Multiscale Modeling of Additively Manufactured Metals: Application to Laser Powder Bed Fusion Process

By Yi Zhang, Yeon-Gil Jung and Jing Zhang

Multiscale Modeling of Additively Manufactured Metals: Application to Laser Powder Bed Fusion Process provides comprehensive coverage on the latest methodology in additive manufacturing (AM) modeling and simulation. Although there are extensive advances within the AM field, challenges to predictive theoretical and computational approaches still hinder the widespread adoption of AM. The book reviews metal additive materials and processes and discusses multiscale/multiphysics modeling strategies. In addition, coverage of modeling and simulation of AM process in order to understand the process-structure-property relationship is reviewed, along with the modeling of morphology evolution, phase transformation, and defect formation in AM parts.

Residual stress, distortion, plasticity/damage in AM parts are also considered, with scales associated with the spatial, temporal and/or material domains reviewed. This book is useful for graduate students, engineers and professionals working on AM materials, equipment, process, development and modeling.

• Includes the fundamental principles of additive manufacturing modeling techniques
• Presents various modeling tools/software for AM modeling
• Discusses various design methods and how to optimize the AM process using these models

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 29, 2020
• ISBN: 9780128225592

Yi Zhang

Dr. Yi Zhang received her BSc & MSc at Ocean University of China, and her PhD & postdoctoral training at McGill University, Canada. She was a Commonwealth Blue Charter Fellow. To-date, Dr. Zhang has published 30 papers and 5 book chapters in the areas of enzymology, agricultural biochemistry, and biological sciences.

Book preview

Laboratory.

Chapter One

Multiscale and multiphysics modeling of metal AM

Abstract

A long-standing challenge is to optimize additive manufacturing (AM) process in order to reduce AM component failure due to excessive distortion and cracking. To address this challenge, a multiscale physics-based modeling framework is required to understand the interrelationship between AM processing parameters and resulting properties. In particular, a multiscale approach, spanning from atomic particle to component levels is needed.

Keywords

Multiscale; Multiphysics; Additive manufacture; Cracking; Fluid flow; Heat transfer

1 Introduction

A long-standing challenge is to optimize additive manufacturing (AM) process in order to reduce AM component failure due to excessive distortion and cracking. To address this challenge, a multiscale physics-based modeling framework is required to understand the interrelationship between AM processing parameters and resulting properties. In particular, a multiscale approach, spanning from atomic particle to component levels is needed.

Modeling the laser powder bed fusion (L-PBF) process requires a comprehensive understanding of the fundamental sciences, including heat transfer, powder mechanics, and modeling of laser source, among others. The sintering process usually occurs very fast (in nanoseconds), which extremely increases the difficulty of experimental measurements. Molecular dynamics (MD) simulation is a powerful tool that is capable of capturing sintering kinetics at very short time and nanosize scale. Discrete element modeling (DEM) is suitable to obtain the information at particle level, e.g., powder flow during the laser heating, as well as the powder movement during the recoating process. With the DEM simulation, localized powder size distribution and the corresponding powder size-based properties can also be applied to the continuum finite element (FE) model for complex-shaped components.

In this chapter, a framework for modeling the L-PBF process is proposed, as illustrated in Fig. 1. Several length scales and multiple physics are included in this framework. Sintering mechanisms, atomic diffusion, and resultant mechanical properties are investigated using a molecular dynamics model at atomistic scale. Powder deposition, powder flow, and laser heating of the powder bed are simulated by a discrete element model at mesoscale. Additionally, a finite element study of thermal stress or distortion evolution at macroscale level is presented. The coupling between different scales is also illustrated in Fig. 1.

Fig. 1 Multiscale multiphysics modeling framework of the laser powder bed fusion process.

2 Physics in the metal AM process

Physical phenomena that occur during the laser powder bed fusion are very complicated. Before the laser sintering actually happens, the metal particles interact with each other, forming a randomly packed layer of powder bed. When the laser heat source is applied, heat transfer, melt pool flow, particle sintering, and thermal stress are the major physical phenomena that must be considered. After the laser heating, the solidified structure in micro scale is also a necessary consideration since it can affect the mechanical properties of the L-PBF fabricated part. Each of these physical phenomena is described in the following.

2.1 Sintering kinetics

The geometric evolution during the initial stage of sintering consists of densification and neck growth. Neck growth forms the bonding between particles and the densification causes volume shrinkage of the sintered part. Several types of mass transport mechanisms may occur during the sintering process, including viscous flow, plastic flow, evaporation-condensation, and material diffusion. Both solid-state and liquid-state sintering occur in the laser powder bed fusion process. For solid-state sintering the mass transportation mechanism includes surface diffusion, volume diffusion, grain boundary diffusion, viscous flow, plastic flow, and vapor transport from solid surfaces. In liquid-phase sintering, the dominant process is diffused through the liquid driven by surface tension.

Sintering is often characterized by the neck size over particle diameter ratio (schematic representation is shown in Fig. 2.

   (1)

where m and n are the factors that depend on the type of mass transportation pathway and B is a term made up of material and geometric constants. With Eq. (1), the configuration changes can be tracked.

Fig. 2 Neck growth in sintering models for spheres of diameter D . The upper drawing shows neck growth measured by the neck diameter X via surface transport mechanisms that do not produce shrinkage. The lower drawing is for bulk transport by mechanisms that move mass from between the particles to allow densification [1].

