# A continuum approach to simulating the sintering of 3D-printed objects with natural anisotropic honeycomb lattice structures

Charles Manière, Theotim Marie, Aymeric Jugan, Loïc Le Pluart, Sylvain Marinel

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Charles Manière, Theotim Marie, Aymeric Jugan, Loïc Le Pluart, Sylvain Marinel. A continuum approach to simulating the sintering of 3D-printed objects with natural anisotropic honeycomb lattice structures. Journal of the European Ceramic Society, 2025, 45 (10), pp.117334. 10.1016/j.jeurceramsoc.2025.117334. hal-05130086

HAL Id: hal-05130086

https://hal.science/hal-05130086v1

Submitted on 25 Jun 2025

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# A continuum approach to simulating the sintering of 3D-printed objects with natural anisotropic honeycomb lattice structures

Charles Manière¹*, Theotim Marie¹, Aymeric Jugan¹, Loïc LePluart², Sylvain Marinel¹

1. Université de Caen Normandie, ENSICAEN, CNRS, CRISMAT, Normandie Univ, 14000 Caen, France

2. Université de Caen Normandie, ENSICAEN, CNRS, LCMT, Normandie Univ, 14000 Caen, France

**Keywords**

Lattice; Honeycomb; Vat Photopolymerization; Anisotropy; Modeling; Sintering

**Abstract**

Honeycomb lattice structures are commonly used to optimize the weight-to-strength ratio in 3D printing, but during high-temperature sintering, these parts are prone to distortions that current tools struggle to predict. To address this, we developed a comprehensive lattice/shell sintering model. Our approach calculates the effective sintering moduli of the honeycomb using virtual shear and isostatic tests, accounting for both material and structural anisotropy. A key challenge was ensuring synchronized sintering behavior between the thick shell and lattice, preventing distortion errors. This continuum simulation method, validated through bar sintering tests, significantly reduces computational demands, making it ideal for design and optimization studies.

**Graphical abstract**

<figure><img src="image_3.png" /></figure>

**Real lattice behavior**

<figure><img src="image_4.png" /></figure>

**Equivalent continuum**

<figure><img src="image_5.png" /></figure>

**Sintering distortions**
**simulations**

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

EXP

SIM

**Highlights**

<figure><img src="image_8.png" /></figure>

Continuum sintering simulating of 3D printed shapes with a honeycomb filling structure.

<figure><img src="image_9.png" /></figure>

Lattice scale virtual test for honeycomb anisotropic sintering behavior assessment.

<figure><img src="image_10.png" /></figure>

Robust and rapid simulation enabling optimization studies.

<footnote>* Corresponding author: CM: Laboratoire de cristallographie et sciences des matériaux (CRISMAT), 6 Bvd du maréchal Juin 14050 CAEN CEDEX 4, France
Ph.: +33.2.31.45.13.69 ; E-mail address: charles.maniere@ensicaen.fr</footnote>
Nomenclature

θ Porosity

θ̇ Porosity elimination rate (s⁻¹)

ρ Relative density

σ Stress tensor (N.m⁻²)

s Deviatoric stress tensor (N.m⁻²)

ε̇ Strain rate tensor (s⁻¹)

η Material viscosity (Pa.s)

η₀ Viscosity pre-exponential factor (Pa.s)

Q Deformability activation energy (J.mol⁻¹)

R Gas constant 8.314 (J.mol⁻¹.K⁻¹)

T Temperature (K)

Ṫ Heating rate (K/s)

φ Shear modulus

ψ Bulk modulus

φ<sub>eff</sub> Effective lattice shear modulus

ψ<sub>eff</sub> Effective lattice bulk modulus

θ<sub>ci</sub> Initial critical porosity

θ<sub>cf</sub> Final critical porosity

P<sub>l</sub> Sintering stress (N.m⁻²)

P<sub>lx</sub>, P<sub>ly</sub>, P<sub>lz</sub> Anisotropic sintering stress terms (N.m⁻²)

ı Identity tensor

α Surface energy (J.m⁻²)

r Particles radius (m)

ε̇ Trace of the strain rate tensor (s⁻¹)

P Hydrostatic stress (N.m⁻²)

I₁ Trace of the stress tensor (N.m⁻²)

σ<sub>r</sub> Hydrostatic applied stress (N.m⁻²)

σ<sub>xz</sub> Shear stress tensor component (N.m⁻²)

ε̇<sub>r</sub> Radial strain rate tensor component (s⁻¹)

ε̇<sub>xz</sub> Shear strain rate tensor component (s⁻¹)

