A Study on the Characteristics of Inherent Strain for Predicting Thermal Deformation in Additive Manufacturing

Seok-Chul Yun; Chunbiao Wu; Si Mo Yeon; Kyunsuk Choi; Jae-Woong Kim

doi:10.5781/JWJ.2025.43.3.8

Abstract

To accurately predict thermal deformation in additive manufacturing using numerical analysis, thermo-elastic-plastic analysis must be performed. This includes thermal flow (or temperature distribution) analysis over time and mechanical (elastic-plastic) analysis that incorporates the results. However, performing thermo-elastic-plastic analysis for laminated processing requires a very large memory and a long computation time, making it difficult to apply to the analysis of actual products. To address this issue, research on predicting thermal stress and thermal deformation using inherent strain has recently been introduced. This method is considered highly efficient for predicting thermal stress and thermal deformation because it simplifies numerical analysis modeling and significantly reduces computation time. In the correction process of determining this inherent strain, it is crucial to consider the fact that the amount of deformation varies significantly depending on the anisotropy of the inherent strain, the application of elastic and plastic material properties, and the laminate thickness modeling. Therefore, this study aims to provide insights into the characteristics of inherent strain by analyzing its effect on the deformation amount in the deformation analysis of a representative cantilever beam test specimen.

Key words: Additive manufacturing, Powder bed fusion, Inherent strain, Thermal deformation

1. Introduction

Additive metal manufacturing (AM) is increasingly recognized as an innovative manufacturing technology across numerous industries, including aerospace, automotive, and medical devices. In particular, its potential is notable in its ability to fabricate complex geometries and high-performance metal components in a relatively short time. However, such additive metal manufacturing processes present a high risk of defects such as distortion and cracking caused by thermal and residual stresses. These thermal stresses and deformations result from repeated and rapid heating and cooling during the AM process, which can negatively impact the mechanical properties and dimensional accuracy of the final product.

The most widely used approach for numerically evaluating thermal stress and thermal deformation is thermal elastic-plastic (TEP) analysis. TEP analysis numerically derives the time-dependent temperature distribution using the thermal conductivity equation, based on the material properties of the base material and the thermal input. This temperature distribution is then used to perform mechanical analysis to compute thermal stress and deformation. Although this method can obtain accurate data, due to the high density of information involved, it requires considerable computational time even for relatively simple numerical interpretations. In particular, when analyzing large structures or high heat input processes, analysis time and data volume increase exponentially¹⁾.

In additive metal manufacturing, various analysis methods have been developed to address such issues. Among them, the inherent strain method drastically shortens computational time by conducting elastic analysis based on minimal material properties without applying time-dependent thermal distribution data, thereby enabling efficient assessment of thermal stress and deformation. A study applying this method to additive manufacturing was published in 2014¹⁾. In this study, the inherent strain was derived from experimental results on cantilever specimens fabricated by powder bed fusion (PBF) additive manufacturing and applied to thermal deformation analysis. Subsequently, systematic experimental methods for determining inherent strain have been developed²^,³⁾. Later, a method to determine inherent strain via thermal elastic-plastic analysis of a reduced domain was proposed⁴^-⁶⁾. This approach is also validated with specimen tests and applied to thermal deformation analysis of actual processes. Currently, major commercial software packages used for analyzing thermal stress and deformation in powder bed fusion additive manufacturing (e.g., ANSYS-additive, Simufact-additive) also employ inherent strain values determined through experimental methods⁵^,⁷⁾. These software tools provide numerical calibration methods for determining the optimal inherent strain⁸^,⁹⁾. However, without a clear understanding of the effect of inherent strain on deformation magnitude, a logical approach to deformation analysis is challenging, and it becomes difficult to examine the limitations of such analyses.

