Finite Element Modeling of Ceramic Green Part Warping Induced by Shrinkage During the Stereolithography Printing Process

Abstract

The shrinkage strain, occurring upon UV curing and aging, leads to non-uniform dimensional changes that can compromise the part’s final geometry. This study investigates the deformation of green parts during the stereolithography process. Based on experimental measurements, a finite element model (FEM) is developed to account for different phenomena contributing to the structural distortion of the part, like polymerization shrinkage and the adhesion between the part and the build platform during printing. In addition, the time dependency of the degree of conversion is also considered to integrate the aging of green parts, and elastoplastic material behavior is also considered to include non-reversible deformations. This novel model makes it possible to predict stress generation during the stereolithography process and simulate part warping over time. The resulting simulations provided a numerical validation for part shapes observed experimentally, as well as insights to better understand the deformation mechanisms and optimize the dimensional fidelity of stereolithography-manufactured components.

1. Introduction

Among the various additive manufacturing methods available, stereolithography stands out for its ability to produce dense ceramic parts with high dimensional precision, greater surface quality, and favorable mechanical characteristics, without the requirement of molds. Stereolithography operates by selectively polymerizing a reactive mixture composed of ceramic particles dispersed in a UV-curable resin, including monomers and/or oligomers along with a photo-initiator. The process entails building up a part layer by layer through exposure of the resin to a UV laser beam, which solidifies the resin according to the desired cross-sectional pattern for each layer [1,2,3].

Upon exposure to UV light, the photo-initiator initiates the release of free radicals, triggering the polymerization of the organic intergranular phase [4,5,6]. These radicals facilitate both the initiation and propagation stages, wherein polymer chains progressively grow by bonding monomers to radicals at the ends of polymer chains. Chain growth ceases when two radicals react, forming stable covalent bonds during termination [7].

Ceramic particles, present in high concentrations within the suspension (50–65% vol), are trapped within the polymer matrix, imparting mechanical strength to the green part. Once the green part is built, it undergoes debinding and sintering processes to achieve the desired properties [8]. Over the past decade, the demand for and applications of stereolithography have significantly expanded, with its utilization spanning various industries, including biomedical, luxury goods, electronics, and casting molds and cores for the aerospace industry [9,10].

Beyond the effect of laser diffusion in a ceramic slurry on the achieved polymerization dimensions [9,11], it is also important to consider the generation of stress during the printing process. Whether they stem from thermal contraction and expansion [12] or chemical shrinkage due to polymerization [13,14,15,16,17], these stresses resulting from the stereolithography printing process may have a significant impact on the mechanical properties of the densified parts and their dimensions. A printed part typically deforms and bends due to the accumulated stresses during the printing process. This phenomenon is commonly referred to as curling or warping, and it is accentuated by low adhesion with the printing surface [13,18,19,20].

In order to enhance predictions and potentially reduce the loss of dimensional accuracy and mechanical strength of both green and sintered parts, it is crucial to account for the presence of these residual stresses. A few studies have already explored stress generation influence on dimensions using numerical approaches, and all described the material behavior with pure elastic properties. Bugeda et al. [21] and Huang et al. [19] both developed a finite-element-method-based model to analyze the part distortion in stereolithography applied to fully organic systems. Wu [22] proposed a model to predict the evolution of residual stresses and distortions and the appearance of cracks or delamination in ceramic green parts built with the Large Area Maskless Photopolymerization (LAMP) technique. More recently, Classens et al. [23] established a 1D multiphysical model to control the curing process and material properties. Westbeek et al. [24] used a homogenization approach to evaluate the influence of filler particles on optical, mechanical, and thermal characteristics of curing in order to use these properties in a process simulation framework.

Consequently, with the help of experimental characterizations, this work aims to develop a finite element method analysis to simulate the warping occurring on green parts and, ultimately, identify key factors for improving the dimensional accuracy of the stereolithography process. To achieve this, this work establishes a novel model to simulate the printing of macro-scale 3D green parts, with two key innovations: It incorporates non-reversible deformations through elastoplastic material behavior and accounts for part aging by considering the time-dependent evolution of the degree of conversion. The conversion of the suspension is associated with linear shrinkage in the three directions, and therefore, it allows the generation of stress depending on contact conditions. Adhesion between the part and the build platform is also integrated into the model using cohesive contact interactions. Experimental measurements are conducted to obtain information on the material properties, polymerization kinetics, and curling magnitude of the green parts and to evaluate the numerical results.

2. Materials and Methods

2.1. Description of SLA Process

In order to design the part, a computer-aided design (CAD) file is generated and then converted to the STL format [9]. Subsequently, the part undergoes digital slicing to determine the pattern for each layer and the corresponding laser beam trajectory to solidify the photosensitive slurry. During the stereolithography process, polymerization is achieved using a pulsed UV laser (Innolas Nanio AIR 355 3W, InnoLas Photonics GmbH, Krailling, Germany) with adjustable power and frequencies, emitting at a wavelength of 355 nm, and it is capable of generating free radicals from the photo-initiator. To mitigate the surface power density of the laser beam, a beam expander is positioned after the laser source. This component maximizes the utilization of the scanning system’s aperture while decreasing the power density on the mirrors [25], thereby preventing their degradation.

