Towards a comprehensive understanding of distortion in additive manufacturing based on assumption of constraining force

ABSTRACT

Additive manufacturing has shown giant potential in advanced manufacturing industry. However, distortion of additive-manufactured parts may interrupt the printing process and decrease manufacturing precision, causing a significant waste of time and money. Distortion, a common problem in additive manufacturing, has not been sufficiently researched. In this study, we provide a comprehensive understanding of distortion in additive manufacturing based on assumption of constraining force. A quantitative model was established to illustrate inconsistent shrinkage during the additive process. A qualitative model was then built for showing the distortion caused by inconsistent shrinkage. It can be founded that inconsistent shrinkage accumulated during the additive process drives the final distortion of part. We concluded most influencing factors of distortion together, forming towards a comprehensive understanding of distortion in additive manufacturing. This study not only links micro melting-solidification process and macro structural mechanics in science, but also provides systematic solutions for decreasing the distortion in additive manufacturing for engineering.

KEYWORDS:

• Distortion
• selective laser melting
• laser powder bed fusion
• shrinkage
• Rapid solidification process

1. Introduction

Additive manufacturing is a promising approach for fabricating complex-shaped and high-performance objects without molds (Debroy et al. 2019; Yang et al. 2020; Morgan et al. 2019; Wang et al. 2019). It has been utilised in producing newly-designed lightweight and integrated structures for aerospace, medical, and automotive applications (Sing and Yeong 2020; John and Mohsen 2016), and has become an area of intense research focus in science and industry (Romero, Yang, and Bolzoni 2018; Yang et al. 2021).

Distortion is the deviation of an additive-manufactured part from its designed shape or dimensions (Anandan and Kumaraguru 2019). It is a common problem in additive manufacturing, such as selective laser melting (SLM or LPBF (Laser powder bed fusion, ISO/ASTM 52900-15)), laser melting deposition (LMD or LDED (Laser directed energy deposition)), fused deposition molding (FDM or ME (material extrusion)), stereolithography (SLA or VP (vat photoploymerization)), and digital light processing (DLP). For instance, a cantilever structure built by SLM bends upwards after being cut from the substrate (C. Li et al. 2017). The shape deviations of a part may interrupt the additive process by hindering the movement of the roller, causing a significant waste of time and money. Distortion also decreases the precision of the built part, which may cause it to be rejected for use in critical applications.

Previous studies have explored the mechanism of distortion. Kruth et al. 2012 explained the micro-process of metal additive manufacturing on the basis of the thermal gradient mechanism (TGM), which fit quite well with the experimental results of in-time distortion measurement from (Denlinger, Heigel, and Michaleris 2015; Denlinger et al. 2017) and (Biegler, Graf, and Rethmeier 2018). TGM provides micro and qualitative insights into mechanism of distortion. It helps to understand the distortion of simple structure fabricated by simple scan strategy. However, it cannot be used to detailedly explain the influences of structure, material property and scan strategy on the distortion from a macro perspective. The gap between the micro melting-solidification process and the macro distortion still exists.

Many efforts have been made to explore the influencing factors and regularity of distortion. Researchers have attempted to predict the distortion of additive-manufactured parts by numerical simulation (Prabhakar et al. 2015; Tawfik, Nasr, and Gamal 2019; Y. Li et al. 2017). Zielinski et al. 2017 investigated the distortion evolution of a bridge-geometry part, and revealed significant changes in distortion once the two piers of the bridge connected. Afazov et al. 2017 and Seidel and Zaeh 2018 simulated the distortion of a blade, and proposed approaches to decrease the distortion by pre-compensation of the blade. Zhang and Zhang 2017 studied the distortion of a bar with several channels via finite element analysis, and found that a thick and low-density porous layer could reduce residual stress and distortion. Li (Li et al. 2016; C. Li et al. 2017) investigated the influence of scan patterns on the distortion via simulation and experiments, and found that vertical scanning results in less distortion than horizontal scanning. Williams, Davies, and Hooper 2018 found that the distortions are almost the same when the same volume of metal is deposited, and it is not necessary to model every laser pass or even every deposited layer to accurately predict the final distortion. Tan et al. 2019 pointed out that solid-state phase transformation should be considered in thermo-metallurgical-mechanical coupling model for selective laser melting. Pre-heating the powder bed and the substrate is regarded as an effective method to decrease the distortion (Zaeh and Brannen 2010; Vilaro, Colin, and Bartovt 2011; Buchbinder et al. 2014). Denlinger et al. 2015 found out that adding dwelling time between layers is beneficial for reducing the distortion. Kruth et al. 2012 stated that post-heat treatment can decrease the maximum distortion to 20%. Nickel, Barnett, and Prinz 2001 and Gao et al. 2013 validated that the scan patterns have a critical impact on distortion. Although studies have shown some influencing factors of distortion by means of numerical simulation and experiments, people may still have to try several times to achieve a successful print. An overall image of distortion is required to be depicted.

