Increase In Bending Stiffness Research Articles

Effect of interlayer bonding strength and bending stiffness on 2-dimensional materials’ frictional properties at atomic-scale steps

Atomic-scale steps generally presented in 2-dimensional materials have important influence on the overall nanotribological properties of surface. Frictional properties at atomic-scale steps of two types of 2-dimensional materials are studied using calibrated atomic force microscopy (AFM) tip sliding against the steps. The lateral force at uncovered step is larger than covered step due to the bending of step edge. The lateral force at monolayer uncovered step edge of h-BN is lower than graphene because h-BN possesses higher interlayer bonding strength than graphene and the bending of h-BN step edge is suppressed to some extent. The high uncovered step exhibits much larger lateral force than low uncovered step, which could be mainly induced by increased bending stiffness of step edge rather than increased step height. The results revealed that interlayer bonding strength and bending stiffness have great influence on the lateral force at atomic-scale steps. The studies can provide a further understanding of frictional properties at atomic scale steps and could be helpful for the applications of 2-dimensional materials as lubricant coating.

Investigating the effects of delamination location and size on the vibration behaviour of laminated composite beams

Composite structures are extensively used in various industries like aerospace, automotive and marine due to their excellent mechanical properties coupled with low density. However the composite structures are very sensitive to processing and working conditions. One of the most common defects that have driven attention of many researchers is the delamination. Present investigation is focussed on studying the effect of delamination location and estimating its effect on natural frequency of composite beams using commercial finite element software, ANSYS. Different ply lay ups are considered for study in both finite elements analysis and analytical formulations. The finite element results are compared with the analytical ones to validate the perfect beam design. The natural frequency is found to reduce with increase in delamination size for both cantilever and simply supported beam. Additionally, the natural frequency in simply supported composite beam is found to be higher than that in cantilever composite beam as expected due to the increased bending stiffness. Further, natural frequency of symmetric laminate is found to be higher than the cross ply laminate for both cantilever and simply supported composite beams.

The effects of pentoxifylline adminstration on fracture healing in a postmenopausal osteoporotic rat model

Previous studies report positive effects of pentoxifylline (PTX) alone or in combination with other drugs on some pathologic bone diseases as well as an ability to accelerate osteogensis and fracture healing in both animal models and human patients. The aim of this present study was to evaluate the effects of PTX administration on Hounsfield unit and bone strength at catabolic response (bone resorbing) of a fracture in an experimental rat model of ovariectomy induced osteoporosis (OVX-D). Thirty adult female rats were divided into groups as follows: 1 (OVX, control, no treatment); 2 (OVX, sham: daily distilled water); 3 (OVX, daily alendronate: 3 mg/kg); 4 (OVX, twice daily 100 mg/kg PTX) and 5 (OVX, PTX+alenderonate). OVX was induced by bilateral ovariectomy in all rats. A complete standardized osteotomy of the right femur was made after 3.5 months. PTX and alendronate treatments were performed for eight weeks. Then, rats were euthanized and had its right femur subjected to computerized tomography scanning for measuring Hounsfield unit; eventually, the samples were sent for a three point bending test for evaluation of the bone strength. Administration of PTX with 200 mg/kg and alendronate alone and in combination showed no significant alteration in Hounsfield unit and biomechanical properties of repairing callus of the complete osteotomy compared with the control group. Results showed increased bending stiffness and stress high load mean values of repairing complete osteotomy in PTX-treated rats compared to the control OVX-D.

Strength and stiffness of glulam beams reinforced with glass and basalt fibres

Abstract Timber has throughout history played an important role in all kinds of construction work as it is easy to work with and has many advantages compared to alternative materials. Through engineered timber products such as glulam, the natural variability in material properties of timber has been reduced, providing stronger, stiffer and more reliable structural elements. In recent years' new techniques have emerged to enhance these properties even further by using FRP materials to strengthen structural timber components. The FRP materials have more tensile strength and higher modulus of elasticity than timber. In recent years' experiments have been conducted in the Structural and Composite Laboratory at Reykjavik University (SEL) in order to investigate the effect of strengthening glulam beams using FRP materials. In these experiments, 24 glulam beams (3,2 m long, 65 mm wide and 167 mm high) were tested. Half of the beams were in strength class GL32h and the second half were in strength class GL30c. The beams were tested in a four-point bending test. The beams were randomly divided into groups with 3 or 4 beams in each group. For each group the bottom laminate of the beams was reinforced with slack basalt fibres, slack glass fibres, pre-stressed basalt fibres or with no reinforcement for comparison. The pre-stressing of the basalt fibre layers was done through cambering, where the beam was cambered to the point where the stresses in the timber reached 90% of authorized stresses. The test results were compared with an analytical model to predict the moment capacity. The numerical model considers the linear elastic behaviour and the plastic behaviour of the timber under compression. The experimental results compare relatively well with the mathematical model. Strengthening glulam beams on the tension side with FRP materials increases bending stiffness and bending strength. Increased reinforcement ratio, in the form of additional fibre cloth layers, had a significant effect on both the strength and the stiffness. Pre-stressing the FRP provides additional strength and higher stiffness compared to slack FRP strengthening. Strengthening the glulam beams on the tension side allows a possible reduction of the cross-section or lower timber grade while maintaining the same bending strength and stiffness as for the unreinforced beam.

