The mechanical response of cellular materials with spinodal topologies

The mechanical response of cellular materials with spinodal topologies is numerically and experimentally investigated. Spinodal microstructures are generated by the numerical solution of the Cahn-Hilliard equation. Two different topologies are investigated: "solid models," where one of the two phases is modeled as a solid material and the remaining volume is void space; and "shell models," where the interface between the two phases is assumed to be a solid shell, with the rest of the volume modeled as void space. In both cases, a wide range of relative densities and spinodal characteristic feature sizes are investigated. The topology and morphology of all the numerically generated models are carefully characterized to extract key geometrical features and ensure that the distribution of curvatures and the aging law are consistent with the physics of spinodal decomposition. Finite element meshes are generated for each model, and the uniaxial compressive stiffness and strength are extracted. We show that while solid spinodal models in the density range of 30-70% are relatively inefficient (i.e., their strength and stiffness exhibit a high-power scaling with relative density), shell spinodal models in the density range of 0.01-1% are exceptionally stiff and strong. Spinodal shell materials are also shown to be remarkably imperfection insensitive. These findings are verified experimentally by in-situ uniaxial compression of polymeric samples printed at the microscale by Direct Laser Writing (DLW). At low relative densities, the strength and stiffness of shell spinodal models outperform those of most lattice materials and approach theoretical bounds for isotropic cellular materials. Most importantly, these materials can be produced by self-assembly techniques over a range of length scales, providing unique scalability

Response of trabecular bone to elevated loading frequencies

Introduction: External mechanical loading stimulates bone to proliferate and mineralize. However, the exact -specification of the loading parameters is unclear and inconsistent due to variations in experimental design and methodology. This study examined the response of trabecular bone to fatigue loading under various frequencies. Materials and methods: Six bovine lumbar (L5) vertebral bodies were sectioned in the axial plane at the midheight to expose a flat surface of cancellous bone. Testing locations identified as 1–7 were subjected to 535 cycles of sinusoidal loading from –2 N to –15 N using a 1-mm diameter indentor and a materials testing machine (Bose ELF 3200, Minnetonka, Minnnesota). Site 1 was located in the center of the vertebra with the six additional sites surrounding the center. Continuous load versus deformation data were recorded at cycle 10 and at subsequent 25 cycle intervals thereafter with the deformation data averaged across the respective cycle number for all vertebral test sites. The testing and deformation extraction process was repeated for subsequent vertebral body specimens using testing frequencies of 1 Hz, 2.5 Hz, 5 Hz, 7.5 Hz, 10 Hz, and 20 Hz. The average deformation over the number of test cycles for each frequency was subjected to nonlinear exponential. Results and Discussion: Results of the nonlinear exponential regression indicated at that at test frequencies \u3e5Hz, a single exponential was sufficient to fit the resulting deformations data, whereas dual exponential functions were required to appropriately fit the deformation data below 5 Hz. (F-test) The dual exponential functions were characterized and compared with K (fast) (rate constant of deformation for fast phase of exponential decay), K (slow) (rate constant of deformation for the slow phase of exponential decay), and Y0 (initial net deformation). The two K values for the dual exponential functions were compared to the K values for the single exponential functions using a one-way ANOVA and Tukey’s posthoc test for multiple comparisons across frequencies. When comparing K (slow) values to the single exponential (higher frequency) K values only the 20-Hz test frequency was not statistically different from the low frequency K (slow) values. The 10-Hz and 7.5-Hz test frequency K values were statistically elevated when compared with the 1 Hz, 2.5 Hz, and 5 Hz K (slow) values (P \u3c 0.05 for all comparisons). When comparing the K (fast) values to the single exponential (higher frequency) K values, there were no statistically significant differences within the high frequencies (7.5 Hz, 10 Hz, and 20 Hz) but the frequencies at or \u3c5Hz were all statistically elevated when compared with the high frequencies. The preferred dual exponential fitting at low frequencies combined with preferred single exponential fitting at higher frequencies may be indicative of altered changes in the flow of the fluid component comprising the bone structure. Conclusions: The results of this study indicate that elevated frequency loading can alter the bone response and lead to stiffening of the structure. Such a finding may explain the propensity toward increased incidence of spinal degeneration often associated with those individuals subjected to elevated frequency loading due to occupational exposures