V<sub>g</sub> Shear velocity (m.s⁻¹)

G Grain size (m)

Ġ grain growth rate (m.s⁻¹)

G₀ Initial grain size (m)

m Densification model grain size exponent

p Grain growth model grain size exponent

K Temperature dependent term of grain growth model (m¹+p·s⁻¹)

k₀ Pre-exponent term of grain growth model (m¹+p·s⁻¹)

Q<sub>G</sub> Grain growth activation energy (J.mol⁻¹)

a, γ, ζ, c, c', c'' Fitting constants

t Time (s)
# 1. Introduction

One of the great advantages of 3D printed object is the possibility to optimize the strength-to-weight ratio[1-5]. Advanced topology optimization and slicing numerical tools are used to design lightweight structures based on various lattice filling strategies[6-11]. Indeed, lattices reduce significantly the printing time and shape weight while efficiently distributing stress in the porous filling zone, without decreasing much the part resistance. In this field, bioinspired honeycomb filling strategies are intensively used as this shape is well supported, easy to print, and highly resistant [7]. For 3D printing, the honeycomb shape anisotropy is particularly interesting when the hexagonal tubes are oriented in the building direction. However, during the sintering stage at high temperatures, these lightweight parts become softer and highly sensitive to distortions [12-14]. Most simulation or topology optimization studies focus on the application and consider the mechanical properties at room temperature [5,15]. There is a need for a model of these complex lattice/shell structures to explore the high temperature sintering behavior and the distortions that develop.

A comprehensive simulation of lattice-filled part sintering may include three main challenging aspects.

* The first aspect involves evaluating reliable parameters for sintering models of the printed material, such as viscosity and moduli. Sintering dilatometry tests are used under different conditions to assess the viscosity, moduli, and the underlying sintering mechanism. For final stage sintering, the identification of the grain growth also plays an important role in correcting the final stage sintering kinetics and identifying the mechanism. Among existing methods, constant heating rate methods like the master sintering curve (MSC) [16,17] or Wang and Raj (kinetic field) [18,19] approaches can be used to assess the viscosity thermal dependence. Afterwards, we can develop combined MSC/regression[20] studies to extract the moduli and the remaining parameters[21,22]. The analytic modeling can be used to verify the reproduction of the
sintering curves before transferring the identified parameters in the finite element simulation (FEM).

* The other important aspect is related to the inherent sintering anisotropy of the 3D printed shape skeleton and the presence of macro-defects such as porosity or cracks [23–25]. Numerous studies on vat photopolymerization, paste/filament extrusion and binder jetting demonstrate that the scraping or printing head motions induce interlayer porosity or particle orientation resulting in more or less significant sintering anisotropy [12,26–32]. This anisotropy may affect the sintering stress, and the shear and bulk viscosities[33–35]. Practically sintering anisotropy is introduced by two approaches, the viscosity[12,29,36] or the sintering stress[33,37]. If the anisotropy originates from a porosity architecture, the anisotropic sintering rate can disappear when approaching the full porosity elimination [26].

* Finally, the last and very important aspect is to take into account the lattices themselves. The lattices are generally located inside the shape and surrounded by a more or less thick shell that ensures the external shape geometry. If the shell can be simulated by the anisotropic sintering model, the more deformable behavior of the lattice porous zone needs to be appropriately simulated too. In addition, if the lattice geometry is anisotropic, as it is for honeycomb structures, different behavior is expected between the honeycomb tubular direction and the 2D hexagonal plane directions. To assess the behavior of the lattice geometry, a direct FEM simulation of the lattice can be conducted. These kinds of simulations require fine meshing and high computational resources if the full shape filling geometry is considered[12,38]. However, if the simulation is limited to the elemental lattice geometry, virtual lattice-scale tests can be conducted to assess the effective lattice moduli through shear and isostatic tests [39].

The method for identifying the effective moduli of lattices has recently been established [39].

It was applied to an isotropic lattice structure, the effective moduli were determined for different
skeleton porosities. In this study, this method is adapted to anisotropic lattice structures, such as honeycomb. The method still relies on virtual shear and isostatic tests, but now assesses the anisotropic modulus properties in both the tubular direction and hexagonal plans of the lattice. Special attention is given to ensuring synchronous anisotropic sintering behavior between the lattice and the printed skeleton. To validate this FEM model on actual printed shapes, deflection bar tests were conducted with varying heights, and shell thicknesses. As illustrated in figure 1, the objective of this study is to enable the sintering simulation of 3D printed shapes with honeycomb filling. The main challenges lie in the anisotropy of both the skeletons and lattice themselves.