The present authors have recently developed a new assumption for discussing distortion in metal additive manufacturing (Xie et al. 2018). A pair of constraining forces was assumed at the interface between the newly added layer and previous added layer or substrate. The distortion was regarded as a result of one constraining force acting on the built structure. The assumption of constraining force provides a macro perspective for the discussion of distortion in additive manufacturing. The aim of the present study was to link the micro melting-solidification process and the macro distortion behaviour, as well as to integrate most influencing factors together.

2. Methodology

2.1. Selective laser melting

The experiments were conducted with a laboratory-assembled SLM machine equipped with a 200-W fibre laser (IPG Co. Ltd., USA), a scanner (Scanlab Co. Ltd., Germany), and an AMCP controller (Materialise Co. Ltd., Belgium). The austenitic stainless steel AISI 316L (Maite Powder Co. Ltd, China) was chosen. The optimised processing parameters of SLM were laser power 180 W, scanning speed 1 m/s, hatch distance 50 μm, layer thickness 30 μm. The scan directions were always parallel to the longitudinal direction of the arm. The built cantilever structures were then cut from the substrate by wire electrical discharge machining (WEDM). The longitudinal section was cut by WEDM and then etched with CuSO₄ (10 g) + HCl (50 ml) + H₂O (50 ml). The displacement of points along the central line on the top surface of each arm were measured by a coordinate measuring machine (Leader Metrology Co. Ltd., USA).

2.2. Selective laser sintering

SLS experiments were completed with a commercial SLS machine (P760, EOS Co. Ltd., Germany) equipped with two 50-W CO₂ lasers. The nylon PA12 (EOS Co. Ltd., Germany) was chosen as the raw material. The material was heated to 145 °C for 1 h before laser sintering. The optimised processing parameters of SLS were laser power 15 W, scanning speed 1.2 m/s, hatch distance 50 μm, layer thickness 100 μm. The laser scan direction was set parallel to the longitudinal direction of the arm. The distortion was characterised by a coordinate measuring machine (CMM; Leader Metrology Co. Ltd., USA).

2.3. Fused deposition molding

FDM experiments were conducted on a commercial FDM machine (Guider II, Flashforge Co. Ltd., China). A 1.75 mm-diameter polylactic acid wire was used as material. The processing parameters were base temperature 80 °C, heating temperature 220 °C, scan speed 30 mm/s, layer thickness 0.1 mm.

2.4. Stereolithography

SLA experiments were conducted on a commercial SLA machine (SPS350B, Hengtong Co. Ltd., China) with the following processing parameters: laser power 220 mW, scanning speed 8 m/s, hatch distance 50 μm, layer thickness 100 μm. The laser scan direction was always parallel to the longitudinal direction of the arm. The liquid resin SPR6000 epoxy (Hengtong Co. Ltd., China) was cured by ultraviolet light at a wavelength of 355 nm. The distortion was characterised by CMM (Leader Metrology Co. Ltd., USA).

2.5. Digital light processing

DLP experiments were conducted on a commercial DLP machine (Prismlab Co. Ltd., China) with the following processing parameters: exposure intensity 50 mJ/cm², exposure time 3 s, layer thickness 0.1 mm. The liquid resin P400 (Prismlab Co. Ltd., China) was used as the raw material.

3. Results

3.1. Distortion in additive manufacturing

A typical cantilever structure was designed as shown in Figure 1a. Cantilever structures of various arm thicknesses were fabricated by SLM. After the arms were cut from the substrate, the arms bent upwards. The distortion can be also found in parts built by other additive manufacturing techniques, including SLS, FDM, SLA, and DLP. The distortion in additive manufacturing reduced the precision of these parts.