Tuning the mechanical properties of vertical graphene sheets through atomic layer deposition

We report the fabrication and characterization of graphene nanostructures with mechanical properties that are tuned by conformal deposition of alumina. Vertical graphene (VG) sheets, also called carbon nanowalls (CNWs), were grown on copper foil substrates using a radio-frequency plasma-enhanced chemical vapor deposition (RF-PECVD) technique and conformally coated with different thicknesses of alumina (Al2O3) using atomic layer deposition (ALD). Nanoindentation was used to characterize the mechanical properties of pristine and alumina-coated VG sheets. Results show a significant increase in the effective Young’s modulus of the VG sheets with increasing thickness of deposited alumina. Deposition of only a 5 nm thick alumina layer on the VG sheets nearly triples the effective Young’s modulus of the VG structures. Both energy absorption and strain recovery were lower in VG sheets coated with alumina than in pure VG sheets (for the same peak force). This may be attributed to the increase in bending stiffness of the VG sheets and the creation of connections between the sheets after ALD deposition. These results demonstrate that the mechanical properties of VG sheets can be tuned over a wide range through conformal atomic layer deposition, facilitating the use of VG sheets in applications where specific mechanical properties are needed.

Influenza M2 Transmembrane Domain Interacting with Lipid Membranes: An Atomic Force Microscopy and Fluorescence Microscopy Study

By clustering at domain edge of host cell membranes, M2 protein of influenza A virus was proposed to participate in virus budding and scission. Despite large interest, the molecular mechanism that is responsible for M2 enhanced membrane curvature is obscure. Previous studies have focused on a water-soluble helix located at the cytoplasmic side of M2. A recent study by Hong and coauthors showed that by adopting a flexible conformation, transmembrane (TM) helix can actively promote curved membrane structures (Yao et al., 2015, Proc Natl Acad Sci USA 112, 10926-10931). Their observation prompts us to ask whether the TM domain of M2 (M2TM) can play a similar role in generating raft dependent membrane curvature. We use high-resolution liquid-compatible atomic force microscopy (AFM) and fluorescence microscopy to study M2TM interacting with planar and vesicular lipid membranes. Specifically, we directly visualize molecular-level structure of M2TM in fluid lipid membranes. We also find that cholesterol and sphingomyelin play an important role in governing M2TM oligomerization. We next investigate M2TM distribution in lipid membranes containing liquid-ordered (Lo) and liquid-disordered (Ld) phases. We find that M2TM exclusively partitions into Ld phase. In addition to topographic information, our AFM measurements also entail mechanical properties of lipid membranes. Surprisingly, we find that M2TM increases bending stiffness of Ld phase, while leaving Lo phase unaltered. In our fluorescence microscopy study of giant unilamellar vesicles (GUVs), we show that M2TM enhances miscibility transition temperature of phase coexisting GUVs. In addition, we observe that M2TM can induce an array of vesicle shapes that have not been seen before. Together, our experimental data highlight that the TM domain of M2 can act as a lipid raft sensor and a membrane curvature enhancer.

An evaluation of the effect of pulsed wave low-level laser therapy on the biomechanical properties of the vertebral body in two experimental osteoporosis rat models.

