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Hierarchical cellular designs for load-bearing biocomposite beams and plates
Materials Science and Engineering A 390 (2005) 178–187
Hierarchical cellular designs for load-bearing biocomposite beams and plates Rigoberto Burgue˜noa,∗ , Mario J. Quagliatab , Amar K. Mohantyc , Geeta Mehtad , Lawrence T. Drzale , Manjusri Misraf a
Department of Civil and Env. Eng., Michigan State University, East Lansing, MI 48824-1226, USA Department of Civil and Env. Eng., Michigan State University, East Lansing, MI 48824-1226, USA c School of Packaging, Michigan State University, East Lansing, MI 48824-1223, USA d Department of Chem. Eng. and Mat. Sc., Michigan State University, East Lansing, MI 48824-1226, USA Department of Chem. Eng. and Mat. Sc., Comp. Mat. and Structures Center, Michigan State University, East Lansing, MI 48824-1226, USA f Comp. Mat. and Structures Center, Michigan State University, East Lansing, MI 48824-1226, USA b
e
Received 19 December 2003; received in revised form 3 August 2004
Abstract Scrutiny into the composition of natural, or biological materials convincingly reveals that high material and structural efficiency can be attained, even with moderate-quality constituents, by hierarchical topologies, i.e., successively organized material levels or layers. The present study demonstrates that biologically inspired hierarchical designs can help improve the moderate properties of natural fiber polymer composites or biocomposites and allow them to compete with conventional materials for load-bearing applications. An overview of the mechanics concepts that allow hierarchical designs to achieve higher performance is presented, followed by observation and results from flexural tests on periodic and hierarchical cellular beams and plates made from industrial hemp fibers and unsaturated polyester resin biocomposites. The experimental data is shown to agree well with performance indices predicted by mechanics models. A procedure for the multi-scale integrated material/structural analysis of hierarchical cellular biocomposite components is presented and its advantages and limitations are discussed. © 2004 Elsevier B.V. All rights reserved. Keywords: Biocomposites; Cellular beams; Cellular plates; Laminate mechanics; Numerical analysis; Mechanical testing
1. Introduction Developments over the past decade in composite materials technology have shown that natural-fiber-reinforced polymer composites, or biocomposites, can be a technical, economical and environmentally conscious alternative to E-glass fiber composites without sacrificing performance [1,2]. In spite of this progress and the environmental appeal of natural-based composites, their lower strength and stiffness compared to conventional structural materials has limited their use to nonprimary or non-load-bearing applications. Increased understanding of natural or biological materials shows that nature has developed ways of achiev∗
Corresponding author. Tel.: +1 517 353 1743; fax: +1 517 353 1743. E-mail address: [email protected] (R. Burgue˜no).
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.08.034
ing high structural efficiency from a rather limited set of non-sophisticated constituents. One of the primary ways in which nature achieves high structural performance with nonexotic materials is through hybrid material combinations assembled in optimized hierarchical strategies [3,4]. Among nature’s most common efficient structures are hierarchical cellular sandwich structures, which consist of a complex arrangement of cells of different sizes arranged across the section such that cells are themselves made from cells (i.e., hierarchical arrangement), leading to dense regions, or skins, integrally connected to regions with lower density, or core [5]. A well-recognized natural material displaying this hierarchical design is human bone. Structural behavior is then dictated by the relative dimensions and topology design of the dense and porous regions, with the material and microstructural hierarchy playing a major role
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in maximizing the efficiency of the resulting material and structure. Hierarchical cellular designs are complex microstructures achieved by placing material where most needed, i.e., in areas of high stress. The process of arranging the microstructure (cellular material) can lead to a hierarchy that maximizes the efficiency of the resulting material or load-bearing component. The above-mentioned concepts motivated the present work to improve the performance of biocomposite materials for use in load-bearing beam and panel components through bio-inspired cellular hierarchical designs. Experimental and analytical feasibility studies on industrial hemp/unsaturatedpolyester hierarchical cellular beams and plates were thus conducted and used to evaluate their potential as load-bearing components. Experimental results verify the structural efficiency that is gained through hierarchical designs, and an analysis approach that permits analyzing the multi-level material and topological hierarchy of these structures is presented and preliminarily validated.
2. Analysis
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performance index is [8]: 2/3
MBf =
σf , ρ
(2)
where the superscript f means failure and σ f is the failure strength of the material. Thus, the best material for the given design requirements is that with the largest value of the performance index. Performance indices work well for comparing materials but they ignore the shape of the component. Thus, for materials with different cross-sectional shapes other factors must be used to quantify efficiency. Sections of shaped material carry load more efficiently than solid sections, where efficiency means that for a given structural demand the section uses as little material as possible, and is therefore as light as possible. To measure the shape and efficiency of a section, for a given mode of structural response, the concept of shape factors has been proposed by Ashby [9]. A shape factor is a dimensionless number that characterizes the efficiency of the shape of a section, regardless of scale, for a given type of structural response. When the design is based on flexural stiffness, the shape factor for elastic bending is [9]:
2.1. Flexural performance The efficiency, or performance, of cellular sandwich structures depends on individual and collective behavior of the dense skins, the core, and their connections [6,7]. The performance for a load-bearing component can be said to depend on three factors [8,9]: (i) the mode of loading, i.e., tension, bending and twisting, (ii) the properties of the material, and (iii) the shape of the section. While performance can be defined in many forms, it is taken here to mean maximum flexural stiffness or flexural strength per weight. Ashby [9] has shown that the performance of materials, shapes and microstructures can be evaluated through different indices. An overview of how these performance indices explain the efficiency of hierarchical materials and structures follow. Material properties can be combined to form performance indices [8], which are groupings of material properties that, when maximized, maximize some aspect of performance. The indices are derived from the design requirements for a component based on the function, objective and constraint. For a beam where the objective is minimum weight for a given flexural stiffness (light stiff beam), the material index can be shown to be [8]: MBe =
E1/2 , ρ
(1)
where the superscript e means elastic and the subscript B means bending, E is the elastic modulus of the material and ρ is the density. For a beam where the objective is minimum weight for a given flexural strength (strong light beam), the
φBe =
4πI , A2
(3)
where the superscript e means elastic, the subscript B means bending, I is the moment of inertia and A is the area of the section. For strength-governed designs (i.e., buckling, fracture, or plastic yielding) the shape factor is [9]: φBf =
√ 4 πS , A3/2
(4)
where the superscript f means failure and S is the section modulus, or I/ym , with ym being the distance from the section neutral axis to the extreme fiber. Shape factors are dimensionless and depend only on the cross-section shape. That is, large and small beams have the same shape factor value if their sections have the same shape and are proportional. Solid symmetric sections (circles, squares, etc.) will have shape factors close to 1, while more efficient shapes will have shape factors larger than 1. Shape factors can be combined with material indices to formulate performance indices that determine the performance-maximizing combination of material and shape. Ashby [9] derived indices for comparing the performance of beams of different material and shape. The best materialshape combination for a light stiff beam is that with the greatest value of the index [9]: M1 =
(EφBe )1/2 , ρ
(5)
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while the best material-shape combination for a light strong beam is that with the greatest value of the index [9]: 2/3
M2 =
(σf φBf ) ρ
.
