Materials and Design 131 (2017) 343–357
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Designing composites with negative linear compressibility Arash Ghaedizadeh a,b, Jianhu Shen a,b, Xin Ren a,c, Yi Min Xie a,b,d,⁎ a
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia Wound Management Innovation Cooperative Research Centre, 25 Donkin Street, West End, Queensland 4101, Australia c Key Laboratory of Traffic Safety on Track, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China d XIE Archi-Structure Design (Shanghai) Co., Ltd., Shanghai 200092, China b
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Two approaches have been developed to design new composite structures with negative linear compressibility. • The effectiveness of the design approaches has been proved by hydrostatic compression experiments. • The designed composites process negative linear compressibility at large deformation. • The negative linear compressibility is dependent on the foam property, the reinforcement pattern, and the stiffness ratio.
a r t i c l e
i n f o
Article history: Received 10 April 2017 Received in revised form 11 June 2017 Accepted 12 June 2017 Available online 15 June 2017 Keywords: Negative linear compressibility Composite material Composite structure Mechanical metamaterial
a b s t r a c t The phenomenon of negative linear compressibility has attracted much interest because of its unusual deformation features with many potential applications. However, the design and fabrication of materials and structures with negative linear compressibility are limited. In this paper, we proposed two approaches to designing and fabricating new composite structures with negative linear compressibility. The effectiveness of the proposed design approaches was validated experimentally by applying uniformly distributed pressure to all surfaces of bulk specimens. The deformation features, strain history, and the effective area reduction of the specimens were analyzed from the experimental data. The results clearly demonstrated the feasibility of the proposed designing and manufacturing approaches for realizing composites with negative linear compressibility. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Materials and structures with uncommon behaviours such as negative Poisson' ratio, negative stiffness, negative compressibility and negative thermal expansion have attracted intense research interest due to a variety of potential applications [1–5]. Negative compressibility is the least studied among them. Compressibility is a measure of the relative
volume change of a fluid or solid as a response to a pressure change. Volumetric compressibility determines the material volume change under applied uniform pressure, whilst linear compressibility defines a change along a specific axis of the material. Conventionally, materials and structures contract three-dimensionally under a positive surrounded uniform pressure. This behaviour is defined as the positive linear compressibility along three main axes. However, some materials and
⁎ Corresponding author at: Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia. E-mail addresses:
[email protected] (A. Ghaedizadeh),
[email protected] (J. Shen),
[email protected] (X. Ren),
[email protected] (Y.M. Xie).
http://dx.doi.org/10.1016/j.matdes.2017.06.026 0264-1275/© 2017 Elsevier Ltd. All rights reserved.
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structures exhibit an unusual behaviour under the application of pressure, resulting in an increase in dimension along one or two directions. This deformation feature is referred to as the negative linear compressibility (NLC) or negative area compressibility (NAC). Such properties need not be associated to unstable systems especially because the overall volume compressibility is always positive [6]. Fig. 1 represents the schematic side view of the behaviour of NLC and conventional materials under the application of uniform pressure. For a material with entirely conventional volume compressibility, the uncommon NLC property may be accompanied by a large positive linear compressibility along the perpendicular directions to the NLC axis. The positive volume compressibility is one of the requirements for stability of an unconstrained block of material. According to elastic theory, the lack of constraint is equivalent to a surface traction boundary condition. This stability condition implies that the elastic modulus tensor is positive definite [7]. Therefore it is concluded that the materials with NLC property could be stable, so the elastic modulus tensor of NLC materials could be positive definite. The negative volume compressibility (NVC) behaviour is forbidden in classical thermodynamics [3]. The elastic modulus tensor of NVC materials is not positive definite, so they are not stable, but it is possible to have NVC performance in a pressure–induced phase transition [7]. Lakes and Drugan [8] presented a foam with strain-dependent NVC, and a negative bulk modulus in a pre-strained foam was observed and interpreted by Lakes and Wojciechowski [7]. Furthermore, metamaterials have the potential to be designed to achieve NVC [9]. NLC behaviour was firstly observed in tellurium along a particular direction, and it was reported by Bridgman in 1922 [10]. Baughman et al. [3] explained a variety of potential applications of the NLC phenomenon such as the development of optical telecommunication systems, artificial muscles [11], infiltrated porous structures and a new generation of pressure sensitive sensors and actuators [3,12,13]. The NLC behaviour was discovered at the molecular level such as LaNbO4 [14], Ag3[Co(CN)6] [15], KMn3[Ag(CN)2]3 [16] and a large giant NLC in the molecular framework of Zn[Au(CN)2]2 [12]. At the micro and macro levels, Baughman et al. [3] proposed a wine-rack mechanism to explain the NLC effect. In such a mechanism, it is assumed that the rods were rigid in both the axial and transverse directions. When the system is subjected to stress, the hinged joints enable the wine-rack system to deform or restore the original