Osteoporosis (OP) increases vertebral fragility as a result of the biomechanical effects of diminished bone structure and composition. This study has aimed to assess the effects of pulsed wave low-level laser therapy (PW LLLT) on cancellous bone strength of an ovariectomized (OVX-d) experimental rat model and a glucocorticoid-induced OP (GIOP) experimental rat model. There were four OVX-d groups and four dexamethasone-treated groups. A group of healthy rats was used for baseline evaluations. The OVX-d rats were further subdivided into the following groups: control rats with OP, OVX-d rats that received alendronate, OVX-d rats treated with PW LLLT, and OVX-d rats treated with alendronate and PW LLLT. The remaining rats received dexamethasone and were divided into four groups: control, alendronate-treated rats, laser-treated rats, and laser-treated rats with concomitant administration of alendronate. PW LLLT (890 nm, 80 Hz, 0.972 J/cm(2)) was performed on the spinal processes of the T12, L1, L2, and L3 vertebras. We extracted the L1 vertebrae and submitted them to a mechanical compression test. Biomechanical test findings showed positive effects of the PW LLLT and alendronate administration on increasing bending stiffness and maximum force of the osteoporotic bones compared to the healthy group. However, laser treatment of OVA-d rats significantly increased stress high load compared to OVA-d control rats. PW LLLT preserved the cancellous (trabecular) bone of vertebra against the detrimental effects of OV-induced OP on bone strength in rats compared to control OV rats.

Finite Element Solutions to the Bending Stiffness of a Single-Layered Helically Wound Cable With Internal Friction

This paper is focused on the bending stiffness of a cable consisting of a straight core wound around by a layer of helical wires, taking friction into account. Depending on the interaction between the cable components, the effective bending stiffness of the cable lies between an upper bound Bmax and a lower bound Bmin according to analytic models in literature. Two finite element models are created. The first aims to determine the maximum obtainable bending stiffness, whereby two contact types are tested: one bonding together all the touching surfaces and the other one only bonding together the wire–core contact surfaces. The numerical results show that Bmax is achieved for the first contact type, while neglecting the wire–wire contact lowers the bending stiffness due to the rotation of wire cross sections. In the second model, the wires are allowed to slip, while the cable is subjected to tension and bending. The effects of the tension level, the friction coefficient, and the contact types are investigated. The numerical results are able to capture the increase of bending stiffness with increasing tension and decreasing curvature, consistent with experimental observations and analytic models. The initial bending stiffness is sensitive to the imperfect contact between the components and is lower than Bmax. The final bending stiffness is higher than Bmin because of the contribution of friction and it increases with the friction coefficient.

Effect of chain stiffness on the competition between crystallization and glass-formation in model unentangled polymers.

We map out the solid-state morphologies formed by model soft-pearl-necklace polymers as a function of chain stiffness, spanning the range from fully flexible to rodlike chains. The ratio of Kuhn length to bead diameter (lK/r0) increases monotonically with increasing bending stiffness kb and yields a one-parameter model that relates chain shape to bulk morphology. In the flexible limit, monomers occupy the sites of close-packed crystallites while chains retain random-walk-like order. In the rodlike limit, nematic chain ordering typical of lamellar precursors coexists with close-packing. At intermediate values of bending stiffness, the competition between random-walk-like and nematic chain ordering produces glass-formation; the range of kb over which this occurs increases with the thermal cooling rate |Ṫ| implemented in our molecular dynamics simulations. Finally, values of kb between the glass-forming and rodlike ranges produce complex ordered phases such as close-packed spirals. Our results should provide a useful initial step in a coarse-grained modeling approach to systematically determining the effect of chain stiffness on the crystallization-vs-glass-formation competition in both synthetic and colloidal polymers.

A new expression for determining the bending stiffness of circular micropile groups

Increased bending stiffness and decreased foundation rotation are two main factors which reduce the rocking motion of foundation. A micropile group in circular arrangement is an innovative technique for reducing the rocking motion of the foundation. In this paper, the effects of several parameters were numerically investigated on the rocking stiffness of circular micropile group foundations. The finite difference software FLAC3D was used to model the foundation, soil, and the structure. The micropiles used in this study varied in the diameter, but had similar length. A total of seven records were selected to cover a wide range of frequency content, duration and amplitude. The results from the numerical simulation were compared and verified against those obtained from an alternative numerical method as well as a set of experimental test results. The model was then used for parametric studies and the effects of relevant parameters were investigated. The results showed that slenderness, inclination angle and distance ratio of a micropile group are main factors which increase the soil stiffness and control the seismic motion of high-rise buildings. Based on the results, a new expression was proposed that can be used to determine the optimal moment of inertia of circular micropile groups. The proposed expression revealed that the primary factors which reduce the effective period of the buildings are the number, diameter, and injection pressure of micropiles in a circular micropile group. Using the proposed relationship, it was found that using circular micropile groups can reduce the destructive seismic effects (i.e. drift demand) in high-rise buildings by up to 60%.