<figure><img src="image_11.png" /><figcaption>Figure 1 3D printing lattice/shell strategy and sintering modeling challenges.</figcaption></figure>
## 2. Experiment and method

The 3D printing experiments utilized the “CERAMAKER100”, a professional ceramic vat photopolymerisation device from 3DCERAM SINTO®. The material is an alumina ceramic cream called “3DMix AL-E02” produced by 3DCERAM SINTO®. We determined [12] this alumina powder is bimodal with smaller grains ranging from 100-600 nm and larger grains from 1-2.5 µm, averaging 0.37 µm. The optimal sintering cycle involved heating at 2 K/min to 1580°C with a 3 hours hold, resulting in a microstructure that was 96.1 % dense with an average grain size of 3.22 µm, and slight orientation of the grains in the layer plane. The anisotropic sintering shrinkage for this cycle was determined using dilatometry (Seteram TMA96) on 5×5×5 mm³ printed cubes. These data have been utilized to develop the anisotropic sintering model for this material.

This study focuses on simulating the sintering of printed shapes featuring both an external shell and internal honeycomb lattice filling. To verify the accuracy of the model in reproducing the sintering behavior of such lattice-shell parts, various bars with 0.5 mm wall thickness honeycomb lattice filling and different shell thicknesses were tested. To accentuate the sintering distortions during sintering, these bars have been sintered in cantilever configuration. As reported in figure 2, the tested bars are referred with an “S#H#” code where the shell thickness in millimeter is indicated after the “S” and the bar height in millimeter is indicated after the “H”. The tested bars are ordered from the S2H5 to S0.5H3.8 for increasing sintering deflections. Traditionally, vat photopolymerization processes containing ~60vol% of organics necessitate long debinding cycles, typically conducted under nitrogen protective atmosphere over the course of a week. However, due to the bars geometry not exceeding 2 mm in wall thickness, muffle furnace debinding is employed. This process involves: heating at 0.2K/min to 200°C and 2 hours hold, subsequent heating at 0.1 K/min to 300°C with a 2 hours hold, final heating at 0.2 K/min to 550°C with a 2 hours hold. Following the debinding cycle, a pre-sintering step involves heating at 2 K/min to 1050°C with a 1 hour hold.
<figure><img src="image_12.png" /><figcaption>Figure 2 Presentation of the printed, debound and sintered bars used to validate the finite element simulation of honeycomb lattice/shell structures. The bars shapes are shown on the left, with the post-sintering results displayed on the right.</figcaption></figure>

In the following, the sintering model of “3DMix AL-E02” printed alumina is described with the identification equations to assess the honeycomb anisotropic behavior from lattice scale simulation.

### 3. Theory and calculations

In this section, the anisotropic sintering model and the corresponding analytical formulations are presented first. Then, the identification of the honeycomb lattice effective moduli by virtual tests is described.

#### 3.1. Sintering model.

In the continuum theory of sintering[33,40], the governing equation of sintering relates the stress ($\underline{\sigma}$), strain rate ($\dot{\underline{\epsilon}}$) tensors by temperature ($T$) and porosity ($\Theta$) variables. This governing equation is defined as follows.

$$ \underline{\sigma} = 2\eta(T) \left(\varphi(\theta)\dot{\underline{\epsilon}} + \left(\psi(\theta) - \frac{1}{3}\varphi(\theta)\right)\dot{\theta}\dot{\underline{i}}\right) + P_l\underline{i}\underline{i} \quad (1) $$

With $\underline{i}$ being the identity tensor, $P_l$ the sintering stress, $\eta$ the viscosity and $\varphi, \psi$ the shear and bulk moduli, respectively.
The sintering stress can be modeled by Skorohod's theoretical expression.

$$P_l = \frac{3\alpha}{r} (1 - \theta)^2 \qquad (2)$$

The term $\dot{e}$ refers to the following strain rate invariant expression.

$$\dot{e} = \dot{\epsilon}_x + \dot{\epsilon}_y + \dot{\epsilon}_z \qquad (3)$$

The viscosity term ($\eta$) was determined for the sintering of 3DCERAM alumina [12]; an average expression is taken in this study (see below).

$$\eta = 0.025[PasK^{-1}]T\exp\left(\frac{360[kJ/mol]}{RT}\right) \qquad (4)$$

Similarly, the modified shear and bulk moduli corresponding to 3DCERAM alumina [12] are expressed as the following porosity functions.

