Materials Process Design and Control Laboratory Finite Element Modeling of the Deformation of 3D Polycrystals Including the Effect of Grain Size Wei Li and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering, Cornell University URL: April 25, 2008

Materials Process Design and Control Laboratory Grain/crystal Inter-grain slip Grain boundary Twinning Macro Meso Mechanical properties of material are extremely essential to the quality of products Preference on material properties requires efficient modeling and designing in virtual environment Considerably advantageous to traditional error-correction method Motivations

Materials Process Design and Control Laboratory Motivations Adequate description of material properties using appropriate mathematical and physical models Use appropriate model to capture the plastic slip in polycrystals and simulate the mechanical properties of the material. Work as a Point Simulator in a multiscale framework

Materials Process Design and Control Laboratory Outlines Problem definition Constitutive Model with homogenization method Grain size effect model Geometric processing techniques Verifications, results and discussions Conclusions

Materials Process Design and Control Laboratory Modeling of realistic 3D polycrystalline microstructure Voronoi Tessellation Conforming grid generation Virtual interrogation of microstructure, constitutive model Mechanical response Deformed microstructure

Materials Process Design and Control Laboratory Generate microstructure (Voronoi Tessellation) Mesh the microstructure Mechanical response and deformed microstructure Constitutive model considering grain size effect Homogenization boundary condition Domain decomposition (Efficient parallel computation) 3D interrogation of microstructure Procedures Geometry processing techniques Physical models implementation Construct the relation between the microstructure and its mechanical properties

Materials Process Design and Control Laboratory Virtual Compression Test The cubic region is compressed in one direction and stretch in the other two uniformly. Initial conditions: (1) Prescribed velocity gradient on boundary; (2) Each grain has a random orientation. L : Velocity gradient r : Strain rate Δt : Time step

Materials Process Design and Control Laboratory Constitutive Model Steps: Boundary conditions Known conditions: all the parameters in previous time step, e.g. deformation gradient and its elastic and plastic components etc. Crystal/latt ice reference frame e1e1 ^ e2e2 ^ Sample reference frame e’ 1 ^ e’ 2 ^ crystal e’ 3 ^ e3e3 ^

Materials Process Design and Control Laboratory Quaternion method A quaternion method is adopted here to transform the orientation expressed in Rodrigues-Frank space to transformation matrix e1e1 e’ 1 e2e2 e’ 2 e3e3 ^ ^ ^ ^ ^ e’ 3 ^

Materials Process Design and Control Laboratory Constitutive model: active slip systems Constitutive Model where

Materials Process Design and Control Laboratory Constitutive model (continued): Constitutive Model Equivalent stress and strain by averaging over all elements

Materials Process Design and Control Laboratory Lattice incompatibility The material deformation gradient is composed of a plastic part due to slips in crystals and an elastic part that accounts for lattice distortion and rotation. This assumes that the lattice only distorts elastically. Elastic distortion generally is not compatible with a regular displacement field, so it is natural to use elastic deformation gradient F e (or (F e ) -1 ) as a measure of lattice incompatibility. Grain size effect

Materials Process Design and Control Laboratory Grain size effect Grain size effect and dislocation density Lattice incompatibility is coupled with the evolution of dislocation density, which is highly intense on grain boundary. This is because dislocation line can not move across grain boundary. Whenever dislocation is generated, it will assemble there and cause lattice distortion, which leads to lattice incompatibility. Due to the restriction on grain boundaries, the grains can not deform as they wanted to and thus no gap or overlap occurs.

Materials Process Design and Control Laboratory Bailey-Hirsch relationship: Lattice incompatibility: Shear strain rate: Dislocation density: Grain size effect Magnitude of lattice incompatibility in a slip system: The first term considers the relation between lattice incompatibility and dislocation density. The k 1 and k 2 are two experience functions coming from experiments.