shape when the stress is removed. In other words, the system deforms solely through the changes in the angles between the rigid rods. The rods can only rotate in the plane of a 2D system that is parallel to the direction of applying load. This assumption is valid when this hinging mechanism is 3D. Also, it is assumed that the stiffness of the structure resulted from resistance to relative rotation of the rods at hinges. Weng et al. [17] presented a list of crystals that exhibited NLC. Barnes et al. [18] proposed a tetragonal network of nodes connected by a network of beams with NLC behaviour. Grima et al. [19] presented a 2D system with a honeycomb structure which exhibited NLC behaviour. In the following, a 3D system with NLC behaviour was generalized from the elongated hexagonal dodecahedron as the 3D equivalence of the honeycomb. The detailed analysis was presented by Grima et al. [6] to assess different uncommon behaviours of negative Poisson's ratio, NLC, and NAC. A new hexagonal truss system with specific
geometric features and NLC behaviour was presented by Grima et al. [20]. This system deformed through non-equal changes in the lengths of the cell walls when deforming through a constrained angle stretching rather than flexure or hinging of joints. Recently Zhou et al. [21] presented three 3D cellular models which exhibited negative compressibility. The new proposed systems have the tuneable compressibility which can be adjusted for different applications with required properties of NLC, NAC, zero and negative Poisson's ratios. There has been extensive work on materials and structures design using the composite techniques, covering various uncommon properties such as auxetic [22–34], negative stiffness [35,36] and negative thermal expansion inclusions [37,38]. The basic idea of the composite technique is to improve the properties of materials and structures through adding a second phase to them in the form of fibre element or particulates [39]. In this study, we developed the simple design approaches to manufacturing a new series of structures with NLC property based on the application of composite techniques. NLC composite materials and structures are rare and so far have been found in a very limited range of applications. Baughman et al. [3] presented a composite material with NLC property that consisted of a framework network with a helical chain configuration in low-density porous solid. Weng et al. [17] investigated the 2D and 3D composite structures with a rhombus-like hinged framework inside the filler component. All the above mentioned NLC composites have interior reinforcements with a wine-rack topology which all the members of the reinforcement were joined. Therefore the NLC property arises from the function of hinged joints. Thus, the corresponding mechanism for NLC can be presented by a wine-rack model which is frequently used to quantify NLC properties. The NLC materials and structures with discrete and non-wine-rack elements are very rare. Recently Miller et al. [40] presented some common materials particularly carbon fibre laminate composites that exhibited NLC. The carbon fibre composites consisted of discrete carbon fibres which reinforced a stable matrix of epoxy. Also, Barnes [41] reported that the responsible mechanism for NLC could be different from a wine-rack model. A generic helical chain structure with discrete and non-wine-rack element was developed with NLC property [41]. In this study, we developed a new composite structure with a discrete reinforcement component which exhibited NLC behaviour. For some specific medical applications such as wound management, specific products with NLC behaviour are required as the wound filler. To transmit the liquids during curing, the void size of the wound filler should be in a specific range, i.e. 200–500 μm. Due to the manufacturing capacity, it is very difficult to produce a large quantity of NLC materials in meters with such a small dimension in its microstructures. A combination of the foam-type material as the filler component and the reinforcement component with NLC behaviour is a possible solution. In such case, the size of voids of new composite structures is determined by the cell size of the filler material. The reinforcement component plays a dominant role in the NLC behaviour of the composite structure. To solve those challenges mentioned above, we proposed the new design approaches to design two types of NLC composite structures, respectively, composite structures with a reinforcement component-like a framework topology and composite structures with a discrete
Fig. 1. Comparison of the behaviour of the conventional material and the NLC material.
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reinforcement component. The newly designed composites consisted of clearly two distinguishable components of foam and reinforcement. It should be noted that the topologies of the reinforcement components of two NLC composites were different. The first NLC composite structure consisted of the reinforcement component with the framework topology. In contrast, the topology of the reinforcement component of the second NLC composite structure consisted of discrete truss stiff elements. The effectiveness of the designed approaches was proved through the uniform pressure loading to all surfaces of a small but sufficient number of bulk specimens. The compressibility of the composite specimens was also compared with the existing pure foam.
approach; the building block was patterned along two normal axes to form the experimental bulk specimen as shown in Fig. 3b. It should be noted that the stiffeners were more rigid and dense than filler component material. The function of stiffeners was affected by the bonding between the filler component and the stiffeners. In order to improve the bonding effect, the surface of the stiffeners was indented to increase the contact areas. In this way, the friction between stiffeners and filler components would provide sufficient resistance to prevent large relative sliding between two different components.