Stiffness response of bone to elevated frequency loading

Introduction: Wolff’s Law qualitatively describes the response of bone to loading conditions. This study examines the interaction and dependence of loading frequency on the fatigue response of bone under axial load. It has been surmised that exposure to elevated frequencies may increase bone stiffness. The resulting stiffness response of bone based on work generated under sinusoidal loading over a range of frequencies was investigated. -Elucidating the response characteristics of bone will help to determine the optimal loading conditions which stimulate bone growth and may be applicable to the improvement of bone fusion regimens as well as design of bioreactor systems. Materials and Methods: A 10-mm trephine was used to extract the central core from 30 frozen thoracic (T9, T10, T11) porcine vertebral bodies (Animal Technologies Inc., Tyler, Texas). The resulting heights from each of the cylindrical specimens of cancellous bone were recorded and used to normalize the resulting deformation data under loading. Cylindrical specimens were milled to achieve parallel surfaces for loading in phosphate-buffered saline. Specimens were subjected to compressive sinusoidal fatigue loading from –2 N to –15 N for 535 cycles at randomly selected rates of 1 Hz, 2.5 Hz, 5 Hz, 7.5 Hz, and 10 Hz using a materials’ testing machine with continuous load versus deformation data acquired at cycle number 10 and at subsequent 25 cycle intervals thereafter. Stiffness at recorded cycle intervals was computed from the elastic region of the load versus deformation curve. At each axial count, the stiffness of the loading phase of the cycle was computed by determining the slope of the load versus deformation curve in the elastic region. A plot of the mean stiffness versus cycle number at each loading frequency tested was subjected to a nonlinear analysis (Prism 5.0, GraphPad Inc., San Diego, California). The resulting curve parameters of rate (K), plateau, stiffness change (span), and half-time were normalized to initial stiffness (Y0) and statistically analyzed using a one-way repeated measures ANOVA test with a Tukey posthoc test. Results and Discussion: The results of compression based on applied loading frequency are provided. The resonant frequency of the spine has been cited at 5 Hz. In addition to this 5 Hz resonant frequency, harmonics at 1 Hz and 10 Hz were also found to increase cycles required to achieve stability, as demonstrated by statistically increased half-life values seen at these loading rates. Among the half-life values, no statistically significant differences were found between the 1 Hz and 5 Hz, 1 Hz and 10 Hz, or 5 Hz and 10 Hz frequencies. All other comparisons were statistically significant (P \u3c 0.05). Conclusions: The mechanical response of bone is highly dependent on loading frequency. This study demonstrates that increased stiffness at elevated loading frequencies may lead to predisposition of pathological clinical conditions. Frequency of loading rate should be considered as stiffness changes display preferential values near 2.5 Hz, 5 Hz, and 10 Hz

ECONOMIC ACTIVITY OF CROATIAN FEMALE POPULATION

U članku se obrađuju odrednice gospodarske aktivnosti ženskoga stanovništva u Hrvatskoj u tijeku posljednjih pedesetak godina. Ta je aktivnost uspoređena sa stanjem u europskim zemljama, a posebno su izdvojena kretanja u tzv. tranzicijskim zemljama. Kao građa poslužili su podaci iz šest popisa stanovništva u Hrvatskoj od godine 1953. do 2001. Uočeno je sve veće starenje, ali i sve veća obrazovna razina stanovništva, i ukupnoga i ženskoga. Neke od zanimljivih pojava koje europski istraživači smatraju bitnima, kao što su odnos aktivnosti i bračnoga stanja, broja i dobi najmlađega djeteta, dostupni su, nažalost, samo u najstarijem popisu iz godine 1953., pa slika o ulozi i stanju žena u Hrvatskoj nije posve cjelovita i usporediva s ostatkom Europe. Promjena gospodarske strukture, među ostalim i zbog sve većega udjela sektora usluga, privlači žensku radnu snagu u aktivnost. Time se odnos među muškarcima i ženama u gospodarskoj aktivnosti mijenja u korist žena, što odgovara pojavama uočenima u razvijenim zemljama svijeta. Najuočljivija je promjena upravo sustavno smanjivanje broja aktivnih žena u poljoprivredi, dakle, i u cjelokupnom primarnome sektoru i povećavanje broja aktivnih žena u različitim kategorijama tercijarnoga sektora, poglavito trgovine, obrazovanja i zdravstvene skrbi. Sve veći udio žena koje brojem čak nešto premašuju aktivno muško stanovništvo, zahtijeva preispitivanje radnih uvjeta i njihovih svojstvenih potreba i njihovo usklađivanje, i to zato da bi one bile cjelovito uspješnije, a društvo u cjelini i bolje i pravednije. Prilagodljivi oblici rada jedan su od pristupa koji vodi tome.The subject of the article is the analysis of the economic activities trends of the Croatian female population during last 50 years. They are compared with situation in other European countries, especially transitional ones. Data from six censuses in Croatia, from 1953 till 2001, had been studied. It is noticeable that overall population, as well as female, became older and better educated. European researchers consider relationship between economic activity and marital status, or number of children and age of the youngest child very important, but they are available in the oldest census from 1953 only. This makes the picture of the female status and role in Croatian society incomplete, so the comparison with the rest of Europe is only partially possible. Changes of the economic structure caused (among other reasons) by growing services industry, attract female workforce to the activity. In this way the ratio between male and female participation in economic activity is changing in favor of female workforce – the trend noticed in developed countries. The most noticeable change is systematic decrease of number of women active in the agriculture, and primary sector in general, and increase of number of women active in different categories of tertiary sector, especially in commerce, education and healthcare. Growing share of the female workforce, which at the moment slightly exceeds active male population, demands reexamination of the working conditions and special needs, and their coordination. By understanding social processes we can find the way toward successful working women having higher quality of life and better overall social environment. Flexible work forms are one of the possible approaches to accomplish this goal