The beam is then precisely deflected by a scanning head, consisting of two galvanometrically controlled mirrors. Subsequently, an F-Theta lens focuses the laser to the focal point. This lens also plays a role in correcting the focal distance as the angle of incidence varies, guaranteeing that the beam is focused on the working plane perpendicular to the optical axis. This design ensures that changes in the targeted area on the build platform do not result in distortion or loss of resolution in the traced patterns. The optical system is illustrated in Figure 1.

Before each layer of photopolymerization, the slurry is added and spread onto a build platform using a scraping blade or scraper. Subsequently, the laser beam scans the selected areas based on the pattern of each layer. Within these patterns, the laser traces lines or hatchings at a chosen laser speed $V_L$ (mm/s), separated by a distance $h_S$ (µm), known as the hatch spacing. After polymerization of the suspension layer, the build platform moves downward by the layer thickness $L_t$ (µm) before the next deposition. These operations are repeated until the completion of the part, followed by a cleaning step.

In order to drive the printing process, a program manages manufacturing parameters and all the steps required for automation. These steps include the movements of the scraper and build platform between layers, adjusting mirror positions during laser insulation, and controlling the piston movement to add suspension before each new layer is spread.

The laser frequency used for all experiments is 100 kHz for a layer thickness $L_t$ of 50 µm. Green parts are printed with a laser speed $V_L$ of 3500 mm/s, a hatch spacing $h_S$ of 20 µm, and a laser power of 225 mW, corresponding to an exposure of about 321 mJ/cm² on one layer. These parameters, as employed in a previous work [11], facilitated the production of a rigid component while minimizing adhesion issues with the build platform.

2.2. Ceramic Slurry

A commercial alumina slurry (3D-Ceram Sinto, Bonnac-la-Côte, France) is used for all experiments. This system, which is able to solidify during UV-induced radical polymerization, consists of a suspension of micrometer-sized alumina particles (58% ceramic by volume) in a matrix of acrylate monomer–oligomers, along with a photo-initiator and a dispersant to stabilize the suspension. The particle size distribution can be described by the following values: $d_{10}=0.5 \mu \mathrm{m}$, $d_{50}=1.61 \mu \mathrm{m}$, and $d_{90}=4.28 \mu \mathrm{m}$ [9].

In order to integrate the shrinkage of this alumina suspension in the developed model, linear shrinkage is measured using a photo-rheological method (IS2M, Mulhouse, France). By exposing the slurry to UV light over time, the variation in height, initially 200 µm, is measured, which allows the determination of linear shrinkage. Figure 2 and Figure 3 illustrate respectively the measurement method and the linear shrinkage evolution during the photopolymerization of the alumina slurry. The UV light has a wavelength of 365 nm and an intensity of 13 mW/cm².

Both measurements exhibit, respectively, a maximum linear shrinkage of about 4 and 5%. The decrease in linear shrinkage once the maximum is achieved may be explained by the appearance of micro-cracks within the material due to the shear stresses. These defects may lead to an increase in volume and thus a height variation. The difference in maximum linear shrinkage between the two measurements may also be explained by these micro-cracks and differences in ceramic particle arrangements or the presence of air bubbles within the suspension.

As shown in Figure 4, neither delamination nor cracks are observed on the polished green part printed with the alumina slurry. The SEM photos only highlight localized grains pulled out due to polishing. Consequently, we assume a maximum linear shrinkage of 4.5% corresponding to the maximum conversion of monomers without defect.

2.3. Tensile Tests

The tensile test is conducted using a micro-device from DEBEN UK Ltd. (Stowmarket, UK) with a maximum load capacity of 2000 N, as shown in Figure 5. Each part is aligned along the $y$-axis using rods inserted into the gripping jaws, which are subsequently removed to avoid any stress concentration. The loading rate of the upper crosshead is set to 0.2 mm/min. The absolute displacement $∆L$ of the crosshead is then recorded, and the strain is determined based on the initial length $L_0$. Tensile stress is calculated using the real cross-sectional area of the gauge section of the specimen and the force measured along the $y$-axis. The stress–strain curve is then obtained for each tested part, with one sample for each aging duration.

2.4. FTIR

The degree of conversion for each sample is calculated from FTIR spectra obtained with a Nicolet 6700 spectrometer (Thermo Fisher Scientific, Waltham, MA, USA) via attenuated total reflection (ATR). The results presented in this work are determined by averaging three measurements, each performed with 64 scans and a resolution of 2 ${\mathrm{c}\mathrm{m}}^{-1}$. This technique allows tracking the evolution of carbon–carbon double bonds and thus the degree of conversion given that the number of C=C bonds decreases during the transformation of acrylate monomers/oligomers into polymer chains.

