relative position 𝑡 = 𝝄 + 𝜽 − 𝝄 Micro- modulus 𝝄 Relative 𝝄 = 𝒚 ′ − 𝒚 function displacement 2𝐹 𝜀 2 𝐵 , 𝝄 ≤ 𝜀 𝑑 𝝄 = ቐ 𝜽 = 𝒗 𝒚 ′ , 𝑢 − 𝒗 𝒚, 𝑢 0, 𝝄 > 𝜀 8
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 X' Bond based PD 𝑔 𝒗 𝒚 ′ , 𝑢 − 𝒗 𝒚, 𝑢 , 𝒚 ′ − 𝒚, 𝑢 𝑒𝑊 𝜍 ሷ u 𝐲, t = න 𝑦 ′ + 𝑐 𝒚, 𝑢 f xx' 𝐼 X Equivalent to second order tensor X' State based PD Force vector state f xx' 𝐔 𝐲, t 𝐲 ′ − 𝐲 − 𝐔 𝐲 ′ , t x − 𝐲 ′ ρ ሷ u 𝐲, t = න dV ξ + 𝐜 𝐲, t H X 𝐔 at each 𝐲 is a function of deformation of all bonds connected at 𝐲 9
ሶ ሷ ሷ Dynamic to static conversion ■ PD is dynamic formulation by nature. 𝑗𝑜𝑢 𝑦, 𝑢 + 𝑐(𝑦, 𝑢) 𝜍 𝑦 𝑣 𝑦, 𝑢 = න 𝑔 𝑞 𝐼 ■ 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 𝑗𝑜𝑢 𝑦, 𝑢 + 𝑐(𝑦, 𝑢) 𝜍 𝑦 𝑣 𝑦, 𝑢 + Λ 𝜍 𝑦 𝑣 𝑦, 𝑢 = න 𝑔 𝑞 𝐼 10
Numerical implementation Start ■ Discretization of domain Discretization (Mesh) ■ Apply boundary conditions Apply BC ■ Carry out static analysis Static analysis Loop over time (t = ∆t, …,n∆t ) ■ For every time increment: – Calculate 𝜏 𝑓𝑟 , 𝜏 1 and 𝜏 𝑠 Calculate stress and 𝜏 𝑓𝑟 , 𝜏 1 and 𝜏 𝑠 – Using Liu-Morakami calculate 𝑑 and D Calculate 𝜁 𝑗𝑘 increment in creep and damage at every point Update E’ = E(1 -D) – Update E Evaluate force vector and apply force – Evaluate force vector Static analysis with updated values – Carry out static analysis again End ■ Update time till D reaches 1 Flow chart showing creep damage model algorithm 11
FEM vs PD Evaluate stress at each gauss point. Evaluate stress at each node. Evaluate increment in creep strain in Evaluate increment in creep strain in damage at each gauss point. damage at each node. Calculate force required to achieve the Calculate force required to achieve the incremented creep strain by: incremented creep strain by: 𝑓𝑞 (Δ𝑔 𝑑 ) 𝑢 = 𝜕𝑫 Damage Δ𝜁 𝑑 𝑳 −1 𝜊 𝑓𝑞 (Δ𝑔 𝑑 ) 𝑢 = න 𝑪 𝑈 𝑫 Damage Δ𝜁 𝑑 dΩ Ω Apply this force at each neighbor node. Apply this force at each node. Repeat the process for all neighbors then repeat for all nodes. Repeat the process for all elements. Solve equilibrium equation again. Solve equilibrium equation again. 12
Influence function and damage ■ Influence function: ω( ξ ) 𝑔 𝒗 𝒚 ′ , 𝑢 − 𝒗 𝒚, 𝑢 , 𝒚 ′ − 𝒚, 𝑢 𝑒𝑊 𝜍 ሷ u 𝐲, t = න 𝑦 ′ + 𝑐 𝒚, 𝑢 𝐼 ■ Bond breaking in peridynamics: ω( ξ ) Influence function – Due to elongation beyond certain limit. ■ Bond breaking in peridynamics creep model: ω ξ , 𝐫, 𝐫 ′ = 𝜕 𝜊 ( ξ )ω d (D, D ′ – Due to elongation beyond certain limit. ෝ ൯ – Damage parameter of point x reached critical value. – Damage parameter of point x’ reached critical value. Damage parameters at ξ 𝒚 and 𝒚 ′ x ω d D, D ′ = 0 if {D = 1 or D ′ = 1} D ≥ 1 D′ ≥ 1 x’ 13
Influence function ■ Influence function is chosen as: = 𝑓 −|𝜊| 2 𝜕 𝜊 ξ 𝑚 2 [1] [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. 14
Selection of damping coefficient ■ Material parameters: – Young’s Modulus: 148 GPa – Poison's ratio: 0.3 ■ Aim: – To check accuracy of code Problem geometry – To select optimum damping coefficient Comparison of results from PD and FEM Parametric study in order to chose optimum value of damping coefficient Λ . 15
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