Torsional Rigidity Research Articles

Elastic properties of 3D printed clavicles are closer to cadaveric bones of elderly donors than commercial synthetic bones

Synthetic bone models have increasing utility in orthopaedic research due to their low cost and low variability and have been shown to be biomechanically equivalent to human bones in a variety of ways. The rise in additive manufacturing (AM) for orthopaedic applications presents an opportunity to construct synthetic whole-bone models for biomechanical testing applications, but there is a lack of research comparing these AM models to cadaveric or commercially available bone surrogates. This study compares the mechanical properties of 3D printed clavicle models to commercially available (4th generation Sawbones) and human cadaveric clavicles via nondestructive cyclic 4-point bending, axial compression, and torsion, and a final axial compression test to failure. Commercially available synthetic clavicles had 57.8–203% higher superior-inferior bending rigidity (p < 0.0001), 80.9–198% higher axial stiffness (p < 0.001), and 314–557% higher torsional rigidity (p < 0.05) on average than AM and cadaveric clavicles. Cadaveric and AM clavicles printed from a BoneMatrix/VeroWhite composite material had similar failure mechanisms under axial compression while AM VeroWhite clavicles experienced catastrophic failure, but these groups did not have significantly different ultimate failure loads. Together, these results demonstrate that current commercially available synthetic clavicles may be too rigid to emulate the mechanical properties of elderly cadaveric clavicles, and that AM bone models can closely mimic these cadaveric bones in a variety of biomechanical loading schemes. These results show promising applications for future work using 3D printed bone surrogates for biomechanical analysis of orthopaedic implants and other surgical repair techniques.

Bending response of cover-plated CFS built-up sections: Testing, numerical parametric analysis and design

This study investigates the bending response of cover-plated cold-formed steel (CPCFS) built-up sections comprising of plain channel members under pure bending achieved through four-point loading, both experimentally and numerically. Channels were appropriately spaced back-to-back and screw-fastened at the flanges using cover-plates on the tension and compression sides. This novel design for the CPCFS built-up section combines the benefits of box-profile (closed sections) and I-profile (open sections), including high strength-to-weight ratio, simplified fabrication, and enhanced torsional rigidity. Eight CPCFS built-up sections with different cross-sectional aspect ratios were tested for bending about the major axis under simply supported end conditions. Flexural strengths, failure modes, and moment-curvature plots were discussed. Subsequently, a finite element model was developed in ABAQUS and validated against the test response obtained. Afterwards, an extensive parametric analysis was conducted to assess the variational effect of aspect ratios, cross-sectional slenderness, and plate slenderness on the ultimate moment capacity of the CPCFS built-up beams. Finally, the adequacy of the codified Continuous Strength Method (CSM) and Direct Strength Method (DSM) for CPCFS built-up beams was evaluated. Generally, it was observed that the flexural strengths predicted by the codified CSM and DSM for CPCFS built-up beams were unconservative and conservative, respectively. Suitable modifications were proposed to both DSM and CSM to accurately predict the flexural strength of CPCFS built-up beams, which were eventually verified through a reliability analysis.