Figure 6 shows the FTIR spectra of both the unpolymerized slurry and the cured material. By comparing the area under the double peak corresponding to the stretching of C=C bonds around 1620 ${\mathrm{c}\mathrm{m}}^{-1}$ with a reference peak around 1750 ${\mathrm{c}\mathrm{m}}^{-1}$ related to the carbonyl group (not influenced by polymerization), the degree of conversion p can be defined by the following equation [25]:

$$p={\displaystyle \frac{\frac{A_0^{1620}}{A_0^{1750}}-\frac{A_{end}^{1620}}{A_{end}^{1750}}}{\frac{A_0^{1620}}{A_0^{1750}}}}$$

(1)

where $A_0$ and $A_{end}$ represent, respectively, the area under the curve obtained for the photosensitive suspension before any UV exposure and for the green parts.

2.5. Curling Evaluation

The part’s curling phenomenon is measured using a chromatic confocal point sensor (Precitec CHRocodile 2 S HS, Precitec Optronik GmbH, Neu-Isenburg, Germany). The measurement probe, positioned perpendicular to the plane of a layer and directly integrated into the scraping blade, transmits a beam of light towards the part. This light is focused, according to its wavelength, at a specific distance along the optical axis. Only one wavelength is perfectly focused on the surface of the part. By determining the wavelength of the reflected light, very precise distance measurements are possible (±1 µm) [26]. Through this technique, the curling profile of an entire surface can be determined by scanning it at multiple points.

In this work, the measurements are performed at one-millimeter intervals along an axis running the entire length of the part, and they are presented in Figure 7.

3. Characterization of Green Parts

In order to provide input data to the model, it is necessary to determine the mechanical behavior of the cured material. In stereolithography, the degree of conversion, curing dimensions, and mechanical properties of the green part depend on the energy density received by the suspension [5,25,27], and they have also shown a dependency on the aging duration [28,29,30]. Therefore, it is necessary to characterize this effect, which has a significant impact on part quality over time. Tensile tests, degree of conversion measurements, and evaluation of the part’s curling phenomenon are then carried out to provide information to the model and enable discussions of the results. These characterizations are performed on samples consisting of 60 layers of 50 µm, with their geometry depicted in Figure 7. Each part ages in a UV-free environment and under ambient conditions.

3.1. Mechanical Properties

Tensile tests are carried out to characterize the mechanical behavior of the cured material. Figure 8 illustrates the evolution of stress/strain curves as a function of aging duration. It appears that green parts printed with the same parameters show an increase in mechanical strength over time. As illustrated by the zoom, Young’s modulus (about 4 GPa) shows little variation with aging time. As the green part ages, the normal stress associated with a given deformation increases, likely due to the cross-linking of polymer chains occurring over time, leading to the progressive reduction in the viscous behavior and creep capability [28,29,30]. This aging process would therefore be associated with higher conversion and shrinkage, leading to stress generation and deformations over time.

3.2. Degree of Conversion

In order to explain the evolution of the material’s mechanical behavior, the degree of conversion on the upper layer of the parts is investigated using FTIR analysis.

Table 1. Influence of aging duration on the degree of conversion at the surface of the upper layer of green parts.

Aging duration (h)	1	48	96	168
Degree of conversion (%)	50.8	63.6	66	66.8

shows an increase in conversion over time, which seems to be significantly faster during the initial day of aging. The decrease in C=C double bonds, as evidenced by an increase in the degree of conversion, could indeed be associated with the cross-linking of polymer chains, thereby reducing mobility and consequently diminishing the viscous behavior of the material over time.

3.3. Printing Surface Adhesion and Curling of Green Parts

In order to characterize the influence of the degree of conversion rise over time on the green part’s deformation, they are observed at various aging durations. The evolution of the curling profile of the upper surface of the samples is observed in Figure 9a. The black curve is obtained before the part is detached from the build platform, while the other curves are acquired after detachment. The 0-day curve corresponds to an aging of 2 h. A notable intensification of the warping phenomenon is observable over time. The maximum displacement increases by more than 50% after 7 days from 175 to 270 µm.

A much slighter displacement of up to 15 µm is also observed before detachment of the sample, suggesting that the part corners separate from the build platform due to stresses accumulated during the process. To confirm this hypothesis, the same measurements are performed on the bottom surface. Figure 9b highlights a higher displacement on this side, mainly located on the corners and ranging from 305 to 395 µm, respectively, for 0 and 7 days. Aging appears to be responsible for an increase of about 90 µm in maximum displacement on each side of the sample between the first and last measurements.

Figure 10 displays the maximum displacement measured on both surfaces as a function of the maximum exposure on one layer. During each scraping step, the upper suspension layer is perfectly horizontal until polymerization. Therefore, no impact of the energy density is noticeable on this side. However, the displacements of the bottom surface emphasize the impact of the bond strength between the part and the build platform on the dimensions of the green parts. This adhesion seems to depend on the exposure used during printing, and it likely degrades during the process due to shrinkage-induced stress. Low adhesion may cause printing issues and should be avoided.

Various techniques are used to lower the impact of this phenomenon or enhance adhesion, such as enhancing the proportion of monofunctional monomers, using adhesive substrates, or applying specific parameters to insulate the first layer [31]. The addition of supports or removable artifacts can also help reduce deformations in the area of interest. Although it is possible to mitigate the issues caused by this phenomenon, it keeps playing a role in green part deformations during the stereolithography process. Therefore, the model developed in this work accounts for adhesion between the part and the build platform.

