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Heterogeneous representation and

fabrication using additive

manufacturing



1. Introduction

A heterogeneous object is referred to a solid component consisting of two or more

material primitives distributed continuously or discontinuously within an object. Modeling and

manufacturing of heterogeneous object (HO) have been paid much attention recently as the

advent of rapid prototyping manufacturing technology, which makes it possible to fabricate the

heterogeneous object. As the continuously variation of material composition produces gradient in

material properties, they are often known as functionally gradient materials (FGM), shown in

Fig. 1(a). For example, a component contains two compositions, metal and heat resistance

material (such as ceramic); the material distribution is illustrated in (b). From the figure we cansee that metal increases its fraction gradually from one side to another (the red line), while the

heat resistance material linearly reduces its fraction (the green line), which can avoid the stress

concentration because of the thermal stress relaxation in transition of two materials, shown in (c).

A discontinuous change in material composition generates distinct regions of material in the

solid, which is usually called multi-material object (MMO) such as composite materials, as

demonstrated in Fig. 2 (ZCorp (2005)). MMO has been extensively used in industry for a long

time, while FGM has shown tremendous potential in many fields, such as aeronautics and

astronautics, biomedical engineering, and nano-technology, etc.[4]

Fig. 1. Model of functionally gradient material.[4]



Solid modeling of objects forms an important task in design and manufacturing. Recent

developments in the field of layered manufacturing have shown potential for the physical

realization of heterogeneous (multi-material) objects. Thus, there is a need to represent material

information as an integral part of the CAD model data. Information models for the representation

of product data are being developed as an international standard informally called STEP (ISO

10303). However, the current application protocols focus on the representation of homogeneous

objects only.

Fig. 2. A multimaterials blade.[4]

In the past few decades, considerable attention has been devoted to FGM representations

(Computer-Aided Design), design validation (Computer-Aided Engineering), fabrication

(Computer-Aided Manufacturing) and material heterogeneity optimization. Recently, several

FGM modeling and representation methods have been reported. These methods have allowed

designers to design not only the geometry, but also the material composition of an object.

However, all of these methods have mainly focused on simple-shaped FGM objects with simple

gradient schemes. It is very difficult (or impossible in some cases) to use these methods to

process arbitrary-shaped FGM objects with authentic 3D gradients.

2. Heterogeneous solid modeling



Heterogeneous solid modeling aims to incorporate material distribution information

along with geometry into the CAD model. This section summarizes results reported in the

literature. The following are several possible representations of the macrostructure of the

material for modeling of heterogeneous objects. [3]

• r m object model,

• Tetrahedral decompositions,

• voxel-based representation,

• R-function method.

2.1 rm object model:

This model is based on the exact representation of geometry and material distribution

function. It forms the basis of the proposed representation to develop a standard for heterogeneous modeling.

The product space T = E3× _ R n forms the mathematical space to model heterogeneous

objects. Material points are restricted to lie in the material space V ⊂ R n. Each point p in the

object S is a combination of n primary materials and is specified by the volume fractions of these

primary materials. The material composition of any point p is represented as a material point v in

R 3, with each dimension representing exactly one primary material. As these volume fractions

must sum to unity, the space of material points (material space) is defined as:

V= {v ∈ R n | || v || 1 ≡ ∑ vi = 1 and vi ≥ 0} [3]

where vi represents the volume fraction of the ith primary material. Thus, each point p ∈ S can be

modeled as a point ( x ∈ E3 ; v ∈ V) in T, where x and v represent the geometric and material

points respectively.

Material r-set (rm set) An r m set is defined as a subset D ≡ ( P , B) of T where P ⊂ E3 is an r-set

and B ⊆ V assigns material to the r-set P .

1. The set B is specified by a material function F . Thus, an r m set can also be defined as the pair

( P , F ) where the subset B is defined implicitly through its material function as F ( P ).

2. An r m set is undefined for all the points lying in the exterior of P .

3. If F =1 & n = 1, then it is a single material r m set.



4. To avoid the need for modeling discontinuities, it is assumed that the material function F is C ∞

continuous in P .

