Warpage Prediction in Wire Arc Additive Manufacturing: A Comparative Study of Isotropic and Johnson–Cook Plasticity Models

Abstract

Wire Arc Additive Manufacturing (WAAM), a specific type of Directed Energy Deposition (DED) additive manufacturing, has recently gained widespread attention for manufacturing industrial components. WAAM has many advantages compared to other metal AM processes such as powder bed fusion. It is not only cost-efficient and easily accessible, but also capable of manufacturing large-scale industrial components in a short period of time. However, due to the inherent layered nature of the process and significant heat accumulation, parts can experience severe warping, often leading to part rejection. Predicting these anomalies prior to manufacturing would allow for process parameter adjustments to reduce or eliminate residual stresses and large deformations. In this study, we develop a simulation-based model capable of accurately predicting final deformations and unintended warpages. A Johnson–Cook plasticity model with isotropic hardening is implemented through a UMAT user subroutine in Abaqus. The proposed model is then utilized to predict the residual stresses and deformations in WAAM-fabricated parts. Simple wall geometries with 4, 8, and 20 layers deposited on build plates of varying thicknesses, are tested to assess the performance of the model. Combined Johnson–Cook plasticity and isotropic hardening for the WAAM process were implemented for the first time in this study, and the model was validated against experimental data, showing a maximum deviation of 4%. Thermal analysis of a four-layer-high wall took 12 min, while structural analysis using the proposed model took 1 h and 40 min. In comparison, thermo-mechanical analysis of the same geometry reported in the literature takes 14 h. The results demonstrate that the proposed model is not only highly accurate in predicting warpage but also significantly faster than other methodologies reported in the literature.

1. Introduction

Wire Arc Additive Manufacturing (WAAM) is technically classified as a Directed Energy Deposition (DED) additive manufacturing process [1]. It is an automated technique for the three-dimensional (3D) manufacturing of large-scale, near-net-shape metallic parts, layer by layer, using arc welding technologies [2]. It offers significant advantages in terms of time and cost for various applications compared to conventional subtractive methods such as CNC machining [3,4]. Unlike many conventional techniques, WAAM can manufacture complex parts in a single step. WAAM has recently been widely used with a broad range of metals, including but not limited to titanium, steel, aluminum, and nickel-based alloys [5].

Industrial components manufactured via WAAM exhibit mechanical properties that are comparable—and sometimes superior—to those produced using traditional manufacturing processes such as CNC machining or casting. The continuous wire feed and high heat input in WAAM result in dense parts with minimal porosity, unlike processes such as powder bed fusion, where porosity is a persistent challenge [6]. Due to the widespread availability of WAAM equipment and easy access to welding materials, it is considered a highly cost-efficient process compared to others, which require the production of metal powders or the use of expensive laser or electron beams, making those processes more challenging and costly [7]. The high deposition rates enabled by a robotic arm—which can move freely—allow WAAM to 3D-print large components in a short period of time, in contrast to other DED technologies or powder bed fusion techniques, which require specialized, often expensive, and sometimes limited setups. As a result, WAAM can be easily integrated into heavy industries [8].

Even though WAAM has many advantages—some of which are stated above—that make it one of the most efficient AM methods for large-scale part production, there are still several limitations and drawbacks. These arise from varying micromechanical properties, accumulated residual stresses, and unexpected warpages caused by high thermal gradients and cooling rates. Due to rapid solidification and uneven cooling rates, the final parts often develop a coarse dendritic microstructure, typically oriented in a specific direction, which causes the material to exhibit noticeable anisotropy [9]. Furthermore, excessive plastic deformation results from the deposition of liquid metal onto a solid build plate and its constrained shrinkage during cooling [1]. The thermal properties of the feed material and its temperature-dependent flow strength determine the magnitude and severity of the resulting residual stresses in the final parts. These accumulated residual stresses may even exceed the material’s yield stress at ambient temperature [10]. Additionally, other process parameters can negatively affect the distribution of residual stress if not properly controlled.