АНАЛІЗ ВПЛИВУ КУТА СТРІЛОПОДІБНОСТІ НА ХАРАКТЕРИСТИКИ СТАТИЧНОЇ АЕРОПРУЖНОСТІ КРИЛА НАВЧАЛЬНО-ТРЕНУВАЛЬНОГО ЛІТАКА

The article is dedicated to the analysis of static aeroelastic phenomena in the wing of a trainer aircraft and the general influence of wing sweep on critical speeds. Aeroelasticity studies the interaction between aerodynamic and elastic forces and the impact of this interaction on the aircraft's structure. There are two types of aeroelasticity: static and dynamic. The article reviews and analyzes publications related to this issue. The analysis of the research shows that when designing an aircraft, it is necessary to determine the aeroelastic characteristics of its aerodynamic surfaces. In addition to structural rigidity, the sweep angle also influences these phenomena. The scientific novelty of the study lies in the fact that, unlike the analyzed research, the designed wing structure of a trainer aircraft is considered.For the designed trainer aircraft, the critical speeds of static aeroelastic phenomena were determined. The calculation of the influence of the wing sweep angle on the divergence speeds of the trainer aircraft confirms previously obtained results regarding this influence. When the sweep is increased or decreased for wings with the same span, the torsional rigidity of the wing decreases due to the increase in the length of the line of centers of rigidity. However, the most significant impact on the aeroelastic characteristics is made by the shift of the bending center when the sweep angle changes.With forward sweep, the arm of the aerodynamic forces relative to the bending center increases, thereby reducing the critical divergence speeds. For straight sweep, as the sweep increases, the bending center moves forward, and at a certain point, the bending center is located in front of the aerodynamic center. This means that the wing twist angle decreases as the aerodynamic force increases, which, in the considered planar case, eliminates the occurrence of divergence.The preliminarily determined maximum flight speed does not exceed the critical speeds of divergence and aileron reversal. This result represents the practical value of the study. To achieve higher flight speeds, wings with straight sweep should be used. More precise computational or experimental methods for determining aeroelastic characteristics should be used in the subsequent design stages to prevent aeroelastic phenomena across the entire operational range of speeds and altitudes of the trainer aircraft.

Microfilm Coatings: A Biomaterial-Based Strategy for Modulating Femoral Deflection.

Wear on the surface of the femoral head increases the risk of hip and femur fractures. Biomechanical experiments conducted on the femur are based on its bending and torsional rigidities. Studies regarding the deflection of the femur bone when the femoral head is coated with microfilms composed of durable and compatible biomaterials are poor. This study aimed to investigate the effects of different biomaterial microfilm coatings over the femoral head on the deflection of the human femur. We utilized 2023 R1 finite element analysis (FEA) software to model the directional deformation on the femoral head and examine the femur's deflection with varying microfilm thicknesses. The deflection of the femur bone was reported when the femoral head was uncoated and coated with titanium, stainless steel, and pure gold microfilms of different thicknesses (namely, 50, 75, and 100 μm). Our results show that the femur's minimum and maximum deflection occurred for stainless steel and gold, respectively. The deformation of the femur was lower when the femoral head was coated with a 50-micrometer microfilm of stainless steel, compared to the deformation obtained with gold and titanium. When the thickness of the microfilm for each of the materials was increased, the deformation continued to decrease. The minimum deformation of the femur occurred for a thickness of 100 μm with stainless steel, followed by titanium and gold. The difference in the directional deformation of the femur between the materials was more significant when the coating was 100 μm, compared to the thicknesses of 50 and 75 μm. The findings of this study are expected to significantly contribute to the development of advanced medical techniques to enhance the quality of life for patients with femur bone-related issues. This information can be used to develop more resilient coatings that can withstand wear and tear.

Multi-phased solutions of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion and the general rule of swapping

Based on the conventional solution of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion, one can derive displacements, shear strains, shear stresses and torsional rigidity. The conventional solution of Prandtl's stress function has a hyperbolic function for the coordinate in the bar's thickness direction, and a trigonometric function for the coordinate in the width direction. This paper raises questions about the solution. Why is the solution not arranged in the opposite way? Why is the hyperbolic function not for the coordinate in the width direction and the trigonometric function for the coordinate in the thickness direction? How is it that these obvious questions have never been addressed? This study rearranges the solution of the conventional Prandtl's stress function using the TSAI technique and finds that the solution is multi-phased, indicating that the coordinates in the width and thickness directions and their corresponding parameters are swappable, a phenomenon proposed as the general rule of swapping.

New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar

New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar

Computed tomography-derived structural analysis for the likelihood of pathologic fracture in canine antebrachial osteosarcoma.