A versatile numerical approach for calculating the fracture toughness and R-curves of cellular materials

We develop a numerical methodology for the calculation of mode-I R-curves of brittle and elastoplastic lattice materials, and unveil the impact of lattice topology, relative density and constituent material behavior on the toughening response of 2D isotropic lattices. The approach is based on finite element calculations of the J-integral on a single-edge-notch-bend (SENB) specimen, with individual bars modeled as beams having a linear elastic or a power-law elasto-plastic constitutive behavior and a maximum strain-based damage model. Results for three 2D isotropic lattice topologies (triangular, hexagonal and kagome) and two constituent materials (representative of a brittle ceramic (silicon carbide) and a strain hardening elasto-plastic metal (titanium alloy)) are presented. We extract initial fracture toughness and R-curves for all lattices and show that (i) elastic brittle triangular lattices exhibit toughening (rising R-curve), and (ii) elasto-plastic triangular lattices display significant toughening, while elasto-plastic hexagonal lattices fail in a brittle manner. We show that the difference in such failure behavior can be explained by the size of the plastic zone that grows upon crack propagation, and conclude that the nature of crack propagation in lattices (brittle vs ductile) depends both on the constituent material and the lattice architecture. While results are presented for 2D truss-lattices, the proposed approach can be easily applied to 3D truss and shell-lattices, as long as the crack tip lies within the empty space of a unit cell.Comment: 40 pages, 14 figure

Catastrophic vs Gradual Collapse of Thin-Walled Nanocrystalline Ni Hollow Cylinders As Building Blocks of Microlattice Structures

Lightweight yet stiff and strong lattice structures are attractive for various engineering applications, such as cores of sandwich shells and components designed for impact mitigation. Recent breakthroughs in manufacturing enable efficient fabrication of hierarchically architected microlattices, with dimensional control spanning seven orders of magnitude in length scale. These materials have the potential to exploit desirable nanoscale-size effects in a macroscopic structure, as long as their mechanical behavior at each appropriate scale – nano, micro, and macro levels – is properly understood. In this letter, we report the nanomechanical response of individual microlattice members. We show that hollow nanocrystalline Ni cylinders differing only in wall thicknesses, 500 and 150 nm, exhibit strikingly different collapse modes: the 500 nm sample collapses in a brittle manner, via a single strain burst, while the 150 nm sample shows a gradual collapse, via a series of small and discrete strain bursts. Further, compressive strength in 150 nm sample is 99.2% lower than predicted by shell buckling theory, likely due to localized buckling and fracture events observed during in situ compression experiments. We attribute this difference to the size-induced transition in deformation behavior, unique to nanoscale, and discuss it in the framework of “size effects” in crystalline strength

Mechanical Performance of 3D Printed Interpenetrating Phase Composites with Spinodal Topologies

The mechanical response of interpenetrating phase composites (IPCs) with stochastic spinodal topologies is investigated experimentally and numerically. Model polymeric systems are fabricated by Polyjet multi-material printing, with the reinforcing phase taking the topology of a spinodal shell, and the remaining volume filled by a softer matrix. We show that spinodal shell IPCs have comparable compressive strength and stiffness to IPCs with two well-established periodic reinforcements, the Schwarz P triply periodic minimal surface (TPMS) and the octet truss-lattice, while exhibiting far less catastrophic failure and greater damage resistance, particularly at high volume fraction of reinforcing phase. The combination of high stiffness and strength and a long flat plateau after yielding makes spinodal shell IPCs a promising candidate for energy absorption and impact protection applications, where the lack of material softening upon large compressive strains can prevent sudden collapse. Importantly, in contrast with all IPCs with periodic reinforcements, spinodal shell IPCs are amenable to scalable manufacturing via self-assembly techniques

Effect of implant design and material on subsidence following dynamic loading of intervertebral devices