Material object (rm-object) an r m-object is then defined as a finite collection of r m-sets ( Pj, Bj)

such that the following conditions hold true:

1. The r m -sets are geometrically interior-disjoint.

2. The r m -sets are minimal.

2.1.1 Processing of Heterogeneous Solid Modeling for SFF:

There are three tasks that typically need to be performed before a design can be process-planned

and fabricated. These are:

i. Selection of a growth direction (or part orientation during fabrication).

ii. Construction of a support structure that enables overhang features in the design.

iii. Decomposition of the supported model into simple features suitable for automatic

tool-path (or deposition-path) generation.[3]

2.1.2 Extension of Modeling Entities:

In this section, we present the entities that are used to describe a decomposed design. These

entities (called SFF-Compacts and SFF-Objects) can be thought of as extensions or “derived

classes” of the heterogeneous modeling entities described earlier.

SFF-Compacts: SFF-Compacts are extended r m-sets that implement the above concept. In

addition to geometry and material information which is inherited from the r m-sets, compacts alsohave a unique growth direction, a build order field (which is used to assemble SFF-Compacts

into SFF-Objects), and a material precedence field (which is used to implement merging and

splitting algorithms that operate on SFF-Objects).[3]

SFF-Objects: SFF-Compacts do not have much significance in and of themselves, and are

partial object prescriptions. In order to represent a real object, they need to be assembled into a

collection called an SFF-Object. An SFF-Object can also be thought of as an extended r m-object,

where each member is an SFF-Compact, and whose minimality condition (C2) has been

dropped. Also, some constraints and dependencies are enforced between the geometry, material,

growth axis, build-order and precedence fields of all SFF-Compacts within a single SFF-Object.

Some of the important constraints are:[3]

i. The SFF-Compacts that constitute an SFF-Object partition the SFF-Object into a finite number

of pair-wise disjoint regions.



ii. The closure of the interior of all the SFF-Compacts in an SFF-Objects is exactly equivalent to

the union of all the compacts (i.e., there are no extraneous “voids” within an object - internal

cavities are explicitly modeled with material type set to “air”, for example).

iii.All SFF-Compacts within an SFF-Object are similarly oriented (i.e. share a common growth

axis direction).

iv. An SFF-Object is fully supported (or encapsulated in support structure) and no two SFF-

Compacts with consecutive positions in the build order have the same material type (i.e. the

compacts are minimal as long as geometric and material restrictions on a solid are not violated).

v. The build order of SFF-Compacts in an SFF-Object is sequential, and decompositions that

enforce cyclic ordering are invalid (i.e. they need to be further decomposed until cyclicity is

destroyed). See Fig. 3.

Fig. 3. A decomposition that violates the cyclicity constraint[3]



2.1.3 Example

RM-OBJECT

RM-SET 1 RM-SET 2

{Wheel geometry, Epoxy} {Shaft geometry, Polyurethane}

Fig. 4. Heterogeneous Solid Model of a Multi-Material Part [3]

The object is modeled as an r m-object (HSM) with two r m-sets. The wheel is modeled as an r m-set,

the material being epoxy. The second r m-set models the shaft made of polyurethane. It is

processed further for fabrication. In Fig. 5, the support structure is generated, and the model is

decomposed into SFF-Compacts that form an SFF-Object. The compacts are ordered according

to their build order. This SFF-Object can be further processed by the deposition and tool path

planner for fabrication by a hybrid process like SDM.



HETEROGENEOUS SOLID MODEL

SUPPORTEDMODEL

DECOMPOSED MODEL

Modeled as r m-objectWith three r m-sets



SFF-Object with 7 SFF-Compacts (in buildorder)

Fig 5. Process planning for the wheel/shaft example using Design Decomposition[3]

In Fig. 6, the decomposed model (SFF-Object) is shown sliced to generate planar layers

(traditional slicing) using an adaptive slicing algorithm . The SFF-Object has been decomposed

into layers which can be manufactured by any purely additive process (e.g. DMD, SLS).

Toolpaths for a sample slice is also shown.