Eliminating or even minimizing residual stress in WAAM parts can significantly enhance the dimensional accuracy, structural stability, and overall quality of the components. Finite Element Analysis (FEA) has long been a powerful and efficient tool for simulating physical processes—such as WAAM—during which multiple physical phenomena occur. It is essential for understanding and optimizing the performance of additive manufacturing (AM) components [11,12,13,14,15]. Sethuraman et al. [16] explored the optimization of WAAM parameters of 316L stainless steel using response surface methodology and validated the results through COMSOL (version 5.6) FEA. Pragana et al. [17] developed a finite element software package to implement a macro-scale thermo-mechanical simulation of WAAM process, considering stainless steel as the candidate material. They emphasize various aspects of computer implementation, such as modeling the heat source, incorporating and element birth approach. Zhao et al. [18] investigated the impact of the deposition pattern and the shell thickness on part distortion in the WAAM process for the reinforcement of a half-cylinder shell geometry using finite element analysis.

In this study, we implement and validate a comprehensive finite element (FE) model designed to predict residual stress and warpage in parts manufactured via WAAM. A Johnson–Cook hardening model, along with J2 plasticity, is implemented using a UMAT user subroutine in Abaqus v2023. First, the model is validated using experimental data available in the literature, and then a thorough investigation is conducted by analyzing the resulting residual stress distribution and deformation in WAAM-manufactured parts. The results demonstrate the accuracy and computational efficiency of the developed model compared to similar models reported in the literature.

2. Materials and Methods

Figure 1 shows a representative schematic of the WAAM process. A metal wire of a certain diameter is fed through a welding torch where the electrical current generates a considerable amount of heat, causing the wire to melt. This molten material is deposited layer by layer onto a base plate to build a component.

The feedstock and build plate materials were selected based on reference [19]. Sample walls and the build plate are fabricated from mild steel. The chemical composition of the material (in wt.%) is 0.08% C, 1.50% Mn, 0.92% Si, 0.16% Cu, 60.040% P, 60.035% S and Fe balance. The thickness of the build plate ranges from 12 mm to 20 mm. The temperature-dependent mechanical properties of the selected material such as thermal conductivity and specific heat can be found in [19,20]. For simulation purposes, the travel speed of the welding torch is set to 8.33 mm/s, and the heat input for the welding process is assumed to be 269.5 J/mm. Assuming an efficiency of 0.9, a water-cooled aluminum backing plate is used to enhance cooling of the sample. A cooling time of 400 s is applied between subsequent layers, allowing the sample to cool below 50 °C before new layers are deposited.

2.1. Thermal Analysis

Heat transfer in the WAAM process is transient, and temperature distribution in a 3D-printed part is non-uniform. Hence, the time-dependent heat conduction equation takes the form [21]:

$$\rho \dot{U}=-\nabla ·(-k\nabla T)+r,$$

(1)

where $k$ is the temperature-dependent thermal conductivity, $\rho$ is the density, $\dot{U}$ is the rate of the internal energy, $r$ is the internal heat supplied into the body per unit volume, and $T$ represents temperature. Radiation and convection are the primary modes of heat dissipation to the surrounding environment, as defined in Equations (2) and (3), respectively.

$$q_{conv}=-h(T-T_0),$$

(2)

$$q_{rad}=-\sigma \left(T^4-{T_0}^4\right),$$

(3)

where $h$ is the convection coefficient, $\sigma$ is Stefan Boltzmann constant, and $T_0$ is the ambient (reference) temperature. The Goldak double ellipsoidal heat source, as defined in Equation (4), is used for heat application in the WAAM process.

$$\begin{array}{l}{q}_f={\displaystyle \frac{6\sqrt{3}Q{f}_f}{\pi \sqrt{\pi }{a}_{f}bc}}\exp \left[-3\left({\displaystyle \frac{x^2}{a_f^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}\right)\right]\\ q_r={\displaystyle \frac{6\sqrt{3}Q{f}_r}{\pi \sqrt{\pi }{a}_{r}bc}}\exp \left[-3\left({\displaystyle \frac{x^2}{a_r^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}\right)\right]\\ f_r+f_f=2.\end{array}$$

(4)

Parameters of the Goldak heat source are defined in

Table 1. Goldak heat source parameters.

