LIGHTWEIGHTING COMPOSITES THROUGH SELECTIVE FIBER PLACEMENT

Transcription

1 LIGHTWEIGHTING COMPOSITES THROUGH SELECTIVE FIBER PLACEMENT Christopher M Pastore Philadelphia University Abstract Lightweighting is essential for the reduction of energy consumption in transportation. The most common approach is through the application of high specific strength and stiffness materials, such as composites and high performance aluminum alloys. One of the challenges associated with the use of advanced materials is the high cost. This paper explores the opportunities of using hybrid composites (glass and carbon, for example) with selective fiber placement to optimize the weight subject to price constraints for given components. The underlying idea is the use the more expensive carbon fiber only where needed thorough the use of a gradient hybrid material that incorporates glass everywhere else. The goal is a process that allows automation while optimizing weight and cost for a given structural element. Through a combination of theoretical and experimental evaluations, a methodology for evaluating the weight-cost efficiency of chopped fiber composites was developed and confirmed experimentally. Optimization is investigated using penalty function and exchange constant methods. Considering the example of a hat-section for hood reinforcement, different material configurations were modeled and developed. For each, the required thickness of the hat section to meet the same bending stiffness as an all carbon composite beam was calculated. It was shown that selective placement of fiber around the highest moments results in a weight savings of around 14% compared to a uniformly blended hybrid with the same total material configuration. From these calculations it is possible to estimate the materials cost of the different configurations as well as the weight of the component. Each solution has an advantage it may be lighter than another, but not less expensive. To determine which is best it is necessary to find an exchange constant that converts weight into cost the penalty of carrying the extra weight. The value of this exchange constant will depend on the particular application. Depending on the value of weight, different materials appear optimal. When weight is valued between $1. and $5.3 per pound, the optimized hybrids offer the best solution to the problem. Background Weight has a cost penalty (or conversely, weight reduction has a value or premium), depending on the application. The relatively high cost of composites is a barrier to their adoption in cost-sensitive (low weight penalty) energy applications. Multi-material preforming enables the production of composite components with increased structural efficiency and reduced cost. This will accelerate the deployment of high performance composites in energy applications. Lightweight design and construction is essential for maximizing performance in many energy applications. For example, a 1% mass reduction in passenger automobiles reduces fuel Page 1

2 demand by about 6 8% 1, hence routes to compliance with 225 U.S. CAFE regulations (54.5 mpg) typically include demanding vehicle weight reduction targets. Similarly, utility scale wind turbines are growing larger with prototype blades approaching 1 m long, requiring a combination of low mass and stiffness that can only be delivered by fiber-reinforced composites, with carbon fibers being very desirable for high stiffness. Carbon fiber composites maximize weight reduction in these applications, but at a very high cost premium. Most structural parts are over-designed with the entire material volume able to withstand the highest stress levels in the part. However, the parts rarely experience uniform stress - some areas are much more stressed than others. An optimized part can be designed such that all of the material approaches the design limit condition in at least one load case while never exceeding that limit in any load case. The use of a high precision, selective fiber placement technique can result in appropriately designed, minimal weight components. In an optimal configuration, lower cost materials can be incorporated in areas where benefits of the carbon fiber do not justify the cost premium, resulting in a reduced overall cost of production while fully satisfying all performance requirements. This method has been realized in continuous fiber applications. Laminated composites often include glass at the core and carbon on the surface for bending resistance, and woven fabrics with glass in one direction and carbon in the other have been developed for applications with directional loading. However with continuous fiber composites there are limitations to the ability to vary the carbon and glass content. The use of short fiber deposition allows variation in carbon content in all dimensions of the part in an arbitrary pattern. Further short fiber deposition offers a significantly lower cost final product. Elastic property predictions Elastic Properties For the purposes of performing the analytical modeling it is necessary to establish some mechanical properties of the materials. For prediction of elastic properties a modified Mori- Tanaka 2 micromechanics model was used to determine equivalent unidirectional composite properties. (1) where C UD is the stiffness matrix of an aligned short filament, C m is the stiffness matrix of the matrix material, C f is the stiffness matrix of the fiber, V f is the fiber volume fraction, I is the identity matrix and A E is given as (2) where E is the Eshelby tensor for ellipsoidal inclusions 3. Page 2