Fig. 6. Sliced model and a sample heterogeneous slice for the wheel shaft example[3]

2.2 Decomposition methods:

2.2.1 The tetrahedral model

In this representation a solid model created on a state-of-the-art CAD system is meshed

into finite elements (tetrahedra). The topology is maintained using the cell-tuple structure as a

graph of cells. Every cell is then associated with information about the composition and the

geometry. Material space is defined as M , spanning the dm materials available to the LM

machine. The material composition of the model is represented as a vector valued function m( x)

defined over the interior of the model. The designer specifies the overall variation in terms of

distance from a particular feature. This expression is used to obtain the volume fractions at the

vertices of each tetrahedron. The composition in the interior of the cell is then obtained in terms

of a set of control points and control compositions blended with barycentric Bernstein

polynomials. One of the major advantages of decomposing the model into tetrahedra is that the

model can directly be used for finite element analysis.

However, the following are the major drawbacks associated with this representation as

compared to the representation in Section 2.1:



1. Any modification in the material distribution function, m( x) will necessitate regeneration of

the mesh because the mesh generation depends on m( x).

2. The exact material distribution function is used to obtain the compositions at the vertices of

very tetrahedron. However, the composition at other points in the tetrahedron is calculated by

interpolation. The advantage of increase in the computational speed may be offset by the

approximation in representation.

3. The approximation in the shape due to meshing may lead to inaccurate dimensions and errors

in the required surface finish. Note that this approach approximates the object geometry and

material distribution function as well.

2.2.2 Voxel-based model

This is a special case of cell decomposition. The cell is cubical in shape and is located in

a fixed grid ( A voxel (x,y,z) in a 3D discrete space is defined by a unit cube centered at (x,y,z)).

Voxelization is the process of converting a geometrically represented 3D object into a voxel

model defined by a set of voxels. The voxelization is such that the voxel size is uniform and

every voxel is small enough to be considered as a homogeneous lump.

It is observed that the voxelization is independent of the material distribution function.

This representation provides to the designer a unique ability to selectively assign materials to

individual voxels. This is also better suited for fabrication using layered manufacturing as each

individual slice can be represented as a collection of voxels.The following constitute some of the limitations associated with this representation as

compared to the representation in Section 2.1:

1. This model also faces limitations like the one in Section 2.2.1 because of the approach of

decomposition.

2. Geometry-dependent function distributions are not easily applicable because of approximation

of the geometry in the voxel object.

3. This method of representation is not compatible for any kind of data transfer amongst CAD

systems. Research is being carried out to construct a solid model from voxel models

4. This technique does not seem to be suited for finite element analysis where tetrahedral

structures are preferred to cubical ones.

2.3 The R-Function approach:



An R-function (which should not be confused with an r-set) is a real-valued function

whose sign is completely determined by the signs of its arguments. Such functions provide

analogies to the Boolean logical functions. Simple examples are provided by min( x1, x2) and

max( x1, x2), which are analogous to Boolean ‘and’ and ‘or’, respectively, if we take + and -

values of the arguments to correspond to the logical values of TRUE and FALSE.

The analogy with Boolean functions allows any closed shape model in 2D or 3D,

expressed in terms of Boolean combinations of half-spaces, to be defined in terms of a single

implicit function, by composition of appropriate R-functions. It can be arranged for this function

to be positive inside the shape, and its value will be zero on the boundary. More generally, an

extension of the R-function approach allows the generation of functions on such a domain that

interpolate continuous distributions of function values or derivatives on its boundary. The

method may therefore be used to model distributions, e.g., of material properties, in the interior

of the shape.

Till now we have seen some of the representation schemes for heterogeneous objects

(functionally graded materials) .In the next session we are going to deal with their fabrication

methods. The RP technologies which are mainly used for making metal– metal or metal–ceramic

FGMare Selective Laser Melting, laser cladding-based techniques, Laminated Object

Manufacturing and Ultrasonic Consolidation (UC) while for fabricating polymer–polymer or

polymer–ceramic, ceramic–ceramic FGM, Selective Laser Sintering, Three Dimensional Printing

and Inkjet Printing have been mainly used. In this paper we are going to focus on two fabrication

techniques for functionally graded (heterogeneous) materials those are Freeze-form extrusion

process.