However, this method is only applicable for single mass transportation mechanism-driven sintering processes and it highly relies on the calibration of those factors. Molecular dynamics (MD) is an alternative method to model sintering of micro- or nanosized particles. For the MD simulation, individual atoms are modeled in the system. The motion of each atom is solved by integrating equations of motion based on Newton’s second law

   (2)

is the acceleration, mi is the mass of atom iis the total force acting on atom i, which is given by the material-dependent force field function.

2.2 Particle mechanics

The movement of particles in the powder bed is based on Newton’s second law. The translational and rotational equations of motion for each metal particle is integrated with time according to Ref. [2]:

   (3)

   (4)

where mi is the mass of particle iis the translational acceleration of particle i, g is the gravity constant, and Fij is the force acting on particle i, given by particle j. For the rotational motion, Ii is the moment of inertia of particle iis the angular acceleration of particle irepresents the total rotational force acting on particle i from the rest of the system.

2.3 Heat transfer

The heat transfer is simulated by solving temperature T in the energy equation in fluids. For the time-dependent energy equation in fluid [3–6]:

   (5)

where the term ∇(ρuCpT) is convection within the fluid due to fluid flow and Q is the energy loss due to various reasons, including convection, radiation, fluid melt and evaporation, and laser heat sources.

2.4 Fluid flow

In the melt pool, the velocity field in the fluid is described by the Navier-Stokes equation [7]:

   (6)

where p is the static pressure; μ is the viscosity; g is the gravitational body force; ρg[1 − β(T − Tmis a Darcy condition that suppress the motion of unmelted metal, where fl is the liquid fraction, ϵ = 0.01 is a small number to avoid zero denominators, and Amesh = 10¹⁴ is the mesh zone constant; and Fs is the additional momentum source term that acts on the surface of the melt pool, which includes surface tension, Marangoni convection (capillary force), and evaporation caused recoil pressure.

2.5 Solidification and microstructure evolution

Microstructure evolution during solidification of the fusion zone represents one of the most important considerations for controlling the properties of L-PBF products. A wide range of microstructural features can form in the fusion zone, depending on the alloy composition, process parameters, and resultant solidification conditions.

The temperature gradient (G), solid/liquid interface growth rate (R), and cooling rate (E) are the important solidification parameters. These three parameters are related by

   (7)

Different microstructure types, including planar, cellular, cellular dendritic and equiaxed dendritic can form at different locations of the fusion zone, depending on G/R ratio. This dependency is not always intuitive and can be understood more clearly with the help of Fig. 3A. The solidification rate is directly related to the laser movement speed, V, by R = V cos θ, where θ is the angle between solid-liquid interface normal and the laser scan direction. Therefore, the solidification rate R = V at the centerline of the fusion zone, and R ≈ 0 at the fusion boundaries. As shown in Fig. 3A, with increasing G/R ratio, the solidification mode may change by the sequence of equiaxed dendritic, cellular dendritic, cellular, and planar. For most metals, during the solidification process, the nuclei will form at the boundary or bulk of the liquid metal, and then each nucleus will grow within an octahedral envelope according to its own orientation as shown in Fig. 3B. Many growing grains will compete with each other and finally occupy the fusion zone. Note that the growth rate is a function of temperature, therefore the actual shape of each grain is not a regular octahedral in most cases due to the nonuniform temperature distribution. Cellular automaton is a computational geometry-based numerical method that can simulate this process. In cellular automaton (CA), the simulation domain is divided into equal sized cells, and the evolution of microstructure is determined by some CA rules.

Fig. 3 (A) Solidification modes in laser fusion zone [8], and (B) grain growth of a nuclei is simplified as a octahedral shape [9].

2.6 Thermal stress and distortion

The high cooling rate during the laser powder bed fusion process often leads to thermal residual stress. This is due to the thermal expansion of metals. In quasi-statics, the equation of motion for the printed part is

   (8)

where σ is the Cauchy stress tensor and Fv is the body force. The stress tensor σ is calculated by three-dimensional Hooks law:

   (9)

where D is the stiffness matrix and ɛ is the total strain tensor, which is a combination of three terms:

   (10)

where ɛel is the elastic strain and ɛie is the inelastic strain calculated from temperature-dependent stress deviation, strain hardening, and plastic yielding. ɛth is the strain due to thermal expansion, which can be obtained by ɛth = αΔT ⋅ 1 (where α denotes the temperature-dependent coefficient of thermal expansion and 1 is the unit tensor).

3 Multiscale and multiphysics modeling and software

3.1 Coupling between different scales

The relationship between different scales and coupling methods between different models are described in Fig. 4 [10, 11].

Fig. 4 Coupling between different scales and experimental data [10 , 11] .

3.2 AM process modeling and simulation software

Additive manufacturing modeling and simulation software has a very broad scope, spanning from design for additive manufacturing (DfAM), improving build setup, and simulation of the print process. Software in additive manufacturing may include geometry design, topology optimization, lattice generation, process modeling, build preparation, and AM part analysis.

The list below is focused on AM process models, the main topic of the book, linking process to properties and performance. Only the commercial packages are included. It should be noted that this not a complete list, and needs to be constantly updated due to rapid development of AM field (Table 1).

Table 1