(6)
For constant shapes the performance indices will reduce to the corresponding material index. The efficiency of a macroscopic shape can be further improved by introducing shape at a small scale, microscopic shape. For example, a rectangular section made of a cellular material (microscopic shape) will have higher efficiency than a solid rectangular section (macroscopic shape). The additional efficiency of the microstructural shape is characterized by micro-structural shape factors [9]. For flexure response, the micro-structural shape factor for elastic bending can be shown to be [9]: ψBe
ρs = ∗, ρ
(7)
while the micro-structural shape factor for flexural strength is given by [9]: 1/2 ρs ψBf = . (8) ρ∗ In other words, the micro-structural shape factor is proportional to the inverse of the relative density. The relative density of a cellular structure is defined as the density of the cellular material (ρ* ), divided by the density of the solid from which the cells are made (ρs ). Relative density is the most important property affecting the mechanical properties of a cellular material [7]. Note that, in the limit, for a solid microstructure (ρ* = ρs ) both ψBe and ψBf take the value of 1, as they should. Both stiffness and strength micro-structural shape factors show that performance is increased by introducing microstructure to a solid shape, resulting in a lighter more efficient structure. This idea of microstructuring can be extended further by introducing cell walls that are themselves microstructured, thus creating a structural hierarchy. Such efficient structural hierarchy is commonly found in natural materials [3,5,7]. However, there are limits on the level of hierarchy that can be achieved in man-made structures due to manufacturing difficulties and the accompanying manufacturing costs. Nevertheless, the use of hierarchical structures can further improve the performance of a structural member resulting in light efficient components. The positive effect of microstructure can also be shown by integrating the microscopic shape factor into the material performance indices. Considering a cellular structure and using the micro-structural shape factor, the performance index for maximum stiffness per weight for a cellular beam can be derived from Eq. (1) to give [10]: 1/2 E1/2 (E∗ )1/2 ρs 1/2 Es e MB = = = , (9) ρ ρ∗ ρs ρ∗
Fig. 1. Potential design of a two-dimensional hierarchical cellular panel.
where the modulus of the cellular material is equal to the modulus of the solid material times the relative density [7]. Eq. (9) indicates that the performance index of a cellular beam is higher than that of a solid one by a factor equal to the square root of the inverse of the relative density. 2.2. Analysis of hierarchical cellular sandwich structures The above concepts show that the efficiency of a structural component, such as a beam or plate, for a given stiffness can be enhanced if composed of a material that features cells or holes strategically distributed along the member crosssection. An example of a two-dimensional hierarchical cellular sandwich panel is shown in Fig. 1. Beams and plates structures with repeating layers of cellular material can be treated as laminated structures where each layer can be defined by a characteristic cell architecture [10]. This idealization can be applied to cellular beams and plates with periodic cells that are oriented in different directions about the longitudinal axis of the element and/or hierarchical cellular structures that feature different levels of cellular architectures along the member depth (Fig. 2). At the material level, Burgue˜no et al. have shown that the properties of biocomposite materials can be adequately predicted using existing micromechanics models for randomly oriented short fiber composites. Based on comparisons with experimental data [12], the quasi-isotropic laminate approximation by Halpin and Pagano [13] together with the Halpin–Tsai [14] equations was found effective in determining the elastic constants of the biocomposite materials. Mechanics of cellular solids are used at the microstructural level to model the properties of the cellular material layers within the hierarchical structure. While there are several approaches to the mechanics of cellular solids, Gibson and Ashby [7] have shown that the stiffness and strength characteristics of honeycombs and foams, i.e., two- and threedimensional cellular solids, can be effectively described using structural mechanics concepts. For two-dimensional cellular structures, as those considered in the experimental part of this study, properties were determined by analyzing the response of a unit hexagonal cell. The mechanical response of the cellular solid is thus, obtained by modeling the cell as an interconnected array of one-dimensional struts. The material elastic constants are then derived by evaluating the force–displacement response of the unit cell under different
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Fig. 2. Overview of the analysis process for hierarchical cellular biocomposite beams and plates.
loading conditions and up to different failure modes depending on the constituent material [7]. Assuming a plane stress condition and that the material response is at most orthotropic, the reduced lamina stiffness for a general single cellular material layer (lamina) is then obtained using conventional mechanics of composite materials [15]. The structural properties of the hierarchical cellular beam or plate can then be found by adding the individual stiffnesses of the different characteristic cellular material layers over the depth of the structure. Each layer may have different properties due to their constituent material, cellular architecture or cell orientation. Addition of all the layer stiffnesses over the section depth through classical lamination theory leads to the section properties of the laminated plate, or [A-B-D] matrix [15]. The section properties for hierarchical cellular beams and plates can thus be obtained by considering the different scales of the material continuum, from the microstructure of the randomly oriented short fiber reinforced biocomposite, to the cellular material microstructure and finally to the arrangement in a hierarchical structure, as shown in Fig. 2. The generality of the method allows determination of section properties along different structural directions. In addition, automation of the process allows the method to be used for parametric and optimization studies that can provide further insight into efficient hierarchical designs. While the analysis concept has thus far only been implemented to determine linear elastic stiffness properties, the same multi-scale approach can be used to determine non-linear response and failure conditions by incorporating appropriate models at each analysis level. Tooling limitations in the experimental program did not allow the manufacturing of beams and plates with a high order of multi-layer cellular topology. Due to the continuity assumptions inherent in the cellular solids models, the analysis approach described above is limited to cellular structures featuring multiple cellular layers. This limitation is discussed later in the paper. The stiffness of cellular components that did
not meet the requirements for the layered analysis method, were determined analytically using a mechanics of materials approach by separately calculating the modulus of elasticity of the material (through the micromechanics models discussed above) and the moment of inertia of the crosssection, and combining the two terms. While this approach allows a simple evaluation of the longitudinal member stiffness, other section properties are not as straight forward to obtain. For example, calculation of the transverse member stiffness requires consideration of the bending of the individual cell walls within the structure, which may be difficult to evaluate analytically. The failure load for all cellular beam and plates was obtained analytically using conventional mechanics of materials by relating the failure load for the specific test set-up to the flexural stiffness. The failure load of the cellular beams under four-point bending was then calculated using: Pf = εf
2EI , aym
(10)
where εf is the failure strain, a is the shear span and ym is the distance from the neutral axis of the beam to the extreme fiber. The failure load of the cellular beams and plates in three-point bending was calculated using: Pf = εf
4EI Lym
(11)
where L is the total span length. The flexural stiffness was determined analytically as previously described. The failure strain was obtained from the measured values in a material characterization study [11,12].
3. Experimental methods Laboratory-scale cellular biocomposite beams and plates were manufactured and tested to verify the improvement in
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Table 1 Average properties of hemp fiber and unsaturated polyester composites Material
Fiber wt. (%)
Green hemp/UPE Raw hemp/UPE
13 20
Fiber (L/d) 60 100
MOE (GPa)
Tensile str. (MPa)
3.81 6.15
14.47 19.49
flexural performance achieved through the use of hierarchical cellular designs and to validate the developed analysis approach. 3.1. Materials Industrial hemp fiber is the stiffest, cheapest and most readily available natural fiber type for use in load-bearing structural applications, while unsaturated polyester, which offers a good balance of cost and performance, was chosen as a proper match for the low-cost natural fibers. Industrial raw hemp (retted hemp, 15–25 mm average length, HempLine, Ontario, Canada) and green industrial hemp (10–15 mm average length, Flaxcraft Inc., Cresskill, New Jersey) were used as natural fiber reinforcement for the biocomposite material. The resin system used was ortho unsaturated polyester (UPE) (Kemlite Co. Inc., Joliet, IL) with methyl ethyl ketone peroxide (MEKP, Sigma–Aldrich) catalyst (1% by weight of resin), and cobalt naphthenate (CoNap, Sigma–Aldrich) promoter (0.03% by weight of resin). Properties of the biocomposite material systems are shown in Table 1. 3.2. Test units The experimental characterization study considered three slender and three deep hierarchical cellular beams, and two cellular plates. The cellular designs featured periodic and hierarchical cell arrangements. Dimensions, cellular architectures and relative densities for the cellular beams and plates are given in Figs. 3 and 4, respectively. Dimensions of the beam and plate components were selected to minimize material use and to parallel a study on the performance of cellular components by Huang and Gibson [10]. The cellular topologies in the beams and plates were determined from the base dimensions of periodic cellular beams and plates and on the available tooling materials to create the different cell sizes. Because tooling used to create the cells (rubber tubing) was only available in standard sizes, the extent of the hierarchical architectures that could be achieved was limited. Ongoing studies are focusing on developing and studying components with higher levels of cellular hierarchy through optimized topologies.