${\mathit{a}}_{\mathit{f}}$ (mm)	${\mathit{a}}_{\mathit{r}}$ (mm)	b (mm)	c (mm)	Q (W)	${\mathit{f}}_{\mathit{f}}$	$\mathit{f}$
2	6	2.5	3	2245.83	0.6	1.4

:

2.2. Mechanical Analysis

The constitutive material model utilized in our study is the Von Mises (J2) yield criteria with an associated flow rule and the Johnson–Cook hardening model. The Johnson–Cook hardening model predicts the evolution of material strength during plastic deformation. Detailed information on this model can be found in [22]. The Johnson–Cook material model involves parameters for material strength, strain hardening, temperature, and strain rate which can be calibrated experimentally for different materials. The Von Mises yield criterion states that a material actively yields when the second invariant of the deviatoric stress tensor ($S_{ij}$) reaches a critical value. The yield function $f$ is defined as [21]

$$f=\overline{\sigma }-{\sigma }_y\left({\overline{\epsilon }}^p,T,\dot{\epsilon }\right)=0,$$

(5)

where ${\sigma }_y$ is the yield stress as a function of equivalent plastic strain (${\overline{\epsilon }}^p$), temperature $T,$ and equivalent plastic strain rate $\dot{\epsilon }$, which will be defined later in more detail. $\overline{\sigma }$ is the equivalent stress defined by

$$\overline{\sigma }=\sqrt{{\displaystyle \frac{3}{2}}S:S.}$$

(6)

In Equation (6), $S$ is the deviatoric stress tensor: $S=\sigma -1/3\left(\sigma :I\right)I$. Following the normality hypothesis of the flow rule for isotropic hardening, one can write

$$d{\epsilon }^p=d\overline{\epsilon }\mathit{n},$$

(7)

where $\mathit{n}$ is the tensor representing the plastic flow direction: $\mathit{n}={\displaystyle \frac{3\mathit{S}}{2\overline{\sigma }}}$. Using this and the integrated strain rate decomposition, the stress at the next increment can be evaluated as

$$\left(1+3G{\displaystyle \frac{∆{\overline{\epsilon }}^p}{\overline{\sigma }}}\right)\mathit{S}={\mathit{S}}^{tr},$$

(8)

where $G$ is the shear modulus, and $S^{tr}$ is the deviatoric component of the trial stress tensor ${\sigma }^{tr}$. After expanding $S^{tr}$ and taking the contracted tensor product of both sides of Equation (8), one obtains

$$\overline{\sigma }+3G∆{\overline{\epsilon }}^p={\overline{\mathit{\sigma }}}^{tr},$$

(9)

where we have

$${\overline{\mathit{\sigma }}}^{tr}=\sqrt{{\displaystyle \frac{3}{2}}{\mathit{S}}^{tr}:{\mathit{S}}^{tr}}.$$

(10)

Using Equations (5) and (9), one obtains

$${\overline{\mathit{\sigma }}}^{tr}-3G∆{\overline{\epsilon }}^p{-\sigma }_y\left({\overline{\epsilon }}^p,T,\dot{\epsilon }\right)=0.$$

(11)

This is a nonlinear equation, and the Newton method can be utilized to solve it as

$$f+{\displaystyle \frac{\partial f}{\partial ∆{\overline{\epsilon }}^p}}d\left(∆{\overline{\epsilon }}^p\right)+\dots =0.$$

(12)

The Johnson–Cook hardening model is defined as [23]

$${\sigma }_y=\left[A+B{\left({\overline{\epsilon }}^p\right)}^n\right]\left[1+C ln\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}}\right)\right]\left[1-{\left({\displaystyle \frac{T-T_0}{T_{melt}-T_0}}\right)}^m\right],$$

(13)

where $A, B, n, C, \mathrm{a}\mathrm{n}\mathrm{d} m$ are material constants that can be calibrated experimentally. According to [24,25,26], the strain rate dependency is ignored for strain rates less than reference strain rate (${\dot{\epsilon }}_0$). Additionally, the last term, which represents the temperature dependency of Johnson–Cook model, is equal to 1 for temperatures less than the reference temperature, while it is 0 for temperatures greater than melting point, as the material behaves like a liquid losing its load-bearing capacity.