4. Development of the Model

4.1. Simulation Process

The model developed in this work aims to consider the chemical shrinkage of polymerization in order to analyze and predict the deformations and stresses of green parts resulting from stereolithography printing. This simulation process, using the finite element method in Abaqus 2024 software, relies on the successive activation of pre-meshed layers and on the progressive rise of the degree of conversion, allowing for a digital reproduction of the steps of suspension spreading, polymerization, and aging of the part. Each layer is activated 100 s after the polymerization of the preceding one, which roughly corresponds to the interval between the exposure of two consecutive layers.

Like the printed samples, the simulated part is composed of 60 layers of 50 µm, resulting from the partition of the initial part. As depicted in Figure 11, a stainless steel build platform also supports the part and consists of a rectangular parallelepiped, for which its bottom surface is fixed as a boundary condition. Both parts are deformable solid 3D parts. A cohesive contact interaction with a traction-separation behavior is also considered between the part and the build platform to account for bond degradation. At the last step of the simulation, this cohesive contact is deactivated to account for the detachment of the part and allows for the aging simulation. To prevent rigid body motion from occurring, the degrees of freedom of three nodes are constrained according to the 3-2-1 method [32]. This technique does not induce any reaction force at the constrained nodes and thus has no impact on the final part shape.

To simulate polymerization shrinkage, this work relies on a functionality [33] capable of modelling curing processes in polymer applications by applying shrinkage depending on the degree of conversion of the material. This functionality requires the use of specific analysis procedures during simulation steps. Therefore, a dynamic implicit analysis procedure [34] is used in all simulation steps. This procedure allows the study of non-linear responses and large deformations and permits better contact management compared to a static procedure.

The mesh is composed of hexahedral linear element C3D8 and is coincident at the interface between layers. Insulation patterns only depend on the STL file provided; thus, elements are activated at a fixed position without considering the strain of previous layers. Therefore, during the activation step, the nodes on the bottom surface slightly move to ensure coincidence with those of the previous layer without inducing any stress. After a mesh size sensitivity analysis, the average element size is set to 400 µm, resulting in a total of about 360,000 elements in the part.

No thermal contraction/expansion is considered in this work. Indeed, our suspension showed a maximum heating of 10 °C during polymerization. This translates into a maximum linear expansion of about a 0.1%, considering the coefficients of thermal expansion of both alumina and polymer [24], likely inducing less stress in the part than chemical shrinkage. However, taking into account volume variations would probably be necessary in the case of a highly exothermic reaction.

4.2. Monomer Conversion

Once a layer is activated, its conversion rate increases according to Kamal’s law, as described by the following equation [35]:

$$\dot{\mathit{p}}={\displaystyle {\sum }_{\mathit{i}=\mathbf{1}}^{\mathit{n}}}{\mathit{Z}}_{\mathit{i}}{\mathit{e}}^{{\displaystyle \frac{-{\mathit{E}}_{\mathit{i}}}{\mathit{R}\left(\mathit{T}-{\mathit{T}}_{\mathbf{0}}\right)}}}({\mathit{b}}_{\mathit{i}}+{\mathit{p}}^{{\mathit{m}}_{\mathit{i}}}){({\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}-\mathit{p})}^{{\mathit{n}}_{\mathit{i}}}$$

(2)

$\mathit{p}$, ${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}$, ${\mathit{Z}}_{\mathit{i}} ({\mathbf{s}}^{-\mathbf{1}})$, $\mathit{R} (\mathbf{J}.{\mathbf{m}\mathbf{o}\mathbf{l}}^{-\mathbf{1}}.{\mathbf{K}}^{-\mathbf{1}})$, ${\mathit{E}}_{\mathit{i}} ({\mathbf{J}.\mathbf{m}\mathbf{o}\mathbf{l}}^{-1})$, and ${\mathit{T}}_{\mathbf{0}} (\mathbf{K})$ represent, respectively, the degree of conversion, the maximum degree of conversion, a rate constant, the universal gas constant, the activation energy, and absolute zero. ${\mathit{m}}_{\mathit{i}}$ and ${\mathit{n}}_{\mathit{i}}$ are reaction constants, and ${\mathit{b}}_{\mathit{i}}$ is the initial degree of conversion. The model considers an initial degree of conversion of zero (${\mathit{b}}_{\mathit{i}}=\mathbf{0}$) and a maximum conversion rate $\dot{\mathit{p}}$ at 0% conversion (${\mathit{m}}_{\mathit{i}}=\mathbf{0}$). The effect of temperature and activation energy is also neglected. Therefore, the previous equation can be simplified to determine the evolution of the degree of conversion depending on the time increment $∆\mathit{t}$ used for iteration:

$$\mathit{p}\left(\mathit{t}+\mathbf{\Delta }\mathit{t}\right)=\mathit{p}(\mathit{t})+\mathbf{\Delta }\mathit{t}{\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}}{\mathit{Z}}_{\mathit{i}}{({\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}-\mathit{p}(\mathit{t}))}^{{\mathit{n}}_{\mathit{i}}}$$

(3)

Laser absorption and differences in exposure between layers may lead to gradients of conversion in each layer and between layers [11,23,25]. Such differences, however, represent a challenge to consider. Therefore, this work makes the assumption that the degree of conversion of each layer is uniform and only depends on the time elapsed since polymerization. This first hypothesis is based on previous work depicting a difference of about 5% between the degree of conversion at the top and the bottom of a layer and good exposure homogeneity in the x and y directions for chosen printing parameters [11].