Fig. 3. Cross-section geometry of cellular beam test units.
beam mold was 25.4 mm × 50.8 mm × 304.8 mm and the cellular plate mold was 101.6 mm × 304.8 mm × 12.7 mm. The removable faceplates allowed different cellular configurations to be implemented with the same mold. The molds were lined with teflon paper to ease sample removal and the cells were created by placing rubber tubing through the faceplates. Steel rods were used to stiffen the rubber tubes during manufacturing. The biocomposite material was placed by hand in alternating layers of material and rods (cells). All cellular beams and plates were oven cured using a cycle of 100 ◦ C for 2 h followed by 150 ◦ C for 2 h. A detailed description of the manufacturing process is given in [11,12].
3.3. Manufacturing 3.4. Testing The cellular beams and plates were manufactured using molds with removable faceplates. The slender cellular beam mold was 25.4 mm × 25.4 mm × 508 mm, the deep cellular
Flexural characterization of the periodic and hierarchical cellular beams and plates was done through flexural test
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point bending with a total span of 267 mm and shear spans of 133 mm. All test units were loading monotonically up to failure in displacement control at a rate of 0.01 mm/s. Deflection of the beams and plates was measured at mid-span with an externally mounted extensometer and the loading frame LVDT. The applied load was measured using the internal load cell of the testing frame.
4. Results 4.1. Performance of hierarchical designs
Fig. 4. Cross-section geometry of cellular plate test units.
set-ups mounted on a loading frame (Fig. 5). The slender cellular beams (L/D = 18.0) were testing in four-point bending with a total span of 457 mm, concentrated loads at 102 mm apart and a shear span of 178 mm. The deep cellular beams (L/D = 5.25) and cellular plates were both tested in three-
Fig. 5. Pictures of flexural test set-ups.
The load–displacement response at mid-span of the slender cellular beams with industrial green hemp/unsaturated polyester are shown in Fig. 6. All cellular beams behaved linear-elastically up to failure. The failure mode in all cases was a sudden tensile rupture of the bottom surface of the beam. The effect of periodic and hierarchical cellular topologies on the performance of the beam units is evident in the force–displacement response (Fig. 6). Notice that while CB1 and CB3 have the same relative density (i.e., use the same amount of material), efficient material arrangement of CB3 in a hierarchical architecture leads to a 15% stiffer element. It is also obvious that CB2, with a higher relative density, i.e., more solid material, had a higher flexural stiffness. The effect of hierarchical structure on the beams flexural performance is, however, better seen by comparing their specific stiffness, as given in Table 2. Thus, the specific stiffness of the two hierarchical cellular beams (CB2 and CB3) is approximately 12% higher than the beam with periodic cells (CB1). The hierarchical arrangement of CB3 thus leads to both absolute and specific flexural performance improvement. Test results of the cellular plates with raw hemp/ unsaturated polyester are shown in Fig. 7. All cellular plates behaved linear-elastically up to failure. The failure mode of the cellular plates was a sudden tensile rupture of the bottom surface of the plate. Again, the effect of hierarchical topology on the performance of the cellular plates can be better evaluated by comparing the specific stiffness of the plates, as
Fig. 6. Load–displacement response of slender cellular biocomposite beams.
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Table 2 Specific bending stiffness of the cellular beams and plates ID
Density (g/cm3 )
Bending stiffness (N/mm)
Specific stiffness ((N/mm)/(g/cm3 ))
CB1 CB2 CB3 CB1-D CB2-D CB3-D CP1 CP2
0.73 0.87 0.75 0.65 0.62 0.66 0.70 0.90
101.9 135.1 117.0 1636.1 1605.7 1616.1 323.2 501.5
140.4 155.5 155.4 2515.1 2589.4 2446.4 461.1 558.5
Fig. 9. Effect of relative density on relative performance index.
Fig. 7. Load–displacement response of cellular biocomposite plates.
shown in Table 2. The hierarchical cellular plate showed an increase in specific stiffness of 13% over the periodic cellular plate. Fig. 8 shows the flexural test results of the deep cellular beams with green hemp/unsaturated polyester, while the specific stiffness of the deep beams are given in Table 2. All deep cellular beams behaved linear-elastically up to failure exhibiting only small deformations. Because of the dimensions of these cellular beams (small span to depth ratio), all the beams failed in a shear mode, with cracking from the bottom of the beam to the point of loading at approximately a 45◦ angle. Further investigation is thus needed to develop
increased reliability on the shear load carrying mechanisms and shear failure modes of cellular structural components. The relative performance index of a micro-structured beam using Eq. (9) divided by the performance of a beam of solid material (Eq. (1)) is shown as a continuous trace in Fig. 9. The figure shows the theoretical curve for the biocomposite material system and is accompanied by the relative performance indices calculated from the flexural tests on the cellular biocomposite beams and plates. Although the experimental data points are only in a limited region of the analytical curve, the data is consistent with the predicted performance and shows that it is possible to produce cellular biocomposite beams with higher performance indices than those made from solid biocomposite sections. It is clear that further improvements in the mechanical efficiency of hierarchical cellular beams and plates can be achieved at lower relative densities (Fig. 9) [10]. For example, at a relative density of 0.20, the performance index of a cellular beam to a fully dense beam is (E1/2 /ρ)/(Es1/2 /ρs ) = 2.2. The measured material performance index, shape factor, and performance index with shape factor for the manufactured biocomposite cellular beams, are shown in Table 3. The measured values show the effect of the hierarchical distribution on beam performance. Recall that cellular beam 1 (CB1) was manufactured with a periodic cellular architecture, while cellular beams 2 and 3 (CB2 and CB3) were manufactured with hierarchical cellular architectures (Fig. 3). In terms of flexural stiffness, the most efficient cellular beam tested was CB3, which had a material index of 2.67, shape factor of 2.66 and a material index with shape factor of 4.35. Conversely, cellular beam CB2 showed the lowest flexural performance. Table 3 Measured flexural performance of the biocomposite cellular beams
Fig. 8. Load–displacement response of deep cellular biocomposite beams.
ID
Material index, (E* )1/2 /ρ* ((GPa1/2 )/(g/cm3 ))
Shape factor, φ
Material and shape factor, φ (E* )1/2 /ρ* ((GPa1/2 )/(g/cm3 ))
CB1 CB2 CB3
2.64 2.33 2.67
2.62 2.04 2.66
4.28 3.33 4.35
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Fig. 10. Accuracy of the proposed analysis procedure for varying number of cellular layers.
Fig. 11. Flexural stiffness evaluation of deep cellular beams by multi-level layered analysis.
