This book explains the fundamental thermomechanical behavior of the 3D printing process in a laser-based powder bed fusion (L-PBF) system

We have now formulated our thermomechanical model for a Kelvin–Voigt material. The plan for the remaining three chapters of the book is as follows. In this chapter, we specialize the strong form of the model into one space dimension, and solve the constant-coefficient case analytically. In one-dimensional and two-dimensional models, the heat escaping via the surrounding air needs special consideration. For completeness, we first solve the one-dimensional model without this additional cooling term—thus corresponding to a thermally insulated domain—and then add in the cooling term, and solve it again to obtain a more realistic result. In Chap. 8 , we consider how the model changes if we make the parameters into linear functions of temperature, which matches physical observations. We will see that doing that will reduce the possibilities to obtain an analytical solution. However, we will take the opportunity to showcase some useful techniques, which allow us to obtain partial analytical solutions. After we are done with the one-dimensional models, in Chap. 9 we move on to a numerical solution in two space dimensions (including the depth direction, i.e. the previously printed layers), using the weak form of the model and the finite element method.

The fundamental physics of continua is governed by the balance laws of four conserved quantities, namely mass, linear momentum, angular momentum, and internal energy.

So far, we have looked at a one-dimensional specialization of our L-PBF printing model, to analyze the behavior of a single line of material being printed as a function of distance from the printing laser, in a steady state where the laser moves at a constant velocity. In this final chapter, to proceed further in understanding the fundamental behavior of the process, we generalize the analysis to two dimensions.

Many manufacturing processes in the process industry are modeled using the theory of axially moving continua. In this book, we apply this perspective to the additive manufacturing of metal products. This chapter briefly reviews the fundamental kinematics, setting up the stage for deriving the governing equations for our model.

We investigate the thermomechanical behavior of 3D printing of metals in the laser-based powder bed fusion (L-PBF) process, also known as selective laser melting (SLM). Heat transport away from the printed object is a limiting factor. We construct a one-dimensional thermoviscoelastic continuum model for the case where a thin fin is being printed at a constant velocity. We use a coordinate frame that moves with the printing laser, and apply an Eulerian perspective to the moving solid. We consider a steady state similar to those used in the analysis of production processes in the process industry, in the field of research known as axially moving materials. By a dimensional analysis, we obtain the nondimensional parameters that govern the fundamental physics of the modeled process. We then obtain a parametric analytical solution, and as an example, illustrate it using material parameters for 316L steel. The nondimensional parameterization has applications in real-time control of the L-PBF process. The novelty of the model is in the use of an approach based on the theory of axially moving materials, which yields a new perspective on modeling of the 3D printing process. Furthermore, the analytical solution is easy to implement, and allows fast exploration of the parameter space.

The paper considers the analysis of a traveling panel, submerged in axially flowing fluid. In order to accurately model the dynamics and stability of a lightweight moving material, the interaction between the material and the surrounding air must be taken into account. The lightweight material leads to the inertial contribution of the surrounding air to the acceleration of the panel becoming significant. This formulation is novel and the case complements our previous studies on the field. The approach described in this paper allows for an efficient semi-analytical solution, where the reaction pressure of the fluid flow is analytically represented by an added-mass model in terms of the panel displacement. Then, the panel displacement, accounting also for the fluid–structure interaction, is analyzed with the help of the weak form of the governing partial differential equation, using a Galerkin method. In the first part of this paper, we represent the traveling panel by a single partial differential equation in weak form, using an added-mass approximation of the exact fluid reaction. In the second part, we apply a Galerkin method for dynamic stability analysis of the panel, and present an analytical investigation of static stability loss (divergence, buckling) based on the added-mass model.

The paper is devoted to the analysis of the axially travelling web supported by a system of fixed rollers and submerged in axially flowing gas medium. In order to accurately model the dynamics and stability of a lightweight moving web, the interaction between it and the surrounding air is taken into account. The light weight of the moving web leads to the inertial contribution of the surrounding air to the acceleration of the material becoming significant. In the context of this paper we apply a Galerkin method for dynamic stability analysis of the moving web based on developed added-mass model.