Figure 1. Distortion of cantilever structures in additive manufacturing. a Distortion of a cantilever structure built by SLM. b Cantilever structures (arm thickness t = 2 mm) built by SLS, FDM, SLA, and DLP. c Distortion of cantilever structures (arm thickness t = 2 mm) built by SLS, FDM, SLA, and DLP measured by a coordinate measuring machine.

3.2. Inconsistent shrinkage during the additive process

In this study, SLM was the main focus because it enables noticeable distortion of cantilever structures after they are cut from the substrate (Cheng et al. 2019). In order to ensure the manufacturability of the arms, a sacrificial support is required because it can provide mechanical fixture and a heat dissipation channel (Cheng et al. 2019). Austenitic stainless steel (AISI 316L) was chosen as the research material.

Unlike the micro mechanism of TGM, the assumption of constraining force offers a method to recognise the distortion mechanism in additive manufacturing from a macro view (Xie et al. 2018). Each layer of the deposited arm was regarded as uniform and underwent the same melting-solidification process in SLM. The laser scan direction was always along the longitudinal direction of the arm. In Figure 2a, a schematic of one layer of a cantilever arm is shown at high temperature (HT). When cooled to low temperature (LT), the free end of the deposition layer shrinks with a shrinkage length of u⁰ if there is no constraint, as shown in Figure 2b. However, the sacrificial support also constrains the shrinkage of the deposited arm at the same time. Hence, the shrinkage of the first layer can be set as $u_1^s$ rather than u⁰. The constraining forces $F_1^s$ can be assumed at the interface between the deposited first layer and the support, as shown in Figure 2c. Similarly, the constraining forces $F_{\mathrm{i}}^s$ can be assumed at the interface between the layer i and the layer i-1, as shown in Figure 2d. The shrinkage of layer i is set as $u_{\mathrm{i}}^s$.

Figure 2. Assumption of constraining force and experimental validation. a Deposited arm at HT without support. b Deposited arm at LT without support. Its shrinkage is u⁰. c Assumption of constraining force between the support and the first layer. d Assumption of constraining force between layer i − 1 and layer i. e The bent support predicted by the assumption of constraining force. f The experimental bent support.

We used a model to predict what will happen if we set constraining force on the cantilever structure. The arm length is 20 mm, while the arm width is 6 mm. The arm thickness is 0.3 mm, which means 10 layers of metal. As shown in Figure 2e, we set 1000 N as an initial value for the constraining force $F_{\mathrm{i}}^s$ (i = 10) in mechanical finite element analysis tools Workbench (ANSYS Ltd., USA). The bottom of support and top surface of substrate is set to be fixed. It can be found that the support can be bent towards the base. The experimental results validated that the support structure was bent towards the base, as shown in Figure 2f.

As shown in Figure 3a, the shrinkage of layer 1 from HT to LT is $u_1^s$. Then, layer 2 at HT is deposited on layer 1. When layer 2 cools to LT, the shrinkage of layer 2 is $u_2^s$. $u_2^s$ is a balanced result of the shrinkage of layer 2 and the constraint of layer 1 together with the support structure. The shrinkage of layer 1 becomes $u_1^s+u_2^s$ after layer 2 cools. Similarly, the shrinkage of layer 1 is $u_1^s+u_2^s+u_3^s$ after layer 3 cools, and the shrinkage of layer 2 become $u_2^s+u_3^s$. Each new added layer shrinks, causing the support and the previous deposition layers to contract towards the fixed end. With the increase in deposition height, the inconsistent shrinkage of the deposited layers emerges. It can be predicted that the temporary shrinkage $u_{\mathrm{i}}^s$ of layer i decreases with the increase in deposition height because the previous deposited structure with greater height has greater shear stiffness. Thus, the increment of shrinkage becomes less when the deposited height increases, as shown in Figure 3b.

Figure 3. Evolution of inconsistent shrinkage during the additive manufacturing process. a Accumulation of shrinkage during the additive process. b Experimental inconsistent shrinkage. c Calculated shrinkage of each layer under the assumption of constraining force. d Comparison of calculated and experimental inconsistent shrinkage of the part (t = 2 mm).