The results indicate that the material distribution in CB3 was more efficient than that in CB2. Thus, the use of hierarchical structures can improve the flexural performance of a cellular beam, provided the hierarchical architecture is correctly designed. Due to manufacturing limitations, the performance improvement seen was not large. Nonetheless, the test results verify the improvement in structural performance that can be achieved by using an efficient hierarchical material arrangement. More detailed and efficient hierarchical layouts should be possible in larger structures that allow more options in cell sizes.
plate with varying numbers of cellular lamina calculated using both the layered analysis method (EIanalysis ) and using conventional mechanics of materials (EImechanics ). The flexural stiffness determined from basic mechanics divided by the flexural stiffness obtained using the layered analysis method is plotted against the number of layers of cellular lamina in the cellular structure. The results show that for cellular structures with four or more layers of cellular material, the proposed analysis procedure agrees with the results from mechanics of materials, but that the procedure is inaccurate when less than four layers exist. Therefore, of the periodic and hierarchical cellular components manufactured in this study, only the deep cellular beams, which contain four or more distinct layers of cellular material, were analyzed using the layered analysis method. The predicted flexural stiffness of the deep cellular biocomposite beams are compared with the experimentally measured values in Fig. 11. The flexural stiffness predicted using the multi-scale layered analysis method agrees well with the measured flexural stiffness for CB1-D (8% error) and CB2-D (3% error) but has a 23% error with the prediction for CB3-D (Table 4). Results from the layered analysis are compared to the flexural stiffness predicted using conventional mechanics of materials and with the experimental values in Table 4. The flexural stiffness predicted using mechanics of materials also agrees well with the measured flexural stiffness, with errors ranging from 1 to 13%. The errors using mechanics of materials are similar to those of the layered analysis approach except for CB3-D. The performance and limitations
4.2. Analysis of hierarchical cellular beams In spite of the advantages provided by the multi-scale layered analysis method previously presented, the approach is limited by assumptions made in the cellular material equations. The equations for the elastic constants of a cellular material are based on analyzing a unit cell and assuming continuity conditions. This assumption is reasonable when there are many layers of cellular lamina in the structure. However, when the component has only a small number of cellular lamina the averaged properties do not account for the actual distribution of material through the section. This leads to errors in the calculated lamina stiffness, and hence the flexural stiffness of the periodic or hierarchical cellular structure. This concept is illustrated in Fig. 10, which shows the flexural stiffness of a periodic two-dimensional cellular
Table 4 Flexural stiffness by layered analysis method and conventional mechanics Beam ID
CB1-D CB2-D CB3-D
EI experimental (GPa mm4 )
EI by analysis
646,597 634,583 638,683
701,417 652,755 788,397
Errors with respect to experimental value.
Value
(GPa mm4 )
EI by mechanics Error (%) 8 3 23
Value (GPa mm4 ) 598,867 550,308 647,086
Error (%) 7 13 1
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Fig. 12. Flexural stiffness and strength of cellular beams and plates.
of the layered analysis method are discussed in the following section. The analytically predicted flexural stiffness and strength of the slender cellular beams and cellular plates using the mechanics of materials approach are compared with the experimental values in Fig. 12. The analytical predictions for the flexural stiffness agree well with the measured results, however, predictions for strength overestimate the measured values. One source of error is the calculation of the moment of inertia, which is determined assuming that the original cellular architecture remained centered with any excess material from manufacturing inconsistencies evenly distributed at the top and bottom of the section. Additional sources of error are the discrepancies from the analytical models used to predict the properties of the biocomposite material systems.
5. Discussion 5.1. Performance of analytical methods The predicted values using the multi-scale layered analysis method compared well with the experimental results of two of the deep cellular biocomposite beams but showed discrepancies for the beam with highest hierarchy. Two reasons for the disagreement are the approximation of the circular cells as a hexagonal honeycomb and the low repeatability of each of the sizes present in the hierarchical beam. The first source of error can be improved by developing cellular solids models specifically for the used cell geometry. The second cause for discrepancy is attributed to the low number of cellular layers of each characteristic size in the hierarchical beam. While it was previously shown that the modeling assumption was adequate for cellular structures featuring four or more layers of periodic cells (Fig. 10), hierarchical cellular components with cellular architectures of different sizes will face the same limitations in the cellular solid equations for each scale. Thus, the predicted response will improve in accuracy as the number of cellular layers for each scale increases. Additionally, the modeling approach does not take into ac-
Fig. 13. Performance biocomposite cellular beams with beams of conventional construction materials.
count partial or multi-sized cells within the same layer. All of these conditions are true for CB3-D, which contained multisized cellular material layers with varying thickness and thus showed the largest error compared to the experimental result. Uncertainty in the analytical models used to predict the biocomposite elastic and failure material properties will cause additional errors. 5.2. Comparisons to conventional material sections Results from the flexural tests of the periodic and hierarchical biocomposite deep cellular beams were compared with beams of equal size (25 mm × 51 mm × 305 mm) made of conventional construction materials (Fig. 13). The conventional materials used for comparison were wood (Douglas fir, E = 10.3 GPa, ρ = 0.56 g/cm3 ) and reinforced concrete (E = 24.8 GPa, ρ = 2.4 g/cm3 ). The reinforced concrete beam (3% longitudinal steel reinforcement ratio) was analyzed both cracked and uncracked. However, reinforced concrete beams typically crack under service loads, reducing the stiffness of the component. The results in Fig. 13 show that the bio-beams compare very favorably with the beams from conventional construction materials. The specific stiffness of a hierarchical biobeam outperforms both the cracked and uncracked concrete beams. Their performance, however, falls short from that of the wood beam. The specific stiffness of the wood beam is very large because of the low density and the moderate elastic modulus of Douglas fir. The cellular bio-beam with a periodic cell arrangement shows a lower specific stiffness than the hierarchical bio-beam but still outperforms the stiffness of the cracked concrete member.
6. Conclusions Results from this study show that hierarchical cellular designs can improve the performance of biocomposite beams and plates and allow them to compete with conventional
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materials. Comparison of experimental data was agreed well with performance indices from mechanics models that predict an increase in performance with decreased relative density of the cellular structure. In addition, it was shown that further improvements in the mechanical efficiency of cellular biocomposite beams and plates can be achieved through optimized microstructures, or hierarchical topological material arrangements. A multi-scale procedure for the integrated material/structural analysis of hierarchical cellular structures was presented and preliminarily evaluated for stiffness calculations using experimental data. While further work is required to fully develop and validate the analytical procedure, particularly with regards to multi-size layers and failure criteria, the approach is considered suitable for the analysis of components with simple hierarchy and can be an efficient tool for performing parametric and optimization studies in the design of hierarchical cellular biocomposite components for load-bearing applications.
Acknowledgements The authors are grateful to National Science Foundation Partnership for Advancing Technologies in Housing (NSFPATH) 2001 Award No. 0122108 for partial financial support. The collaborating efforts and many fruitful discussions with our industrial partners from Flaxcraft Inc., Cresskill, NJ, HempLine, Ontario, Canada and Kemlite Inc., Joliet, IL are gratefully acknowledged. Authors R. Burgue˜no and M.J.