Materials Process Design and Control Laboratory Shear resistance: Updated shear stress: Grain size effect is used as the plastic resistance on the slip systems to judge whether plastic deformation occurs

Materials Process Design and Control Laboratory Voronoi Tessellation method and microstructures (a)(b)(c) Steps: 1.Sample a set of points, say 5, in the domain; 2.Calculate the grain boundaries with V.T.; 3.Generate the grains.

Materials Process Design and Control Laboratory Methods – Mesh generation Advantages: No restriction on grains Fully adaptive to microstructure geometries Element numbers manageable Simulate the “real” microstructures without assuming unrealistic grain boundaries

Materials Process Design and Control Laboratory Methods – Mesh generation Conforming grids with 4097 elements Pixel grids with 20×20×20 elements Pixel grids with 70×70×70 elements

Materials Process Design and Control Laboratory Mesh Generation and Domain Decomposition Mesh the grains Split into brick elements Domain decomposition

Materials Process Design and Control Laboratory Domain decomposition The whole region is decomposed into continuous sub-regions Each sub-region is individually processed by one processor in parallel computation Faster than using the indices to assign the elements to the processors Divide the region into 32 parts, use 8 nodes (32 processors) Speed: 33,000 time steps ~ 6hr, comparing with 17,000 time steps~ 6hr previously, approximately 50% increased Each separate region is “continuous”, when integrating the local matrices, processors just need to communicate when doing calculation on the boundaries.

Materials Process Design and Control Laboratory Verifications – comparison with simulated result This work (Taylor) This work (Homogenization) Anand and Kothari (1996) Constitutive model, without considering grain size effect

Materials Process Design and Control Laboratory Verifications – comparison with experimental result Constitutive model, grain size effect included

Materials Process Design and Control Laboratory Grains configuration in non-conforming grid Since the adopted constitutive model is a non-scale model, grain size can not be altered by changing the calculated region. In non-conforming grids, grain size is determined by the way of specifying the grains. If each 1 element is seen as a grain, the average grain size is just the size of a single element. If each 8(=2 3 ) elements are seen to compose a grain, the size will be twice as much. Larger grain sizes can be obtained with similar method.

Materials Process Design and Control Laboratory Results and discussions – nonconforming grids Comparison with the experimental results (Narutani and Takamura, 1991)

Materials Process Design and Control Laboratory Results and discussions – nonconforming grids Comparison with the experimental results (Narutani and Takamura, 1991) Stress-strain curves of different grain sizes 1/6, 1/8, 1/12, 1/24mm 24×24×24 elements 1/36mm 36×36×36 elements 1/48mm 48×48×48 elements

Materials Process Design and Control Laboratory Mechanical response Equivalent stress field Grain size: 1/36mm Results and discussions – nonconforming grids

Materials Process Design and Control Laboratory Results and discussions – conforming grids Microstructure deformation Equivalent Stress field Mechanical response

Materials Process Design and Control Laboratory (a) (c) (e) (b) (f) (d) Results and discussions Displacement field Equivalent stress field

Materials Process Design and Control Laboratory 20, 50 and 100 grains Results and discussions

Materials Process Design and Control Laboratory Results and discussions Mechanical responses of three different grain sizes Comparison with experimental results

Materials Process Design and Control Laboratory Conclusions (1) A finite element analysis of large deformation of 3D polycrystals is presented. The effect of grain size is included by considering a physically motivated measure of lattice incompatibility. (2) A domain decomposition method, Voronoi Tessellation method and conforming grids generation technique are developed. (3) Calculated mechanical properties of polycrystals are shown to be consistent with experimental results. (4) Conforming grids method is adopted to investigate the strengthening effect of grain sizes.

Materials Process Design and Control Laboratory Information Relevant publication W. Li and N. Zabaras, “A virtual environment for the interrogation of 3D polycrystals including grain size effects”, Computational Materials Science, to be submitted Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 101 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Prof. Nicholas Zabaras Contact information