From the schematic in Figure 2d, the assumption of constraining force makes it possible to calculate the temporary shrinkage $u_{\mathrm{i}}^s$ of layer i. On one hand, the constraining force $F_{\mathrm{i}}^s$ causes the previous structure, i.e. the support and layer i-1 deposition, to contract. According to finite element analysis, a certain shrinkage u* can be obtained if forces F* are applied to the arms of the cantilever structure with i-1 layers’ deposition. The Young's modulus is reported as 193 GPa (Yakout, Elbestawi, and Veldhuis 2019). A relationship between u* and F* can be fitted and then depicted, as shown in Figure 3c. There will be N curves if the maximum number of deposition layers is N. In Figure 3c, several curves concerning u* and F* are shown when the deposition number of layers varies from 1 to 60. For example, the red dotted line in Figure 3c reflects the changes of u* when the forces F* with different values act on the arm with nine layers.

On the other hand, the deposited layer i shown in Figure 2d is regarded as an elastic rod. The shrinkage of layer i should satisfy Hooke's law: (1) ${\displaystyle \frac{F_i^s}{F_{max}}={\displaystyle \frac{u^0-u_i^s}{u^0}}}$(1) (2) $u_i^s=u^0-{\displaystyle \frac{u^0}{F_{max}}\cdot F_i^s}$(2) Here, $F_{\mathrm{i}}^s$is the assumed the constraining force at the interface between layer i and layer i-1, as shown in Figure 2d. F_max can be calculated as (Xie et al. 2018) (3) $F_{max}={\displaystyle \frac{u^0}{l_d}\cdot E_d\cdot A_d=({\alpha }_d\cdot \mathrm{\Delta }T+{\beta }_{pt})\cdot E_d\cdot A_d}$(3) E_d and A_d are the Young's modulus and cross-sectional area of the deposition at room temperature, respectively. α_d is the mean thermal expansion coefficient of the deposition, and ΔT represents the temperature differences between solidus temperature and temperature of substrate or previous deposition. β_pt is the shrinkage rate caused by phase transformation. The values of α_d, ΔT, and β_pt are 20.02×10⁻⁶/°C, (1371 − 20) °C, and 2.78×10⁻², respectively (Xie et al. 2018). In this experiment, the arm width w was 6 mm. The thickness of each layer was 30 μm. Thus, the value of F_max can be calculated as (4) $F_{max}=(20.02\times {10}^{-6}\times (1371-20)+2.78\times {10}^{-2})\times 1.93\times {10}^{11}\times 6\times {10}^{-3}\times 3\times {10}^{-5}=1905.4N$(4) The length of the arm was 20 mm; the maximum shrinkage u⁰ can be calculated as (5) $u^0=({\alpha }_d\cdot \mathrm{\Delta }T+{\beta }_{pt})\cdot l_d=(20.02\times {10}^{-6}\times (1371-20)+2.78\times {10}^{-2})\times 20=1.1mm$(5) By applying Equation (2) to Figure 3c, the shrinkage of $u_{\mathrm{i}}^{\mathrm{s}}$ can be obtained. Thus, the final shrinkage of each layer can be calculated, as shown in Figure 3d. The designed height of the arm was 2 mm. Thus, there should be 67 layers in total when the thickness of each layer is 30 μm. The shrinkage of the top layer is $u_{67}^s$, whereas the shrinkage of the bottom layer is the summation $\sum _{\mathrm{i}=1}^{67}{u}_i^s$. The connection of each layer's calculated shrinkage denotes the end of the arm. The experimental shrinkage lengths agree closely with the calculated shrinkage lengths. They share almost the same tendency from top to bottom. The differences between the calculated and experimental maximum shrinkage lengths may due to the anisotropy of the microstructure, changes in local properties caused by thermal elastoplastic behaviour, the contraction in the deposition direction, and the imprecise estimation of shrinkage caused by thermal variation and phase transformation. It is reported that the accumulated heat can maintain the previous deposited structure at 500–1200 K (Luo and Zhao 2019). The constraining force can be distinctly decreased with ΔT, which means the difference between the solidus temperature and the previous deposition's temperature. Although accurate calculation of the constraining force requires further study, the idea of constraining force is critical to quantitatively recognising the shrinkage of the deposited structure layer by layer. In this study, it revealed the evolution of inconsistent shrinkage of the cantilever structures during the additive manufacturing process.