Determining the risk of pathologic fracture in dogs with a primary bone tumor would aid in case selection for in-situ treatment options. Prior research found strong relationships between in vitro strength of canine antebrachii with primary bone tumors and CT-derived metrics. This study assesses the prognosis for pathologic fracture in dogs with distal radial bone tumors using CT-derived structural analysis metrics. CT images of the antebrachium in dogs with aggressive osseous lesions of the radius were used to calculate structural rigidity and failure forces, including axial rigidity (AR), craniocaudal bending rigidity (BR), torsional rigidity (TR), and failure forces for a slightly-curved/asymmetric beam (Fs) or a curved beam (Fc). Metrics were compared with the clinical outcome of radial fracture. Eight of 19 dogs with CT-derived metrics developed a radial fracture. The prognostic potential of the metrics to discriminate fractured and nonfractured bones was analyzed using receiver operating characteristic curves (area under the curve), stepwise logistic regression, and classification regression (CART) analyses. Fc was the most sensitive and specific metric for prognosing fracture occurrence (AUC=0.864). When dog body weight (BW) was included, all five metrics had AUC>0.705. Fc was the best predictor of fracture using stepwise logistic regression and CART analysis, followed by BR. An indication of fracture probability can be determined by normalizing Fc or BR with dog BW or by using the logistic regression equation of either metric with dog BW. Results warrant further analysis of a larger cohort to evaluate fracture likelihood in dogs with antebrachial bone neoplasia.

Validation of the Chicken Femur as a Model for the Human Metacarpal: An In-Vitro Analysis.

Background: The aim of this study was to evaluate the chicken femur as a laboratory model for the human metacarpal by comparing the bone microarchitecture and mechanical properties of chicken femurs to human cadaveric metacarpals. Methods: Sixteen fresh chicken femora and 20 fresh frozen cadaveric human metacarpals were imaged using a micro computed tomography scanner. The bones were then mechanically tested using four-point-bending and torsional testing. Results: There were no significant differences in macroscopic features between chicken femora and human metacarpals, including overall length, external radius, internal radius, cortical width and cross-sectional area of the diaphyseal cortex (p > 0.05). There were no significant differences in the trabecular number and spacing in the distal metaphysis of both groups (p > 0.05). The diaphysis and proximal metaphysis did not share any microarchitectural similarities. Four-point bending tests resulted in significantly higher yield forces, ultimate force, failure points and stiffness in human metacarpals (p < 0.05). Torsion tests resulted in significant higher ultimate torque and torsional rigidity in human metacarpals (p < 0.05). Conclusions: The chicken femur has structural and biomechanical differences to the fresh frozen human metacarpal despite the similarity in their macroscopic features.

Pure distortion of symmetric box beams with hinged walls

This paper is concerned with the analysis of the pure torsion and distortion of straight box beams with trapezial cross-sections and hinged walls. A one-dimensional mechanical model for this kind of system subjected to anti-symmetric loads on the end cross-sections and no warping constraints is developed. The distortional stiffness of the system is provided by the torsional rigidity of the wall panels. The cross-sectional kinematic condition for which torsion and distortion are uncoupled has been determined. Novel explicit expressions of the internal and external distortional moments, the distortion constant, and the distortional warping pattern have been deduced; they can be directly translated to the classical distortion theory. Results of representative test cases with different section shapes and loads show excellent agreement with finite element models using shell elements. The model is a first step to analyse bridge decks with a distortionable central cell for wind engineering applications. Finally, an extension of the model, including the distortional stiffness provided by the frame bending stiffness of the cross-section walls, is presented. The extended model is applicable to assess the large-scale torsional–distortional effects in long beams with closed cross sections.