3. Freeze-form extrusion:

Freeze-form Extrusion Fabrication (FEF) is a novel, environmentally friendly, additive

manufacturing process that builds a 3D part layer-by-layer by computer controlled extrusion and

deposition of aqueous based colloidal pastes. Unlike most other extrusion freeform fabrication

processes, which use organic binders to bond the ceramic powders, the organic binder content is

only 2–4 vol% in this process, while the paste solids loading is 45–50 vol% or higher. Also,

unlike Robocasting which fabricates parts at room temperatures, FEF builds a ‘green’ (before

postprocessing) part in an environment below the freezing point of water to solidify the paste

after the deposition of each layer during the fabrication process. This enables relatively large



parts to be built compared with Robocasting. Cones and other monolithic ceramic components,

including alumina (Al2O3) and zirconium diboride (ZrB2) parts, have been fabricated using the

FEF process; for example, see Fig. 7.[1]

Fig. 7. Sintered ceramic cones fabricated using the FEF process for (a) Al2O3 and (b)

ZrB2.[1]

The part fabrication process involves computer control of flows of multiple aqueous pastes (each

controlled separately), the mixing of these pastes, and the extrusion of the mixed paste to

fabricate a 3D part layer-by-layer according to a CAD model with pre-specified material

compositions.

3.1. Process and system concept:

An FEF machine equipped with three servo controlled extruders and an inline static

mixing unit to merge the pastes through a single orifice has been designed and developed. This

mixing technique results in a natural transition between composition changes; however, it

introduces a transport delay. This delay must be repeatable and accurately predicted in order for

the path planning algorithm to deposit material in the desired location properly. The system

transport delay t is modeled using linear relationships between the paste volumetric flow rate Q

and the combined internal volume of each segment of the static mixer V:

t =V/ (A1v1+ A2v2 + A3v3) [1]

Ai is the cross sectional area of the i th cylinder, and vi is the velocity of the i th plunger. The

combined flow rate from all three extruders, Q, is equal to the sum of the individual flow rates,

Q1, Q2 and Q3. The ratio of Q1:Q2:Q3 represents the ratio of the three pastes in the material

composition.



The three pastes are extruded simultaneously by a triple extruder mechanism, as

illustrated in Fig. 8. Continuous control over the material compositions and their gradients during

the part building process can be achieved by planning (with time delay taken into consideration)

and controlling the relative flow rates of the different pastes. As an example, assuming that the

three cylinders containing the three different pastes have the same cross-sectional area, a desired

paste mixture consisting of 20% paste A, 30% paste B, and 50% paste C can be achieved by

controlling the three plunger velocities with the ratios of v1:v2:v3 = 2:3:5, where v1, v2, and v3 are

the plunger velocities for pastes A, B, and C, respectively.[1]

Fig. 8. Triple-extruder mechanism design.[1]

3.2 System design and development:

The triple-extruder mechanism is designed using three stainless steel cylinders, each

containing a paste driven by an individual plunger whose movement is controlled by a DC servo

motor; see fig 9, The paste flow rate in each cylinder is controlled by the plunger’s velocity, and

the force exerted on the plunger is measured by a load cell. The FEF system uses a static mixer

to blend the three different pastes and mixes them into a homogeneous stream as they pass a

series of mixing blades positioned at alternating angles.[1]



Fig. 9. The triple-extruder FEF system in a temperature-controlled enclosure: three

Servo motors control linear cylinders for paste extrusion and a three-axis gantry

system controls nozzle movement.[1]

The triple-extruder mechanism is mounted on a gantry system, which consists of three

orthogonal linear drives, each with a 508 mm travel range. The X-axis consists of two parallel

slides and is used to support the Y-axis. The Z-axis is mounted on the Y-axis, and the extrusion

mechanism is mounted on the Z-axis. Four DC servo motors, each with a resolver for position

feedback at a resolution of 1000 counts per revolution, drive the various axes.