3.3. Distortion caused by inconsistent shrinkage after cutting

Distortion of cantilever structures is reported to occur after the arm is cut from the support and substrate (Anandan and Kumaraguru 2019; Li et al. 2016). However, the distortion mechanism has not been clarified. In this paper, we propose a distortion mechanism concerning the relationship between distortion and inconsistent shrinkage, as shown in Figure 4a and b. Rod A and rod B are regarded as ideal elastic bodies with the same original lengths in free state. First, they are stretched by forces F_A and F_B, respectively. Then, the two rods are tightly connected, such as by welding, which is similar to the connection between two layers in SLM and other additive manufacturing processes. As shown in Figure 4a, if the forces F_A and F_B are equal in value, the two rods will contract to the original state with no distortion after the forces are removed. However, as shown in Figure 4b, if F_A is greater than F_B, the movement of the two rods will be composed of shrinkage and distortion after the removal of the two forces. There will be compressive strain in rod A and tensile strain in rod B, resulting in strain equilibrium at the interfaces between the two rods. The difference in length between rod A and B is thought to offer the strain energy of distortion. It can also be deduced that a greater difference between F_A and F_B results in greater distortion of rods A and B.

Figure 4. Distortion caused by inconsistent shrinkage. a Consistent shrinkage causes no distortion. b Inconsistent shrinkage drives the distortion. c Edges of SLM parts with various arm thicknesses, showing the inconsistent shrinkage of these parts. d Comparison of the edges (inconsistent shrinkage) of SLM parts (t=1 mm and t=5 mm) when the arm thicknesses are the same. e Inconsistent shrinkage of various arm thicknesses built by SLM causes various distortions. f Distortion of FDM parts with various arm thicknesses. The arm length is 30 mm, and the arm width is 15 mm. g The distortions change with arm thicknesses built by SLA, FDM, DLP, and SLS.

The inconsistent shrinkage of the 67-layer deposition is shown in Figure 3d. The shrinkage of each layer is less than the maximum shrinkage u⁰; thus, each layer further contracts when the arm is cut from the substrate. By combining Figure 3d and Figure 4b, the 67 layers as built with different shrinkage lengths can be regarded to be stretched by 67 different forces, which are in accordance with the assumed 67 constraining forces. The distortion emerges after the arm is cut from the substrate, which means that the 67 forces are set to be free. In a word, the inconsistent arm shrinkage caused by the different shrinkage of each deposition offers the strain energy of distortion after the arm is cut from the substrate.

The relationship between the inconsistent shrinkage and distortion of part will be illustrated as follows. Figure 4c demonstrates the edges of SLM parts with various arm thicknesses, which show the inconsistent shrinkage of these parts. If we want to compare the distortion caused by different inconsistent shrinkage, we should let the arm thicknesses (i. e., volume of arm) be the same. As shown in Figure 4d, we magnify the edge (i. e., inconsistent shrinkage) of part with 1-mm thickness to 5 times. It can be found that the inconsistent shrinkage of part with 1-mm thickness is much bigger than that of part with 5-mm thickness. It can be deduced that the distortion of part with 1-mm thickness is bigger than that of part with 5-mm thickness, which can be validated in Figure 4(e). The result is also consistent with those in the literature (Anandan and Kumaraguru 2019; Li et al. 2016; Buchbinder et al. 2014). The same tendency can also be found in parts built by FDM, SLA, and DLP, as described in Figure 4 g. However, the distortion of SLS parts does not change distinctly because of the high-temperature preheating process in SLS.

4. Discussions

4.1. The mechanism of distortion in selective laser melting

It can be found that the shrinkage of each layer and inconsistent shrinkage of all layers can be calculated based on the assumption of constraining force, as shown in Figure 2 and 3. The link between the inconsistent shrinkage and distortion of part is illustrated in Figure 4. Thus, we are able to illustrate the mechanism of distortion and its overall influencing factors, as shown in Figure 5a. Firstly, there is temporary shrinkage of each layer. As the additive process progresses, the temporary shrinkage accumulates, causing inconsistent shrinkage of all layers. After the built part is cut from the substrate, the inconsistent shrinkage drives the part to distort. Thus, the temporary shrinkage, especially in the first few layers, should be decreased so as to decrease the inconsistent shrinkage from top layer to bottom layer and the final distortion of built part. The temporary shrinkage of layer N is determined by the stiffness of the structure and constraining force F_N. The structure is composed of deposited structures (N − 1 layers) and support structure. The constraining force is a vector with magnitude and direction.