Buckling behavior and failure mechanism of cold-formed steel built-up special shape cross-section columns

This paper introduces a new type of cold-formed steel (CFS) built-up closed cross-section (CBC-C) column to connect multi-directional wall panels easily. Axial compression experiments and numerical analysis on 18 columns as well as theoretical calculations are carried out to evaluate the stability and bearing capacity of the proposed column. The effect of factors such as width-to-thickness, slenderness ratios, and screw spacing on the buckling behavior and failure mechanism was studied. As the slenderness ratio increased, the ultimate bearing capacity of specimens with the same section decreased. For specimens with a web width of 140 mm, the ultimate strength increased by 7.35 % when the slenderness ratio was reduced from 26.7 to 3.2. It was observed that the failure modes of short and intermediate long columns were generally local and distortion-related buckling, while the long columns experienced local, distortional, and finally global buckling when the slenderness ratio reached 36.73. The local buckling load and ultimate strength of specimens of the same length and section were reduced with the growth of the width-to-thickness ratio. On average, as the web width-to-thickness ratio increased by 10 %, the local buckling load decreased and the ultimate bearing capacity decreased by 22.37 % and 14.79 %, respectively. It was found that changing the section thickness had a greater effect on the stability and strength of the specimens compared to changing the web width of the CBC-C column. The mechanical properties of the CBC-C were not affected by the screw spacing and the differences among data were less than 5 %. Finally, the effective width method (EWM) and the direct strength method (DSM) in the AISI S100-16 wereassessed for CBC-C. The novel built-up column proposed in this paper has advantages in modular structures for their flexibility in configurations as well as high torsional rigidity and good stability.

Structural Evaluation of a Three-Wheeler VehicleUsing Simulation Tools: A Case Study in Mexico

The mototaxi, a three-wheeled vehicle equipped with a roof, is a widely utilized mode of transportation in Mexico. Typically, it is employed for short-distance journeys in exchange for payment, similar to the operation of a conventional taxi. This study conducts astructural analysis of a mototaxi-type vehicle utilized in Mexico to assess its performance and safety. It underscores the significance of this mode of transportation, widely relied upon by numerous individuals. A product design and development methodology was employed, utilizing torsional deformation simulations to validate the new geometry. The objective was to minimize torsions as much as possible, thereby enhancing the motorcycle taxi’s safety and ensuring the vehicle’s correct positioning. Through computeraided design, the prevailing torsions within the casing were assessed, establishing the operating conditions to which the system is commonlysubjected. The findings from the chosen vehicular structure reveal a flexural rigidity of 6,508.15 N/mm, torsional rigidity of 27.35 KNm/°, and a range of natural frequencies between 8-21 Hz. These values indicate favorable resistance against bending forces and operational frequency. However, the torsional results exhibit deficiencies, suggesting an unsafe structure for all motorcycle taxi occupants. Consequently, technology developers and national legislators should prioritize enhancing the structural integrity of such vehicles.

Computed tomography-based structural rigidity analysis can assess tumor- and treatment-induced changes in rat bones with metastatic lesions