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Quagliata are thankful for a 2002 Michigan State University’s Research Excellence Funds award. References [1] A.K. Mohanty, M. Misra, G. Hinrichsen, Macromol. Mater. Eng. 276/277 (2000) 1–24. [2] A.K. Mohanty, M. Misra, L.T. Drzal, Compos. Interfaces 8 (2001) 313–343. [3] S.L. Gunderson, K.E.G. Thorp, Proceedings of the ASC 8th Annual Technical Conference, Cleveland, OH, 1993, pp. 431–440. [4] F. Nogata, Composites and Functionally Graded Materials, ASME, MD, 80 Dallas, TX, 1997, pp. 11–18. [5] R. Lakes, Nature 361 (1998) 511–515. [6] S. Evans, J.W. Hutchinson, N.A. Fleck, M.F. Ashby, H.N.G. Wadley, Proceedings of the 7th International Conference on Aluminum Alloys, Charlottesville, VA, 2002. [7] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, first ed., Pergamon Press, Oxford, 1988. [8] M.F. Ashby, Materials Selection in Mechanical Design, Butterworth Heinemann, Oxford, 1999. [9] M.F. Ashby, Acta Metall. Mater. 39 (1999) 1025–1039. [10] J.S. Huang, L.J. Gibson, Acta Metall. Mater. 43 (1995) 1643–1650. [11] R. Burgue˜no, M.J. Quagliata, A.K. Mohanty, G. Mehta, L.T. Drzal, M. Misra, Composites—Part A 35 (2004) 645–656. [12] M.J. Quagliata, Development and Characterization of Biocomposite Cellular Beams and Plates for Load-Bearing Components, Masters Thesis, Michigan State University, East Lansing, MI, August 2003. [13] J.C. Halpin, N.J. Pagano, J. Compos. Mater. 3 (1969) 720–724. [14] J.C. Halpin, S.W. Tsai, Effects of Environmental Factors on Composite Materials, Technical Report AFML-TR-67-423, June 1969. [15] R.M. Jones, Mechanics of Composite Materials, second ed., Taylor and Francis, Philadelphia, 1999.
Hierarchical cellular designs for load-bearing biocomposite beams and plates Rigoberto Burgue˜noa,∗ , Mario J. Quagliatab , Amar K. Mohantyc , Geeta Mehtad , Lawrence T. Drzale , Manjusri Misraf a
Department of Civil and Env. Eng., Michigan State University, East Lansing, MI 48824-1226, USA Department of Civil and Env. Eng., Michigan State University, East Lansing, MI 48824-1226, USA c School of Packaging, Michigan State University, East Lansing, MI 48824-1223, USA d Department of Chem. Eng. and Mat. Sc., Michigan State University, East Lansing, MI 48824-1226, USA Department of Chem. Eng. and Mat. Sc., Comp. Mat. and Structures Center, Michigan State University, East Lansing, MI 48824-1226, USA f Comp. Mat. and Structures Center, Michigan State University, East Lansing, MI 48824-1226, USA b
e
Received 19 December 2003; received in revised form 3 August 2004
Abstract Scrutiny into the composition of natural, or biological materials convincingly reveals that high material and structural efficiency can be attained, even with moderate-quality constituents, by hierarchical topologies, i.e., successively organized material levels or layers. The present study demonstrates that biologically inspired hierarchical designs can help improve the moderate properties of natural fiber polymer composites or biocomposites and allow them to compete with conventional materials for load-bearing applications. An overview of the mechanics concepts that allow hierarchical designs to achieve higher performance is presented, followed by observation and results from flexural tests on periodic and hierarchical cellular beams and plates made from industrial hemp fibers and unsaturated polyester resin biocomposites. The experimental data is shown to agree well with performance indices predicted by mechanics models. A procedure for the multi-scale integrated material/structural analysis of hierarchical cellular biocomposite components is presented and its advantages and limitations are discussed. © 2004 Elsevier B.V. All rights reserved. Keywords: Biocomposites; Cellular beams; Cellular plates; Laminate mechanics; Numerical analysis; Mechanical testing
1. Introduction Developments over the past decade in composite materials technology have shown that natural-fiber-reinforced polymer composites, or biocomposites, can be a technical, economical and environmentally conscious alternative to E-glass fiber composites without sacrificing performance [1,2]. In spite of this progress and the environmental appeal of natural-based composites, their lower strength and stiffness compared to conventional structural materials has limited their use to nonprimary or non-load-bearing applications. Increased understanding of natural or biological materials shows that nature has developed ways of achiev∗
Corresponding author. Tel.: +1 517 353 1743; fax: +1 517 353 1743. E-mail address: [email protected] (R. Burgue˜no).
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.08.034
ing high structural efficiency from a rather limited set of non-sophisticated constituents. One of the primary ways in which nature achieves high structural performance with nonexotic materials is through hybrid material combinations assembled in optimized hierarchical strategies [3,4]. Among nature’s most common efficient structures are hierarchical cellular sandwich structures, which consist of a complex arrangement of cells of different sizes arranged across the section such that cells are themselves made from cells (i.e., hierarchical arrangement), leading to dense regions, or skins, integrally connected to regions with lower density, or core [5]. A well-recognized natural material displaying this hierarchical design is human bone. Structural behavior is then dictated by the relative dimensions and topology design of the dense and porous regions, with the material and microstructural hierarchy playing a major role
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in maximizing the efficiency of the resulting material and structure. Hierarchical cellular designs are complex microstructures achieved by placing material where most needed, i.e., in areas of high stress. The process of arranging the microstructure (cellular material) can lead to a hierarchy that maximizes the efficiency of the resulting material or load-bearing component. The above-mentioned concepts motivated the present work to improve the performance of biocomposite materials for use in load-bearing beam and panel components through bio-inspired cellular hierarchical designs. Experimental and analytical feasibility studies on industrial hemp/unsaturatedpolyester hierarchical cellular beams and plates were thus conducted and used to evaluate their potential as load-bearing components. Experimental results verify the structural efficiency that is gained through hierarchical designs, and an analysis approach that permits analyzing the multi-level material and topological hierarchy of these structures is presented and preliminarily validated.
2. Analysis
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performance index is [8]: 2/3
MBf =
σf , ρ
(2)
where the superscript f means failure and σ f is the failure strength of the material. Thus, the best material for the given design requirements is that with the largest value of the performance index. Performance indices work well for comparing materials but they ignore the shape of the component. Thus, for materials with different cross-sectional shapes other factors must be used to quantify efficiency. Sections of shaped material carry load more efficiently than solid sections, where efficiency means that for a given structural demand the section uses as little material as possible, and is therefore as light as possible. To measure the shape and efficiency of a section, for a given mode of structural response, the concept of shape factors has been proposed by Ashby [9]. A shape factor is a dimensionless number that characterizes the efficiency of the shape of a section, regardless of scale, for a given type of structural response. When the design is based on flexural stiffness, the shape factor for elastic bending is [9]:
2.1. Flexural performance The efficiency, or performance, of cellular sandwich structures depends on individual and collective behavior of the dense skins, the core, and their connections [6,7]. The performance for a load-bearing component can be said to depend on three factors [8,9]: (i) the mode of loading, i.e., tension, bending and twisting, (ii) the properties of the material, and (iii) the shape of the section. While performance can be defined in many forms, it is taken here to mean maximum flexural stiffness or flexural strength per weight. Ashby [9] has shown that the performance of materials, shapes and microstructures can be evaluated through different indices. An overview of how these performance indices explain the efficiency of hierarchical materials and structures follow. Material properties can be combined to form performance indices [8], which are groupings of material properties that, when maximized, maximize some aspect of performance. The indices are derived from the design requirements for a component based on the function, objective and constraint. For a beam where the objective is minimum weight for a given flexural stiffness (light stiff beam), the material index can be shown to be [8]: MBe =
E1/2 , ρ
(1)
where the superscript e means elastic and the subscript B means bending, E is the elastic modulus of the material and ρ is the density. For a beam where the objective is minimum weight for a given flexural strength (strong light beam), the
φBe =
4πI , A2
(3)
where the superscript e means elastic, the subscript B means bending, I is the moment of inertia and A is the area of the section. For strength-governed designs (i.e., buckling, fracture, or plastic yielding) the shape factor is [9]: φBf =
√ 4 πS , A3/2
(4)
where the superscript f means failure and S is the section modulus, or I/ym , with ym being the distance from the section neutral axis to the extreme fiber. Shape factors are dimensionless and depend only on the cross-section shape. That is, large and small beams have the same shape factor value if their sections have the same shape and are proportional. Solid symmetric sections (circles, squares, etc.) will have shape factors close to 1, while more efficient shapes will have shape factors larger than 1. Shape factors can be combined with material indices to formulate performance indices that determine the performance-maximizing combination of material and shape. Ashby [9] derived indices for comparing the performance of beams of different material and shape. The best materialshape combination for a light stiff beam is that with the greatest value of the index [9]: M1 =
(EφBe )1/2 , ρ
(5)
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while the best material-shape combination for a light strong beam is that with the greatest value of the index [9]: 2/3
M2 =
(σf φBf ) ρ
.
