Fluid-structure interaction approach to thermal residual stress development in laser materials

However, laser processing inevitably produces thermal residual stresses and product defects such as deformation and breakage may occur. Fluid-structure interaction has a significant influence on residual stress development, but has been simplified or neglected in traditional numerical models.

Background and motivation

Fluid structure interaction (FSI)

FSI coupling method
Moving interface
Unified momentum equation

On the other hand, fluid problems are described in an Eulerian way and a fixed grid is used. In this method, the fluid is described in the Eulerian way and the structure is represented in the Lagrangian way.

Residual stress during laser processing

Traditional residual stress prediction method
Residual stress computation using unified momentum equation

We extend the unified momentum equation approach for FSI to a numerical model for residual stress calculation. The residual stress is measured using the contour method and microstructures are examined by optical microscopy.

Figure 5. Schematic drawing of residual stress calculation. (a) sequential calculation, (b) simultaneous calculation

Unified momentum equation approach for stationary interface FSI problems

Mathematical model

Fluid equation
Structure equation

To obtain a velocity-based momentum equation for structures, the displacement vector d is expressed in terms of the velocity. Note that the lubricated interface has a thickness of 2α, and the pressure term of the fluid and the displacement term of the structure coexist within the lubricated interface region.

Numerical algorithm

In the same way, the equation of the y-component of the moment is discretized and written in the following form. This pressure correction equation is solved in the fluid and smeared interface regions because the continuity constraint is only required for the fluid.

Figure 7. The SIMPLE algorithm applied to the unified momentum equation

Numerical examples

Lid-driven cavity flow inside a solid container (square cavity)
Lid-driven cavity flow inside a solid container (semi-circular cavity)
Flow over a circular cylinder

Magnified stress fields around the cylinder: (a) normal stress in the x direction and (b) normal stress in the y direction. Comparison between the present study and the COMSOL simulation along the central horizontal line: (a) Normal stress in the x direction, (b) normal stress in the y direction.

Figure 11 presents shear stress distributions. Similar to the normal stress distributions, shear stress is the largest at the upper right corner of the interface, where the flow suddenly changes direction at the wall and a large shear stress is developed

Conclusions

Unified momentum equation approach for moving interface FSI problems

Mathematical model
Numerical algorithm
Numerical examples

Falling disk in a fluid
Oscillation of a flexible rod in a channel
Bouncing ball

Conclusions

Since the drag force is proportional to the speed, the acceleration of the disk gradually decreases. As a second example, a flexible rod was simulated that oscillates due to the pressure oscillation, where the deformation of the rod is relatively large. The rod also bends in the flow direction due to the increasing flow, and the stress concentration is observed at the bottom of the rod where it is attached.

Then the direction of motion of the rod begins to reverse, although the fluid still flows from left to right. In case D, the inlet pressure period is reduced to half that of case A. Normalized rod tip displacement in the x direction versus normalized time for case A with four different interface thicknesses.

At the bottom of the container is placed a 0.02 m thick plate of the same material as the sphere. Due to the symmetry, only the right half of the domain was used for the simulation. However, due to the different material model used on the structure, the bounce heights of the ball differ from each other.

Figure 27. Schematic drawing of a moving fluid–structure interface, where the yellow and green regions represent a moving structure at time steps t 0 and t, respectively.

Unified momentum equation approach for laser melting problem

Mathematical model

Solid-phase strain analysis
Thermal analysis

The volumetric strain εv is the strain caused by the volume change when the material undergoes phase changes in the solid state. However, this transformation does not have much influence on the development of residual stress [63] and was ignored in this chapter. To simplify the phase transformation phenomena in the solid state, this chapter considers only the martensite transformation, which is the most dominant mechanism and influences the volumetric strain the most.

In this study, the Koistinen-Marburger equation for carbon steel [64] is used to calculate the volume fraction of martensite (fm), which is written as . Here, σY is the yield stress, K is the isotropic hardening parameter, and a is the equivalent plastic strain. In this algorithm, the yield criterion is first checked with Eq. 47) with the experimental stress, which is calculated by the plastic deformation of the previous time step.

If the trial yield function is equal to or less than zero, there is no update to the plastic strain. Here σtests is the test stress, εop is the plastic deformation of the previous time step, and a.o. the equivalent plastic strain of the previous time step. In this study, an enthalpy-based energy equation with a convection term was used to calculate the temperature field, which is written as [66].

Figure 46. Enthalpy versus temperature for a material with a solid-liquid phase change

Numerical algorithm

Computational cells near the fluid-mushy zone interface
Energy equation
Extending the displacement field to newly solidified regions

In this figure, the shaded area on the right represents a mushy zone (or solid area) and the white colored area on the left is a liquid area. To solve the energy equation in a finite volume framework, Eq. 53) is first integrated over a scalar control volume and the following equation is obtained. Level setting function ϕ is defined with the zero level setting (ϕ = 0) located at the liquid-mushy zone interface.

To solve this problem, we extended the displacement field of the near rigid or stiff zone to the newly stiffened regions using a level set-based method [46]. We defined a level set function ϕ as a signed normal distance from the liquid-mushy region interface (T=Tl) as follows: Using the level set function, the displacement field was copied from the neighboring solid regions and solid in the fluid region by solving the following equation [53]:.