The parameters ${\mathit{Z}}_{\mathit{i}}=\mathbf{0.0065}$, ${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}=\mathbf{0.7}$, and ${\mathit{n}}_{\mathit{i}}=\mathbf{3.9}$ allow us to impose an evolution of the degree of conversion very similar to the experimental measurements. Figure 12 shows the evolution of the degree of conversion in a layer after its activation at t = 0.

4.3. Polymerization Shrinkage

In the developed model, the deformation ε resulting from polymerization shrinkage directly depends on the degree of conversion $p$ through the introduction of a shrinkage coefficient $\gamma$, and it is considered isotropic [33]:

$$\left(\begin{array}{ccc}\dot{\epsilon }& 0& 0\\ 0& \dot{\epsilon }& 0\\ 0& 0& \dot{\epsilon }\end{array}\right)=-\left(\begin{array}{ccc}\gamma & 0& 0\\ 0& \gamma & 0\\ 0& 0& \gamma \end{array}\right)\ast {\displaystyle \frac{\delta p\left(x,y,z,t\right)}{\delta t}}$$

(4)

The maximum linear shrinkage was previously (2.2) estimated to 4.5%, and it is hypothesized that this value corresponds to the maximum degree of conversion of the material. Therefore, the model considers a shrinkage coefficient $\mathit{\gamma }$ of 0.064, allowing for a linear shrinkage of 4.5% at the maximum conversion of 70%, as determined previously with ${\mathit{p}}_{\mathit{m}\mathit{a}\mathit{x}}$.

4.4. Mechanical Behavior

The introduction of permanent deformations may be essential in the case of polymer parts, often showing high plastic deformations due to polymer chain stretching and alignment [36]. Therefore, this work considers an elastoplastic material for the part and elastic stainless steel for the build platform.

Based on the stress–strain curve of the green part at zero day of aging (Figure 13), also represented in Figure 8, Young’s modulus, yield points, hardening points, and the corresponding plastic strains are determined and given in

Table 2. Mechanical properties of materials used in the model.

Parameter	Value	Unit	Description
${\mathrm{E}}_{\mathbf{b}\mathbf{u}\mathbf{i}\mathbf{l}\mathbf{d} \mathbf{p}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{e}}$	200	$\mathrm{G}\mathrm{P}\mathrm{a}$	Stainless steel Young modulus [37]
${\mathsf{\nu }}_{\mathbf{b}\mathbf{u}\mathbf{i}\mathbf{l}\mathbf{d} \mathbf{p}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{e}}$	0.3		Stainless steel Poisson ratio [37]
${\mathrm{E}}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}}$	4	$\mathrm{G}\mathrm{P}\mathrm{a}$	Part’s Young modulus
${\mathsf{\nu }}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}}$	0.4		Part’s Poisson ratio [27]
${\mathsf{\sigma }}_{\mathbf{0}}^{\mathit{p}\mathit{l}}$	0.4	MPa	True stress at the yield point
${\mathsf{\sigma }}_{\mathbf{1}}^{\mathit{p}\mathit{l}}$	1.8	MPa	True stress at the first hardening point
${\mathsf{\sigma }}_{\mathbf{2}}^{\mathit{p}\mathit{l}}$	4.8	MPa	True stress at the second hardening point
${\mathsf{\sigma }}_{\mathbf{3}}^{\mathit{p}\mathit{l}}$	8	MPa	True stress at the third hardening point
${\mathsf{\epsilon }}_{\mathbf{0}}^{\mathit{p}\mathit{l}}$	0		Plastic strain at the yield point
${\mathsf{\epsilon }}_{\mathbf{1}}^{\mathit{p}\mathit{l}}$	0.00065		Plastic strain at the first hardening point
${\mathsf{\epsilon }}_{\mathbf{2}}^{\mathit{p}\mathit{l}}$	0.00265		Plastic strain at the second hardening point
${\mathsf{\epsilon }}_{\mathbf{3}}^{\mathit{p}\mathit{l}}$	0.0056		Plastic strain at the third hardening point

.

It is important to note that the model does not account for viscous relaxation over time and material property evolution during polymerization. It also relies on two important assumptions: (i) the hardening curve follows a constant extrapolation after the last hardening point, and the slope becomes zero if the maximum stress is reached, resulting in fully plastic deformation; (ii) once the part is completely detached from the platform, plastic strain remains constant, and any new strain caused by aging is fully elastic.

4.5. Cohesive Contact

To account for the adhesion between the build platform and the part, the model includes a cohesive contact interaction at the interface (Figure 11). This cohesive contact interaction is created at the very beginning of the simulation and deactivated after the activation of all layers to simulate the detachment of the part from the platform. The part is then able to deform freely and relax some of the stresses accumulated during the process.