$$\varphi(\theta) = \left(1 - \frac{\theta}{0.5}\right)^2 \qquad (5)$$

$$\psi(\theta) = \frac{2(0.5 - \theta)^3}{3(\theta - 0.02)} \qquad (6)$$

The mass conservation equation is then used to relate the trace of strain rate tensor ($\dot{e}$) to the porosity elimination rate ($\dot{\theta}$).

$$\frac{\dot{\theta}}{1-\theta} = \dot{e} \qquad (7)$$

Equations (1) to (7) are implemented in COMSOL Multiphysics® to simulate the natural sintering of alumina. For this implementation, the strain rate formulation can also be used (see below).

$$\dot{\epsilon} = \frac{1}{2\eta} \left( \frac{s}{\phi} + \frac{(P-P_l)}{3\psi} \dot{\mathbf{I}} \right) \qquad (8)$$

### 3.2. Sintering anisotropy.

As discussed in the introduction, the sintering anisotropy originates from the layered architecture, interlayer porosity, and the preferential grain orientation in the layer plane. The sintering anisotropy can be implemented either by the “viscosity” [12,29,36] or by the “sintering stress” [33,37,39]. Implementing the sintering anisotropy through the sintering stress
$(P_{lx}, P_{ly}, P_{lz})$ proved more effective [39], as it separates the sintering anisotropy from the final calculation involving different moduli for lattice and shell zones. Additionally, this approach enables the prior assessment of the viscosity and moduli from densification curves. Afterwards, the $P_l$ anisotropy can be imposed to adjust the shrinkage curves in the different directions. Inspired by previous works on sintering anisotropy of 3D printed green samples[12,29,36,39], the sintering rate anisotropy originating from the porosity architecture vanishes at the end of the sintering as total porosity elimination is approached. Consequently, the anisotropy in the $P_l$ factor is a function of the porosity that tends to 1 at the end of sintering, see below.

$$ \begin{cases} P_{lx} = Pl (1 + c \theta) \\ P_{ly} = Pl (1 + c' \theta) \\ P_{lz} = Pl (1 + c'' \theta) \end{cases} \qquad (9) $$

The parameters $c, c'$ and $c''$ are used as fitting constant to adjust the porosity correction curve and increase or decrease the sintering stress in different directions (with the z-axis being the building direction). For isotropic case, all terms $c, c', c''$ are equal to zero. This serves as the starting point of the model. Using equation (8) and (2$\eta$) for the viscosity expression in equation (4), the sintering analytical formulation are as follows.

$$ \begin{cases} \dot{\epsilon}_z = \frac{-P_{lz}}{(2\eta)3\psi} \\ \dot{\epsilon}_y = \frac{-P_{ly}}{(2\eta)3\psi} \\ \dot{\epsilon}_x = \frac{-P_{lx}}{(2\eta)3\psi} \end{cases} \qquad (10) $$

### 3.3. Lattices effective modeling strategy.

The effective lattice behavior aims to simulate the deformation caused by the empty lattice zones due to the weight of the shape it supports or the temperature gradient developed in the lattice zones. Knowing the anisotropic sintering of the printed alumina, the remaining parameter to identify is the effective lattice sintering behavior. This is modeled using the Representative Volume Element (RVE) method[41]. The latter can be used to apply virtual mechanical tests on a microstructure[29,42–44], grains[45,46], for sintering behavior [47–52]
or in the present case, on the lattice cell volume [53,54] to identify the effective properties of the medium. Previously, [39] the lattice filling zone has been simulated by identifying the effective shear and bulk modulus through shear and isostatic virtual tests on the lattice periodic geometry (RVE). In the latter, the sintering properties have been used to describe the behavior of the skeleton and the test has been conducted at different skeleton porosities. The effective shear and bulk moduli consequently exhibit significantly lower values compared to the equivalent moduli of a fully filled volume (100% filling density). Their lower moduli help in modeling the higher deformation sensitivity of the lattices zones compared to the shell zones.

Based on the shear and isostatic strain rate responses observed in the two lattice-scale virtual tests, the effective shear and bulk modulus can be determined. The analytical reduction of the lattice-scale shear and bulk tests yields the following formulations for shear and isostatic strain rates [39].

$$ \begin{cases} \dot{\epsilon}_{xz} = \frac{|V_g|}{2e} = \frac{\sigma_{xz}}{2\eta\phi_{eff}} & \text{Shear test} \\ \dot{\epsilon}_r = \frac{\sigma_r}{(2\eta)3\psi_{eff}} & \text{Isostatic test} \end{cases} \qquad (11) $$

Consequently, the equations for identifying the effective shear and bulk lattice moduli are as follows..