Only a few iterations are required for Eq. 69), because in one small time step, squishy zone (or solid) cells can only appear very close to the interface. In this study, the actual fluid-mushy zone interface was not moved using the level set method, but by updating the fluid volume fraction (Eq. 55)) after solving the energy equation. However, when the liquid-mushy zone interface moves due to melting and solidification, the level set function must still be re-initialized according to its definition of signed distance function.

Figure 48. A problem where a mushy zone exists between solid and liquid regions. Level set function ϕ is defined with the zero level set (ϕ = 0) located at the liquid-mushy zone interface.

Numerical examples

Effect of solid strain terms and fluid flow on residual stress development
Effect of laser heating time on residual stress development
Effect of mushy zone size on residual stress development
Validation of numerical algorithm

Stress and temperature fields for case A immediately after completion of laser heating (laser pulse). a) Normal stress in the x direction, (b) Normal stress in the y direction, (c) Temperature. Stress and temperature fields for case A after the material has completely cooled to ambient temperature. a) Normal stress in the x direction, (b) Normal stress in the y direction, (c) Temperature. Stress and temperature fields for case B immediately after completion of laser heating (laser pulse). a) Normal stress in the x direction, (b) Normal stress in the y direction, (c) Temperature.

Stress and temperature fields for case B after the material has completely cooled to ambient temperature. a) Normal stress in the x-direction, (b) Normal stress in the y-direction, (c) Temperature. Voltage and temperature fields for case C immediately after the laser heating (laser pulse) is completed. a) Normal stress in the x-direction, (b) Normal stress in the y-direction, (c) Temperature. Voltage and temperature fields for case D immediately after the laser heating (laser pulse) is completed. a) Normal stress in the x-direction, (b) Normal stress in the y-direction, (c) Temperature.

Stress, temperature and martensite distributions for Case D after the material has completely cooled to ambient temperature. a) Normal stress in the x direction, (b) Normal stress in the y direction, (c) Temperature, (d) Martensitic phase distribution (represented as a red colored area). Stress and temperature fields just after the laser heating (laser pulse) is completed. a) Normal stress in the x direction, (b) Normal stress in the y direction, (c) Temperature. Stress, temperature and martensite distributions after the material has cooled down to ambient temperature. a) Normal stress in the x direction, (b) Normal stress in the y direction, (c) Temperature, (d) Martensitic phase distribution (represented as a red colored area).

Table 3. Material properties of POSTEN80

Conclusions

However, we can say that residual stresses predicted by the presented numerical algorithm are physically realistic and at least qualitatively accurate.

Unified momentum equation approach for laser heat treatment

Experimental procedure
Mathematical model

Mechanical analysis
Thermal analysis
Metallurgical analysis

Numerical and experimental results
Conclusions

A symmetry boundary condition is applied to the (x, 0, z) plane in Fig. 65, and the right half is considered in the numerical analysis. In the temperature field, we observe much higher temperatures in the numerical result without fluid flow, since no convection due to fluid flow occurs. Numerical results with consideration of fluid flow (left) and without consideration of fluid flow (right) for case A on the top surface during the laser process. a) normal stress in x-direction, (b) solid phase, (c) temperature.

Numerical results considering the fluid flow (left) and without considering the fluid flow (right) for case A at the top surface after the material has completely cooled to ambient temperature. a) Normal stress in the x-direction, (b) Solid phase, (c) Temperature. The temperature distribution is further increased to accurately observe the liquid flow in the melt pool. The cross-sectional figures show more clearly the differences between the numerical results with and without fluid flow.

Numerical results considering liquid flow (left) and without considering liquid flow (right) for case A in the cross-section during the laser process. a) Normal stress in the x direction, (b) Solid state phase, (c) Temperature. Numerical results considering liquid flow (left) and without considering liquid flow (right) for case A in the cross-section after the material has completely cooled to ambient temperature. a) Normal stress in the x direction, (b) Solid state phase, (c) Temperature. Consequently, in this case, the transformation of austenite to bainite also takes place, and martensite and bainite coexist in the HAZ.

Table 6. Process conditions of laser heat treatment experiment Case Laser power (P) Scanning speed (U s )

Summary and future works

Kim, Laser transformation hardening of carbon steel sheet using heat sink, Journal of Materials Processing Technology. Matsumoto, A complete Eulerian finite difference approach for solving fluid-structure coupling problems, Journal of Computational Physics. Nave, Reference chart technique for finite strain elasticity and fluid-solid interaction, Journal of the Mechanics and Physics of Solids.

Ki, A uniform momentum equation approach for fluid-structure interaction problems involving linear elastic structures, Journal of Computational Physics. Kermandis, Numerical simulation of the laser welding process of butt joint specimens, Journal of Materials Processing Technology. Davies, FEM prediction of welding residual stresses in fiber laser welded AA 2024-T3 and comparison with experimental measurements, The International Journal of Advanced Manufacturing Technology.

Osher, A level set approach for computational solutions for incompressible two-phase flow, Journal of Computational Physics. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, Journal of Computational Physics. Na, Prediction of weld residual stress with real-time phase transformation by CFD thermal analysis, International Journal of Mechanical Sciences.