2. Design and fabrication of new NLC composite structures
Six experimental specimens were fabricated to examine the NLC property through the uniform pressure loading test. For the first specimen, the 3D printing technique was used to create the NLC frameworks by using the silicone base rubber material. The dimensions of the printed frameworks were selected as height × width × depth = 40 × 160 × 320 mm as shown in Fig. 4. The supporting material in the original printed models was carefully removed by a drilling process followed by flushing using water. Various drills with different diameters (3, 8, 11 mm respectively) were used to remove the supporting material from the small holes along the width and depth directions. After the initial cleaning process, in order to reduce the bonding strength between supporting material and the framework structure, the printed models were compressed 20% along out-of-plane direction by using Shimadzu compression testing machine and then released the load. The sample restored to its original shape. The same process was repeated for the other two directions. According to the dimension of the printed framework, it was compressed 8 mm, 32 mm and 64 mm along the height, width and depth direction respectively. In the following, the printed models were flushed by water. The second specimen consisted of the black reticulated polyurethane foam specimen without any reinforcement component and apertures as shown in Fig. 5b. The behaviour of the second specimen was examined under the uniform pressure loading to show the lack of NLC property in the absence of reinforcement components. The third specimen was manufactured based on the first design approach. A conventional foam casting method was employed to create the third specimen with a reinforcement component-like framework topology. The reinforcement phase of this composite specimen was exactly similar to the first specimen with the framework topology. Two liquid agents from Polytech Ltd. were used to produce the white crushable polyurethane foam. The stiffness of the polyurethane foam could be adjusted through changing the foaming temperature and the ratio of liquid agents. The weight ratio of two liquid agents was changed from 100:55 (Part A: Part B) to 100:30, which decreased the strength of the crushable polyurethane foam. Thus, the produced foam would be too weak for the proposed application if the weight of part B was b30. Also, the foaming temperature was controlled by putting the foaming container in an incubator. The foam density was inversely proportional to the foaming temperature. Accordingly, the temperature was kept at 80° Celsius to manufacture the second specimen as shown in Fig. 5c. The dimensions of the fourth to sixth specimens were selected according to the availability of existing reticulated polyurethane foam. The fourth specimen with the discrete reinforcement component and the set of apertures was shown in Fig. 5d. The stiffeners were common polyethylene, and the acute angle between them and the vertical Z axis was 10°. A hot wire cutting technique was employed to make some slits for housing the stiffeners inside the foam as illustrated in Fig. 6. The apertures were cut in a similar manner, and their dimensions were selected as length × width = 10 × 12 mm. The fifth and sixth specimens were fabricated in the same way but without apertures. The angle between stiffeners and the vertical Z axis was 10° for the fifth specimen and 80° for the sixth specimen as shown in Fig. 6c and d. The fifth specimen was used to examine the influence of apertures on NLC property, and the sixth specimen with 80° angled stiffeners was fabricated to
In this study, two design approaches were developed to produce a new series of NLC composite structures with a simplified manufacturing process. The main objective of these design approaches was to reinforce the crushable foam with the stiff members. The void size of composite structures was determined by the crushable foam cells, and the NLC behaviour was dominated by the deformation feature of the stiff members. 2.1. Designing the NLC composite structure with a reinforcement component-like a framework topology In the first design approach, a framework with NLC property was designed which the corresponding mechanism for NLC was similar to the wine-rack model. The topology design of the microstructure was created through the simplification of the results of our previous work [42] on the application of the bi-directional evolutionary structural optimization (BESO) to find the microstructure of metamaterial with negative compressibility. As a result, the unit building cell of the NLC framework was generated as shown in Fig. 2a. According to the deformation pattern after loading, the representative volume element (RVE) of the NLC framework contains eight building cells as depicted in Fig. 2b. The NLC framework was constructed with 8 RVEs along Y direction and 4 RVEs along X direction as shown in Fig. 2d. In order to form the building block, the RVE of the NLC framework was placed inside the filler component cube as shown in Fig. 2c. The building block was patterned along X and Y direction to form the bulk specimen of NLC composite structure as illustrated in Fig. 2e. 2.2. Designing the NLC composite structure with a discrete reinforcement component and a set of apertures According to the classification of NLC structures mentioned in Section 1, the interior reinforcement of most known NLC composites have the framework topology, and the NLC property arises from a mechanism like the wine-rack model [3]. The goal of second design approach was to create a new pattern of discrete truss elements which enables the composite structure to exhibit NLC property. The initial geometry of the building block was very simple and was formed by embedding individual angled stiffeners in the highly compressible foam as shown in Fig. 3a. The key point of this design was the rotation of discrete stiffeners under applying the uniform pressure loading which led to a vertical or height increase of the whole specimen and NLC behaviour. The definition of NLC requires a positive compressibility along two other axes perpendicular to the NLC axis. Under the uniform pressure, the volume compressibility must remain positive. Thus, increasing the PLC along two normal directions leads to the higher capacity of NLC along the third direction. The stiffeners were arranged in a particular pattern together with embedded apertures to facilitate their designed deformation mode inside the crushable foam as shown in Fig. 3a. The apertures were positioned between the stiffeners to assist the lateral movement of the stiffeners toward each other on the top surface. The apertures also enhanced the in-plane PLCs. Similar to the previous
2.3. Fabrication of specimens for experiments
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Fig. 2. Topology design of the NLC composite structure with the reinforcement component-like framework topology. (a) The unit building cell of the NLC framework; (b) the representative volume element of the NLC framework; (c) the building block of the NLC composite with the reinforcement component-like framework topology; (d) the full-size model of the NLC framework; (e) the NLC composite structure with the reinforcement component-like framework topology (LBC = 20 mm).