$$ \begin{cases} \varphi_{eff} = \frac{\sigma_{xz}}{(2\eta)\dot{\epsilon}_{xz}} = \frac{2e \sigma_{xz}}{(2\eta)|V_g|} & \text{Shear test} \\ \psi_{eff} = \frac{\sigma_r}{(2\eta)3\dot{\epsilon}_r} & \text{Isostatic test} \end{cases} \qquad (12) $$

These equations will yield different calculated effective moduli data points for each tested skeleton porosity. However, to implement the effective lattice behavior in a finite element code, a mathematical fitting of these points is required using the following porosity equations that are inspired by the real moduli function [21,36,39,55].

$$ \begin{cases} \varphi_{fit} = \left(1 - \frac{\theta}{\theta_{ci}}\right)^2 & \text{Shear test} \\ \psi_{fit} = a \frac{(\theta_{ci} - \theta)^\gamma}{(\theta - \theta_{cf})^\zeta} & \text{Isostatic test} \end{cases} \qquad (13) $$

As reported in figure 3a, for the case of honeycomb lattice the RVE can be a square for shear testing and a cylinder for the isostatic test. In contrast to the first study[39], the natural
honeycomb shape exhibits a strong inherent anisotropy. Consequently two shear tests are required to identify the effective shear modulus, depending on whether the shear stress is directed on the hexagonal plane (radial direction $\phi_{eff}RR$) or along the tubular direction (axial direction $\phi_{eff}ZR$). For the isostatic test, different strain rates will result in radial and axial directions (see figure 3b). For the shear test, a 5 hexagons cell has been selected to ensure a representative volume. For the isostatic test, a single hexagon element with half of the adjacent hexagons has been used to have a representative simulation of the shape periodicity, the RVE is cylindrical.

The final parameter to adjust for simulating the lattice zones is the sintering synchronicity between the lattices and shell behaviors. Indeed, if a small lattice piece is sintered and compared to a small shell piece without an external load, the sintering shrinkage should be similar, as both are made from the same printed material and differ only in geometry. Like for the moduli, the resulting lattice sintering stress ($P_{l lattice}$) on the external boundary of the equivalent continuum is significantly reduced. If we determine the effective lattice moduli while maintaining the sintering stress at 100% density ($P_l$), the sintering kinetics would be erroneously accelerated. Consequently, the lattice sintering stress ($P_{l lattice}$) needs to be reduced and adjusted to ensure that the lattice sintering shrinkage follows the shrinkage of the shell ($\dot{\epsilon}_{shell} = \dot{\epsilon}_{lattice}$ for any of the components). Based on equation (10), the initial adjustment of the sintering stress in the lattice can be formulated by equalizing the strain rates of the lattice and shell zones.

$$ \dot{\epsilon}_{shell} = \frac{-P_{l shell}}{(2\eta)3\psi_{shell}} = \dot{\epsilon}_{lattice} = \frac{-P_{l lattice}}{(2\eta)3\psi_{lattice}} \quad (14) $$

This resulted in the initial expression of $P_{l lattice}$ that will still require final adjustment through finite element simulation to perfectly match the shell shrinkage curves (shell/lattice synchronicity).

$$ P_{l lattice} = P_{l shell} \frac{\psi_{lattice}}{\psi_{shell}} \quad (15) $$
(A) Lattice scale simulation & identification equation

<figure><img src="image_13.png" /></figure>

(B) Lattice simulation configuration & geometry

<figure><img src="image_14.png" /><figcaption>Figure 3 Identification of the effective lattice sintering behavior; (a) scheme of the lattice scale simulation and the corresponding identification equation for shear and isostatic tests. The shape of the representative volume element (RVE) is shown for both tests; (b) Simulation configuration in the RVE for the shear (left) and isostatic (right) tests.</figcaption></figure>
### 3.4. Method of identification and validation

To summarize the previously described stages, a diagram representing the main stages of identification and validation is shown in Figure 4. The goal is to assess both the printed anisotropic sintering behavior and the lattice's effective sintering behavior, which is also anisotropic. The final application aims to predict the sintering distortions of parts with a shell/lattice architecture using the simplest possible configuration. The identification process is divided into four main stages:

I. The first stage involves identifying the anisotropic sintering model ($\eta, \varphi, \psi, P_l$) from dilatometry curves.

II. The second stage is the FEM-based identification of the honeycomb's effective moduli ($\varphi_{lattice}, \psi_{lattice}$) using lattice-scale (RVE) shear and isostatic tests.

III. The third stage focuses on identifying the effective lattice sintering stress ($P_{lattice}$) through synchronicity conditions between the lattice and shell shrinkage kinetics. This condition ensures that the sintering curves of the lattice and the dense shell are the same in the absence of external loads.