We present a novel approach for identifying a multiaxial thermodynamic magneto-mechanical constitutive law by direct bi- or trivariate spline interpolation from available magnetization and magnetostriction data. Reference data are first produced with a multiscale model in the case of a magnetic field and uniaxial and shear stresses. The thermodynamic model fits well to the results of the multiscale model, after which the models are compared under complex multiaxial loadings. A surprisingly good agreement between the two models is found, but some differences in the magnetostrictive behaviour are also pointed out. Finally, the model is fitted to measurement results from an electrical steel sheet. The spline-based constitutive law overcomes several drawbacks of analytical approaches used earlier. The presented models and measurement results are openly available.

In this chapter, we present some prototype bifurcation problems that arise in the mechanics of rigid and deformable structural elements. These problems are typical for engineering applications and characterize the approaches that can be applied in the investigation of stability. Some methods of bifurcation theory will be presented in the context of stability studiesStability analysis of the considered one-dimensional mechanical problems.

In this chapter, using analytical approaches, we consider the problems of dynamics and stability of moving elastic rods and strings, axially traveling between two supports at a constant velocity. Transverse, longitudinal and torsional vibrations of the moving structure are reduced to the same mathematical form, a hyperbolic second-order partial differential equation. The analysis is then extended to the axially traveling stringString with damping. An analytical free-vibration solution is obtained. It is seen external friction leads to stabilization, whereas internal friction in the traveling material will destabilize the system in a dynamic mode at the static critical point. Finally, we consider the effects of bending rigidity, which in the case of paper materials introduces a singular perturbation to the governing equation. We consider an implicit exact eigensolution for beams, the effect of elastic supports at the boundaries to the vibration behavior of a long traveling beam, and the stability of a beam traveling in a homogeneous gravitational field.

This paper utilizes a thermodynamic approach based on Helmholtz free energy density and a finite element (FE) model to analyze a galfenol-based magnetostrictive energy harvesting concept device. An analytical energy density function is first presented assuming an isotropic material for the identification of a magneto-mechanical constitutive law. The model utilizes the magnetic flux density and mechanical strain as state variables. Compared to some earlier approaches, this simplifies the implementation of FE models based on magnetic vector potential and mechanical displacement, since time-consuming inversion of the constitutive law is not required. The Maxwell and mechanical balance equations are then solved utilizing the constitutive law in an axisymmetric FE model. A prototype device is developed and tested under uniaxial cyclic compressive loading of 100 Hz at different preload and dynamic loading cases. Finally, the results from the simulations are compared with the experimental results for validation. The comparison shows that the analytical constitutive model fits well to the magnetization curves measured under static loading. Furthermore, the FE model closely predicts the measured power with some discrepancies under different preload values. The model is able to predict the behavior of the device with respect to preload, load resistance and magnetization of the sample, proving to be an effective tool in the design of such devices.

In this chapter, we consider three thermoelastic optimization problems. We look at the optimal thickness distribution for a beam of variable thickness, when the goal is to maximize its resistance to thermoelastic buckling, or in other words, to maximize the critical temperature at which buckling occurs. In the second problem, we allow the beam to be constructed inhomogeneously, looking for an optimal distribution of materials that maximizes the critical temperature. The third and final problem concerns heat conduction in locally orthotropic solid bodies. By locally orthotropic, we mean a particular type of inhomogeneity, where the principal directions (axes of orthotropy) may vary as a function of the space coordinates. We derive a guaranteed double-sided estimate for energy dissipation that occurs in heat conduction in a locally orthotropic body, without assuming anything about the material orientation field. This yields guaranteed lower and upper bounds for energy dissipation that always hold regardless of how the local material orientation is distributed in the solid body.

In this chapter we present some results on the stability and bifurcations of the systems with a finite number of degrees of freedom. We consider damping-induced destabilization in nonconservative systems. We start with a general theoretical treatment of the topic. As the model problem, we consider the double pendulum subject to both a follower force and gravitational loading. A special case of interest is treated with the theoretical framework. The chapter finishes with a thorough presentation and analysis of the model problem including the nonlinear dynamics, quasistatic equilibrium paths and their stability, and special cases of interest. In numerical examples, we show equilibrium paths and trajectory density visualizations of the time evolution of the nonlinear system. Sample-based uncertainty quantification is employed to capture both branches of a bifurcation in the same visualization.