Archetype-Blending Multiscale Continuum Method

Transcription

1 Archetype-Blending Multiscale Continuum Method John A. Moore Professor Wing Kam Liu Northwestern University Mechanical Engineering 3/27/

2 Outline Background and Motivation Archetype-Blending Continuum (ABC) Theory Computational Fatigue ABC microplasticity simulation ABC fatigue simulations Conclusions 3/27/

3 Motivation and Background 2 1 Microstructure Macrostructure 2 1 Porter, Easterling, Toro et. al, J. Mater. Eng. Perform Pelton, et al J. Mech. Behav. Biomed Mater., /27/

4 Alternating Strain % Fatigue in Biomedical Stents Stents Wire Specimens Stent Cycles to Failure Duerig, T., A. Pelton, and D. Stöckel, An overview of nitinol medical applications. Materials Science and Engineering: A, /27/

5 Stress, S Stress, S Notch Effects Un-notched Notched K t Fatigue Notch Factor K f K t Log Life N Log Life N 3/27/

6 Stress, S Neuber s Equation Notched Neuber s Equation K f K t = K t A ρ 1 + A ρ ρ K f Log Life N K t Non-linear Material dependent Schijve, Fatigue of Structures and Materials, 2001 Non-linear material dependent behavior indicates microstructural dependence 3/27/

7 FIP Computational Fatigue Fatemi-Socie in 1988 proposed a fatigue indicating parameter (FIP) to account for discrepancies in ε-n curves due to loading condition Empirical Constant Material Parameter From FEM/Experiment Plastic strain increment Δγ p. Torsion.... Axial Log Life N Fatemi and Socie Fatigue & Fracture of Engineering Materials & Structures, 1988 Log Life N 3/27/

8 Fatigue Regimes Total fatigue life is broken into three regimes 1 N Total = N Inc. + N MSC + N LC Incubation (N Inc. ): nucleation and growth of crack beyond influence of microstructural notch 1 Characterized by microscale plastic strain and nonlocal damage parameters Microstrucurally Small Cracks (N MSC ) : growth of crack from incubation size a i, such that a i < a < kgs, where k is (1-3) and GS the lengthscale of a grain or other prominent microstructural features 1 Characterized by elasto-plastic fracture mechanics Long Cracks (N LC ) : macroscopic crack growth 1 Characterized by linear elastic fracture mechanics This slide was not originally presented on 3/4/ Horstemeyer, ICME for Metals, /24/2014 3/27/2014 Steel Research Group 30 8 th Annual Meeting 8

9 Treatment of Fatigue Regimes Total fatigue life is broken into three regimes N Total = N Inc. + N MSC + N LC The following work will address only N Inc. as it accounts for a large % of fatigue life for many alloys The ABC theory will be able to model the N MSC and N LC regimes by: Studying several (5-10) initial cycles and determining N Inc. from a FIP Using this as an initial state for explicit modeling of N MSC and N LC N MSC region could be considered 1 element (kgs = 1 element) and growth modeled with methods such as XFEM 1 Once crack grows beyond 1 element N LC can be modeled based on basic damage models, strain gradients in ABC will aide in regularization (reducing mesh sensitivity) and localization 1 Menouillard, Thomas, et al. "Time dependent crack tip enrichment for dynamic crack propagation." Int. J. of Frac (2010): This slide was not originally presented on 3/4/ /24/2014 3/27/2014 Steel Research Group 30 9 th Annual Meeting 9

10 Direct Numerical Simulation of FIP Zhang s Mesh Zhang et al. Eng. Fract,Mech, /27/

11 Stress, S Fatigue Theory Summary Linear Elastic Theory Overly conservative No material information No microscale information Neuber s Theory Require extra material tests Only works for simple notches No microscale information Fatemi-Soci (FIP) Theory No macroscale notch information Linear Elastic Neuber Log Life N FIP 3/27/