3 The composite properties were generated using a stiffness averaging method 4 to account for the various fiber orientations within the part. (3) where C c is the stiffness matrix of the entire part, f(θ, φ) is the probability distribution function associated with a fiber having in-plane orientation θ and elevation orientation φ, T(θ, φ) is the Hamiltonian transformation matrix of stress and C UD is the stiffness matrix of a unidirectional composite. For the hybrid materials, the stiffness of the composite was determined by volumetric proportional averaging of the stiffness matrices of carbon and glass composites. (4) where C H is the stiffness matrix of the uniform hybrid, k glass is the volumetric portion of the fiber which is glass, k carbon is the volumetric portion of the fiber which is carbon (k glass + k carbon =1), C glass is the stiffness matrix of a glass reinforced composite and C carbon is the stiffness matrix of a carbon reinforced composite. Fiber orientations and probability distribution function The probability distribution was approximated through examination of photomicrographs of materials produced using the P4 robotic deposition system. Surface micrographs were taken, such as shown in Figure 1. Page 3

4 Frequency Figure 1 Photomicrograph of surface of composite produced using P4 robot with 2.5 cm carbon and glass fibers. Fiber paths were traced using Image-J to determine approximate in-plane fiber orientation distribution, which was then plotted as shown in Figure 2. The mean fiber orientation was 3, reflecting a fairly good interpretation of the images, as an ideal distribution would have a mean fiber orientation of. One observation is that the robot does not have a fully uniform distribution of fibers - there appears to be a slight bias towards the longitudinal direction In-Plane orientation of fiber (deg) Figure 2 Distribution of in-plane fiber orientations produced using P4 robot with 2.5 cm carbon fibers. To get angle of elevation (φ) out of the plane, the same process was performed using edge cut samples, as illustrated in Figure 3. These edge cuts are from consolidated Page 4

5 Frequency composites, so it is expected that the angle of elevation will be relatively low. Figure 3. Photomicrograph of edge of composite produced with 2.5 cm carbon fibers. Again the fiber paths were traced using Image-J and a distribution of angle of elevation was established, as shown in Figure 4. The mean fiber elevation angle was less than 1, suggesting fairly good measurements Out-of-Plane orientation of fiber (deg) Figure 4 Distribution of out of plane orientation produced using P4 robot with 2.5 cm carbon fibers. Experimental Results Panels were fabricated for evaluation of elastic properties. The constituent materials used for experimental studies were fiberglass (Advantex ), carbon (Panex 35) and Epoxy (Epikote ). The manufacturer provided properties are shown in Table.2 Page 5

6 Fiber Volume Fraction Fibers/yarns were chopped to 2.5 cm length prior to fabrication, and this fiber length was incorporated into the micromechanics model. Panels were produced in a fixed thickness mold with dimensions of 15 cm x 2 cm x.2 cm. Five configurations of fiber blending were evaluated, with proportion of carbon (relative to total fiber) ranging from to 1 in increments of.25. The proportion of carbon is described in terms of volume ratio. For the hybrid combinations, the fibers (both carbon and glass) were first chopped, the appropriate amount of fiber determined, and the fibers were then mixed by hand in a large bowl. The fibers were placed in a 15C oven for 1 hour to drive off moisture. Then the fibers were placed into the mold, resin applied, and the mold was closed. The mold was placed in a hot press and closed with a pressure of approximately 3 MPa or until the frame was completely closed. The mold was held closed at a temperature of 9C for 1 hour, then cooled and demolded. Thickness of the parts was measured to confirm fiber volume fraction. There was some variation from the target 35%, as it was discovered that the mold thickness had decreased to 1.8 mm. The calculated fiber volume fraction is shown in Figure 5. It can be seen that the glass fibers had slightly higher fiber volume fraction than the carbon fibers, partly due to the reduced bulkiness. 45% 4% 35% 3% 25% 2% 15% 1% 5% % % 25% 5% 75% 1% Proportion of Fiber that is carbon Figure 5 Fiber volume fraction of tensile specimens Samples were cut into tensile coupons and tested in tension. The results of the tensile modulus are presented in Figure 6, compared with the theoretical predictions. For the purposes of comparison, the modulus values are normalized to 35% fiber volume fraction. The relatively large error bars are associated with the hybridization process, as the fiber distribution was not completely uniform. Page 6