IV. The last stage consists of calibrating boundary conditions to reproduce the deformation resistance of the shell at varying thicknesses, while accurately matching the experimental deformation.

In this work, the sintering model for 3DCERAM alumina has already been assessed. In the first stage, this process is replaced with an adaptation of the existing anisotropic model, transitioning from a viscosity-based approach to a sintering stress-based model, which is more stable when applied to lattice structures.
Stage I: printed specimen sintering model assessment.

<figure><img src="image_15.png" /></figure>

Stage II: FEM lattice RVE → lattice effective moduli assessment.

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

Stage III: $P_{\text{lattice}}$ adjustment → Lattice/shell synchronicity.

<figure><img src="image_18.png" /></figure>

Stage IV: Cantilever test → Effective shell boundary conditions.

Creep deformation correction, capacity to simulate shell/lattice complex structure with fully continuum simulation

<figure><img src="image_19.png" /></figure>

Figure 4 Schematic representation of the identification and validation stages for anisotropic sintering and lattice behavior.
## 4. Results and discussions

In this section, the identification of the anisotropic sintering and honeycomb effective moduli is presented first. Subsequently, the finite element simulation is tested through the sintering of cantilever bars with honeycomb filling and varying shell thicknesses.

### 4.1. Sintering anisotropy.

As highlighted in the introduction, the sintering anisotropy can be implemented by the viscosity [12,29,36] or the sintering stress[33,37]. However, as previously shown [39], implementation through the sintering stress is preferable for the identification process of the lattice moduli. This approach allows for the separation of the porous creep component used to simulate the lattice skeleton resistance from the sintering component, which can be applied subsequently. Based on the anisotropic sintering model of 3DCERAM alumina [12] it is possible to use a single average viscosity (as per equation (4)) and introduce anisotropy at the level of the sintering stress ($P_{lx}$, $P_{ly}$, $P_{lz}$) using equation (9). To achieve this, the isotropic sintering stress $P_l$ in Skorohod's expression (2), which is initially isotropic, is transformed into an anisotropic form by adjusting the coefficients $c, c', c''$ to fit the dilatometry shrinkage curves. Since the anisotropy exists only between the building direction (called Z axis) and in the in-plane (layer) directions (R axis), the parameters $c$ and $c'$ are identical. This approach is referred to as the “sintering stress” approach, whereas the previous approach is known as the “viscosity approach”. The comparison of shrinkage curves and the anisotropic models using the “viscosity” and “sintering stress” approaches is plotted in figure 5 for verification. In this figure, the linear shrinkage is calculated as $100(l-l_0)/l_0$, where “l” is the specimen length in one direction. It is clear from figure 5 that both approaches can accurately reproduce the sintering shrinkage. The obtained expression for the anisotropic sintering stress is as follows:

$$ \begin{cases} P_{lz} = Pl (1 + 0.52\theta) \\ P_{lr} = Pl (1 - 0.52\theta) \end{cases} \qquad (16). $$
<figure><img src="image_20.png" /><figcaption>Figure 5 Comparison of the printing specimen relative density (a) and shrinkage (b) curves with anisotropic sintering models using viscosity and sintering stress approaches. The calculated shrinkage is extracted from a probe located at the corner of the cube (red spot).</figcaption></figure>
## 4.2. Effective lattice sintering behavior assessment.

The determination of the honeycomb lattice effective moduli is a crucial aspect of the modeling assessment. As shown in figure 3b, this is carried out through shear and hydrostatic tests. Due to the anisotropic nature of the honeycomb structure, different moduli are identified for the in plane directions (R axis) and building direction (Z axis). These lattice scale simulations are presented in figure 6. The two shear tests are depicted in figure 6a, with an example of the total displacement distribution shown below. Compared to the shell shear modulus (represented by the dashed curve), the identified shear modulus in the ZR plane exhibits lower values, with more than 30-fold decrease compared to the shell shear modulus. This is expected because the resistance is significantly reduced due to the substantial volume of voids in the honeycomb structure, compared to the shell which is completely filled (100% solid). In the RR plane, the shear moduli is approximately 10 times lower than the shell modulus because the resistance to shear is greater in the plane of the hexagonal lattice structure.