Figure 5. Mechanism of distortion. a Mechanism of distortion and its influencing factors. b physical model for calculating temporary shrinkage when scan direction is parallel to arm. c physical model for calculating temporary shrinkage when scan direction is vertical to arm.

4.2. Application of the distortion mechanism in additive manufacturing

Although different processes can generate different distortion, the distortion mechanism can also be applied in most additive manufacturing techniques. For example, Huang and Jiang 2003 said that short raster scanning produces less distortion than long raster scanning in SLA. It can be explained by using Figure 5(b-c). The long raster scanning (as shown in Figure 5b) generate more shrinkage because of its relative lower stiffness of deposited structure. Liu et al. 2015 studied the influence of layer thickness, nozzle temperature, and scan strategy on the distortion built by FDM. It is said that ‘with the decreasing of layer thickness, the distortion of PLA thin-plate part increases obviously.’ It can be comprehended that the less layer thickness, the less stiffness of deposited structure in the first several layers. The shrinkage in the first several layers is larger when using less layer thickness, causing a bigger distortion of part. It is also said that ‘the low nozzle temperature is helpful to reduce the distortion.’ Lower nozzle temperature means a less temperature difference between nozzle and environment or room. So the distortion can be decreased by lowering the nozzle temperature. Wu et al. 2019 stated that the distortion of a DLP printed object depends on the gradient of addition DoC (degree of conversion) in the material and stiffness of structure. The conclusions are consistent with the mechanism shown in Figure 5(a). Different degree of conversion means different shrinkage of material and different distortion of part. And stiffness of structure is an important factor in distortion of part built by additive manufacturing, which is also emphasised in Figure 5a.

According to engineering experience, the distortion in SLS is much less, because the pre-heating temperature is quite close to the glass-transition temperature of the sintered material (Zeng et al. 2019). So the shrinkage of SLS material is quite small, which means a small distortion of part. As a result, the distortion mechanism shown in Figure 5 can be applied in most additive manufacturing processes including SLM, SLS, SLA, FDM, and DLP.

In addition, many researchers have used the cantilever structure to study the distortion, because the distortion of cantilever structure is more obvious and easy to be measured (Buchbinder et al. 2014; Li et al. 2015; Li et al. 2016; Mugwagwa et al. 2018.). In fact, the distortion mechanism can also be applied in other models. For example, Wu et al. 2014 depicted the distortion of prism structure built by SLM. Both the horizontal specimen and vertical specimen generate the distortion with edges ‘peeling up’ away from substrate. The authors attributed this behaviour to constraint against in-plane shrinkage imposed by sub-surface layers. The idea is in good consistent with the proposed distortion mechanism in this study. And we can use constraining force models to clearly illustrate the distortion in both horizontal specimen and vertical specimen. Because of the island scan strategy, the integrated constraining forces on the deposition plane (x-y plane) point to the centroid of figure. So the distortion of both horizontal prism and vertical prism can be easily understood. Khouzani et al. 2017 studied the distortion of disk-shaped part built by SLM. The built disk was distorted as a ‘dish’ after removal of the substrate and support structure. The scan direction was rotated by 67° between the consecutive layers, so after many layers the integrated constraining forces are centrosymmetric with directions pointed to the centroid (centre of a circle). So the distortion of ‘disk to dish’ can be easily comprehended based on the distortion mechanism shown in Figure 5.

4.3. Approaches for decreasing the distortion

Based on the Figure 5a, the approaches to decrease the inconsistent shrinkage and distortion are summarised as follows.

(1) Enhance the stiffness of the structure.

The stiffness of the deposited structure can be enhanced by model optimisation based on morphology and dimension. Furthermore, a part's laying angle in the additive process enables variation in the deposited structure's stiffness, because it can change the slice of the model and the relevant structural stiffness layer-by-layer.

The stiffness of the support structure is believed to have a significant influence on the distortion. We compared seven support structural patterns to validate the link between distortion and the shear stiffness of the support, as shown in Figure 6. The shear stiffness of a cantilever structure with various support structures is partially calculated in the same way as shown in Figure 2e. The shrinkage u* was for F* = 1000 N and a deposited arm height of 0.3 mm. As shown in Figure 6b, the cross-X support offers the smallest shear stiffness, whereas the diamond support affords the largest. The inconsistent shrinkage (shown by the ends of the arms) and the distortion of built parts present the same tendency. Therefore, a higher shear stiffness of support structure can reduce the inconsistent shrinkage and the distortion of built part.