The effect of foam and PA6 reinforcement on bending performance of 6063-T5 tube

The influence of cast polyamide (PA6) and polyvinyl chloride (PVC) foam, which have different lengths, on the bending performance of a thin-walled aluminium tube was investigated experimentally. Three-point bending test was performed to determine load carrying and energy absorbing capacity (EAC) of composite tubes. The experimental results indicated that the reinforcement of PVC foam relatively enhanced buckling displacement but its effect on load carrying capacity (LCC) and EAC is negligible. PA6 and PVC foam-reinforced composite beam shows superior bending resistance with higher buckling displacement. The improvement in bending performance of composite specimens arises from the hindering of local buckling as well as increasing bending stiffness. Accordingly, LCC and EAC increased a maximum of 3.6 and 3.9 times, respectively. The developed composite structure is promising for parts serving as support member of vehicles for which light weight is a critical design consideration.

Small changes in bone structure of female α7 nicotinic acetylcholine receptor knockout mice.

BackgroundRecently, analysis of bone from knockout mice identified muscarinic acetylcholine receptor subtype M3 (mAChR M3) and nicotinic acetylcholine receptor (nAChR) subunit α2 as positive regulator of bone mass accrual whereas of male mice deficient for α7-nAChR (α7KO) did not reveal impact in regulation of bone remodeling. Since female sex hormones are involved in fair coordination of osteoblast bone formation and osteoclast bone degradation we assigned the current study to analyze bone strength, composition and microarchitecture of female α7KO compared to their corresponding wild-type mice (α7WT).MethodsVertebrae and long bones of female 16-week-old α7KO (n = 10) and α7WT (n = 8) were extracted and analyzed by means of histological, radiological, biomechanical, cell- and molecular methods as well as time of flight secondary ion mass spectrometry (ToF-SIMS) and transmission electron microscopy (TEM).ResultsBone of female α7KO revealed a significant increase in bending stiffness (p < 0.05) and cortical thickness (p < 0.05) compared to α7WT, whereas gene expression of osteoclast marker cathepsin K was declined. ToF-SIMS analysis detected a decrease in trabecular calcium content and an increase in C4H6N+ (p < 0.05) and C4H8N+ (p < 0.001) collagen fragments whereas a loss of osteoid was found by means of TEM.ConclusionsOur results on female α7KO bone identified differences in bone strength and composition. In addition, we could demonstrate that α7-nAChRs are involved in regulation of bone remodelling. In contrast to mAChR M3 and nAChR subunit α2 the α7-nAChR favours reduction of bone strength thereby showing similar effects as α7β2-nAChR in male mice. nAChR are able to form heteropentameric receptors containing α- and β-subunits as well as the subunits α7 can be arranged as homopentameric cation channel. The different effects of homopentameric and heteropentameric α7-nAChR on bone need to be analysed in future studies as well as gender effects of cholinergic receptors on bone homeostasis.

The effect of intramedullary pin size and monocortical screw configuration on locking compression plate-rod constructs in an in vitro fracture gap model.

To investigate the effect of intramedullary pin size in combination with various monocortical screw configurations on locking compression plate-rod constructs. A synthetic bone model with a 40 mm fracture gap was used. Locking compression plates with monocortical locking screws were tested with no pin (LCP-Mono) and intramedullary pins of 20% (LCPR-20), 30% (LCPR-30) and 40% (LCPR-40) of intramedullary diameter. Locking compression plates with bicortical screws (LCP-Bi) were also tested. Screw configurations with two or three screws per fragment modelled long (8-hole), intermediate (6-hole), and short (4-hole) plate working lengths. Responses to axial compression, biplanar four-point bending and axial load-to-failure were recorded. LCP-Bi were not significantly different from LCP-Mono control for any of the outcome variables. In bending, LCPR-20 were not significantly different from LCP-Bi and LCP-Mono. The LCPR-30 were stiffer than LCPR-20 and the controls. The LCPR-40 constructs were stiffer than all other constructs. The addition of an intramedullary pin of any size provided a significant increase in axial stiffness and load to failure. This effect was incremental with increasing intramedullary pin diameter. As plate working length decreased there was a significant increase in stiffness across all constructs. A pin of any size increases resistance to axial loads whereas a pin of at least 30% intramedullary diameter is required to increase bending stiffness. Short plate working lengths provide maximum stiffness. However, the overwhelming effect of intramedullary pin size obviates the effect of changing working length on construct stiffness.