study the influence of the angle of stiffeners on NLC property. Theses composite specimens with discrete truss elements and without apertures were shown in Fig. 5e and f. For the third and fifth specimens, the colour difference between embedded black reinforcement component and the white foam led to that the ends of the black framework and angled stiffeners appeared a little in the top view of these specimens as shown in Fig. 5c and e. Also, for the sixth specimen with 80° angled stiffeners, the similarity between the black colour of angled stiffeners and foam caused that the black stiffeners were not distinguished in Fig. 5f.
strain rate and complete load-displacement information was recorded for each of the specimens by Shimadzu machine. A typical example of a nominal stress-strain curve for a cylinder specimen was shown in Fig. 7. Accordingly, it was found that the linear behaviour of foam was different from solid incompressible materials. The base material of composite specimens was the crushable foam. Before densification, the foam materials exhibited a long plateau stress similar to the progressive crushing mechanism of metallic foams. However, the deformation could be fully recovered after the load was removed. This behaviour originated from the buckling and post-buckling deformation of cell walls of the foam but it is elastic.
3. Testing the performance of designed composite structures through the uniform pressure loading test
3.2. Experimental set-up on designed NLC composites
3.1. Experiments on the base foam material The material properties of the base foam were measured through the standard compression test using six cylinder specimens with the dimensions of diameter × height = 50 × 70 mm. The standard compression tests were conducted by a Shimadzu machine (Shimadzu Company, Kyoto, Japan) at a fixed strain rate of 3 mm/min. The cylinder specimen were compressed to 85% of the foam thickness at the fixed
In order to prove the effectiveness of proposed design approaches, six fabricated specimens with rectangular shape were tested. The proposed test set-up was different from the standard uniaxial compression test to obtain material properties. In the proposed test method, the pressure was uniformly applied to all six surfaces of the specimens rather than two surfaces in the standard uniaxial compression test. The specimen of the pure base foam with identical sizes to the NLC specimens was tested to illustrate the difference in their performance. A camera was used to capture the in-plane deformation. Each specimen was put
Fig. 3. Topology design of NLC composite structure with the discrete reinforcement component and the set of apertures. (a) The unit building block with two angled stiffeners and two vertical apertures; (b) NLC composite structure model with the discrete reinforcement component and the set of apertures (LBB = 30 mm).
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Fig. 4. The experimental specimen of the NLC framework. (a) The specimen of 3D printed NLC framework before removing supporting material; (b) the specimen of 3D printed NLC framework after removing supporting material (overall dimensions of the printed frameworks: height × width × depth = 40 × 160 × 320 mm).