Introduction: Subsidence is not the consequence of a single loading event. More specifically, subsidence may be interpreted as the continuous sinking due to continuous loading. As such, a single static load may not be appropriate. Few studies involving cage subsidence have employed continuous cyclic loading. The goal of this study was to address the mechanical subsidence performance via prediction of the final subsidence depth and the rate of subsidence. It is hypothesized that those spacer designs, which engage the stronger vertebral body periphery, enable endplate stress distribution through increased contact area, and reduced stress concentrations would display more favorable performance characteristics with respect to subsidence. Materials and Methods: Three intervertebral spacer designs were evaluated; threaded titanium, endplate-sparing titanium, and Polyetheretherketone (PEEK). Devices were randomly but equally assigned to porcine L4 and L5 vertebral bodies with endplates prepared as per recommended surgical procedure. Specimens were loaded from –50 N to –350 N at 1 Hz for 600 cycles with continuous load versus deformation acquired at cycle 10 and at 25 cycle intervals thereafter. For each cycle interval, the net deformation between the maximum and minimal applied load (or subsidence) was computed. The deformation for all six samples of each design were averaged across each cycle interval and subjected to a nonlinear exponential analysis. More specifically, the subsidence is represented by the independent variable Y. Initial subsidence is calculated at the 10th loading cycle and was represented by the variable Y0. From the subsidence versus cycle data, the rate of subsidence (K) was determined as well as the plateau, or asymptotic limit, of the subsidence. All parameters were compared using a one way ANOVA with a Tukey posthoc test for determination of statistical difference (α \u3c 0.05) between designs. Results and Discussion: All implants displayed an exponential relationship with respect to the number of applied cycles. Significant differences among all three designs were determined for initial subsidence Y0 (P \u3c 0.001) and subsidence limit (plateau) (P \u3c 0.001). For both parameters, the endplate-sparing titanium device displayed the least subsidence when compared with the other designs. The subsidence rate K displayed a statistically reduced rate for the endplate-sparing titanium device when compared with the threaded or PEEK designs. (P \u3c 0.001). Clinically, such a condition results in a slow and gradual settling of the titanium implant upon the endplate surface. The PEEK implant displayed a more rapid and greater subsidence than either the endplate-sparing or threaded titanium designs. Conclusions: Under continuous loading, an endplate-sparing titanium device displayed significantly reduced initial and final subsidence and subsidence rate when compared with threaded and PEEK designs. The clinical implication of these results is that implant material modulus is not the sole determinant for subsidence. Acknowledgements: This work is the result of a sponsored research grant from Titan Spine LLC, 6140 Executive Drive, Mequon, WI 52092

Single transducer for measurement of small displacements or forces

Introduction: One of the limits to characterization of tissues at the microscopic level is the substantial costs associated with the instrumentation required for such investigations. The use of strain gages for displacement or force measurements can be adapted to this microscopic scale provided the gages are incorporated into a transducer consisting of a curved geometry. Materials and Methods: Uniaxial strain gages 13 mm (L) × 6 mm (W) were secured to 0.2-mm thick brass shim stock fabricated to yield a 12.5-mm diameter semicircle with mounting tabs on either end. The terminal ends of the strain gage mounting pads were connected to an adjustable strain gage amplifier with adjustable gain and offset. For displacement calibration, the transducer was secured to the jaws of a digital caliper. Caliper displacement was set to ±0.25 mm increments from 0 to a maximum of ±1.5 mm. In this configuration, positive represents tension. Amplifier Gains were set to 500, 1000, and 5000 to observe nonlinearity. For each of three displacement calibration runs, output voltage from the transducer was recorded at each distance with the mean output at absolute distances averaged across each of three runs for each of six transducers. Using the same transducers, a mass balance was used to identify the unique individual mass associated with a total of 10 masses to within ±10 μg. The masses were sequentially secured to the transducer and the respective output voltage recorded. Amplifier Gains were set to 500, 1000 and 5000. For each of three force calibration runs, output voltage from the transducer was recorded with respect to the force generated by the suspended masses. The mean output across each of three runs for each of six transducers was subjected to linear regression. Results and Discussion: All transducers displayed good linearity when calibrated for displacement with regression R2 values of 0.9994, 0.9967, and 0.9941 for amplifier gains of 500, 1000, and 5000, respectively. Further, in displacement mode, the transducers provided a mean output of 0.66, 1.01, and 3.86 V/mm at amplifier gains of 500, 1000, and 5000, respectively. In force mode, regression R2 in excess of 0.9995 were observed over all amplifier gain settings examined. When employed as force transducers, the devices provided mean outputs of 0.60, 1.15, and 5.85 V/N at amplifier gains of 500, 1000, and 5000, respectively. The response of these devices to either applied displacement or force permits the use of a single device type to be used as either a displacement or force transducer. The electrical output in either mode at modest amplification gains of 5000 combined with excellent linearity affords the use of these devices for studies where characterization of tissues requires mN and μm level resolution. Conclusions: A transducer employing a strain gage had been configured so as to function as either a displacement or force measuring instrument. The resulting device displays high linearity and electrical output even at modest amplifier gains