Insight into the fractional calculus via a spreadsheet

Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

Spectroscopic vibrations of austinite (CaZnAsO4⋅OH) and its mineral structure : Implications for identification of secondary arsenic-containing mineral

Austinite (CaZnAsO4⋅OH) is a unique secondary mineral in arsenic-contaminated mine wastes. The infrared and Raman spectroscopies were used to characterize the austenite vibrations. The IR bands at 369, 790 and 416 cm−1 are assigned to the ν2, ν3 and ν4 vibrations of AsO43− unit, respectively. The Raman bands at 814, 779 and 403 cm−1 correspond to the ν1, ν3 and ν4 vibrations of AsO43− unit respectively. The sharp bands at 3265 cm−1 for IR and 3270 cm−1 both reveals that the structural hydroxyl units exist in the austenite structure. The IR and Raman spectra both show that some SO4 units isomorphically replace AsO4 in austinite. X-ray single crystal diffraction provides the arrangement of each atom in the mineral structure, and also confirms that the conclusions made from the vibrational spectra. Micro-powder diffraction was used to confirm our mineral identification due to the small quantity of the austenite crystals.

Deformation of a single red blood cell in a microvessel

Red blood cells (RBCs) are the most common type of cells in human blood and they exhibit different types of motions and deformed shapes in capillary flows. The behaviour of the RBCs should be studied in order to explain the RBC motion and deformation mechanism. This article presents a numerical simulation method for RBC deformation in microvessels. A two dimensional spring network model is used to represent the RBC membrane, where the elastic stretch/compression energy and the bending energy are considered with the constraint of constant RBC surface area. The forces acting on the RBC membrane are obtained from the principle of virtual work. The whole fluid domain is discretized into a finite number of particles using smoothed particle hydrodynamics concepts and the motions of all the particles are solved using Navier--Stokes equations. Minimum energy concepts are used to simulate the deformed shape of the RBC model. To verify the model, the motion of a single RBC is simulated in a Poiseuille flow and the characteristic parachute shape of the RBC is observed. Further simulations reveal that the RBC shows a tank treading motion when it flows in a linear shear flow.

Dynamic simulation of CFRP strengthened steel column under impact loading

Carbon fibre reinforced polymer (CFRP) strengthening of metallic structures under static loading has shown great potential in the recent years. However, steel structures are often experienced natural (e.g. earthquake, wind) as well as man-made (e.g. vehicular impact, blast) dynamic loading. Therefore, there is a growing interest among the researchers to investigate the capability of CFRP strengthened members under such dynamic conditions. This study focuses on the finite element (FE) numerical modelling and simulation of CFRP strengthened steel column under transverse impact loading to predict the behaviour and failure modes. Impact simulation process and the CFRP strengthened steel column are validated with the existing experimental results in literature. The validated FE model of CFRP strengthened steel column is then further used to investigate the effects of transverse impact loading on its structural performance. The results are presented in terms of transvers e impact force, lateral and axial displacement, and deformed shape to evaluate the effectiveness of CFRP strengthening technique. Comparisons between the bare steel and CFRP strengthened steel columns clearly indicate the performance enhancement of strengthened column under transverse impact loading.

A deformable Brodmann area atlas

Functional MRI studies commonly refer to activation patterns as being localized in specific Brodmann areas, referring to Brodmann’s divisions of the human cortex based on cytoarchitectonic boundaries [3]. Typically, Brodmann areas that match regions in the group averaged functional maps are estimated by eye, leading to inaccurate parcellations and significant error. To avoid this limitation, we developed a method using high-dimensional nonlinear registration to project the Brodmann areas onto individual 3D co-registered structural and functional MRI datasets, using an elastic deformation vector field in the cortical parameter space. Based on a sulcal pattern matching approach [11], an N=27 scan single subject atlas (the Colin Holmes atlas [15]) with associated Brodmann areas labeled on its surface, was deformed to match 3D cortical surface models generated from individual subjects’ structural MRIs (sMRIs). The deformed Brodmann areas were used to quantify and localize functional MRI (fMRI) BOLD activation during the performance of the Tower of London task [7].

Automated counting of the Northern Pacific Sea Star in the Derwent using shape recognition

A novel shape recognition algorithm was developed to autonomously classify the Northern Pacific Sea Star (Asterias amurenis) from benthic images that were collected by the Starbug AUV during 6km of transects in the Derwent estuary. Despite the effects of scattering, attenuation, soft focus and motion blur within the underwater images, an optimal joint classification rate of 77.5% and misclassification rate of 13.5% was achieved. The performance of algorithm was largely attributed to its ability to recognise locally deformed sea star shapes that were created during the segmentation of the distorted images.