When analyzing thermal deformation through the inherent strain approach, one must account for variations in deformation that arise from the inherent strain’s magnitude and anisotropy, material property application methods, and layer thickness modeling. Accordingly, the objective of this study is to present the basic concept of inherent strain and evaluate its effect on deformation through numerical thermal deformation analysis of a standard cantilever specimen, offering insights into its defining characteristics.

2. Inherent Strain

2.1 Definition of Inherent Strain

The inherent strain method was developed to analyze residual stress and deformation in welded structures. It encapsulates the concept that the permanent strain generated in weld zones due to welding thermal cycles is the inherent value responsible for residual stresses and deformations. In general, thermal expansion in the weld zone produces compressive stress exceeding the yield strength, causing substantial plastic deformation. After cooling, the residual plastic strain gives rise to stress and deformation that persist in equilibrium with the applied forces.

The total strain generated during the thermal cycle can be expressed as in Equation (1), where the permanent strain component corresponds to the inherent strain defined in Equation (2)¹⁰^,¹¹⁾.

(1)

εtotal =εelastic +εthermal +εplatic +εcreep

(2)

εtotal −εelastic =εinherent

Equation (2) shows that inherent strain is the sum of all non-elastic strain components, including plastic, thermal, transformation-induced, and creep strains, obtained by subtracting elastic strain from the total strain.

2.2 Experimental Determination of Inherent Strain

The application of the inherent strain concept to analyze thermal deformation in the powder bed fusion (PBF) process was first reported in 2014¹⁾. A specific method for determining inherent strain was introduced in 2016²⁾, which assumed thermal strain as inherent strain based on the fact that thermal expansion is the main cause of plastic deformation. The effectiveness of the inherent strain method was demonstrated by analyzing deformation corresponding to inherent strain at various peak temperatures. In 2019, a method for estimating inherent strain based on the measured deformation of a symmetric cantilever specimen was proposed³⁾.

To determine the value of the inherent strain in the fabricated specimen, FEM analysis is performed. The value of the inherent strain is defined as the one for which the calculated deformation-obtained by inputting the estimated inherent strain as thermal strain-matches the measured deformation. This method, in which the inherent strain is substituted by thermal strain for computation, is referred to as the equivalent thermal strain method¹¹^-¹³⁾. It involves entering a coefficient of linear thermal expansion and a temperature corresponding to the inherent strain value at each node in commercial FEM software to perform numerical calculations. Thermal deformation analysis using inherent strain does not require temperature distribution data; instead, it performs a simple elastic mechanical analysis by using room-temperature material properties and the equivalent thermal strain, thereby enabling rapid computation.

However, due to the diversity of analysis conditions in FEM and the various influences that the magnitude and characteristics of inherent strain exert on deformation, understanding these relationships is essential for accurately determining the optimal inherent strain.

3. Thermal Deformation Analysis Using Inherent Strain

3.1 Specimen Fabrication and Experimental Results

A cantilever structure, as shown in Fig. 1, is fabricated on a base plate using the powder bed fusion (PBF) additive manufacturing process with a laser beam. The layered structure is then separated using electrical discharge machining (EDM), and the vertical displacement at the cantilever end is measured. The specimen material is Ti6Al4V, and the primary process parameters used during the additive manufacturing process for specimen fabrication are listed in Table 1. To analyze the effect of the laser beam scanning direction under process conditions, specimens were fabricated with the scan directions set as longitudinal stripes and transverse stripes, as shown in Fig. 2, and the resulting displacements were measured.

Fig. 1

Cantilever specimen built by powder bed fusion process

Table 1

Process parameters used in the fabrication of specimens

Parameter	Condition
Laser power [W]	145
Scan speed [mm/s]	1000
Hatch spacing [mm]	0.082
Layer thickness [mm]	0.03

Fig. 2

Laser beam scan directions used in specimen building

The displacement of the cantilever, measured after separation from the base plate, is shown in Fig. 3. When the laser beam was scanned in the longitudinal direction, a greater amount of deformation was observed, indicating that the inherent strain exhibits anisotropy. In other words, since the amount of deformation changes with the scan direction, the inherent strain values, which cause the deformation, may also differ based on the scan direction. However, previous studies have demonstrated that in cases where the scan directions are mixed or the geometry of the additively manufactured part is bulk-shaped, the inherent strain can exhibit isotropic behavior⁵^,⁷⁾.