BackgroundBreast cancer (BrCa) is a predominant malignancy, with metastasis occurring in one in eight patients, nearly half of which target the bone, leading to serious complications such as pain, fractures, and compromised mobility. Structural rigidity, crucial for bone strength, becomes compromised with osteolytic lesions, highlighting the vulnerability and increased fracture risk in affected areas. Historically, two-dimensional radiographs have been employed to predict these fracture risks; however, their limitations in capturing the three-dimensional structural and material changes in bone have raised concerns. Recent advances in CT-based Structural Rigidity Analysis (CTRA), offer a promising, more accurate non-invasive 3D approach. This study aims to assess the efficacy of CTRA in monitoring osteolytic lesions’ progression and response to therapy, suggesting its potential superiority over existing methodologies in guiding treatment strategies.MethodsTwenty-seven female nude rats underwent femoral intra-medullary inoculation with MDA-MB-231 human breast cancer cells or saline control. They were divided into Control, Cancer Control, Ibandronate, and Paclitaxel groups. Osteolytic progression was monitored weekly using biplanar radiography, quantitative computed tomography (QCT), and dual-energy X-ray absorptiometry (DEXA). CTRA was employed to predict fracture risk, normalized using the contralateral femur. Statistical analyses, including Kruskal-Wallis and ANOVA, assessed differences in outcomes among groups and over time.ResultsBiplanar radiographs showed treatment benefits over time; however, only certain time-specific differences between the Control and other treatment groups were discernible. Notably, observer subjectivity in X-ray scoring became evident, with significant inter-operator variations. DEXA measurements for metaphyseal Bone Mineral Content (BMC) did not exhibit notable differences between groups. Although diaphyseal BMC highlighted some variance, it did not reveal significant differences between treatments at specific time points, suggesting a limited ability for DEXA to differentiate between treatment effects. In contrast, the CTRA consistently demonstrated variations across different treatments, effectively capturing bone rigidity changes over time, and the axial- (EA), bending- (EI), and torsional rigidity (GJ) outcomes from the CTRA method successfully distinguished differences among treatments at specific time points.ConclusionTraditional approaches, such as biplanar radiographs and DEXA, have exhibited inherent limitations, notably observer bias and time-specific inefficacies. Our study accentuates the capability of CTRA in capturing real-time, progressive changes in bone structure, with the potential to predict fractures more accurately and provide a more objective analysis. Ultimately, this innovative approach may bridge the existing gap in clinical guidelines, ushering in enhanced Clinical Decision Support Tool (CDST) for both surgical and non-surgical treatments.

Variational Worn Stones

We introduce an evolution model à la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, a principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of the existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball.

Assessment of structural mechanical effects related to torsional deformations of propellers

Lifting propellers are of increasing interest for Advanced Air Mobility. All propellers and rotors are initially twisted beams, showing significant extension–twist coupling and centrifugal twisting. Torsional deformations severely impact aerodynamic performance. This paper presents a novel approach to assess different reasons for torsional deformations. A reduced-order model runs large parameter sweeps with algebraic formulations and numerical solution procedures. Generic beams represent three different propeller types for General Aviation, Commercial Aviation, and Advanced Air Mobility. Simulations include solid and hollow cross-sections made of aluminum, steel, and carbon fiber-reinforced polymer. The investigation shows that centrifugal twisting moments depend on both the elastic and initial twist. The determination of the centrifugal twisting moment solely based on the initial twist suffers from errors exceeding 5% in some cases. The nonlinear parts of the torsional rigidity do not significantly impact the overall torsional rigidity for the investigated propeller types. The extension–twist coupling related to the initial and elastic twist in combination with tension forces significantly impacts the net cross-sectional torsional loads. While the increase in torsional stiffness due to initial twist contributes to the overall stiffness for General and Commercial Aviation propellers, its contribution to the lift propeller’s stiffness is limited. The paper closes with the presentation of approximations for each effect identified as significant. Numerical evaluations are necessary to determine each effect for inhomogeneous cross-sections made of anisotropic material.

Seismic performance of recentering energy dissipation bracing with pendulum in prefabricated steel frame structure systems

This research introduces the recentering energy dissipation bracing with a pendulum in prefabricated steel frame (REBP-PSF) structures. The primary objective of this study is to mitigate potential severe damage to steel frames during seismic events and to expedite postearthquake rehabilitation. This innovative system harnesses energy dissipation through friction between structural members and facilitates recentering through the cable pretensioning force. Consequently, this may make postearthquake repairs minimal or even unnecessary. Three REBP-PSF structures and a prefabricated steel frame with cantilever beam splices (PSF-CBS) were meticulously modeled using the ABAQUS finite element method to evaluate the seismic performance of this novel system. The study investigated the effects of pretensioning parameters and varied structural morphologies on seismic performance metrics, including failure modes, hysteresis performance, energy dissipation capacity, and residual displacements. This paper introduced a simplified method to simulate the restoring force model of REBP using the connection element method. Subsequently, a model for the REBP-RSF overall structure was established, enabling nonlinear dynamic time-history analysis, which was then compared with the rigid steel frame (RSF) structure. The results showed that the REBP-PSF structure outperformed the PSF-CBS in terms of initial stiffness, yield load, ultimate load, and energy dissipation capability. The bearing and energy dissipation capacities of the REBP-PSF structure were enhanced by augmenting the cable pretensioning force within this structure. The proposed connection element method could accurately simulate the hysteretic behavior of REBP. Compared with the RSF structure, the REBP-RSF demonstrated superior seismic performance, characterized by small story drift and residual displacements during seismic activities. Furthermore, a notable augmentation was observed in the lateral stiffness and torsional rigidity of the proposed structure.