The hardening slope h can be derived by taking the derivative of Equation (13) with respect to equivalent the plastic strain as

$${\displaystyle \frac{{\partial \sigma }_y}{\partial {\overline{\epsilon }}^p}}=h=\left[nB{\left({\overline{\epsilon }}^p\right)}^{n-1}\right]\left[1+C ln\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}}\right)\right]\left[1-{\left({\displaystyle \frac{T-T_0}{T_{melt}-T_0}}\right)}^m\right].$$

(14)

By substituting Equation (11) into Equation (12) and using Equation (14), we obtain

$$d\left(∆{\overline{\epsilon }}^p\right)={\displaystyle \frac{{\overline{\mathit{\sigma }}}^{tr}-3G∆{\overline{\epsilon }}^p{-\sigma }_y\left({\overline{\epsilon }}^p,T,\dot{\epsilon }\right)}{3G+h}}.$$

(15)

In Equation (15), $∆{\overline{\epsilon }}^p$ is updated until the yield function is satisfied. Once Newton iterations are converged, the plastic strain tensor can be updated as

$$\mathsf{\Delta }{\epsilon }^p={\displaystyle \frac{3{\mathit{S}}^{tr}}{2{\overline{\mathit{\sigma }}}^{tr}}}\mathsf{\Delta }{\overline{\epsilon }}^p.$$

(16)

Once the plastic strain increment is calculated, the stress tensor can be updated for the next increment. A detailed derivation of the above equations can be found in [21]. A schematic representation of the implemented UMAT user subroutine is shown in Figure 2.

From a simulation perspective, the WAAM process is comparable to the multi-pass welding process of metals [27]. Hence, WAAM-manufactured components possess similar properties to those of welded materials. The Johnson–Cook material model constants for mild steel are taken from [28] and listed in

Table 2. Johnson–Cook parameters of mild steel.

A (MPa)	B (MPa)	n	C	m	${\mathit{T}}_0$ (°C)	${\mathit{T}}_{\mathit{m}\mathit{e}\mathit{l}\mathit{t}}$ (°C)	${\dot{\mathit{\epsilon }}}_0 \mathbf{(}\mathbf{1}\mathbf{/}\mathbf{s}\mathbf{)}$
363	792.7	0.5756	0.0054	1.6456	20	1525	0.6

. It should be noted that the proposed methodology can be readily applied to any metal with isotropic hardening behavior and calibrated material properties.

3. Thermal–Mechanical Finite Element Models

A sequentially coupled thermal–mechanical analysis is performed to simulate the WAAM process of simple 4-, 8-, and 20-layer walls. First, the thermal analysis is completed to determine the nodal temperatures. Then, this temperature history is applied to the structural model as a thermal pre-load. Figure 3 shows the CAD model of geometry used in our simulations. The build plate is a rectangular plate that is 60 mm wide, 500 mm long, and 12 mm thick.

The layer height is 2 mm, and the bead width is 5 mm. Since the model is symmetrical to the XZ plane, only half of the model is considered for the simulation (Figure 3c) to reduce time and computational costs.

3.1. Thermal Simulation

Heat dissipation to the surrounding environment ($T_0=20 °\mathrm{C}$) occurs through radiation and convection from the exterior surfaces of the wall and base plate. Radiation and convection coefficients are considered temperature-independent and are $0.2$ and $5.7{\displaystyle \frac{\mathrm{w}}{{\mathrm{m}}^2\mathrm{K}}}$, respectively. A base water-cooled aluminum plate is attached to the bottom of the build plate for fast cooling. A large convection coefficient ($300{\displaystyle \frac{\mathrm{w}}{{\mathrm{m}}^2\mathrm{K}}}$) is considered for the bottom surface of the build plate to simulate the water-cooling process.