inside a sealed plastic bag connected with a plastic tube. The sealed specimen was left on a smooth surface while the plastic tube was connected to the valve of the vacuum pressure pump. The internal pressure of sealable bag was gradually increased from 0 to 26 kPa with a small increment to monitor the deformation pattern, the top surface area reduction and the out-of-plane deformation at different levels of uniform pressure. The pressure was held at 0, 5, 6, 8, 9, 10, 12, 13, 16, 18, 21, 24 and 26 kPa. A camera (digital) recorded the in-plane deformation of specimens at each level of uniform pressure. The images were calibrated by using Adobe® Photoshop® software. For the calibration, the number of pixels of one known area on the scale and one unknown area on the image was counted. These two numbers were compared together to find the nearest known area on to unknown area. This information was used to determine the number of pixels per square mm for resolution. In the next step, the pressure resistance of all six composite specimens was examined to quantify the merits of the proposed design approaches. The experimental values of in-plane area reduction were measured from the calibrated photos at different levels of uniform pressure. The relation between the level of applied pressure and the area reduction was presented in Fig. 11. The pressure resistance of NLC designs can be quantified from the pressure-area curves. The experimental values of in-plane strains along X and Y directions were measured through image processing from the nodes in the central parts of specimens. The central area consisted of 9 marked red points indicating 1 ≤ i ≤ 3 and 1 ≤ j ≤ 3 as shown in Fig. 8. The centroid to centroid distances along two main directions of X and Y were calculated by using coordinates (xi , j, yi , j), Δxi , j = xi + 1 , j − xi , j and Δyi , j = yi , j + 1 − yi , j. The
centre to centre distances between these points for the undeformed specimen were defined as ΔX(0) = ΔY(0) = 50 mm. Eq. (1) was employed to calculate the local values of the in-plane strains along two main directions of X and Y. Eventually, six local values of in-plane strains along X and Y directions were calculated and the average was computed at each level of uniform pressure. Also, the out-of-plane deformation was measured using a dial gauge at 9 marked locations in Fig. 8. ϵx ¼
Δxi; j Δxð0Þ
ϵy ¼
Δyi; j Δyð0Þ
ð1Þ
3.3. Experimental results The deformation patterns, the strain-uniform pressure history, and the area reduction- uniform pressure history of all six composite specimens were presented in Figs. 9, 10 and 11 respectively. Also, the deformation patterns of all experimental specimens at more level of pressures can be found in the appendix. It is worth to note that during the test, no large scale out-of-plane bending and wrinkling were observed. From the Fig. 9 and Fig. A1 in the Appendix, it was found that the deformation process of all specimens was not similar and their shapes did not remain the same. The in-plane and out-of-plane deformations of all six composite specimens were not uniform. It should be noted that within the range of the applied loading pressure up to 26. kPa, all deformation of our composite samples, was elastic.
Fig. 5. Six experimental specimens. (a) Specimen of 3D printed NLC framework; (b) specimen of reticulated polyurethane foam without any reinforcement component and apertures; (c) composite specimen with the reinforcement component-like framework topology (overall dimensions of the first to sixth specimens: height × width × depth = 40 × 160 × 320 mm); (d) composite specimen with the discrete reinforcement component and the set of apertures (angle of stiffeners is 10°); (e) composite specimen with the discrete reinforcement component without apertures (angle of stiffeners is 10°); (f) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 80°), (overall dimensions of specimens with the discrete reinforcement component: height × width × depth = 30 × 120 × 270 mm), (Scale bar: 20 mm).
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Fig. 6. The fabrication process of the experimental specimen with the discrete reinforcement component using the hot wire cutters technique. (a) Polyethylene stiffeners (height × width × depth = 31 × 10 × 2 mm); (b) passing the hot wire through the foam to make slits; (c) placing the stiffeners inside the foam with 10° angle; (d) placing the stiffeners inside the foam with 80° angle.
As can be seen in Fig. 10a and c, the first and third specimens exhibited NLC property. However, the first specimen with the pure framework structure exhibited the larger in-plane and out-of-plane deformations than the third experimental specimen with the composite structure and the framework reinforcement. Therefore, it was found that adding the crushable foam led to reducing the in-plane deformation and weakening the NLC property. The second specimen that consisted of pure reticulated polyurethane foam without any reinforcement component and apertures did not exhibit NLC property as shown in Fig. 10b. In contrast, the Fig. 10d and e show the NLC property of the fourth and fifth specimens with
Fig. 7. A typical stress-strain curve of the crushable foam used as the filler component of the NLC composite specimens.