A method for optimal fit of patient-specific fracture fixation plates

Anatomically pre-contoured fracture fixation plates are a treatment option for bone fractures. A well-fitting plate can be used as a tool for anatomical reduction of the fractured bone. However, recent studies showed that some plates fit poorly for many patients due to considerable shape variations between bones of the same anatomical site. Therefore, the plates have to be manually fitted and deformed by surgeons to fit each patient optimally. The process is time-intensive and labor-intensive, and could lead to adverse clinical implications such as wound infection or plate failure. This paper proposes a new iterative method to simulate the patient-specific deformation of an optimally fitting plate for pre-operative planning purposes. We further demonstrate the validation of the method through a case study. The proposed method involves the integration of four commercially available software tools, Matlab, Rapidform2006, SolidWorks, and ANSYS, each performing specific tasks to obtain a plate shape that fits optimally for an individual tibia and is mechanically safe. A typical challenge when crossing multiple platforms is to ensure correct data transfer. We present an example of the implementation of the proposed method to demonstrate successful data transfer between the four platforms and the feasibility of the method.

Numerical investigation of motion and deformation of a single red blood cell in a stenosed capillary

It is generally assumed that influence of the red blood cells (RBCs) is predominant in blood rheology. The healthy RBCs are highly deformable and can thus easily squeeze through the smallest capillaries having internal diameter less than their characteristic size. On the other hand, RBCs infected by malaria or other diseases are stiffer and so less deformable. Thus it is harder for them to flow through the smallest capillaries. Therefore, it is very important to critically and realistically investigate the mechanical behavior of both healthy and infected RBCs which is a current gap in knowledge. The motion and the steady state deformed shape of the RBCs depend on many factors, such as the geometrical parameters of the capillary through which blood flows, the membrane bending stiffness and the mean velocity of the blood flow. In this study, motion and deformation of a single two-dimensional RBC in a stenosed capillary is explored by using smoothed particle hydrodynamics (SPH) method. An elastic spring network is used to model the RBC membrane, while the RBC's inside fluid and outside fluid are treated as SPH particles. The effect of RBC's membrane stiffness (kb), inlet pressure (P) and geometrical parameters of the capillary on the motion and deformation of the RBC is studied. The deformation index, RBC's mean velocity and the cell membrane energy are analyzed when the cell passes through the stenosed capillary. The simulation results demonstrate that the kb, P and the geometrical parameters of the capillary have a significant impact on the RBCs' motion and deformation in the stenosed section.

A new approach to kimberlite facies terminology using a revised general approach to the nomenclature of all volcanic rocks and deposits: Descriptive to genetic

Although kimberlite pipes/bodies are usually the remains of volcanic vents, in-vent deposits, and subvolcanic intrusions, the terminology used for kimberlite rocks has largely developed independently of that used in mainstream volcanology. Existing kimberlite terminology is not descriptive and includes terms that are rarely used, used differently, and even not used at all in mainstream volcanology. In addition, kimberlite bodies are altered to varying degrees, making application of genetic terminology difficult because original components and depositional textures are commonly masked by alteration. This paper recommends an approach to the terminology for kimberlite rocks that is consistent with usage for other volcanic successions. In modern terrains the eruption and emplacement origins of deposits can often be readily deduced, but this is often not the case for old, variably altered and deformed rock successions. A staged approach is required whereby descriptive terminology is developed first, followed by application of genetic terminology once all features, including the effects of alteration on original texture and depositional features, together with contact relationships and setting, have been evaluated. Because many volcanic successions consist of both primary volcanic deposits as well as volcanic sediments, terminology must account for both possibilities.

Using axially loaded MRI to investigate the biomechanics of adolescent idiopathic scoliosis

Introduction. Spinal flexibility measurement is an important aspect of pre-operative clinical assessment in the treatment of Adolescent Idiopathic Scoliosis (AIS). Clinically, curve flexibility is a combined measure for all vertebral levels. We propose that in vivo flexibility for individual spinal joints could provide valuable additional information in planning treatment for scoliosis. Methods. Individual spinal joint flexibility in the coronal plane was measured for a series of AIS patients using axially loaded magnetic resonance imaging. Each patient underwent magnetic resonance imaging in the supine position, with no axial load, and then following application of an axial compressive load equal to half the patient’s bodyweight. Coronal plane disc wedge angles in the unloaded and loaded configurations were measured. Joint moments exerted by the axial compressive load were used to derive estimates of individual joint compliance. Results. Fifteen AIS patients were included in the study (mean clinical Cobb angle 46 degrees, mean age 15.3 years). Mean intra-observer measurement error for endplate inclination was 1.6˚. The mean increase in measured major Cobb angle between unloaded and loaded scans was 7.6˚. For certain spinal levels (+2,+1,-2 relative to the apex) there was a statistically significant relationship between change in wedge angle under load and initial wedge angle, such that initially highly wedged discs demonstrated a smaller change in wedge angle than less wedged discs. Highly wedged discs were observed near the apex of the curve, which corresponded to lower joint compliance in the apical region. Conclusion. Approaches such as this can provide valuable biomechanical data on in vivo spinal biomechanics in AIS. Knowledge of individual joint flexibility may assist surgeons to determine which spinal procedure is most appropriate for a patient, which levels should be included in a spinal fusion and the relative mobility of individual joints in the deformed region of the spine.