Figure 4 represents the meshed part prepared for the thermal simulation. A total of 2400 heat transfer DC3D8 elements are used, with dimensions of $2\times 2\times 2 {\mathrm{m}\mathrm{m}}^3$ for the wall and $4\times 4\times 4 {\mathrm{m}\mathrm{m}}^3$ for the base plate, for a four-layer-high wall. The thermal analysis consists of a printing step and a cooling step, which takes 1444 s and 400 s, respectively, making the total analysis last 1844 s.

The nodal temperature distribution of the first layer obtained from the thermal simulations is depicted in Figure 5. As seen in Figure 5, heat dissipates very rapidly from the bottom face due to large convection coefficient considered in this area. This reduces heat accumulation in the bottom layers, which will ultimately reduce residual stress.

3.2. Structural Simulation

In the structural analysis, the temperature history obtained from the previous section is applied as a thermal pre-load. Since the analysis is sequentially coupled, the number, shape, and size of elements must exactly match those used in the thermal analysis. Hence, the same mesh is used for structural analysis. The only difference is the element type, which is the 3D stress brick C3D8 element.

Like the thermal analysis, the structural analysis is also composed of two steps: the printing step and the cooling step. During the printing step, the four corners of the base plate and the middle section are fixed in all three directions. During the cooling step, the four corners are unclamped, while the base plate remains fixed in the middle to ensure convergence. A symmetry boundary condition is applied on the XZ plane, and a mechanical tie is defined between the wall and the build plate. A hard contact is considered between the build plate and a thin rigid plate, which only allows the bottom nodes to move in the positive Z direction but limits their downward motion in the negative Z direction. Figure 6 shows the Von Mises stress distribution in the printed wall at the end of the cooling step, after the boundary conditions are removed.

The temperature-dependent thermal conductivity of the material decreases as temperature increases. However, low thermal conductivity values at temperatures above the melting point pose challenges for the convergence of the structural model. Therefore, high thermal conductivity must be assumed for temperatures exceeding the melting point to ensure convergence.

3.3. Model Validation

Using available experimental data from the literature [19], we verify the accuracy of the proposed methodology. Figure 7 shows the deflection of the printed part along the bottom edge of the build plate after the clamps are removed. Warpage predictions are verified by comparison with the experimental results from laser scanners, and residual stresses are verified using the neutron diffraction strain scanner ENGIN-X [19].

It is clear from Figure 7 that the simulation results of the deflected part agree well with the experimental data. The maximum deviation is observed at both ends of the plate, which remains below four percent, indicating that our model is sufficiently accurate for deflection prediction in the WAAM process. The deformation is zero at the middle of the part because the middle clamps remain in place during the cooling step. However, the accumulated residual stress in the part results in upward deflection at both free ends of the beam after the clamps are removed.

4. Results and Discussion

Table 3. Thermo-mechanical simulation time.

Type of Analysis	Analysis Time (Hour)	Maximum Increment Size (Second)
Thermal	0.2	5
Structural	1.7	5

shows the time and computational efficiency of the proposed methodology. The thermal analysis of a four-layer-high wall took 12 min, while structural analysis using the proposed model took 1 h and 40 min. In comparison, thermo-mechanical analysis of the same geometry reported in the literature takes 14 h.

This clearly demonstrates that the model can readily be used to predict residual stress and distortions before the printing process begins. Therefore, parameter adjustments can be made to minimize warpage and distortion, thereby maximizing the likelihood of producing a defect-free part. In the following section, we discuss the stress distribution and deformation of walls with different heights. The effects of wall height and build plate thickness are also investigated.

4.1. Residual Stress

The WAAM process involves repeated heating and cooling of metals, which induces residual stress throughout the part. If not properly predicted, these stresses may exceed the yield stress of the material and cause complete failure of the printed part. Even lower levels of residual stress can lead to unintended warpage and dimensional inaccuracies. Figure 8 shows the longitudinal stress along Path 1 (see Figure 3) after the clamps are removed and the part has cooled to ambient temperature.