10° angled stiffeners. Therefore it was concluded that reinforcing of the crushable foam through embedment of discrete truss elements led to changing the crush pattern of pure foam and generating a new type of NLC composite structure. Also, comparison of strain–pressure history of fourth and fifth specimens in Fig. 10d and e demonstrates that the fourth specimen with angled stiffeners and vertical apertures exhibited the larger in-plane and out-of-plane deformations compared with the fifth specimen. Thus, modification of filler foam through several embedded apertures led to a larger in-plane deformation and improved the NLC behaviour. Experimental result in Fig. 10f reveals that the sixth specimens with 80° angled stiffeners did not exhibit NLC property under the uniform pressure loading. Comparison of strain–pressure history of fifth and sixth specimens indicates that increasing the angle of stiffeners with vertical Z axis led to changing the mechanism of rotation of stiffeners and removing the NLC property. Consequently, it could be concluded that the NLC property of composite specimens depends mainly on three factors, i.e., the compressibility characterization of the filler foam, filling foam with apertures and the performance of stiff elements. Hence, it is possible to control the out-ofplane deformation of NLC composite specimen by adjusting the compressibility of filler foam and the performance of stiffeners. The comparison of area reduction–pressure history of all experimental specimens was presented in Fig. 11a. In general, the deformation processes of all samples demonstrated area reduction as a result of inplane deformation. Therefore, for all specimens except the third one, the majority of area reduction occurred at the uniform pressure level of 6 kPa. The results indicated that by increasing the uniform pressure level from 6. to 26 kPa, only minor reduction in width and length was observed as shown in Fig. 11a. This sequence was repeated for the out-of-plane deformation which led to NLC behaviour. Therefore, for all specimens except the second and sixth ones, the majority of NLC behaviour was observed in the loading range of 0 to 6 kPa and out of this
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Fig. 8. Experimental setup of the uniform pressure loading test. (a) Sealed specimen; (b) schematic diagram of central region with nine marked red points (Δx0 =Δy0 =50 mm); (c) dial gauge for measuring the out-of-plane deformation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
pressure range, the change of NLC was minor. This finding was presented in Fig. 11b by the corresponding measured strains along Z direction for all specimens. 3.4. Estimation of the out-of-plane deformation of NLC composite structure with the discrete reinforcement component based on the specially ordered patterns without apertures The NLC property of the composite structure with the angled stiffeners is originated from a simple mechanism. The alignment of stiffeners under the uniform pressure causes the compaction of filler components and tensile stresses on the top and bottom surface of the foam. This stress forces each stiffener to undergo rotation to reduce its angle with out-of-plane direction. This rotation leads to vertical or height increase of the whole specimen as shown in Fig. 12. In this figure a schematic view of the position of angled stiffeners and the mechanism of rotation were illustrated. The initial acute angle between the individual stiffeners and the vertical axis of Z can be adjusted to design a composite structure with the expected strain along NLC axis (εNet). For this purpose, a simple equation was derived based on simple assumptions. For the sake of simplicity, it was supposed that all stiffeners had the same size, section area, and stiffness. Also, the compressive stiffness of filler materials as compared to stiffeners is negligible. Consequently, the force due to the pressure was mostly applied to the stiffeners in equal amounts where P is the applied pressure in Pascal. The net strain of composite structure in out-of-plane direction (z in Fig. 12) presents the limit of NLC property of new composite structure and can be expressed as: 0
εNet ¼
H S−H H
ð2Þ
where: 0
H S−H ¼ H S ;
cosðΘÞ ¼
H H S −H S cosðΘÞ ; ε Net ¼ HS HS cosðΘÞ
The resulting equation is given below: 1 Θ ¼ arccos ε Net þ 1
ð3Þ
By using this equation, the range of variation of θ for a magnitude of expected strain along NLC axis (εNet) can be adjusted. The out-of-plane deformation of the fifth experimental specimen with discrete truss elements (10°) without apertures was predicted by using Eq. (3). When θ = 10°, the predicted net strain of specimen with angled stiffeners was about εNet = 0.015. This represented a conservative estimate of maximum out-of-plane deformation. The average
measured strain form Fig. 9e is about 0.02 for the fifth specimen that shows an acceptable agreement with the experimental result. The out-of-plane deformation of the fourth experimental specimen with discrete truss elements and the set of apertures (angle of stiffeners is 10°) was larger than predicted value of equation (εNet = 0.015) as shown in Fig. 9d. It is indicated that the plate specimen were undergo a small amount of local wrinkling along the out-of-plane direction of the specimen. It resulted in large measuring errors of strain in the outof-plane direction. It would be an inherent feature for specimen with apertures. However, this measuring error will not influence our major conclusion. The sharp difference between the black reticulated polyurethane foam and our designed composites confirmed the effectiveness of our design approach 4. Discussions This research demonstrated the effectiveness of two proposed approaches to designing new composite structures with negative linear compressibility (NLC). The first approach was to simplify the topology optimization results from previous research [42] to generate the framework reinforcement for a new NLC composite structure. The second approach was to use the discrete truss elements of a special pattern to reinforce the crushable foam for generating a new type of NLC composite structure. Those two design approaches have been validated by experiments. The key performance of designed NLC composite structures was captured under the application of different levels of uniform pressure. The size of the specimens was selected as rectangular plates due to their intended applications in which the in-plane area contraction ratio and out-of-plane deformation are the major factors. The experimental set-up was different from standard uniaxial compression test to obtain material properties. The pressure was uniformly applied to all six surfaces of the specimens rather than two surfaces in the standard uniaxial compression test. It should be noted that the main aim of our experiments is to verify the effectiveness of our design approaches to achieve NLC with a limited number of specimens. Due to the variation of properties of foams and stiffeners, a large number of specimens are required to obtain reliable material properties for the designed and manufactured composite materials. It should be noted that the conventional definition of negative compressibility is based on the linear concept. In our work, we used it to describe a special deformation feature of the composite under hydrostatic pressure, i.e., the direction of the deformation is against the direction of applied pressure. In the experimental data presented in Figs. 10 and 11, there are no data points in the region of small strain due to the limitation of the test device used to apply and maintain negative pressure, whose lowest pressure value is 5 kPa. For most of our design, the large amount of deformation occurred within that pressure. However, it will not influence our conclusion to prove the effectiveness of our design approach of our NLC composites. Based on the deformation features of the specimen under large deformation, the NLC behaviour only exhibits
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Fig. 9. Deformed shapes of experimental specimens under the uniform pressure loading. (a) Specimen of 3D printed NLC framework; (b) specimen of reticulated polyurethane foam without any reinforcement component and apertures; (c) composite specimen with the reinforcement component-like framework topology; (d) composite specimen with the discrete reinforcement component and the set of apertures (angle of stiffeners is 10°); (e) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 10°); (f) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 80°), (Scale bar: 20 mm).