12 Archetype-Blending Continuum (ABC) Theory Combines generalized continuum mechanics and constitutive modeling Degrees of freedom represent partitions of microstructure x x x x Each degree of freedom is similar to assembly of Eshelby problems in micromechanics but strain are determined by solving equations of motions Virtual Power δp int = Ω Intrinsic stress relative stress DOF1 Interaction relative strain strain gradient σ: δl + σσ δl + s n : δλ n + ss n δλ n dω DOF2 DOFn Elkhordary et al., Comput. Methods Appl. Mech. Engrg., /27/

13 Stress, S Notched Fatigue and ABC Non-linear reduction in S-N curve is : Material dependent. A function on macroscale strain.. gradients 1... A function of microscale strain gradients 1 ABC Log Life N matrix interphase inclusion K t Micro K t Implicit Microstructure 1 McDowell, Mat. Sci. Eng. A, 2007 δp int = σ: δl + σσ δl + s n : δλ n + ss n δλ n dω 3/27/ Ω

14 Models 10% volume fraction 300MPa 1% volume fraction y Ramped Velocity: 7571 m/s x 3/27/

15 Implicit Microstructure Interphase Inclusions Matrix Matrix: Young s Modulus: 200 GPa Yield Strength : 250 MPa matrix interphase Interphase: Young s Modulus: 200 GPa Yield Strength : 250 MPa Inclusion: Young s Modulus: 2000 GPa Linear Elastic inclusion Degree-of-Freedom 1 ε 1 Matrix Degree-of-Freedom 2 ε 2 Interphase E (ε 2 ) Inclusion E = mapping of strain using Eshelby s solution 03/24/2014 3/27/2014 Steel Research Group th Annual Meeting 15

16 Microplasticity 300MPa matrix interphase inclusion 3/27/

17 Response of Notched Sample Linear Elastic Macro Stress Equivalent Plastic Interphase Strain Stress w/o notch is 300 MPa, K t MPa x10-3 3/27/

18 Notched Sample Fatigue Prediction 3/27/

19 Summary and Conclusions ABC uses a multiscale multicomponent formulation rooted in micromechanics to predict material behavior ABC can predicted notch sensitivity of notched devices, giving information of geometric effects and statistics Goal is that device designers can optimize microstructure and geometry concurrently 3/27/

20 Questions / Comments Robertson, et al., Int. Mater. Rev /27/

21 Backup 3/27/

22 Constitutive Modeling Simple Partition Realistic Partition 1 Partition 1 = matrix Partition 2 = matrix carbide Partition 3 = matrix oxide Archetype A = matrix Archetype B = Inclusion Partition 1 = matrix Partition 2 = matrix inclusion Archetype A = matrix Archetype B = carbide Archetype C = oxide Partition 4 = matrix oxide interphase Archetype D = damaged matrix/oxide interphase Toro et. al, J. Mater. Eng. Perform Volume 18(5 6) August /27/

23 Eshelby s Problem Then the inclusion strain is ε 1 = {I + SL 0 1 L 1 L 0 }ε 0 This maps the applied strain to the inclusion strain based on the material properties of the matrix and inclusion and Eshelby s tensor Homogenization with Eigenstrain ε L 1 ε 0 + Sε = L 0 (ε 0 + Sε ε ) Where L 0 and L 1 are stiffness tensors, S, Eshelby s tensor and ε 0 the applied strain Solving for Eigenstrain ε gives : ε = [ L 1 L 0 S + L 0 )] 1 L 1 L 0 ε 0 If the inclusion strain ε 1 is given by: ε 1 = ε 0 + Sε For an arbitrary complex problem a region around the inclusions (r) can be considered with a material properties L and applied strain ε, then the inclusion strain is ε r = {I + S r L 1 L r L }ε Different assumptions for L and ε yield many popular micromechanics models Dilute Model : L = L 0 ; ε = ε Mori-Tanaka : L = L 0 ; ε = ε 0 Self Consistent : L = L ; ε = ε 3/27/