Regarding the isostatic lattice test reported in figure 6b, the anisotropy of the honeycomb prevents the achievement of an isotropic strain rate, despite the isostatic stress imposed on the equivalent RVE cylindrical shape. As with the shear tests, different bulk modulus are obtained in the Z axis (building direction and hexagonal tube direction) and the R axis (layer plane). In the hexagon plane (R axis), the bulk modulus is about a quarter lower than the shell bulk modulus. In the Z axis, which is the hexagonal tube direction, the honeycomb structure is highly resistant, and the bulk modulus is only half that of the shell bulk modulus. A similar decrease of the lattice moduli compared to the shell has been observed for isotropic lattices [39]. These points have been fitted by the equation indicated below for an easy implementation in the finite element code.
(A) Shear modulus lattice assessment

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

(B) Bulk modulus lattice assessment

<figure><img src="image_23.png" /></figure>

$$ \psi_{effZ} = 0.3 \frac{(0.5 - \theta)^3}{\theta - 0.02} \qquad \psi_{effR} = 0.08 \frac{(0.5 - \theta)^{2.6}}{\theta - 0.02} $$

<caption>Figure 6 Honeycomb lattice scale simulation and identification of effective lattice moduli, (a) shear tests and (b) isostatic tests. An example of lattice scale simulation is provided, illustrating the total displacement distribution.</caption>
4.3. Identification of lattice sintering stress by adjustment of lattice/shell sintering synchronicity.

After determining the lattice effective moduli, the sintering stress used in the shell needs to be adjusted to ensure sintering synchronicity conditions. The latter ensures that the sintering behavior of a small printed lattice piece matches the shrinkage of a shell piece made from the same material (with only the shape differing). If these synchronicity conditions are not satisfied, modifying only the moduli can lead to erroneous sintering kinetics, as the model does not take into account the reduction of effective lattice capillary forces. To achieve this, the shell sintering stress (16) needs to be modified using equation (15) and the identified lattice bulk moduli equations fitted in figure 6. In order to verify the synchronicity of the lattice with the shell, a simulation with two cubes is conducted. In the first cube, the shell anisotropic behavior is applied. In the second cube, the effective lattice moduli and sintering stress are tested. The objective is to adjust the lattice sintering stress using equation (15). The objective is to identify the sintering stress at which the lattice cube shrinkage in the building direction (Z axis) and in the layer plane (R axis) matches the shrinkage of the first cube, where the shell behavior is applied as a reference. The results are reported in figure 7a. To perfectly mimic the Z and R displacement shrinkage of the shell, the $P_I$ coefficient of ± 0.52 in the shell (16) is reduced to ± 0.26 for the lattice.

As will be discussed in the next section, a modified lattice model will be applied to better reproduce the sintering behavior observed in the bar deflection test where the effective shear viscosity must be adapted. In this version of the model, the diagonal shear modulus terms, whose average values are decreased by a factor of 5.5, aim to increase deformability without affecting densification. This necessitates readjusting the lattice/shell behavior synchronicity. Consequently, $P_I$ coefficient of ± 0.26 in the lattice is decreased to ± 0.046 for $P_{I,Z}$ and ± 0.042 for $P_{I,R}$. See validation curves in figure 7b.
(A) Shell/Lattice sintering synchronicity verification: identified model

<figure><img src="image_24.png" /></figure>

(B) Shell/Lattice sintering synchronicity verification: modified model

<figure><img src="image_25.png" /><figcaption>Figure 7 Lattice sintering stress identification through adjustment and verification of lattice/shell sintering synchronicity for the identified (a) and modified (b) sintering models.</figcaption></figure>