when the strain of the specimen is large than a critical value. Determine this critical value and the variation of compressibility from small strain to large strain would be an interesting research topic which will be explored in the future In this study, three parameters of the compressibility characterization of the filler foam, the size of apertures and the performance of stiff elements were identified to control the NLC property of our designed composites. However, investigation the influence of these dominant parameters on NLC performance of new designed composites is out of the scope of this research. It should be mentioned that a set of systematic parameters using validated finite element simulation is
required to show the possibility of providing an effective control method to tune the NLC of the composite specimens during the designing process. For example, it was experimentally shown that under the uniform pressure the composite specimen with discrete truss element exhibited NLC behaviour when the angle of stiffeners is 10°. However, the similar specimen with 80° angled stiffeners did not exhibit NLC performance. Therefore, the numerical simulations as a systematic parametric study are conducting to identify the critical domain of angle of stiffeners for NLC behaviour. All samples with the angle of stiffeners from this critical domain will exhibit the NLC behaviour, and the results of corresponding numerical simulations can be used to investigate the
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Fig. 10. Strain–pressure history for experimental specimens under the uniform pressure loading. (a) Specimen of 3D printed NLC framework; (b) specimen of reticulated polyurethane foam without any reinforcement component and apertures; (c) composite specimen with the reinforcement component-like framework topology; (d) composite specimen with the discrete reinforcement component and the set of apertures (angle of stiffeners is 10°); (e) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 10°); (f) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 80°).
effects of changes of angle on NLC behaviour as well. Currently, we are conducting more numerical simulations to investigate those controlling parameters, and they will be reported in our future paper. It is also worth noting the limitations of the present study. Firstly, the in-plane deformations of NLC composite specimens were not uniform. This non-uniform shape change could be caused by two factors. The first factor was the design of composite samples. The composite specimens consisted of two components of the crushable foam and the reinforcement component. The standard compression test of foam revealed that the crushable foam had the non-linear behaviour also the interaction between the foam and the reinforcement could make the problem more nonlinear. The second factor of non-uniform shape change was the effect of boundary condition on experimental samples in the plastic
bag during the applying negative pressure. Under applied negative pressure, the top and bottom sheets of a plastic bag around experimental specimen stuck together. Therefore, the non-uniform lateral load due to the stuck regions of the plastic bag led to non-uniform in-plane deformation of NLC composites while the distribution of pressure inside the plastic bag was uniform. Secondly, much of the area reduction occurred at the lower levels of uniform pressure except for the third specimen with the framework reinforcement and filler foam, and the in-plane deformation was minor at the higher levels of negative pressure. Thirdly, the increase of stiffness of the crushable foam led to a reduction of the in-plane deformation and weakening of the NLC effect. These issues will be addressed in the subsequent research. Fourthly, it should be noted that the plastic cover bag on the top and bottom surface of the
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the deformation of the composite specimen toward NLC behaviour. However, if the thickness of the plastic bag is large than a certain value, i.e., 0.5 mm comparing to 0.1 mm in our experiments, increasing the thickness further will hinder the rotation of the stiffener toward the NLC behaviour. Investigation of the influence of thickness of plastic bag on NLC performance requires a set of systematic parametric studies using validated finite element simulation. It is out of the scope of this paper but will be an interesting future work
5. Conclusions Two proposed design approaches have been validated by experiments. The negative strain in one direction and the compressibility in the other two directions of designed NLC composite structures was captured under the application of different levels of uniform pressure. From the obtained results, the following conclusions can be drawn: 1. The new designed composite specimens exhibited NLC behaviour in the out-of-plane direction and had a significant area reduction in the in-plane directions. Those results proved the effectiveness of our design approaches.