7 Experimental Tensile Modulus (GPa) 35% Vf Normalized Tensile Modulus (GPa) % 25% 5% 75% 1% Proportion of Fiber that is carbon Figure 6 Experimental Tensile Modulus of Hybrid Composites Normalized to 35% Fiber Volume Fraction and Compared with Theoretical Predictions (dashed line). Another way of viewing the experimental and theoretical modulus values is presented in Figure 7 without any volume fraction normalization % Carbon Carbon 25 2 Glass 75% Carbon 15 25% Carbon Theoretical Tensile Modulus (GPa) Figure 7 Comparison of Experimental and Theoretical Tensile Modulus of Hybrid Composite Specimens. Solid line indicates perfect agreement. Page 7

8 35% Vf Normalized Tensile Strength (MPa) The strength of the hybrid composites is presented in Figure 8. Again the values are normalized to 35% fiber volume fraction for comparison purposes. The theoretical curves are developed by applying a theoretical strain to failure criteria for the glass and carbon fibers % 25% 5% 75% 1% Proportion of Fiber that is carbon Figure 8 Experimental Tensile Strength of Hybrid Composites Normalized to 35% Fiber Volume Fraction and Compared with Theoretical Predictions (dashed line). Hat Section Beam Bending Model For the purposes of structural analysis, a hat section beam, representative of a hood stiffener, was chosen for evaluation. The profile is shown in Figure 9, and is 3 cm long. Figure 9 Profile of Hat Section beam used for evaluation The effect of wall thickness on the moment of inertia is shown in Figure 1. Page 8

9 Moment of Inertia (mm 4 ) 6, 5, 4, 3, 2, 1, Wall Thickness (mm) Figure 1 Effect of wall thickness on moment of inertia of hat section. Prediction of Beam Behavior The first comparison is to evaluate the deflection of the baseline profile using different materials. The maximum deflection of the homogenous materials happens at the center of the beam, and can be calculated as: (5) where δ is the deflection at the center point, P is the load, L is the span of the beam, E is the elastic modulus and I is the bending moment of inertia. For the hybrid beams, because the tensile modulus changes as a function of location, it is necessary to derive the displacement. We know that the slope of the beam can be determined by (6) where θ(x) is the instantaneous slope of the beam at position x, M(x) is the internal moment on the beam at x, and E(x) is the elastic modulus of the beam at x. Page 9

10 The constant of integration can be solved in this case as zero slope at the center of the beam: θ(l/2) =. Then the deflection can be calculated as (7) The constant of integration can be solved for δ() = δ(l) = due to the supports. The maximum deflection will occur at the midpoint of the beam (x=l/2). Materials Considered Four different material models were considered for the analysis, as shown in Table.1. Table.1 Material Configurations used in this study Name Description 1% Carbon All carbon for the entire part Linear Hybrid Quadratic Hybrid Starts as 1% glass at end and changes to 1% carbon at center, then back again. The rate of material change is linear with respect to volumetric proportion. Starts as 1% glass at end and changes to 1% carbon at center, then back again. The rate of material change is quadratic with respect to volumetric proportion. 1% Glass All glass for the entire part The relevant material properties are presented in Table.2. Page 1

11 Proportion of Fiber that is Carbon Table.2 Material Properties used for Analysis of Hat Section Beams Property Glass (Advantex ) Carbon (Panex 35) Epoxy (Epikote ) Tensile Modulus (GPa) Yield Strength (MPa) 3,5 4,1 67 Density (g/cm 3 ) Cost ($/kg) $1.65 $22. $3.3 Fiber Length (cm) Fiber Diameter (µm) For the linear hybrid, the volumetric distribution of carbon fiber (proportionate to total fiber content) follows a linear path, with no carbon (all glass) at the ends and all carbon (no glass) at the center, as show in the distribution model in Figure 11. This follows the distribution of moment along the length of the beam, placing carbon proportionately to the magnitude of the bending moment Volume Weight Position along beam (m) Figure 11 Proportion of total fiber that is carbon (in carbon/glass hybrid) as a function of position along the beam for the linear hybrid. Both volume and weight proportions are shown. In the case of the quadratic distribution, the carbon content varies with the square of the Page 11