The cohesive contact interaction follows a traction–separation law, illustrated in Figure 14. This interaction allows each node at the interface to be virtually linked to the platform with a spring of stiffness $\mathit{K}$, until reaching a maximum contact stress ${\mathit{t}}_{\mathit{n}}^{\mathbf{0}}({\mathit{t}}_{\mathit{s}}^{\mathbf{0}},{\mathit{t}}_{\mathit{t}}^{\mathbf{0}})$ at the initiation of separation ${\mathit{\delta }}_{\mathit{n}}^{\mathbf{0}}({\mathit{\delta }}_{\mathit{s}}^{\mathbf{0}},{\mathit{\delta }}_{\mathit{t}}^{\mathbf{0}})$. Once this distance is reached, the spring’s stiffness gradually degrades until it reaches zero at the failure separation ${\mathit{\delta }}_{\mathit{n}}^{\mathit{f}}({\mathit{\delta }}_{\mathit{s}}^{\mathit{f}},{\mathit{\delta }}_{\mathit{t}}^{\mathit{f}})$.

The relationship between the tensile stress and the separation is defined by the following expression [38,39,40]:

$$\mathit{t}=\left\{\begin{array}{c}{\mathit{t}}_{\mathit{n}}\\ {\mathit{t}}_{\mathit{s}}\\ {\mathit{t}}_{\mathit{t}}\end{array}\right\}=\left(\begin{array}{ccc}{\mathit{K}}_{\mathit{n}\mathit{n}}& {\mathit{K}}_{\mathit{n}\mathit{s}}& {\mathit{K}}_{\mathit{n}\mathit{t}}\\ {\mathit{K}}_{\mathit{n}\mathit{s}}& {\mathit{K}}_{\mathit{s}\mathit{s}}& {\mathit{K}}_{\mathit{s}\mathit{t}}\\ {\mathit{K}}_{\mathit{n}\mathit{t}}& {\mathit{K}}_{\mathit{s}\mathit{t}}& {\mathit{K}}_{\mathit{t}\mathit{t}}\end{array}\right)\left\{\begin{array}{c}{\mathit{\delta }}_{\mathit{n}}\\ {\mathit{\delta }}_{\mathit{s}}\\ {\mathit{\delta }}_{\mathit{t}}\end{array}\right\}$$

(5)

where $\mathit{K}(\mathbf{P}\mathbf{a}/\mathbf{m})$ denotes the spring stiffness matrix, and $\mathit{t}$ is the contact stress vector for which its three components ${\mathit{t}}_{\mathit{n}}$, ${\mathit{t}}_{\mathit{s}}$, and ${\mathit{t}}_{\mathit{t}}$ (Pa) are oriented, respectively, along the $\mathit{x}$, $\mathit{y}$, and $\mathit{z}$ axes. The corresponding separations along these three axes are denoted by ${\mathit{\delta }}_{\mathit{n}}$, ${\mathit{\delta }}_{\mathit{s}}$, and ${\mathit{\delta }}_{\mathit{t}}$ (m). In order to simplify the model, two assumptions are made about the contact: (i) the coefficients are uncoupled, ${\mathit{K}}_{\mathit{s}\mathit{n}}={\mathit{K}}_{\mathit{s}\mathit{t}}={\mathit{K}}_{\mathit{n}\mathit{t}}=\mathbf{0}$, and (ii) the contact stiffness is isotropic, ${\mathit{K}}_{\mathit{n}\mathit{n}}={\mathit{K}}_{\mathit{s}\mathit{s}}={\mathit{K}}_{\mathit{t}\mathit{t}}$.

The contact stress damage initiation criterion is evaluated using the contribution of each component. Degradation starts when this quadratic stress criterion, represented as follows, reaches a value of one:

$${\left\{{\displaystyle \frac{〈{\mathit{t}}_{\mathit{n}}〉}{{\mathit{t}}_{\mathit{n}}^{\mathbf{0}}}}\right\}}^{\mathbf{2}}+{\left\{{\displaystyle \frac{{\mathit{t}}_{\mathit{s}}}{{\mathit{t}}_{\mathit{s}}^{\mathbf{0}}}}\right\}}^{\mathbf{2}}+{\left\{{\displaystyle \frac{{\mathit{t}}_{\mathit{t}}}{{\mathit{t}}_{\mathit{t}}^{\mathbf{0}}}}\right\}}^{\mathbf{2}}=\mathbf{1}$$

(6)

Compressive normal contact stress does not lead to any contact degradation; thus, only traction normal contact stress is able to initiate damage in this direction. Once damage initiates, contact stress is reduced with the help of a damage variable D, increasing from 0 to 1 upon degradation:

$${\mathit{t}}_{\mathit{n}}=\left(\mathbf{1}-\mathit{D}\right) \overline{{\mathit{t}}_{\mathit{n}}} \mathit{f}\mathit{o}\mathit{r} \overline{{\mathit{t}}_{\mathit{n}}}>\mathbf{0} \left(\mathit{t}\mathit{r}\mathit{a}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\right)$$