(6)
For constant shapes the performance indices will reduce to the corresponding material index. The efficiency of a macroscopic shape can be further improved by introducing shape at a small scale, microscopic shape. For example, a rectangular section made of a cellular material (microscopic shape) will have higher efficiency than a solid rectangular section (macroscopic shape). The additional efficiency of the microstructural shape is characterized by micro-structural shape factors [9]. For flexure response, the micro-structural shape factor for elastic bending can be shown to be [9]: ψBe
ρs = ∗, ρ
(7)
while the micro-structural shape factor for flexural strength is given by [9]: 1/2 ρs ψBf = . (8) ρ∗ In other words, the micro-structural shape factor is proportional to the inverse of the relative density. The relative density of a cellular structure is defined as the density of the cellular material (ρ* ), divided by the density of the solid from which the cells are made (ρs ). Relative density is the most important property affecting the mechanical properties of a cellular material [7]. Note that, in the limit, for a solid microstructure (ρ* = ρs ) both ψBe and ψBf take the value of 1, as they should. Both stiffness and strength micro-structural shape factors show that performance is increased by introducing microstructure to a solid shape, resulting in a lighter more efficient structure. This idea of microstructuring can be extended further by introducing cell walls that are themselves microstructured, thus creating a structural hierarchy. Such efficient structural hierarchy is commonly found in natural materials [3,5,7]. However, there are limits on the level of hierarchy that can be achieved in man-made structures due to manufacturing difficulties and the accompanying manufacturing costs. Nevertheless, the use of hierarchical structures can further improve the performance of a structural member resulting in light efficient components. The positive effect of microstructure can also be shown by integrating the microscopic shape factor into the material performance indices. Considering a cellular structure and using the micro-structural shape factor, the performance index for maximum stiffness per weight for a cellular beam can be derived from Eq. (1) to give [10]: 1/2 E1/2 (E∗ )1/2 ρs 1/2 Es e MB = = = , (9) ρ ρ∗ ρs ρ∗
Fig. 1. Potential design of a two-dimensional hierarchical cellular panel.
where the modulus of the cellular material is equal to the modulus of the solid material times the relative density [7]. Eq. (9) indicates that the performance index of a cellular beam is higher than that of a solid one by a factor equal to the square root of the inverse of the relative density. 2.2. Analysis of hierarchical cellular sandwich structures The above concepts show that the efficiency of a structural component, such as a beam or plate, for a given stiffness can be enhanced if composed of a material that features cells or holes strategically distributed along the member crosssection. An example of a two-dimensional hierarchical cellular sandwich panel is shown in Fig. 1. Beams and plates structures with repeating layers of cellular material can be treated as laminated structures where each layer can be defined by a characteristic cell architecture [10]. This idealization can be applied to cellular beams and plates with periodic cells that are oriented in different directions about the longitudinal axis of the element and/or hierarchical cellular structures that feature different levels of cellular architectures along the member depth (Fig. 2). At the material level, Burgue˜no et al. have shown that the properties of biocomposite materials can be adequately predicted using existing micromechanics models for randomly oriented short fiber composites. Based on comparisons with experimental data [12], the quasi-isotropic laminate approximation by Halpin and Pagano [13] together with the Halpin–Tsai [14] equations was found effective in determining the elastic constants of the biocomposite materials. Mechanics of cellular solids are used at the microstructural level to model the properties of the cellular material layers within the hierarchical structure. While there are several approaches to the mechanics of cellular solids, Gibson and Ashby [7] have shown that the stiffness and strength characteristics of honeycombs and foams, i.e., two- and threedimensional cellular solids, can be effectively described using structural mechanics concepts. For two-dimensional cellular structures, as those considered in the experimental part of this study, properties were determined by analyzing the response of a unit hexagonal cell. The mechanical response of the cellular solid is thus, obtained by modeling the cell as an interconnected array of one-dimensional struts. The material elastic constants are then derived by evaluating the force–displacement response of the unit cell under different
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Fig. 2. Overview of the analysis process for hierarchical cellular biocomposite beams and plates.
loading conditions and up to different failure modes depending on the constituent material [7]. Assuming a plane stress condition and that the material response is at most orthotropic, the reduced lamina stiffness for a general single cellular material layer (lamina) is then obtained using conventional mechanics of composite materials [15]. The structural properties of the hierarchical cellular beam or plate can then be found by adding the individual stiffnesses of the different characteristic cellular material layers over the depth of the structure. Each layer may have different properties due to their constituent material, cellular architecture or cell orientation. Addition of all the layer stiffnesses over the section depth through classical lamination theory leads to the section properties of the laminated plate, or [A-B-D] matrix [15]. The section properties for hierarchical cellular beams and plates can thus be obtained by considering the different scales of the material continuum, from the microstructure of the randomly oriented short fiber reinforced biocomposite, to the cellular material microstructure and finally to the arrangement in a hierarchical structure, as shown in Fig. 2. The generality of the method allows determination of section properties along different structural directions. In addition, automation of the process allows the method to be used for parametric and optimization studies that can provide further insight into efficient hierarchical designs. While the analysis concept has thus far only been implemented to determine linear elastic stiffness properties, the same multi-scale approach can be used to determine non-linear response and failure conditions by incorporating appropriate models at each analysis level. Tooling limitations in the experimental program did not allow the manufacturing of beams and plates with a high order of multi-layer cellular topology. Due to the continuity assumptions inherent in the cellular solids models, the analysis approach described above is limited to cellular structures featuring multiple cellular layers. This limitation is discussed later in the paper. The stiffness of cellular components that did
not meet the requirements for the layered analysis method, were determined analytically using a mechanics of materials approach by separately calculating the modulus of elasticity of the material (through the micromechanics models discussed above) and the moment of inertia of the crosssection, and combining the two terms. While this approach allows a simple evaluation of the longitudinal member stiffness, other section properties are not as straight forward to obtain. For example, calculation of the transverse member stiffness requires consideration of the bending of the individual cell walls within the structure, which may be difficult to evaluate analytically. The failure load for all cellular beam and plates was obtained analytically using conventional mechanics of materials by relating the failure load for the specific test set-up to the flexural stiffness. The failure load of the cellular beams under four-point bending was then calculated using: Pf = εf
2EI , aym
(10)
where εf is the failure strain, a is the shear span and ym is the distance from the neutral axis of the beam to the extreme fiber. The failure load of the cellular beams and plates in three-point bending was calculated using: Pf = εf
4EI Lym
(11)
where L is the total span length. The flexural stiffness was determined analytically as previously described. The failure strain was obtained from the measured values in a material characterization study [11,12].