12 Proportion of Fiber that is Carbon position on the beam, resulting in 2/3 glass and 1/3 carbon volume content for the entire part. The distribution of fiber along the length of the beam is as shown in Figure 12. This follows the moment distribution from a uniformly distributed load on the beam, and provides less carbon, thus less cost to the part Volume Weight Position along beam (m) Figure 12 Proportion of total fiber that is carbon (in carbon/glass hybrid) as a function of position along the beam for the quadratic hybrid. Both volume and weight proportions are shown. Predicted Beam Deflections The maximum deflection of a 2 mm wall thickness hat section subject to a central point load of 1 N was calculated for each of the six material models, as shown in Figure 13. Note that the magnitude of the load is not crucial in the comparison - only the comparison of the various beam deflections for a given point load. Page 12

13 Maximum Deflection (mm) Maximum Deflection (mm) Glass Quad Hybrid Linear Hybrid Carbon Steel Aluminum Figure 13 Maximum deflection of simply supported 2 mm thick, hat sections with 3 cm span subject to central point load of 1, N made of different materials. To grasp the benefits of the selective fiber placement associated with the Quad Hybrid and Linear Hybrid, it is instructive to compare their performance with a uniformly blended hybrid of the same fiber content. Figure 14 shows the maximum deflection of 2 mm beams for different amounts of carbon fiber. The solid line indicates the behavior of a uniform hybrid..4 Glass.3 Uniform Hybrid.2.1 Quad Hybrid Linear Hybrid Carbon Proportion of Fiber that is Carbon Figure 14 Maximum deflection of 2mm thick hat-section beam subject to central load, comparing uniform hybrid with selective placement hybrids. It can be seen that the use of selective fiber placement results in a increased stiffness of the Page 13

14 Wall Thicckness needed to match Carbon Stiffness (mm) part approximately 15% less deflection. Designing for Equivalent Structural Stiffness In order to compare equal performance components, the wall thickness necessary to achieve the same maximum deflection (structural stiffness) for each material was determined. The all carbon composite was chosen as the baseline. As can be seen from the previous discussion, this means the composites containing glass will need to be thicker and the metal parts can be thinner. The wall thickness of each material was parametrically changed, resulting in different moments of inertia as illustrated in Figure 1 above, and subsequently in different maximum deflections. The thickness value at which the maximum deflection was the same as the 2 mm carbon baseline was recorded as the stiffness optimized wall thickness. These results are shown Figure 15. Note that the wall thicknesses are proportional to the total weight of the part because perimeter and length are constant Glass Quad Hybrid Linear Hybrid Carbon Steel Aluminum Figure 15. Wall thickness required to get same maximum deflection as simply supported 2mm thick carbon composite hat section with 3 cm span. Comparing the selective placement schemes (Quad and Linear Hybrids) to a uniform hybrid, the thicknesses required to meet the same bending stiffness as the carbon beam are shown in Figure 16. It can be seen that there is about 13-14% thickness (thus weight) savings achieved by selective placement compared to uniform blending. Page 14

15 Thickness Needed for Equal Stiffness (mm) 5 4 Glass Uniform Hybrid 3 2 Quad Hybrid 14% weight savings Linear Hybrid 13% weight savings Carbon Proportion of Fiber that is Carbon Figure 16 Thickness needed to match bending stiffness of carbon composite beam. Designing for Equivalent Bending Strength Another important measure of performance is the strength of the beam. To consider equal performance in this area the strength of the materials was brought into consideration. For the composites this was the fracture strength and for the metals, the yield strength, using the data presented in Table.2 and determined experimentally for the hybrids. Note that factors of safety were not included in the following calculations. In flexure, the maximum stress on the beam is given as (8) For the 2 mm carbon part, we can calculate the maximum moment that it can withstand, and this is found to be 292 N-m. The design for equivalent strength thus means finding the appropriate section modulus (S=I/c) for each material so that the material reaches its design stress level at that same maximum moment of 292 N-m. For the hybrid parts, it is necessary to check the stress level at each point along the length as the tensile strength varies with position. Then the necessary section modulus for each point can be found based on the local moment and strength, and subsequently the requisite thickness. In the case of the linear hybrid, the maximum section modulus needed for the hybrid parts occurs at the midpoint, as shown in Figure 17. At this location the fiber is all carbon, thus 2 mm is the necessary wall thickness. However for the quadratic hybrid, the maximum section modulus need occurs at approximately one third of the beam length, requiring 2.4 mm wall Page 15