Figure 6. Influence of support structural stiffness on distortion. a The seven designed support–structure patterns. b The relationship between experimental distortion and support structural stiffness (showing by shear displacement in ABAQUS structural mechanics finite element analysis).

(2) Decrease the value of the constraining force.

As shown in Figure 5a, the value of the constraining force is determined by the constraining coefficient k_ds, temperature differences ΔT, cross-sectional area A_d, and properties of the built materials, including mean thermal expansion coefficient α_d, shrinkage caused by phase transformation β_pt, and Young's modulus E_d.

The constraining coefficient k_ds is believed to be closely related to the micro mechanism i. e., classical thermal gradient mechanism (TGM), as illustrated in (Kruth et al. 2012). According to TGM, the molten pool expanded when it is heated and expanded, which has a compressive stress on the surrounding material. After it cools down and shrinks, the solidified molten pool will generate a tensile stress with the surrounding material. We believe that the k_ds is mainly influenced by the micro process of ‘expansion to shrinkage’. In other words, the k_ds may be related to the dimension and energy distribution of heat source, the heating time heat source (speed of heat source), the cooling time (the temperature gradient of heat source), the thermal physical properties of the material, and the cooling condition of the surrounding material. For example, we can conclude from engineering experience that the part built by laser may generate less distortion than that built by arc, because laser has a sharper temperature gradient. The material can be heated and cooled more rapidly, causing a less constraining coefficient k_ds and distortion of part.

The temperature differences ΔT can be decreased by improving the pre-heating temperature, especially when printing the first several layers. The structure of first several layers is prone to shrinkage due to its weak shear stiffness. Studies (Buchbinder et al. 2014; Zaeh and Brannen 2010; Vilaro, Colin, and Bartovt 2011) showed that pre-heating the powder bed and substrate can effectively decrease the final distortion of the cantilever structure. The parts built by SLS show little distortion, as shown in Figure 4d. This result is attributed to the pre-heating process and the material properties. The pre-heating temperature in SLS is quite close to the glass-transition temperature of the sintered material (Zeng et al. 2019). The temperature difference ΔT is decreased dramatically. Furthermore, there is no phase transformation in the cooling process of nylon. The shrinkage tendency illustrated in Equation (3) is quite low. The inconsistent shrinkage during the additive process and the final distortion of the built part can be small. Thus, improving the pre-heating temperature close to the solidus temperature is effective for decreasing distortion of parts built by additive manufacturing techniques involving heating–cooling process, such as SLM, LMD and FDM.

Proper processing parameters, such as heat input, are necessary to reduce distortion. Excessive heat input is believed to increase the constraining coefficient and the corresponding distortion. Materials with lower thermal expansion coefficients, such as Invar 36, exhibit less distortion than those with high thermal expansion coefficients (Yakout et al. 2020). Further study is suggested on decreasing the thermal expansion coefficient of metal and thermoplastic polymer, the shrinkage ratio of curing resin, and phase transformation shrinkage of most printing materials. It would be great that the expansive phase transformation during cooling or curing process can be developed for the printing materials.

(3) Optimise the scan strategy.

The direction of constraining force is mainly parallel to the laser scan direction. The inconsistent shrinkage can be decreased by optimising the constraining force direction, which is determined directly by the laser scan direction. Different scan strategies can change the directions and working areas of constraining forces, and thus can change the shrinkage and final distortion of the part. As shown in Figure 5b and c, different scan direction means different magnitude and direction of constraining force, thus can cause different temporary shrinkage. The rational laser scan strategy (Yang et al. 2019; Li et al. 2015), fractal scan strategy (Gao et al. 2013; Dai and Shaw 2002), and island scan strategy (Yan et al. 2018) for a certain structure can effectively decrease distortion. It is reported that the distortion can be decreased by almost 50% if the scan vectors was reduced from 20 mm to 2.5 mm (Salem et al. 2020). Novel scan strategies are required for structures with different characteristics, under the guidance of Figure 5. Good scan strategies for decreasing the distortion in SLM can be shared with other techniques including SLA, FDM, and LMD. The island scan strategy can be referred by the optimisation of exposure sequence in DLP.