3. Experimental methods Laboratory-scale cellular biocomposite beams and plates were manufactured and tested to verify the improvement in
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Table 1 Average properties of hemp fiber and unsaturated polyester composites Material
Fiber wt. (%)
Green hemp/UPE Raw hemp/UPE
13 20
Fiber (L/d) 60 100
MOE (GPa)
Tensile str. (MPa)
3.81 6.15
14.47 19.49
flexural performance achieved through the use of hierarchical cellular designs and to validate the developed analysis approach. 3.1. Materials Industrial hemp fiber is the stiffest, cheapest and most readily available natural fiber type for use in load-bearing structural applications, while unsaturated polyester, which offers a good balance of cost and performance, was chosen as a proper match for the low-cost natural fibers. Industrial raw hemp (retted hemp, 15–25 mm average length, HempLine, Ontario, Canada) and green industrial hemp (10–15 mm average length, Flaxcraft Inc., Cresskill, New Jersey) were used as natural fiber reinforcement for the biocomposite material. The resin system used was ortho unsaturated polyester (UPE) (Kemlite Co. Inc., Joliet, IL) with methyl ethyl ketone peroxide (MEKP, Sigma–Aldrich) catalyst (1% by weight of resin), and cobalt naphthenate (CoNap, Sigma–Aldrich) promoter (0.03% by weight of resin). Properties of the biocomposite material systems are shown in Table 1. 3.2. Test units The experimental characterization study considered three slender and three deep hierarchical cellular beams, and two cellular plates. The cellular designs featured periodic and hierarchical cell arrangements. Dimensions, cellular architectures and relative densities for the cellular beams and plates are given in Figs. 3 and 4, respectively. Dimensions of the beam and plate components were selected to minimize material use and to parallel a study on the performance of cellular components by Huang and Gibson [10]. The cellular topologies in the beams and plates were determined from the base dimensions of periodic cellular beams and plates and on the available tooling materials to create the different cell sizes. Because tooling used to create the cells (rubber tubing) was only available in standard sizes, the extent of the hierarchical architectures that could be achieved was limited. Ongoing studies are focusing on developing and studying components with higher levels of cellular hierarchy through optimized topologies.
Fig. 3. Cross-section geometry of cellular beam test units.
beam mold was 25.4 mm × 50.8 mm × 304.8 mm and the cellular plate mold was 101.6 mm × 304.8 mm × 12.7 mm. The removable faceplates allowed different cellular configurations to be implemented with the same mold. The molds were lined with teflon paper to ease sample removal and the cells were created by placing rubber tubing through the faceplates. Steel rods were used to stiffen the rubber tubes during manufacturing. The biocomposite material was placed by hand in alternating layers of material and rods (cells). All cellular beams and plates were oven cured using a cycle of 100 ◦ C for 2 h followed by 150 ◦ C for 2 h. A detailed description of the manufacturing process is given in [11,12].
3.3. Manufacturing 3.4. Testing The cellular beams and plates were manufactured using molds with removable faceplates. The slender cellular beam mold was 25.4 mm × 25.4 mm × 508 mm, the deep cellular
Flexural characterization of the periodic and hierarchical cellular beams and plates was done through flexural test
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point bending with a total span of 267 mm and shear spans of 133 mm. All test units were loading monotonically up to failure in displacement control at a rate of 0.01 mm/s. Deflection of the beams and plates was measured at mid-span with an externally mounted extensometer and the loading frame LVDT. The applied load was measured using the internal load cell of the testing frame.
4. Results 4.1. Performance of hierarchical designs
Fig. 4. Cross-section geometry of cellular plate test units.
set-ups mounted on a loading frame (Fig. 5). The slender cellular beams (L/D = 18.0) were testing in four-point bending with a total span of 457 mm, concentrated loads at 102 mm apart and a shear span of 178 mm. The deep cellular beams (L/D = 5.25) and cellular plates were both tested in three-
Fig. 5. Pictures of flexural test set-ups.
The load–displacement response at mid-span of the slender cellular beams with industrial green hemp/unsaturated polyester are shown in Fig. 6. All cellular beams behaved linear-elastically up to failure. The failure mode in all cases was a sudden tensile rupture of the bottom surface of the beam. The effect of periodic and hierarchical cellular topologies on the performance of the beam units is evident in the force–displacement response (Fig. 6). Notice that while CB1 and CB3 have the same relative density (i.e., use the same amount of material), efficient material arrangement of CB3 in a hierarchical architecture leads to a 15% stiffer element. It is also obvious that CB2, with a higher relative density, i.e., more solid material, had a higher flexural stiffness. The effect of hierarchical structure on the beams flexural performance is, however, better seen by comparing their specific stiffness, as given in Table 2. Thus, the specific stiffness of the two hierarchical cellular beams (CB2 and CB3) is approximately 12% higher than the beam with periodic cells (CB1). The hierarchical arrangement of CB3 thus leads to both absolute and specific flexural performance improvement. Test results of the cellular plates with raw hemp/ unsaturated polyester are shown in Fig. 7. All cellular plates behaved linear-elastically up to failure. The failure mode of the cellular plates was a sudden tensile rupture of the bottom surface of the plate. Again, the effect of hierarchical topology on the performance of the cellular plates can be better evaluated by comparing the specific stiffness of the plates, as
Fig. 6. Load–displacement response of slender cellular biocomposite beams.
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Table 2 Specific bending stiffness of the cellular beams and plates ID
Density (g/cm3 )
Bending stiffness (N/mm)
Specific stiffness ((N/mm)/(g/cm3 ))
CB1 CB2 CB3 CB1-D CB2-D CB3-D CP1 CP2
0.73 0.87 0.75 0.65 0.62 0.66 0.70 0.90
101.9 135.1 117.0 1636.1 1605.7 1616.1 323.2 501.5
140.4 155.5 155.4 2515.1 2589.4 2446.4 461.1 558.5
Fig. 9. Effect of relative density on relative performance index.
Fig. 7. Load–displacement response of cellular biocomposite plates.
shown in Table 2. The hierarchical cellular plate showed an increase in specific stiffness of 13% over the periodic cellular plate. Fig. 8 shows the flexural test results of the deep cellular beams with green hemp/unsaturated polyester, while the specific stiffness of the deep beams are given in Table 2. All deep cellular beams behaved linear-elastically up to failure exhibiting only small deformations. Because of the dimensions of these cellular beams (small span to depth ratio), all the beams failed in a shear mode, with cracking from the bottom of the beam to the point of loading at approximately a 45◦ angle. Further investigation is thus needed to develop
increased reliability on the shear load carrying mechanisms and shear failure modes of cellular structural components. The relative performance index of a micro-structured beam using Eq. (9) divided by the performance of a beam of solid material (Eq. (1)) is shown as a continuous trace in Fig. 9. The figure shows the theoretical curve for the biocomposite material system and is accompanied by the relative performance indices calculated from the flexural tests on the cellular biocomposite beams and plates. Although the experimental data points are only in a limited region of the analytical curve, the data is consistent with the predicted performance and shows that it is possible to produce cellular biocomposite beams with higher performance indices than those made from solid biocomposite sections. It is clear that further improvements in the mechanical efficiency of hierarchical cellular beams and plates can be achieved at lower relative densities (Fig. 9) [10]. For example, at a relative density of 0.20, the performance index of a cellular beam to a fully dense beam is (E1/2 /ρ)/(Es1/2 /ρs ) = 2.2. The measured material performance index, shape factor, and performance index with shape factor for the manufactured biocomposite cellular beams, are shown in Table 3. The measured values show the effect of the hierarchical distribution on beam performance. Recall that cellular beam 1 (CB1) was manufactured with a periodic cellular architecture, while cellular beams 2 and 3 (CB2 and CB3) were manufactured with hierarchical cellular architectures (Fig. 3). In terms of flexural stiffness, the most efficient cellular beam tested was CB3, which had a material index of 2.67, shape factor of 2.66 and a material index with shape factor of 4.35. Conversely, cellular beam CB2 showed the lowest flexural performance. Table 3 Measured flexural performance of the biocomposite cellular beams
Fig. 8. Load–displacement response of deep cellular biocomposite beams.