The part fabrication process is conducted in a freezing environment, which could becontrolled to as low as -20 oC using a liquid nitrogen injection system. This enables the aqueous

paste to solidify at temperatures below the freezing point of water after it is extruded to solidify

the paste, thus avoiding part deformation during the fabrication process and enabling fabrication

of larger parts. A heating jacket is used to keep the paste temperature above the freezing point of

water until it is deposited.

3.3 Test Results:

Control of grading between two pastes:

A cylindrical part with a 50 mm diameter was fabricated using two extruders filled with

limestone (CaCO3) pastes, one with a green color and the other with a pink color. Fig.10

demonstrates that a part can be built with desired material gradients by varying plunger velocities

for extrusion of different pastes. The color of the fabricated part starts at pink (A) and shifts to



brown (B), then green (C), then brown (D), then pink (E), and finally green (F). The color

distribution of the part is consistent with the velocity profiles of the two plungers.[1]

Fig. 10. Extrusion of pink and green colored CaCO3 pastes for tests with and without

mixing. [1]

4.0 Ultrasonic Consolidation

It is another important technique to fabricate heterogeneous objects. Out of all RP

techniques, UC gives an advantage (such as solid state bonding, low temperature processing and

surface texture retaining) to fabricate precisely a graded layer metal–metal FGM which has

potential to furnish a metallic product with pre-defined properties gradient. Foils are joined by

ultrasonic welding using a UC machine named Formation, manufactured by Solidica Inc., USA.

Formation is a hybrid additive layer manufacturing machine which consists of following main

parts: (1) a computer program to generate toolpaths for welding and cutting as per an STL file of

a 3D CAD, (2) a base plate and attached heater, (3) an automatic foil-feeding system to lay foils,

(4) an ultrasonic welding system to join foils, (5) a 3-axis CNC milling machine to shape and

trim joined foils.[2]

Process description:

The machine works by joining two foils (layers) using mechanical vibrations of a

welding head (sonotrode) at 20 kHz and removing the extra part of the foil/build by a milling

machine. For joining foils, a foil is laid on a base plate which could be maintained at a higher

temperature, if needed. Another foil is manually fixed on top of it as automatic feeding is used

only for aluminium foils. A rolling welding head, forced on the foils travels along the length of



the foils. During travel, the head vibrates ultrasonically with small amplitude normal to the

direction of the travel. This vibration plus normal force helps foils bond metallurgically. Fig.11

shows the principle of the bonding of foils in a UC machine. The machine is equipped with a

sonotrode of size 5.8’’ of titanium alloy; Foils are taken from the following materials: stainless

steel (SS) 316L, annealed, thickness 0.00400; Cu 110 annealed, 0.00500; Al 3003 H19, 0.00600

and Al 1100 annealed, 0.00200. Al 1100 foil is used only to facilitate joining of two SS foils by

placing it between the two foils. [2]

Fig. 11. Principle of the bonding of foils in a UC machine[2]

Summary:

In this paper we studied new approach of various representation schemes of heterogeneous

objects with an example of wheel and shaft. A Freeze-form Extrusion Fabrication (FEF) process

and ultrasonic consolidation process aimed at fabricating 3D parts with functionally graded

materials are presented in this paper.

References

1) Ming C. Leu , Bradley K. Deuser , Lie Tang , Robert G. Landers , Gregory E. Hilmas ,

Jeremy L. Watts, “Freeze-form extrusion fabrication of functionally graded materials”,

CIRP Annals - Manufacturing Technology,61, pp. 223–226, 2012.



2) S. Kumar, “Development of Functionally Graded Materials by Ultrasonic

Consolidation”, CIRP Journal of Manufacturing Science and Technology, 3, pp. 85–87,

2010.

3) Vinod Kumar, Sanjay Rajagopalan, Mark Cutkosky and Debasish Dutta, “Representation

and Processing of Heterogeneous Objects for Solid Freeform Fabrication”, IFIP WG5.2

Geometric Modelling Workshop, Tokyo, 1998.

4) Xiao J. Wu, Wei J. Liu and Michael Y. Wang, “Modeling Heterogeneous Objects in

CAD”, Computer-Aided Design & Applications, Vol. 4, No. 6, pp. 731-740, 2007.

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