4.4. Honeycomb bar sintering simulation.

Now that the sintering model is fully identified to reproduce the experimental shrinkage and ensure the effective lattice respects synchronicity with the shell behavior, this model is validated on FEM simulations using the deflection bar test. Different bars with various shell thicknesses and honeycomb lattices are printed, debound, and sintered in the deflection configuration as shown in figure 2. Across all these bars, varying sintering deflections are observed, from thick
shells to a shell thickness equal to the honeycomb thickness, which behaves similarly to the
lattice with maximum deflection. The shell and identified lattice sintering models are first tested
and reported in figure 8a. Since the bar deflection is perfectly symmetric, only half of the bar
geometry is simulated, applying a normal displacement constraint at the symmetry plane
section. The graph on the right represents the FEM results of the sintered bar, with the initial
shape depicted by the black bars. On the left, the graph indicates the simulated sintering
deflection curves. These curves represent the displacement of point probes at the edge of each
bar, measuring displacement in the building direction. This displacement corresponds to
warping deformation not originating from sintering, which we define as "deflection". The post-
sintered deflection was measured and is indicated on the graph by markers at the end of the
cycle. The sintering model is expected to replicate the deformation behavior of the lattice/shell
structure when the simulated deflection curves closely match the experimental final deflection.
As shown in figure 8a, while the sintering behavior perfectly reproduces the anisotropic
sintering shrinkage (see figure 7a), it fails to replicate the specimen deformation behavior at
high temperatures. This discrepancy has been noted previously, and it has been found that
modifying the shear modulus can increase or decrease the sensitivity of high temperature
deformation without affecting the sintering behavior [29]. Indeed, the identified model
estimates viscosity by assuming that the diffusion mechanism responsible for sintering is
similar to the one governing creep deformations. Undesired deformations, such as deflection in
this case, primarily occur due to shear contribution. Consequently, shear viscosity can be treated
as an independent behavior responsible for deformation. In other models, such as Riedel *et al*
[56], shear and bulk viscosities, which can be seen as a combination of viscosity and moduli
[33], are identified separately. Consequently, modifying the shear modulus to correct the creep
deformation is similar than identifying the shear viscosity. As shown in figure 8b, the sintering
stress also needs to be adjusted due to the interdependence of the parameters needing to account
for the anisotropy and lattice behavior. This way, the model reproduces a reliable creep
deformation behavior that faithfully reflects the sintering behavior. This new version of the
model, which maintains lattice/shell synchronicity, referred to as the “modified model”. As previously explained, the diagonal shear modulus was divided by 5.5 to increase the bar deflection at high temperatures to match the experimental deflection. The results are reported in figure 8b. This approach is successful for the two bars with the same shell thickness as the honeycomb wall thickness (S0.5H3.8 and S0.5H5). These bars are modeled using a continuum approach, assuming that the shell has no impact because it is the as the lattice thickness. However, for the two bars with thicker shells (S1H5 and S2H5), it appears that the two behaviors (shell and lattice) hinder the deflection, which remains very sensitive despite their perfect sintering synchronicity. This can be explained by multiple factors such as anisotropic meshing, localized interfaces stresses, etc.

To address this issue, one possible solution could be to replace the thicker shell with boundary conditions that enhance resistance to deformation until they match the experimental deformation observed with both shell thicknesses. This can be modeled in Comsol® unig a “spring foundation” boundary condition applied to the surface where the thick shells are present. To adjust the boundary conditions spring stiffness parameters of 5 and 15 N/m were applied for the 1 and 2 mm shells, respectively. With this approach, all bar tests successfully reproduced the sintering and high-temperature deflection, as shown in the results in figure 8c. This method avoids the need to model separate materials zones for the shell and lattice, and it circumvents the need for high local mesh refinement at the shell locations, significantly reducing calculation time. It enables the simulation of sintering in printed parts within a fully continuum framework, ensuring high stability and robustness of the simulation.
<figure><img src="image_26.png" /><figcaption>Figure 8 Finite element simulation of the cantilever bar sintering deflection test with (a) using the identified model, (b) using the modified model, and (c) using the modified model using the continuum shell approach.</figcaption></figure>
**5. Conclusion**

Simulating the sintering deformation of 3D printed shapes is crucial for designing shapes and determining the filling strategy in slicing software. Including the lattice filling of 3D printed objects in simulations represents a significant challenge. For honeycomb lattice structures, this challenge is heightened by their inherent anisotropy. The solution proposed in this study involves using finite element simulations of lattice elements to evaluate the effective anisotropic behavior of the lattice at high temperatures. These effective lattice parameters are then incorporated into a simulation capable of predicting the lattice/shell complex structure using a two zones simulation. This method employs a continuum model instead of simulating the intricate geometry of the high-lattice structure, which would be unstable and exceedingly time consuming. This approach has been tested on alumina honeycomb structure with a wall thickness of 0.5mm printed using vat photopolymerization. Various shell thicknesses have been tested, resulting in different sintering distortions simulated by the numerical tool. This study highlights the importance of ensuring that the sintering shrinkage curve of the effective lattice zone mimics that of the shell. Specifically, achieving synchronization between the sintering of the shell and lattice requires careful adjustment due to the anisotropy of both the printed material and the lattice structure itself. Finally, it has been demonstrated that the most efficient approach involves a one-zone continuum simulation with an effective boundary condition that simulates the strengthening of the thick printed shell. This method allows for fast and highly stable simulations that accurately replicate sintering shrinkage and high temperature deformations. This numerical approach can quickly predict whether a shape intended for printing will withstand the high-temperature sintering stage of production, making it well-suited for optimization studies during the design phase.

**Acknowledgements**

The help and support of Jérôme Lecourt and Christelle Bilot are gratefully acknowledged.

**Data Availability**

All experimental data are present in the graphs. If needed, numerical data will be made available on request.
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