ID
Material index, (E* )1/2 /ρ* ((GPa1/2 )/(g/cm3 ))
Shape factor, φ
Material and shape factor, φ (E* )1/2 /ρ* ((GPa1/2 )/(g/cm3 ))
CB1 CB2 CB3
2.64 2.33 2.67
2.62 2.04 2.66
4.28 3.33 4.35
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Fig. 10. Accuracy of the proposed analysis procedure for varying number of cellular layers.
Fig. 11. Flexural stiffness evaluation of deep cellular beams by multi-level layered analysis.
The results indicate that the material distribution in CB3 was more efficient than that in CB2. Thus, the use of hierarchical structures can improve the flexural performance of a cellular beam, provided the hierarchical architecture is correctly designed. Due to manufacturing limitations, the performance improvement seen was not large. Nonetheless, the test results verify the improvement in structural performance that can be achieved by using an efficient hierarchical material arrangement. More detailed and efficient hierarchical layouts should be possible in larger structures that allow more options in cell sizes.
plate with varying numbers of cellular lamina calculated using both the layered analysis method (EIanalysis ) and using conventional mechanics of materials (EImechanics ). The flexural stiffness determined from basic mechanics divided by the flexural stiffness obtained using the layered analysis method is plotted against the number of layers of cellular lamina in the cellular structure. The results show that for cellular structures with four or more layers of cellular material, the proposed analysis procedure agrees with the results from mechanics of materials, but that the procedure is inaccurate when less than four layers exist. Therefore, of the periodic and hierarchical cellular components manufactured in this study, only the deep cellular beams, which contain four or more distinct layers of cellular material, were analyzed using the layered analysis method. The predicted flexural stiffness of the deep cellular biocomposite beams are compared with the experimentally measured values in Fig. 11. The flexural stiffness predicted using the multi-scale layered analysis method agrees well with the measured flexural stiffness for CB1-D (8% error) and CB2-D (3% error) but has a 23% error with the prediction for CB3-D (Table 4). Results from the layered analysis are compared to the flexural stiffness predicted using conventional mechanics of materials and with the experimental values in Table 4. The flexural stiffness predicted using mechanics of materials also agrees well with the measured flexural stiffness, with errors ranging from 1 to 13%. The errors using mechanics of materials are similar to those of the layered analysis approach except for CB3-D. The performance and limitations
4.2. Analysis of hierarchical cellular beams In spite of the advantages provided by the multi-scale layered analysis method previously presented, the approach is limited by assumptions made in the cellular material equations. The equations for the elastic constants of a cellular material are based on analyzing a unit cell and assuming continuity conditions. This assumption is reasonable when there are many layers of cellular lamina in the structure. However, when the component has only a small number of cellular lamina the averaged properties do not account for the actual distribution of material through the section. This leads to errors in the calculated lamina stiffness, and hence the flexural stiffness of the periodic or hierarchical cellular structure. This concept is illustrated in Fig. 10, which shows the flexural stiffness of a periodic two-dimensional cellular
Table 4 Flexural stiffness by layered analysis method and conventional mechanics Beam ID
CB1-D CB2-D CB3-D
EI experimental (GPa mm4 )
EI by analysis
646,597 634,583 638,683
701,417 652,755 788,397
Errors with respect to experimental value.
Value
(GPa mm4 )
EI by mechanics Error (%) 8 3 23
Value (GPa mm4 ) 598,867 550,308 647,086
Error (%) 7 13 1
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Fig. 12. Flexural stiffness and strength of cellular beams and plates.
of the layered analysis method are discussed in the following section. The analytically predicted flexural stiffness and strength of the slender cellular beams and cellular plates using the mechanics of materials approach are compared with the experimental values in Fig. 12. The analytical predictions for the flexural stiffness agree well with the measured results, however, predictions for strength overestimate the measured values. One source of error is the calculation of the moment of inertia, which is determined assuming that the original cellular architecture remained centered with any excess material from manufacturing inconsistencies evenly distributed at the top and bottom of the section. Additional sources of error are the discrepancies from the analytical models used to predict the properties of the biocomposite material systems.
5. Discussion 5.1. Performance of analytical methods The predicted values using the multi-scale layered analysis method compared well with the experimental results of two of the deep cellular biocomposite beams but showed discrepancies for the beam with highest hierarchy. Two reasons for the disagreement are the approximation of the circular cells as a hexagonal honeycomb and the low repeatability of each of the sizes present in the hierarchical beam. The first source of error can be improved by developing cellular solids models specifically for the used cell geometry. The second cause for discrepancy is attributed to the low number of cellular layers of each characteristic size in the hierarchical beam. While it was previously shown that the modeling assumption was adequate for cellular structures featuring four or more layers of periodic cells (Fig. 10), hierarchical cellular components with cellular architectures of different sizes will face the same limitations in the cellular solid equations for each scale. Thus, the predicted response will improve in accuracy as the number of cellular layers for each scale increases. Additionally, the modeling approach does not take into ac-
Fig. 13. Performance biocomposite cellular beams with beams of conventional construction materials.
count partial or multi-sized cells within the same layer. All of these conditions are true for CB3-D, which contained multisized cellular material layers with varying thickness and thus showed the largest error compared to the experimental result. Uncertainty in the analytical models used to predict the biocomposite elastic and failure material properties will cause additional errors. 5.2. Comparisons to conventional material sections Results from the flexural tests of the periodic and hierarchical biocomposite deep cellular beams were compared with beams of equal size (25 mm × 51 mm × 305 mm) made of conventional construction materials (Fig. 13). The conventional materials used for comparison were wood (Douglas fir, E = 10.3 GPa, ρ = 0.56 g/cm3 ) and reinforced concrete (E = 24.8 GPa, ρ = 2.4 g/cm3 ). The reinforced concrete beam (3% longitudinal steel reinforcement ratio) was analyzed both cracked and uncracked. However, reinforced concrete beams typically crack under service loads, reducing the stiffness of the component. The results in Fig. 13 show that the bio-beams compare very favorably with the beams from conventional construction materials. The specific stiffness of a hierarchical biobeam outperforms both the cracked and uncracked concrete beams. Their performance, however, falls short from that of the wood beam. The specific stiffness of the wood beam is very large because of the low density and the moderate elastic modulus of Douglas fir. The cellular bio-beam with a periodic cell arrangement shows a lower specific stiffness than the hierarchical bio-beam but still outperforms the stiffness of the cracked concrete member.
6. Conclusions Results from this study show that hierarchical cellular designs can improve the performance of biocomposite beams and plates and allow them to compete with conventional
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materials. Comparison of experimental data was agreed well with performance indices from mechanics models that predict an increase in performance with decreased relative density of the cellular structure. In addition, it was shown that further improvements in the mechanical efficiency of cellular biocomposite beams and plates can be achieved through optimized microstructures, or hierarchical topological material arrangements. A multi-scale procedure for the integrated material/structural analysis of hierarchical cellular structures was presented and preliminarily evaluated for stiffness calculations using experimental data. While further work is required to fully develop and validate the analytical procedure, particularly with regards to multi-size layers and failure criteria, the approach is considered suitable for the analysis of components with simple hierarchy and can be an efficient tool for performing parametric and optimization studies in the design of hierarchical cellular biocomposite components for load-bearing applications.
Acknowledgements The authors are grateful to National Science Foundation Partnership for Advancing Technologies in Housing (NSFPATH) 2001 Award No. 0122108 for partial financial support. The collaborating efforts and many fruitful discussions with our industrial partners from Flaxcraft Inc., Cresskill, NJ, HempLine, Ontario, Canada and Kemlite Inc., Joliet, IL are gratefully acknowledged. Authors R. Burgue˜no and M.J.
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