Microstructural characterization of ultrafine-grain interstitial-free steel by X-ray diffraction line profile analysis

This paper highlights the microstructural features of commercially available interstitial free (IF) steel specimens deformed by equal channel angular pressing (ECAP) up to four passes following the route A. The microstructure of the samples was studied by different techniques of X-ray diffraction peak profile analysis as a function of strain (epsilon). It was found that the crystallite size is reduced substantially already at epsilon=2.3 and it does not change significantly during further deformation. At the same time, the dislocation density increases gradually up to epsilon=4.6. The dislocation densities estimated from X-ray diffraction study are found to correlate very well with the experimentally obtained yield strength of the samples.

Mechanical processing and microstructural control in hot working of hot isostatically pressed P/M IN-100 superalloy

The hot deformation behavior of hot isostatically pressed (HIPd) P/M IN-100 superalloy has been studied in the temperature range 1000-1200 degrees C and strain rate range 0.0003-10 s(-1) using hot compression testing. A processing map has been developed on the basis of these data and using the principles of dynamic materials modelling. The map exhibited three domains: one at 1050 degrees C and 0.01 s(-1), with a peak efficiency of power dissipation of approximate to 32%, the second at 1150 degrees C and 10 s(-1), with a peak efficiency of approximate to 36% and the third at 1200 degrees C and 0.1 s(-1), with a similar efficiency. On the basis of optical and electron microscopic observations, the first domain was interpreted to represent dynamic recovery of the gamma phase, the second domain represents dynamic recrystallization (DRX) of gamma in the presence of softer gamma', while the third domain represents DRX of the gamma phase only. The gamma' phase is stable upto 1150 degrees C, gets deformed below this temperature and the chunky gamma' accumulates dislocations, which at larger strains cause cracking of this phase. At temperatures lower than 1080 degrees C and strain rates higher than 0.1 s(-1), the material exhibits flow instability, manifested in the form of adiabatic shear bands. The material may be subjected to mechanical processing without cracking or instabilities at 1200 degrees C and 0.1 s(-1), which are the conditions for DRX of the gamma phase.

Some Considerations on the Occurrence of Intergranular Fracture during Fatigue Crack growth in Steels

The occurrence of a maximum in the percentage of intergranular fracture on the fracture surface during the transition from intermediate to low fatigue crack growth rates has been observed for a high strength steel. It is suggested that transgranular planar slip leading to slip localization is essential in promoting intergranular fracture when the cyclic plastic zone size becomes equal to the prior austenite grain size.

Evolution of crystallographic texture and microstructure in the orthorhombic phase of a two-phase alloy Ti-22Al-25Nb

Evolution of crystallographic texture in the orthorhombic phase of a two-phase alloy Ti–22Al–25Nb (at%), consisting of orthorhombic (O) and bcc (β/B2) phases, was studied. The material was subjected to deformation in two-phase field as well as in the single β phase field. The resulting evolution of microstructure and crystallographic texture were recorded using scanning electron microscopy and X-ray diffraction. The orthorhombic phase underwent change in morphology (from platelets to equiaxed) on rolling in the two-phase field with the texture getting sharper with the amount of deformation. Rolling above β transus temperature led to hot deformation of single β phase microstructure and its subsequent cooling produced transformed coarse platelets of orthorhombic phase with texture in orientation relation with the high temperature deformed β phase.

Role of free volume in strain softening of as-cast and annealed bulk metallic glass

Plasticity in amorphous alloys is associated with strain softening, induced by the creation of additional free volume during deformation. In this paper, the role of free volume, which was a priori in the material, on work softening was investigated. For this, an as-cast Zr-based bulk metallic glass (BMG) was systematically annealed below its glass transition temperature, so as to reduce the free volume content. The bonded-interface indentation technique is used to generate extensively deformed and well defined plastic zones. Nanoindentation was utilized to estimate the hardness of the deformed as well as undeformed regions. The results show that the structural relaxation annealing enhances the hardness and that both the subsurface shear band number density and the plastic zone size decrease with annealing time. The serrations in the nanoindentation load-displacement curves become smoother with structural relaxation. Regardless of the annealing condition, the nanohardness of the deformed regions is similar to 12-15% lower, implying that the prior free volume only changes the yield stress (or hardness) but not the relative flow stress (or the extent of strain softening). Statistical distributions of the nanohardness obtained from deformed and undeformed regions have no overlap, suggesting that shear band number density has no influence on the plastic characteristics of the deformed region.