Theoretical Modeling and Experimental Characterization of Stress Development in Parts Manufactured Through Large Area Maskless Photopolymerization

Tao Wu, Suman Das

Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30318

Abstract

This paper aims at investigating the evolution of stresses in parts manufactured through large area maskless photopolymerization (LAMP). A theoretical model was established to understand the curing process for LAMP and a finite element analysis was performed to model the dynamic evolution of stresses during the layer-by-layer fabrication process using Abaqus software. This model serves to suggest strategies for reducing stresses, part warpage, and crack development in parts made through LAMP.

Introduction Large Area Maskless Photopolymerization (LAMP) is a direct digital manufacturing technology that can build three-dimensional objects layer-by-layer with both high speed and fine feature resolution. A schematic of the LAMP process being developed by the direct digital manufacturing (DDM) laboratory at Georgia Tech is shown in Fig 1. The core part of LAMP system is the spatial light modulator (SLM), which is a digital micro-mirror device (DMD). It is an optical chip with more than 1.3 million mirrors which can be turned on or off according to the color (white or black) of pixels in the corresponding bitmap image. A UV source is used to project light onto the DMD and expose the photocurable ceramic-loaded liquid resin according to the input image. The optical imaging head, with the UV lamp and DMD inside, moves in a serpentine path and raster scans the building area. Using this exposure mechanism, LAMP can realize massively parallel scanning lithography and its patterning speed is much higher than the traditional stereolithography in which only one single beam is used for scanning. In industry, extremely complex interior cooling passages of turbine airfoils are usually produced by investment casting, which typically involves the creation of over thousand tools needed for fabricating the cores, patterns, mold, and setters. The DDM group in Georgia Tech is using the LAMP process to produce the integral ceramic cored molds directly. In this way, the production rate, casting yield and costs can be improved dramatically when compared to the conventional investment casting procedures. Therefore, direct digital manufacturing using LAMP is a kind of technology that disrupts the current state-of-the-art investment casting process, not only in the manufacturing of airfoils but also in many other applications involved complex components.

748

Thithrough by inducresults feach curthe devethis papalleviati

1. Cur

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cing defectsfrom the difrrently expoelopment ofper is to ining residual

re Dependenis model dical propertical properts is stronglyd by Dillmakinetic and

onvenient α-neous isotrion degree α

Eα0 and Eα

ure 1. Schem

inly focusesprocess. Re

s such as disfferent degreosed layer wf residual stnvestigate th

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orgia Tech’

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0) and full

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machine

s in a part bthe quality ol stress in LArs. The contfied previoutory. The obo infer stra

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(1)

α=1) resin,

749

respectively. The parameter γ (-1<γ<1) is introduced to quantify the competing mechanisms between stress relaxation and chemical hardening [1]. Increasing γ will make the modulus increase more rapidly at a relatively low conversion degree. In this paper, a zero value of γ is used for the simulation.

The material used in LAMP is silica particles loaded resin composite. The effective bulk modulus of this composite can be derived according to the self-consistent method [4], shown in equation (2).

(2)

Where K0 and K1 are the bulk moduli of matrix (resin) and inhomogeneity (particle) respectively. ̅ is the effective shear modulus of composite and c1 is the concentration of particles loading. The particles can be considered rigid compared to resin, thus we obtain:

(3) ̅ can be expressed as a function of and the effective Poisson’s ratio ν, which is considered as constant here.

(4)

Substitute into equation (3), we obtain:

(5)

The relation between bulk modulus and Young’s modulus can be expressed as:

(6)

Make use of equations (5) and (6), we obtain the effective Young’s modulus of composite as a function of resin’s modulus E0, effective Poisson’s ratio ν and particles concentration c1.

(7)

It can be seen that the effective modulus is proportional to resin’s modulus. So, the effective modulus of the composite is also related to the conversion degree α of resin.

(8) Equation (8) is similar with (1), but here , ∝ , ∝ are moduli of the composite, not resin.

The experimental data obtained from tensile testing according to ASTM D638 is shown in Table 1.

1 1 00

1

( )(3 4 )3 4

c K K KK KK

μμ

− += +

+

0 1

1

3 43(1 )K cK

cμ+

=−

3 (1 2 )2(1 )K νμ

ν−

=+

01 1

11 3 3

K Kc cν

ν ν+

=+ − +

00,

3(1 2 ) 3(1 2 )EEK K

ν ν= =

− −

01 1

11 3 3

E Ec cν

ν ν+

=+ − +

0 1 1 0(1 ) (1 )( )E E E E Eα α α αα α γα α= − + + − −

750

Properties Value ∝ (MPa) 0.829 ∝ (MPa) 829

ν 0.437 Table 1. Mechanical properties of acrylic-based composite

Here the modulus of uncured composite is chosen arbitrarily small due to the negligible stiffness. 2. Cure Dependent Shrinkage

This model proposes a theoretical relationship between the shrinkage strain and degree of conversion. The volumetric shrinkage occurring during LAMP process is related to the photopolymerization mechanism. The linkage of small monomer units produces large polymer chains and the corresponding intermolecular spacing is reduced from Van der Waals distance (~104Å) to covalent bond (C=C) lengths (~ 1 Å). This results in density changes and thus bulk contraction in the cured resin, which accumulates as the part is fabricated layer-by-layer. Thus, the volumetric shrinkage is a direct measure of the number of covalent bonds formed (degree of conversion) and an exact semi empirical relationship can be derived. Experimentally, the volume change per mole of acrylate groups (C=C) is 22.5cm3/mol [5] when acrylate monomer is polymerized. For a general case of multiacrylates with ceramic filler particles, the volumetric shrinkage value can be estimated through the following equation [6]:

( )(%) 22.5 (1 ) 100( ) 100i i i

mixi mi i

fV FLV M

χα ρχ

ΣΔ= × × − ×

Σ (9)

Where fi is the functionality of monomer (i), χi is mole fraction of monomer (i), Mmi is molecular weight of monomer (i) and ρmix is the density of the mixture.

For the material system used in LAMP, the values of above parameters are shown in Table 2.

Components fi Mmi ρ(g/cm3) v/o (%) w/o (%) χi ρmix(g/cm3)Hexanediol diacrylate, HDDA 2 226 1.02 3.4 18.6 .95 1.68 Ethoxylated Penta erythryitol

tetra acrylate, EPETA 4 528 1.12 31 2.3 .05

Filler (ceramic powder) 2.2 FL=55 72

Table 2. Data of the material system

Assuming a uniform strain contraction for all principle strain components and considering the volumetric strain to be smaller than 10 percent, the value of linear shrinkage strain is approximately one third of the volumetric shrinkage value. Thus the linear shrinkage strain εT can be expressed as:

( )1 22.5 (1 ) 1003 ( ) 100

i i iT mix

i mi i

f FLM

χε α ρχ

Σ= × × × − ×

Σ (10)

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Fig 14. Com

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is paper preprocess. Sim

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work will be

mparison for

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e conducted

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The effects ocases. It is fess and distol. To minim

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ases

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759

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