PSI - Issue 71

A Shivnag Sharma et al. / Procedia Structural Integrity 71 (2025) 469–476

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challenging for this class of problems. This study tries to explore the possibility of using ConvLSTM as an alternative of CPFEM while highlighting on the associated error when compared to solutions obtained from latter. 2. Theory and Methodology This section provides an overview of the Crystal Plasticity (CP) constitutive model, the dataset generation process and the ConvLSTM machine learning model. 2.1. Crystal Plasticity Constitutive Model The CP model to capture the plastic deformation in metals (polycrystals) proposed by (\cite{Kalidindi}) was employed in this work. The stress-strain constitutive relation is expressed as = : (1) Where C is the fourth order elasticity tensor and E e is the strain tensor given as = 1 2 ( − ) (2) where F e is the elastic deformation gradient tensor. In the CP model, the deformation gradient is multiplicatively decomposed such that = − 1 (3) where F p is the plastic deformation gradient and its evolution can be related to plastic velocity gradient by = − 1̇ =∑ γ α ̇ α=1 0 ⊗ 0α (4) where ̇ α is the slip rate on system α , N s is the total number of slip systems, and m 0 α and n 0 α are the slip direction and plane normal respectively. A phenomenological relation for the evolution of plastic shear is used following (Asaro and Needleman, Kalidindi) γ α̇ =γ 0α̇ | τ α α | 1 sign(τ α ) (5) where ̇ 0 α is the reference slip rate on slip system α , τ α and g α are the resolved shear stress and effective slip system resistance on slip system α respectively and m is the slip rate sensitivity. The evolution of slip system resistance is given by α̇ =∑ ℎ αβ |γ β̇ | β (6) with slip hardening rate as ℎ αβ = αβ ℎ 0α (7) where h αβ are the components of the hardening matrix that captures the effect of slip on the β th slip system on the resistance of the α th slip systems, q αβ is taken as 1 when α is equal to β or 1.4 otherwise. The CP model is implemented as user material subroutine in ABAQUS. The various equations are implicitly integrated using the scheme provided in Balasubramanian’s thesis (1998) . 2.2. Dataset generation For dataset generation, a Voronoi tessellated 2D microstructure of dimensions 1 x 1 with 64 x 64 elements was created with the help of a MATLAB code. 5 grains were considered in the RVE and was assigned random Euler angles. The Voronoi tessellation allowed for the spatial representation of grains, while the Euler angles provided orientation information, capturing the crystallographic texture of the material. CPFEM simulations based on section 2.1 were performed on the RVE with the help of the UMAT code in FEM software ABAQUS. The boundary conditions were applied such that it replicated the uniaxial tension state and the microstructure was stretched upto 20% engineering strain. The time increment for the simulation was applied such that the frames proceeded with a constant time step size of Δt = 0.001 secs. Table [1] presents the parameters used for the CP simulations. The user material subroutine of ABAQUS allows the CP variables such as slip system resistances g α , and plastic deformation gradient F P ij and 2 nd Piola-Kirchhoff stress S ij tensors to be easily accessed via the state variables, hence, allowing the data to be written for every increment. The variables such as components of displacement (u and v)