Fig. 3

Measured cantilever deflection after separation from substrate

3.2 Numerical Analysis Modeling

A numerical analysis model for the thermal deformation of a cantilever identical to the experimental specimen was constructed, as illustrated in Fig. 4. Here, the support section was modeled as a bulk (solid) structure, while the cantilever section was modeled as a multi-layer structure with 30 layers. First-order mode’s hexahedral elements were adopted, with a mesh size of 0.5 × 0.5 × 0.1 mm and a total element count of 124,992. To simulate the fabrication process, each layer’s finite elements were defined using the birth and death element, sequentially activating each layer to perform mechanical analysis. In the numerical simulation, the material Ti6Al4V was assigned mechanical properties of an elastic modulus (E) of 103.95 GPa and yield strength (Y) of 768.15 MPa. The material was modeled as elastic-perfectly plastic.

Fig. 4

Numerical model for predicting thermal deformation by using inherent strain

3.3 Numerical Analysis Results

After applying inherent strain and conducting a numerical simulation of the layer-by-layer build process, the cutting stage was simulated by deactivating the elements connecting the support and the substrate. All numerical analyses were performed using the commercial software MSC-Marc. The results of the numerical analysis applying arbitrary inherent strain are shown in Fig. 5. Through repeated simulations, the effect of inherent strain characteristics on thermal deformation was analyzed, and the model was validated for determining the optimal inherent strain by comparing the results with experimental results.

Fig. 5

Typical simulation result for cantilever deflection after cutting process

4. Effect of Inherent Strain on Deformation

To investigate the impact of inherent strain characteristics and numerical analysis methods on thermal deformation, the anisotropy and magnitude of inherent strain, along with the effects of modeling and analytical approaches, were analyzed.

4.1 Effect of Anisotropy in Inherent Strain

As confirmed by the experiment, variations in cantilever deformation occurred depending on the laser beam’s scanning direction. This indicates that the inherent strain may exhibit anisotropic behavior. Accordingly, the variation in deformation with respect to the in-plane anisotropy (k) of the inherent strain is illustrated in Fig. 6.

Fig. 6

Results of numerical simulations according to plain anisotropic inherent strains

As shown in the figure, when the ratio of inherent strain in the scan direction (longitudinal) to that in the perpendicular direction (transverse) is 2:1 or 3:1 (k = 2 or 3), the numerical analysis results for the cantilever beam closely match the experimental values. For a more detailed comparison, the displacement measured at the 11th measurement point and the root mean square (RMS) error of the numerical results are presented in Table 2(a) and (b). Comparing both cases, it was found that the error was smallest when the anisotropy of inherent strain (k) was 3. The anisotropic effect of inherent strain observed in this study is consistent with the research findings of Inaki Setien et al.³⁾

Table 2

Error estimation of FEM analysis cases (Fig. 6)

FEM case #	L1	L2	L3	L4
Anisotropy(k)	0.9	0.7	2	3
RMS error [mm]	0.348	0.444	0.137	0.153
(a) Fig. 6(a)
FEM case #		T1		T2
Anisotropy(k)		2		3
RMS error [mm]		0.247		0.078
(b) Fig. 6(b)

Furthermore, Fig. 7 presents the numerical analysis results of deformation according to inherent strain in the layup direction (vertical), and Table 3 compares the root mean square error between simulation and experimental results.