Besides, the deflection compensation is also an effective way to decrease the distortion of part. The deflection compensation means that the FEA predicted displacements or optical 3D scanned displacements were inverted in the opposite direction, changing the sizes of designed model (Afazov et al. 2017). The optical 3D scanned displacements is more reliable for deflection compensation. However, it costs more time and money.

(4) Balance the part quality and distortion

The part built by additive manufacturing (especially SLM or LPBF) is required to have good quality and precision. The priority of distortion is suggested to be higher than that of part quality, because the part quality can be improved by parameters optimisation and post processing including re-melting, polishing, and Hot Isostatic Pressing (HIP). However, the distortion is mainly determined by the factors in additive manufacturing process including part's laying angle, support structure, pre-heating, scan strategy. If given a designed model and required material, the followings factors are suggested to be determined in order: part's laying angle, support structure, pre-heating, scan strategy, parameters optimisation, and post processing. The influence of part's laying angle, support structure, pre-heating, and scan strategy on the distortion have been discussed before. There will be much work on parameters optimisation mainly for improving the density of part, for example laser profile optimisation (Huang et al. 2021) and parameter development (Kuo et al. 2020). The re-melting is beneficial for improving the density of part (Yu et al. 2019). However, it may have complicated impact on part's distortion. In one hand, the higher relative density may lead to a high distortion of part (Mugwagwa et al. 2018). In the other hand, the re-melting may function as a pre-heating method, which can decrease the temperature difference and distortion of part. Polishing (Zhang, Chaudhari, and Wang 2019) can improve the surface roughness, which may have little effect on density and distortion. HIP (Schneller et al. 2019) can effectively improve the relative density of part, however it may also increase the distortion according to (Mugwagwa et al. 2018).

5. Conclusions

The influencing factors of distortion in additive manufacturing haven't been systematically researched. In this study, the macro mechanism and influencing factors of distortion have been comprehensively discussed. We established a quantitative model for illustrating inconsistent shrinkage during the additive process. This model links the micro heating–cooling process and the macro structural mechanics. We also developed a qualitative model for showing the distortion caused by inconsistent shrinkage.

It can be concluded that the shrinkage tendency of additive deposition is the intrinsic reason of the final distortion of built parts. Each layer has different temporary shrinkage because of the different shear stiffness of previously deposited structure and support structure. The inconsistent shrinkage, which is the accumulation of all layers’ temporary shrinkage, drives the disengaged part to distort. We also discussed approaches for decreasing the distortion of cantilever structures from the perspective of structural, material, and processing. This study offers a comprehensive understanding of part distortion in most additive manufacturing techniques for scientific research and industrial application.

Acknowledgement

This work was supported by the National Key Research and Development Program ‘Additive Manufacturing and Laser Manufacturing’ (Nos. 2018YFB1105400 and 2018YFB1105801), the National Natural Science Foundation of China (No. 51475238), the Key Research and Development Program of Jiangsu Provincial Department of Science and Technology of China (No. BE2019002), and the China Post Doctoral Fund (No. 2020M671475).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China: [Grant Number 51475238]; National Key R&D Program of China [Grant Number 2018YFB1105400, 2018YFB1105801]; State Key R&D Program of Jiangsu Province, China [Grant Number BE2019002]; China Postdoctoral Science Foundation [Grant Number 2020M671475].

Notes on contributors

Deqiao Xie

Deqiao Xie is a post doctor at Nanjing University of Aeronautics and Astronautics.

Fei Lv

Fei Lv is a graduate student at Nanjing University of Aeronautics and Astronautics.

Huixin Liang

Huixin Liang is a post doctor at Drum Tower Hospital affiliated to Medical School of Nanjing University.

Lida Shen

Lida Shen, Zongjun Tian, Jianfeng Zhao, and Yingdong Song are professors at Nanjing University of Aeronautics and Astronautics.

Zongjun Tian

Lida Shen, Zongjun Tian, Jianfeng Zhao, and Yingdong Song are professors at Nanjing University of Aeronautics and Astronautics.

Jianfeng Zhao

Lida Shen, Zongjun Tian, Jianfeng Zhao, and Yingdong Song are professors at Nanjing University of Aeronautics and Astronautics.

Yingdong Song

Lida Shen, Zongjun Tian, Jianfeng Zhao, and Yingdong Song are professors at Nanjing University of Aeronautics and Astronautics.

Cijun Shuai

Cijun Shuai is a professor at Central South University.