(7)

$${\mathit{t}}_{\mathit{s}}=\left(\mathbf{1}-\mathit{D}\right) \overline{{\mathit{t}}_{\mathit{s}}}$$

(8)

$${\mathit{t}}_{\mathit{t}}=\left(\mathbf{1}-\mathit{D}\right) \overline{{\mathit{t}}_{\mathit{t}}}$$

(9)

where $\overline{{\mathit{t}}_{\mathit{n}}}$, $\overline{{\mathit{t}}_{\mathit{s}}}$, and $\overline{{\mathit{t}}_{\mathit{t}}}$ are the contact stress components predicted by the elastic traction-separation behavior for the current separations without damage. The damage variable $\mathit{D}$ evolves with linear softening from the initiation of separation ${\mathit{\delta }}^{\mathbf{0}}$ to the given failure separation ${\mathit{\delta }}^{\mathit{f}}$.

4.6. Simulation Steps

The different stages of the simulation are described in Figure 15. After an initialization step, this model simulates additive manufacturing in stereolithography through the repetition of a loop. This loop consists of (i) the instantaneous activation of the printed top layer and (ii) a progressive increase in the degree of conversion and shrinkage of this layer and the previous ones over 100 s, based on kinetics provided by Kamal’s law. After the activation of all layers and an aging step of 30 min on the build platform, the deactivation of the cohesive interaction with the platform and the application of constraints on three nodes allow the part to deform freely without rigid body motion. The stresses generated during the printing process and aging then cause the part to warp over time. The time increment $∆\mathit{t}$ is fixed at 10 s until part detachment (step n + 2), and it progressively increases upon the last step.

5. Results and Discussions

First, this work compares the case of a part completely tied to the build platform until contact deactivation (step n + 2) with the case of cohesive interactions. As a first approach to this phenomenon and in order to prove the impact of adhesion, this work considers only one contact stiffness $\mathit{K}$ of 5 $\mathbf{M}\mathbf{P}\mathbf{a}.{\mathbf{\mu }\mathbf{m}}^{-1}$ and maximum contact stresses ${\mathit{t}}^{\mathbf{0}}$ of 5 $\mathbf{M}\mathbf{P}\mathbf{a}$ in each direction, corresponding to damage initiation at a separation ${\mathit{\delta }}^{\mathbf{0}}$ of 1 µm and a complete failure of the contact at a distance ${\mathit{\delta }}^{\mathit{f}}$ of 5 µm. These low separation values express a nearly instantaneous break in the part–platform bond once the maximum contact stress is reached.

Figure 16 displays the $\mathit{z}$-displacement over time of the simulated parts, from the end of the printing process to 2 h of aging, for (a) a part tied to the platform and (b) a part with cohesive contact. During printing, any bottom surface tied or still in cohesive contact with the build platform cannot shrink in the $\mathit{x}$ and $\mathit{y}$ directions.

The upper layers that do not meet this boundary condition are able to slightly shrink, but the $\mathit{z}$-displacement remains low and inferior to the layer thickness until activation of all layers. However, it is important to note that this displacement, mainly observed around the borders of the top surface, increases with vertical distance to the plate, eventually reaching a layer thickness for sufficiently high or unsupported parts and causing scraping issues. This model may then be helpful in determining suitable supports.

Curling of the part over time is monitored using the displacement over the $\mathit{z}$-direction along the measurement axis, as shown in Figure 7. The simulated displacements for both contact conditions are then compared to the experimental measurements in Figure 17. To facilitate observations, the displacement on the top surface is slightly shifted by 30–40 µm towards a positive $\mathit{z}$ in order to set the minimum to zero.

Though the curve has the same shape, the simulated curling phenomenon appears to be much higher than the experimental one. These discrepancies can be attributed to several unmodeled physical factors. First, viscoelastic relaxation is not considered in the current model, whereas it could be responsible for a substantial reduction in long-term residual stresses and warping. Furthermore, localized material damage due to mechanical fatigue may develop during the printing process, which would contribute to relieving residual stresses and reducing the overall curling magnitude [41]. Another significant simplification is the omission of light irradiation from the upper layers; this additional exposure strongly influences the conversion kinetics of the underlying material and the subsequently generated stresses. Finally, a more complex consideration of the continuous evolution of mechanical properties upon polymerization could also lead to lower final stresses in the printed part. It should be noted that it is highly challenging to estimate the contribution of each of these factors to the observed discrepancies between the experimental and numerical displacement values.

Just after detachment, the degree of conversion is lower in the last printed layers than in the first ones, but it keeps increasing and causing shrinkage all along the aging step, as illustrated in Figure 18. With the rise being higher in the last layers, the associated shrinkage is also higher and is therefore responsible for the accentuation of the curling phenomenon over time.

The displacement quickly increases during the first hours of aging, until stabilizing when the degree of conversion differences in the part approaches zero. With shrinkage being the source of over-time deformations, aging highly depends on polymerization kinetics, and thus, it may induce substantial post-detachment warping with respect to slow kinetics or almost no effect with respect to already complete polymerization.