Fig. 7

Results of numerical simulation for the effect of vertical inherent strains

Table 3

Error estimation of FEM analysis cases(Fig. 7)

FEM case #	L1	L2
Vertical inherent strain	0.0	0.012
RMS error [mm]	0.153	0.140

As shown in the figure and table, inherent strain in the vertical direction has little influence on cantilever deformation. However, when the vertical inherent strain is 0.012, a smaller error is observed, and this satisfies condition of ε^L+ ε^T+ ε^V=0, making it reasonable in terms of volume conservation.

4.2 Effect of Inherent Strain Magnitude

With the horizontal anisotropy of inherent strain fixed at 3:1 and vertical inherent strain neglected, the deformation was numerically analyzed while gradually increasing the inherent strain in the negative direction. The results are shown in Fig. 8.

Fig. 8

Relationship between the cantilever deflection and inherent strains

As the absolute value of inherent strain increased, the cantilever deformation initially increased proportionally, but beyond a certain value, the deformation remained constant without increasing further. This is because as the magnitude of the inherent strain increases, the elastic strain surpasses the yield point, leading to an increase in plastic strain, as shown in Fig. 9, while the elastic strain ceases to increase and remains constant.

Fig. 9

Calculated cantilever deflection and longitudinal plastic strains with increasing inherent strains

4.3 Effect of Elastic/Elastic-Plastic Material Properties

In FEM numerical analyses, depending on the objective of the analysis, one may choose between elastic and elastic-plastic analyses for modeling mechanical behavior. Elastic analysis is based on the assumption that the material remains elastic even as the strain increases. Therefore, the model assumes no plastic strain development even when inherent strain or thermal strain increases. Conversely, elastic-plastic analysis reflects the material behavior in which plastic strain arises once the inherent strain or thermal strain surpasses the yield strength. The relationship between the inherent strain and deformation under elastic and elastic-plastic analysis methods is presented in Fig. 10.

Fig. 10

Comparison between elastic and elastic-plastic analyses

Fig. 10 displays the analysis results with isotropic inherent strain, used to observe the effect of different material modeling approaches. The results show that elastic analysis yields unrealistic outcomes; therefore, detailed input of material properties such as modulus of elasticity, yield stress, and strain hardening index is required for applying elastic-plastic analysis.

4.4 Effect of Layer Thickness Modeling

A deformation analysis was performed based on the thickness of a single layer in the model. Specifically, the 3 mm thick cantilever was divided into 3, 6, and 30 layers, and the simulation results are presented in Fig. 11. An increase in the number of layers means a decrease in the thickness of each layer in the analysis model.

Fig. 11

Results of numerical simulation for the effect of layer thickness in model

As the layer thickness in the analysis decreases-that is, as the number of layers increases-the deformation initially increases, then decreases. In this analysis, the model with 30 layers demonstrated a more detailed analysis.

5. Conclusion

To investigate the influence of inherent strain on deformation during thermal deformation in additive manufacturing processes, experimental and numerical analyses were conducted to evaluate the effects under various conditions. The key findings of this research can be summarized as follows:

1) Experiments using cantilever-shaped specimens showed that the amount of deformation differed based on the laser scan direction, with greater deformation occurring when scanned along the length of the beam.
2) The numerical analysis results extracted anisotropic inherent strain values that effectively predicted the actual deformation. The inherent strain anisotropy was found to have a longitudinal-to-transverse ratio of approximately 3.
3) Under FEM elastic analysis conditions, the deformation increased proportionally with increasing inherent strain. However, in FEM elastic-plastic analysis, the deformation increased with inherent strain only up to a certain value, beyond which no further increase was observed. This is attributed to the elastic-plastic behavior of the material and indicates that incorporating elastic-plastic material properties is essential in thermal stress analysis.
4) In thermal deformation analysis using inherent strain, deformation magnitude varies with layer thickness; hence, modeling should employ a layer thickness below an appropriate threshold. Therefore, it is necessary to select an optimal layer thickness based on numerical analysis.

This study has analyzed how inherent strain influences deformation, offering useful information for selecting appropriate inherent strain values in additive manufacturing analysis.