Vibration characteristics of rotor system with tilting-pad journal bearing of elastic and damped pivots

In view of an entire dynamic model of tilting-pad journal bearing (TPJB) in which the pads swing and vibrate along geometric direction of preload, a TPJB of elastic and damped pivots was designed and manufactured. Vibration experiments were carried out under the conditions of different rotor bending stiffness and oil supply pressure to find out the relationship between the new bearing’s vibration depression effect and other dynamic parameters of the rotor. The result shows that critical amplitudes can be efficaciously reduced while system’s stability can be remarkably improved by this bearing. Besides, the bearing’s effect of vibration depression weakens as the rotor bending stiffness increases, but heightens it as the oil supply pressure increases.

The Conformance Analysis on Bending Forming of Honeycomb Carton

Geometry constructions of honeycomb core are finding increasing use because of their relative advantages over other structural materials in terms of improved stability, high stiffness to weight and strength to weight ratios. It provides an efficient solution to increase bending stiffness without significant increase in structural weight. However, honeycomb carton, as a new product of paper honeycomb, has been made by bending forming and failure process of paper honey- comb. Theoretical analysis on bending mechanical properties of paper honeycomb has been done. Then a model has been built on finite element analysis (FEA) software and analyzed its nonlinear bending process. The shear deformation process of honeycomb carton may be approximately categorized into four phases: linear elastic phase, elastic-plasticity phase, plasticity collapse phase, densification and fracture phase. In order to get precise results, not only shear force but also the failure process of paper-core should be considered. Therefore, the curve of external force and the shear deformation proc- ess has been analyzed. In addition, stress concentrations due to bending load and cohesive force have been eliminated by drying process of honeycomb carton. At last, the results are introduced by experiment and put into practice.

Fracture healing in mice lacking Pten in osteoblasts: a micro-computed tomography image-based analysis of the mechanical properties of the femur

In the United States, approximately eight million osseous fractures are reported annually, of which 5–10% fail to create a bony union. Osteoblast-specific deletion of the gene Pten in mice has been found to stimulate bone growth and accelerate fracture healing. Healing rates at four weeks increased in femurs from Pten osteoblast conditional knock-out mice (Pten-CKO) compared to wild-type mice (WT) of the same genetic strain as measured by an increase in mechanical stiffness and failure load in four-point bending tests. Preceding mechanical testing, each femur was imaged using a Skyscan 1172 micro-computed tomography (μCT) scanner (Skyscan, Kontich, Belgium). The present study used µCT image-based analysis to test the hypothesis that the increased femoral fracture force and stiffness in Pten-CKO were due to greater section properties with the same effective material properties as that of the WT. The second moment of area and section modulus were computed in ImageJ 1.46 (National Institutes of Health) and used to predict the effective flexural modulus and the stress at failure for fourteen pairs of intact and callus WT and twelve pairs of intact and callus Pten-CKO femurs. For callus and intact femurs, the failure stress and tissue mineral density of the Pten-CKO and WT were not different; however, the section properties of the Pten-CKO were more than twice as large 28 days post-fracture. It was therefore concluded, when the gene Pten was conditionally knocked-out in osteoblasts, the resulting increased bending stiffness and force to fracture were due to increased section properties.

Quantitative Study of Parathyroid Hormone (1-34) and Bone Morphogenetic Protein-2 on Spinal Fusion Outcomes in a Rabbit Model of Lumbar Dorsolateral Intertransverse Process Arthrodesis