16 Thickness Required for equivalent Strength to Carbon (mm) thickness to meet the strength requirements LH QH Position along beam (m) Figure 17. Required wall thickness to prevent flexural strength failure as a function of position along the beam for the hybrid configurations. For the uniform materials, the calculations can be performed at the mid-section. The wall thicknesses required to achieve the same flexural strength as the 2 mm carbon beam are shown in In Figure 18. Combined Design In Figure 18, a comparison of the minimum thickness needed to achieve the same structural stiffness is shown with the minimum thickness needed to achieve the same structural strength. If both of these criteria are applied simultaneously, it is necessary to choose the larger value from each pair. Page 16

17 Wall Thickness need for carbon equivalent properties (mm) Stiffness Strength Glass Quad Hybrid Linear Hybrid Carbon Steel Aluminum Figure 18 Wall thickness required to get same structural stiffness and same flexural strength as simply supported 2mm thick carbon composite hat section with 3 cm span. Knowing the minimum acceptable wall thickness, it is possible to calculate the mass of the part and the material cost of the part. Using the densities and cost per unit weight, the fiber volume fraction, and the proportion of carbon/glass, it is possible to calculate the mass and material cost of the parts made from each material. The mass and cost needed to make equivalent performing parts are shown in Figure 19 and Figure 2. For the purposes of comparison with a carbon composite, it can be seen that the hybrid composites have a cost savings of approximately 3-4% less than carbon, with a weight increase of approximately 23-36%. Page 17

18 Material Cost of equivalent part ($) Mass of equivalent hat section (g) Glass Quad Hybrid Linear Hybrid Carbon Steel Aluminum Figure 19 Mass needed to achieve both stiffness and strength criteria comparable to 2 mm carbon hat section. $1.2 $1. $.8 $.6 $.4 $.2 $. Glass Quad Hybrid Linear Hybrid Carbon Steel Aluminum Figure 2 Material cost to achieve both stiffness and strength criteria comparable to 2 mm carbon hat section. As can be seen here, it is difficult to determine which product is best. Carbon is the lightest, but steel is the least expensive. If the solution space is considered to be mass and cost, the potential material solutions can be plotted on this coordinate space, as shown in Figure 21. The optimal solution has a mass of zero and a cost of zero, which is unachievable, but gives insight into how to read this chart. The goal is to approach the origin. Page 18

19 Material Cost of Component ($) $1.25 Carbon $1. $.75 Linear Hybrid 613-T6 Al Quad Hybrid $.5 Glass $.25 AHSS Mass of Component (g) Figure 21 Plot of mass and material cost for parts optimized to meet both stiffness and strength criteria. There is a trade-off between weight and cost, and the value of that trade-off, called the exchange constant, depends on the particular application and market demands. When a solution space such as that shown in Figure 21 does not show a clear optimal solution, it makes sense to look at a penalty function that calculates the effective cost of the component. In this case, the effective cost would be: C E = C M + M (9) where C E is the effective cost, C M is the cost of materials, M is the mass of the component, and is the exchange constant (cost/mass). A plot of effective cost as a function of the exchange constant can demonstrate over what ranges a given solution is optimal. In Figure 22 the effective cost is calculated for a range of exchange constants comparing the 4 composite solutions. The material with the lowest cost at some exchange constant would be a locally optimal solution for this space. Page 19

20 Effective Cost ($) Glass Quad Hybrid Linear Hybrid 3. Carbon Glass Quad Hybrid Linear Hybrid Carbon Weight Penalty ($/lb) Figure 22 Effective Cost of 4 Composite Material Solutions as a Function of Exchange Constant In Figure 22, it can be seen that for very low exchange constants (less than $.99/lb) glass is the best choice. After that, Quad Hybrid is best, followed by Linear Hybrid and finally carbon at high exchange rates. This is summarized in Table.3 Range of Exchange Constants for which each Composite is optimal Material Exchange Constant Range Glass $. - $.99 Quad Hybrid $.99 - $2.27 Linear Hybrid $2.27 $5.3 Carbon $5.3 - When the metal choices are added to the analysis, steel becomes the low cost optimal choice. Steel is best between $. and $1.62/lb, aluminum next between $1.62 and $8.2/lb and carbon the best for high value applications. Effect of Carbon Fiber Cost The limiting factor here is the cost of carbon, a topic of much research. As the cost of carbon decreases, assuming no property losses, the composites become more cost competitive with aluminum, and even potentially AHSS steel. The maximum cost of carbon fiber required to make the parts cost the same as 613-T6 aluminum is shown in Figure 23. Page 2