Fig. 11. Comparison of the behaviour of experimental specimens under the uniform pressure loading. (a) Area reduction–uniform pressure history for experimental specimens under the uniform pressure loading; (b) strain along Z direction–uniform pressure history for experimental specimens under the uniform pressure loading.
composite structure during the hydrostatic compression test may contribute to the NLC behaviour of the specimens. Especially for the composites with angled stiffeners as reinforcement component, the membrane tension between two adjacent legs of stiffeners will facilitate
2. The results of experiments confirmed that the NLC behaviour of the designed composite structures depended on three main factors: the compressibility of the filler foam, the pattern of the network element, and the stiffness ratio between the two constituent materials of the composites. 3. The modification of filler foam through embedding some vertical apertures led to larger in-plane deformation and enhanced NLC property. The in-plane deformation of the NLC composite structures could be tuned by the shape and the pattern of the apertures to meet specific compressibility requirements. 4. A simple formula was derived to guide the adjustment of the angle of the stiffeners so that the strain along the NLC axis would reach the desired magnitude. Author contributions Yi Min Xie and Jianhu Shen initiated and supervised this study. Arash Ghaedizadeh and Jianhu Shen carried out the analysis and experiments. Arash Ghaedizadeh wrote the draft manuscript. Yi Min Xie, Jianhu Shen, and Xin Ren edited and revised the manuscript.
Fig. 12. Schematic deformation mechanism of stiffener inside the foam under the applied uniform pressure. (a) Perspective view of the undeformed experimental specimen with the discrete reinforcement without apertures; (b) the perspective view of the idealized deformed pattern with stiffeners becoming vertical, resulting in NLC property; (c) side view of the undeformed experimental specimen with discrete reinforcement without apertures; (d) side view of the idealized deformed pattern with stiffeners becoming vertical, resulting in NLC property.
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Conflicts of interest The authors declare no conflict of interest.
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Acknowledgments The authors would like to acknowledge the support of the Australian Government's Cooperative Research Centres (CRC) Programme. This research was supported by the Wound Management Innovation CRC (project no. CRC2-16) and the Australian Research Council (project nos. DP140100213 and DP160101400). Appendix A Fig. A1Deformed shapes of experimental specimens under the application of uniform pressure. (a) Specimen of 3D printed NLC framework; (b) specimen of reticulated polyurethane foam without any reinforcement component and apertures; (c) composite specimen with the reinforcement component-like framework topology; (d) composite specimen with the discrete reinforcement component and the set of apertures (angle of stiffeners is 10°); (e) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 10°); (f) composite specimen with the discrete reinforcement component without any apertures (angle of stiffeners is 80°).
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Appendix B. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.matdes.2017.06.026.
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P= 0 kPa
P= 9 kPa
P= 16 kPa
P= 5 kPa
P= 6 kPa
P= 8 kPa
P= 10 kPa
P= 12 kPa
P= 13 kPa
P= 18 kPa
P= 21 kPa
P= 26 kPa
P= 24 kPa
Unloaded
(a) P= 0 kPa
P= 5 kPa
P= 6 kPa
P= 8 kPa
P= 9 kPa
P= 10 kPa
P= 12 kPa
P= 13 kPa
P= 16 kPa
P= 18 kPa
P= 21 kPa
P= 26 kPa
Unloaded
(b)
P= 24 kPa
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P= 0 kPa
P= 5 kPa
355
P= 6 kPa
P= 8 kPa
P= 13 kPa
P= 9 kPa
P= 10 kPa
P= 12 kPa
P= 16 kPa
P= 18 kPa
P= 21 kPa
P= 24 kPa
Unloaded
P= 26 kPa
(c) P= 0 kPa
P= 9 kPa
P= 16 kPa
P= 5 kPa
P= 6 kPa
P= 10 kPa
P= 12 kPa
P= 13 kPa
P= 21 kPa
P= 24 kPa
P= 18 kPa
Unloaded
P= 26 kPa
(d)
P= 8 kPa
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P= 0 kPa
P= 9 kPa
P= 16 kPa
P= 5 kPa
P= 6 kPa
P= 10 kPa
P= 12 kPa
P= 18 kPa
P= 21 kPa
P= 26 kPa
P= 8 kPa
P= 13 kPa
P= 24 kPa
Unloaded
(e) P= 0 kPa
P= 9 kPa
P= 16 kPa
P= 5 kPa
P= 6 kPa
P= 10 kPa
P= 12 kPa
P= 18 kPa
P= 21 kPa
P= 26 kPa
Unloaded
(f)
P= 8 kPa
P= 13 kPa
P= 24 kPa
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