The longitudinal component of stress is tensile (positive) on the bottom surface of the base plate and gradually decreases to zero at the center of the build plate. It must be noted that the maximum stress in all three cases occurs within the first layer, where the part and the build plate meet. This result clearly supports the practice of using higher voltages in the bottom layers during experiments to achieve better adhesion. For shorter walls with four-layer and eight-layer heights, the longitudinal stress remains positive along the vertical Path 1. However, for the taller wall with 20-layer height, the stress becomes compressive around the middle of the wall. A similar reduction in stress level is observed in the eight-layer wall, although it remains tensile. In all cases, the longitudinal stress decreases near the top edge of the wall but remains tensile.

Figure 9, on the other hand, represents the longitudinal stress distribution along Path 1 when the build plate is kept clamped during the cooling step.

The most significant difference compared to the case with an unclamped build plate is the noticeable compressive (negative) stress on the bottom surface of the base plate. This occurs because the clamps restrict the shrinkage of the build plate during the cooling phase. Another important observation is that the magnitude of these compressive stresses increases with the size of the built part. Furthermore, similar to the unclamped case, the tensile stresses reach their maximum in the first layer and remain nearly constant during the cooling period. Once again, near the center of the build plate, the stress is effectively negligible.

Figure 10 shows the effect of build plate thickness on the longitudinal stress distribution when clamps are removed. The maximum tensile stress, which occurs at the top section of the printed part, decreases from 380 MPa for a 12 mm-thick build plate to 350 MPa for 16 mm-thick plate, and increases to 375 MPa for a 20 mm-thick build plate. Compressive stress, on the other hand, occurs in the middle of the build plate and is equal to −98 MPa and −115 MPa for the 16 and 20 mm-thick base plates, respectively. No compressive stress is observed for the 12 mm-thick build plate. These results indicate that even though using thicker plates may reduce the deflection of the printed part significantly, the increase in compressive stresses within the build plate as the thickness increases may result in base plate failure. GTN-derived fracture models, known as pressure-dependent failure models, support this conclusion.

4.2. Deflection

In this section, we analyze the effect of residual stress on geometric accuracy, as discussed in the previous section, and its negative effect on the final deformation and unpredictable warpages. Figure 11 shows the deformation of the final printed part in the stacking direction (Z) for three different wall heights.

Maximum deflection occurs at both ends of the build plate for all three geometries. Increasing the number of layers from four to eight results in a 67% increase in deflection. However, the pattern is reversed when the number of layers increases from 8 to 20. The maximum deflection of the 20-layer-high wall is 25% less than that of the 8-layer-height wall. This reduction in deflection is likely related to the compressive stresses developed in the middle portion of the wall, as discussed in the previous section and shown in Figure 8. These negative stresses resist further deflection in the printed part. It is worth mentioning that the strong agreement between our simulation results and the physical interpretation of the residual stress distribution offers meaningful insights into the underlying mechanisms, even in the absence of additional numerical metrics.

Figure 12 depicts the part distortion in the longitudinal direction for the different geometries. As can be seen in the contour plot, material points tend to move from both ends toward the center of the wall, which causes contraction in the whole geometry and curves the wall upward.

5. Conclusions

In this research work, we developed a model to predict stress and deformation in the Wire Arc Additive Manufacturing (WAAM) process. The proposed model is based on Von Mises plasticity and the Johnson–Cook hardening model. The model was found to be not only accurate but also computationally efficient compared to existing techniques reported in the literature.

We validated our methodology with available experimental data in the literature. Excellent agreement was found between the simulation results and the experiments. The validated model was then used to analyze residual stress distribution and deformation in parts manufactured via WAAM.

It is worth mentioning that the strong agreement between our simulation results and the physical interpretation of residual stress distribution offers meaningful insight into the underlying mechanisms, even in the absence of additional numerical metrics.

6. Future Work

There are many process parameters involved in the WAAM process that have significant effects on the printed parts. Among these parameters are heat input, travel speed, wire feed rate, and cooling time. In the continuation of our study, we aim to conduct a parameter sensitivity analysis and optimize the WAAM process using the proposed methodology.