Macroscopic Failure Analysis Based on a Random Field Representations Generated from Material Microstructures

Transcription

1 Macroscopic Failure Analysis Based on a Random Field Representations Generated from Material Microstructures Reza Abedi Mechanical, Aerospace & Biomedical Engineering University of Tennessee Knoxville / Space Institute

2 Random fracture response and patterns Random features of fracture (mostly brittle fracture) Fracture patterns greatly depend on microscale defects. Al-Ostaz 1997: Epoxy sheets with holes Fracture and failure initiates from initial defects in material (new defects may also nucleate from the applied loads). Material micro-defects are stochastic in nature.

3 Some consequences of random fracture: Size effect in (quasi)-brittle materials Genet, Ritchie 2014, quasibrittle fracture of cellular ceramic structures Larger samples have 1. higher variability in failure strength 2. Lower (mean) fracture strength

4 Thermodynamics: Representative Volume Elements (RVE) A Representative Volume Element (RVE) can represent material microscale features by homogenized values obtained by RVE in thermodynamics. Stochastic Volume Elements (SVE) are used when assumption of RVE are no longer valid: 1. Unit cell (RVE) is not fully periodic 2. When RVE size is not appropriate: too large or small Ostoja-Starzewski 1998 Sakata:2013

5 Where to use SVEs? When RVEs cannot be used? 1. A quasi-periodic structure 2. Domain size constraints: examples MEMS/NEMS: composites/laminates with large inclusion sizes. 3. When fracture is sensitive to initial defect distribution: For a macroscopically homogeneous material, homogenization with RVEs is: Appropriate when computing displacements, stresses, etc. when there is no major material failure. Inappropriate when fracture is concerned and when fracture is sensitive to microstructural defects. While Stochastic Models are relatively mature there is clear lack of micromechanically-motivated random material characterization. SVEs provide a natural way to characterize random media

6 Computational moving window method: A novel and simple way to characterize random media SVE analysis on randomly generated domains and at different locations fully characterizes random material: 1. PDF of all point are obtained. 2. Covariance function is also obtained: Computational approach Experimental measurements Moving window auto- and cross-correlation functions Baxter, Lori Graham-Brady, Characterization Of Random Composites Using Moving-window Technique (2000)

7 Random realizations of microstructure, Generation of SVEs Ongoing research

8 Random realizations of microstructure, Generation of SVEs w Index 1, 0 x s s Index 0, 0 Realization 1 Realization 2

9 shear displacement Effective elastic properties of SVEs von Mises stress Stress intensity number

10 Point-wise statistics 4 x 4 window size 1 x 1 window size

11 Covariance function Average crack length 0.2 crack density 0.05

12 Karhunen-Loeve method to generate random field realizations for Stochastic PDEs Covariance function can be used to generate sample random media e.g. by Karhunen-Loeve (KL) method: Once random material is characterized, classical stochastic methods such as Monte-Carlo Collocation method Stochastic spectral Galerkin method

13 Sample random field realizations Covariance function and point-wise PDE (obtained from SVEs) are used with KL method to generate random field realizations log( E തE ) σ E: Elastic modulus തE, σ mean and standard deviation Length scale for Cov function = 1/16 domain length # KL terms: 100

14 Sample results with random microstructure A. Implicit representation of microstructure Movie: Click here to play movie

15 B. Explicit representation of microstructure (microcracks) Movie: Click here to play movie Initial mesh

16 Two different meshes (solution and random field) discrete meshes / fracture analysis (size sensitivity) Finite Element Method Fracture Consistency (FEMFC) Adaptive operations ensure that failure probability is independent of the initial mesh

17 Computational aspects: Adaptive spacetime discontinuous Galerkin Method time Unstructured grids in spacetime Adaptive operations permit capturing complex fracture patterns

18 Random realizations of microstructure, Generation of SVEs Future / proposed research

19 Stochastic thermodynamics & constitutive laws Motivation Microstructure features can be incorporated as initial macroscopic damage/fracture parameters: 1. Microscale structures to effective elasticity 2. Effective elasticity to initial macroscopic damage: Damage reduces elasticity 3. Once macroscopic damage field is characterized fully macroscopic models can be employed 4. Randomness is incorporated through solving stochastic partial differential equations (SPDEs) by Monte Carlo method, stochastic collocation FEM, spectral stochastic Galerkin FEM, etc. While this framework is rigorous it is much better to keep microstructural characterization for the entire analysis (e.g. not borrow phenomenological/micromechanically-motivated macroscopic damage models)

20 Stochastic constitutive laws Microscale parameters: probabilistic parameters of micro defects (e.g. inclusion/void volume density, minimum distance, inclusion/void average radius, etc.) Sensitivity analysis: Which parameters are the most important? Stochastic constitutive equations: From microscale parameters to macroscale parameters (e.g. yield stress) Wing Kam Liu (Northwestern) Stochastic constitutive laws for high strength steel with voids Computational uncertainty analysis in multi-resolution materials via stochastic constitutive theory (Greene 2011) Statistical volume element method for predicting microstructure-constitutive property relations (Yin 2008)

21 Micromechanically-based stochastic thermodynamic and constitutive modeling Important micromechanical parameters a: Based on simulation results and sensitivity analysis find the most important parameters for a given type of material (e.g. microcrack density & shortest microcrack distance(s) for brittle material). Stochastic thermodynamics: Stochastic thermodynamic energy and potential functions enable the expression of the following stochastic evolution laws: Macroscopic parameter evolution (e.g. stress) Derivation of stochastic constitutive equations: One dynamic solution Several data points for computational evolution law characterization. SDG solver is ideal for such dynamic homogenization (characterization) Question: How to incorporate randomness w? Microscopic internal parameter evolution laws (e.g. microcrack density) Microcrack density increases by applied load

22 Sample microstructures Image source: Chen, Case, Lattimer (2015) Creep damage Quantification and post-fire residual strength of 5083 Aluminum alloy Goal: Forming point-wise statistics from microstructure and subsequent simulation of random fields generated based on the underlying statistical representation. Image Source: Dynamic Failure Modeling, Espinosa (Northwestern)