A posterolateral rabbit spinal fusion model was used to evaluate the effects of recombinant human bone morphogenetic protein-2 (rhBMP-2) and teriparatide (PTH [1-34]) used individually and in combination on spinal fusion outcomes. To test the efficacy of parathyroid hormone on improving spinal fusion outcomes when used with BMP-2. Of the more than 250,000 spinal fusion surgical procedures performed each year, 5% to 35% of these will result in pseudarthrosis. Growing controversy on the efficacy and cost of rhBMP-2 for improving spinal fusion outcomes has presented a challenge for clinicians. Research into PTH as an adjunct therapy to rhBMP-2 for spinal fusion has not yet been investigated. Forty-eight male New Zealand white rabbits underwent bilateral posterolateral intertransverse process arthrodesis surgery at the L5-L6 level. Animals were divided into 6 groups. Two groups were treated with autograft alone or autograft and PTH (1-34), whereas the other 4 groups were treated with low-dose rhBMP-2 alone, high-dose rhBMP-2 alone, or either dose combined with PTH (1-34). All animals were euthanized 6 weeks after surgery. The L4-L7 spinal segment was removed and assessed using manual palpation, computed tomography (CT), and biomechanical testing. CT assessments revealed fusion in 50% of autograft controls, 75% of autograft PTH (1-34) animals, 87.5% in the 2 groups treated with low-dose rhBMP-2, and 100% in the 2 groups treated with high-dose rhBMP-2. CT volumetric analysis demonstrated that all groups treated with biologics had fusion masses that were on average significantly larger than those observed in the control group (P < 0.0001). Biomechanical data demonstrated no statistical difference between controls, PTH (1-34), and low-dose rhBMP-2 in any testing orientation. PTH (1-34) did not increase bending stiffness when used adjunctively with either low-dose or high-dose rhBMP-2. Although intermittent teriparatide administration results in increased fusion mass volume, it does not improve biomechnical stiffness over use of autograft alone. When delivered concurrently with high- and low-dose rhBMP-2, teriparatide provided no statistically significant improvement in biomechanical stiffness. N/A.

Evaluating the stability of slope reinforced with one row of free head piles

In this paper, the stability analysis of slope reinforced with one row of free head piles is studied by shear strength reduction method, with software of ABAQUS. The effects of pile location, pile length, pile spacing, pile bending stiffness, and slope angle on the performance of reinforced slope are also analyzed. The optimal pile position is found to be located slightly upper of the middle of the slope. The critical pile length, after which the factor of safety will be remained constant, is dependent on the pile spacing, pile bending stiffness, and the slope angle. The results show that the factor of safety increases with decreasing pile spacing and increasing bending stiffness of the piles. Furthermore, stabilizing piles can influence more on the increase of the factor of safety of the reinforced slope as the slope angle is reduced. The equivalent shear strength parameters of the soil for unreinforced slope, with the factor of safety similar to the mode of reinforced slope, are also proposed in the present study that can be advantageous before conducting a detailed design of a reinforced slope with piles.

Mechanical and Dynamical Properties of Racing Snowboards and their Modification by Different Binding Plates

One-piece binding plates in snowboard racing have initially been introduced at the Olympics in 2006 by the later gold medalist. Today different types of such binding plates are used and play an important role to setup the equipment according to individual preferences. This study aims to quantify to what extent different binding plates modify the mechanical and dynamical properties of racing snowboards. The mechanical and dynamical properties of five racing snowboards with and without binding plates were characterized by laboratory measurements of the following parameters: Overall bending stiffness, bending stiffness distribution, torsional stiffness of rear and front body, force distribution along the running base, natural frequencies and their damping ratios. An increase in bending stiffness of 3.5% to 10.7% was measured for the different binding plates. Concerning torsional stiffening, large differences were found between the tested items of 0% to 19.7% for the front body and 2.6% to 35.1% for the rear body of the snowboard-binding plate systems. Free vibration tests showed a strong increase in damping for 4 of 5 binding plates while one plate damped distinctively less, which was also the only binding plate causing a clear change of the force distribution along the running base. While all plates cause relatively low bending stiffening, indicating a consensus about what a binding plate in snowboard racing should provide, the role of torsional stiffening and damping is probably considered controversially among manufactures and athletes as strong differences were found for these properties between the tested binding plates. It could be shown that current binding plates do partly modify the mechanical and dynamical properties of the snowboard - binding plate system to an extent that is larger than the differences between the analyzed racing snowboards itself.

Quasi-analytical solution for the stable system of the multi-layer folded graphene wrinkles

A quasi-analytical solution on the minimum length and its corresponding system energy is proposed for the stable multi-layer folded graphene wrinkles (FGWs). The quasi-analytical solution shows that: (1) at a certain threshold height, a single-layer FGW becomes energetically favorable compared to a standing graphene wrinkle. (2) All the geometrical properties of single-layer FGW reproduce in the double-layer FGWs, which is considered as the typical configuration for predicting the multi-layer folded FGWs. (3) Parametric studies show that the increased bending stiffness per length promotes the minimum graphene length while the case is reversed for the increased adhesion energy density. Both of the increased bending stiffness per length and adhesion energy density lead to the decreased system energy for the stable folded structure, while the system energy is less sensitive to the variation of adhesion energy density compared to that of the bending stiffness per length. Besides, molecular mechanics simulation shows that the present model has high accuracy on evaluating the system energy and the configuration for multi-layer FGWs.