The appearance of curling can also be justified with normal stress in the $\mathit{y}$-direction. Figure 19 shows the stress tensor component for both simulated parts before and after the detachment of the platform. A cross-section at $\mathit{x}=\mathbf{0}$ allows for internal observations. A positive value means that the material is under tension along this direction, while a negative value indicates a compressive state. Before detachment, both parts mainly exhibit tensile stress, with higher stress ${\mathit{\sigma }}_{\mathit{y}\mathit{y}}$ around the top center of each part. This stress gradient in the $\mathit{z}$-direction leads to the curling of the part just after detachment, allowing for the balance of residual stress. By opening the part–platform interface during printing, the part printed with cohesive interactions is able to shrink with fewer constraints, and thus, it exhibits higher compressive stresses in these areas.

The progressive opening of the interface during printing and the curling evolution over time are illustrated in Figure 20, showing the $\mathit{z}$-displacement of a node located at the very end of the measurement axis for both simulated parts. No opening of the interface occurs for the tied part. In comparison, cohesive contact allows the node to move up to 400 µm along the $\mathit{z}$-axis before deactivation of the contact. Contact condition deactivation leads to elastic springback of the part, leading to instantaneous warping upon detachment from the platform. This instant displacement is observed for both parts, but it is higher for the tied part since its interface stored more elastic energy during printing [42]. This translates into higher tensile stresses before detachment (Figure 19).

The introduction of the cohesive contact allowed the reproduction of the displacement difference between the bottom and top surfaces, as observed experimentally (Figure 9). The cohesive contact also induces a larger curling phenomenon compared to the tied part. These differences between the two contact conditions are justified by the plastic strain field, especially around the borders of the part. Figure 21 shows the equivalent plastic strain field (PEEQ) in the two parts just after detachment. Stress indeed accumulates faster in the tied part due to contact conditions, resulting in increased plastic hardening. Therefore, the tied part that accumulated more plastic strain before detachment undergoes less elastic strain, resulting in a slightly different curling profile.

The influence of subsequent insulations on the degree of conversion may result in both faster polymerization kinetics and a higher overall degree of conversion, as well as a larger gradient in the printing direction [25]. Such a different field would cause more plastic strain before part detachment. Along with the neglect of mechanical property evolution and viscous relaxation, this may be another factor explaining the difference between the experimental and simulated dimensions. Since the aging duration on the build platform has an influence on the degree of conversion, it also impacts the proportion of plastic strain and thus the final part’s dimensions, as evidenced in Figure 22, which shows the influence of aging duration on the build platform on both the post-stabilization dimensions and the equivalent plastic strain. Indeed, aging while the part remains tied to the build platform results in higher plastic strain. This induces a much smaller curling phenomenon, comparable to the experimental measurements (Figure 23). This result highlights the critical role of the degree of conversion kinetics and plastic strain in understanding green part deformation and in improving the precision of the stereolithography process.

6. Conclusions and Perspectives

Predicting the deformation of a ceramic green part printed by stereolithography is one of the key points for optimizing the process. A finite element model was developed in this work to provide insights for improving the dimensional accuracy of parts produced by stereolithography.

Accounting for polymerization shrinkage, the time dependency of the degree of conversion to integrate the aging of green parts and the elastoplastic behavior of the material enabled the simulation of the curling phenomenon. The progressive increase in the degree of conversion and shrinkage led to an intensification of curling upon aging. Although showing differences in the simulated displacement magnitudes, this model provided a numerical validation for deformations often observed experimentally.

It was experimentally observed that the adhesion force between the part and the build platform has a direct impact on the shape and dimensions of the green part after printing. This adhesion force, depending on the exposure used during printing, degrades during the process due to shrinkage-induced stress. Low adhesion may cause printing issues and must therefore be avoided. The consideration of a rigid cohesive contact to model part–platform adhesion allowed for a better replication of the experimental observations. The model confirms that applying strong irradiation (overcuring) to the first layer helps reduce the final deformation of the printed part. This increased exposure ensures stronger adhesion to the build platform, which is mechanically represented in this study by the “tied” contact condition.

Improving platform adhesion, accelerating polymerization kinetics, or reducing the volumetric shrinkage of the photosensitive suspension could be viable solutions to help minimize dimensional issues of green parts in stereolithography. More significant plastic behavior could also help increase plastic strain during printing and thus limit the effective deformations primarily caused by the elastic component. In addition to these material-related approaches and for complex geometries, the strategic addition of support structures forces the part being printed to undergo higher plastic deformations during the printing process. Increased constraints lead to higher localized plastic strains, which effectively limit the residual elastic deformations responsible for springback and warping after printing.

Stereolithography involves numerous chemical, thermal, and mechanical phenomena that are challenging to fully capture. This work does not yet aim to precisely predict part dimensions but rather offers insights for better understanding the mechanical behavior during printing. Accounting for other factors, such as viscous relaxation, the effect of prior layer exposure, the heterogeneity of the degree of conversion within a layer, or even the anisotropy and evolution of mechanical properties upon polymerization and aging, could lead to more precise predictions of residual stresses and final dimensions in future versions of this model.