21 Cost of Carbon Fiber ($/lb) $9 $8 $7 $6.93 $8.18 $6 $5 $4.87 $4 $3 $2 $1 $ Carbon Linear Hybrid Quad Hybrid Figure 23 Cost of Carbon Fiber Needed to Make Optimized Composite Hat Section Component with Same Cost as Aluminum with Equivalent Stiffness and Strength In order to make the composites cost effective with AHSS steel, the cost of carbon needs to be extremely low approximately $2.5 per pound. If exchange rates are included, depending on the particular cost of weight, the needed cost of carbon fiber will vary with the specific exchange rate. Figure 24 shows the minimum carbon fiber cost needed to achieve equivalency with 613-T6 aluminum as a function of exchange rate. For the carbon composite, as the exchange rate increases, it is acceptable to pay more for the carbon fiber. The Linear Hybrid has a slight increase in minimum fiber cost, but is generally flat due to the 5% glass content. For the Quad Hybrid, because of the greater amount of glass and thus higher overall mass, increasing the exchange constant makes it less attractive, and requires less expensive carbon fiber to be competitive. Page 21

22 Minimum Carbon Fiber Cost ($/lb) Carbon Linear Hybrid Quad Hybrid Exchange Rate ($/lb) Figure 24 Minimum Carbon Fiber Cost needed to Make Optimized Composite Hat Section Component with Same Cost as Aluminum with Equivalent Stiffness and Strength as a Function of the Exchange Rate Effect of Fiber Volume Fraction The analysis performed above assumed a total fiber volume fraction of 35% for all calculations. During experimental work, it was observed that 4% fiber volume fraction is achievable with the short fiber composites. Higher fiber volume fraction will result in increased stiffness and strength for all the composite materials, and subsequently a reduced wall thickness for the hat section under consideration. This results in lower weight and lower cost composite parts. The solution space showing the 4% fiber volume fraction parts along with the previously analyzed parts is shown in Figure 25. The weight of the components drops significantly, whereas the cost drops less dramatically, primarily because the amount of fiber needed is roughly the same and the cost reduction comes from the lesser amount of resin used in the parts. Page 22

23 Material Cost of Component ($) $1.25 Carbon $1. 4% Vf 35% Vf $.75 $.5 $.25 Linear Hybrid Quad Hybrid 613-T6 Al AHSS Glass Mass of Component (g) Figure 25 Cost and Mass Solution Space for Hat-Section Composites showing both 4% and 35% Total Fiber Volume Fraction Composites Future Work The solution space optimization presented herein addressed only material cost. This needs to be expanded to include full manufacturing costs, including the amortization of tooling over the life of the part. Adding embodied energy from a life cycle assessment might also prove useful, depending on the end use application. The use of a robotic fiber placement system is preferable to hand-layup to reduce the errors. It becomes clear that any specific component will have a unique solution. The stress distributions, manufacturing issues, and the number parts expected all affect the total calculations. Subsequent steps should focus on a generic framework that includes manufacturing costs, and a focus on a particular component of interest, whether automotive, wind power, or other. Acknowledgements Hat-section samples were produced using 3D printed molds created at the Manufacturing Demonstration Facility of ORNL by Drs. Kunc and Love. Testing was performed at Oak Ridge National Laboratories. Fibers and resin were provided by Owens Corning. This research was sponsored by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office, under contract DE-AC5-OR22725 with UT-Battelle, LLC. Page 23

24 Bibliography 1. 1 Lynette W. Cheah (21), Cars on a Diet: The Material and Energy Impacts of Passenger Vehicle Weight Reduction in the U.S., Massachusetts Institute of Technology Ph.D. Thesis, Sept Mori, T. and Tanaka, K. (1973). "Average Stress in the Matrix and Average Elastic Energy of Materials with Misfitting Inclusions". Acta Metallurgica 21: J.D. Eshelby (1957), The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proceedings of the Royal Society of London Part A, 2 August 1957 vol. 241 no Kreger, A.F and Teters, G. A. (1979) "Use Of Averaging Methods to Determine the Viscoplastic Properties of Spatially Reinforced Composites", Mechanics of Composite Materials, 15,(4). Page 24