Elastic medium and torsional spring effects on the nonlocal dynamic of functionally graded porous nanotubes

In this study, Eringen’s nonlocal elasticity theory that applies the small size effects in functionally graded porous nanotubes embedded in an elastic matrix is discussed. The material properties of functionally graded porous nanotubes are taken into account to vary over the radius direction with a rule of mixture. The free torsional vibration relation according to nonlocal elasticity theory, via Hamilton’s principle, is obtained and an eigenvalue solution is constructed for the free torsional vibration response of the presented work. The presented analytical model is validated by comparing the calculated mathematical results for homogeneous nanotubes with rigid and non-rigid boundary conditions. Special attention is given to deformable boundary conditions, porosity coefficient, material grading coefficient and also to the influence of elastic medium on the free torsional vibration frequencies. In this paper, it has been proven that the influence of length, elastic medium, elastic torsional spring rigidities, material grading and porosity coefficients on the vary in the torsional vibration frequency of the functionally graded nanotube is not small.

Exploring Tunable Torsional Mechanical Properties of 3D‐Printed Tubular Metamaterials

Additive manufacturing has proven to be a reliable and quick method for creating devices and tools. One example is 3D‐printed nasal swabs that require robust mechanical strength while maintaining adequate flexibility. However, there is a lack of detailed understanding on the mechanical response of tubular structures under torsion loading. The goal of this work is to understand the torsional behavior of a group of 3D‐printed tubular metamaterials. The idea of mechanical metamaterials is used and it is started with a rectangular lattice. Two main design methods are used: reinforced lattices and compliant lattices. First, diagonal reinforced beams are added to enhance the torsional rigidity. Furthermore, pre‐buckling is added to the straight beam, creating a flexible structure that increases the maximum torsional resistance. The experiments demonstrate that the sponge‐inspired design exhibits enhanced stiffness approximately 21 times that of baseline tubular designs. Moreover, the strength and toughness of the sponge‐inspired design are approximately 2.5 and 3 times higher. By curving the beam, compliant tubulars are created. The torsional angle is increased by at least 2.7 times, while the torsional toughness can reach up to 17 times. Using these two design strategies helps in creating tubular metamaterials with varied applications.

Torsion constants and virtual mechanical tests are valid image-based surrogate measures of ovine fracture healing.

In large animal studies, the mechanical reintegration of the bone fragments is measured using postmortem physical testing, but these assessments can only be performed once, after sacrifice. Image-based virtual mechanical testing is an attractive alternative because it could be used to monitor healing longitudinally. However, the procedures and software required to perform finite element analysis (FEA) on subject-specific models for virtual mechanical testing can be time consuming and costly. Accordingly, the goal of this study was to determine whether a simpler image-based geometric measure-the torsion constant, sometimes known as polar moment of inertia-can be reliably used as a surrogate measure of bone healing in large animals. To achieve this, postmortem biomechanical testing and microCT scans were analyzed for a total of 33 operated and 20 intact ovine tibiae. An image-processing procedure to compute the attenuation-weighted torsion constant from the microCT scans was developed in MATLAB and this code has been made freely available. Linear regression analysis was performed between the postmortem biomechanical data, the results of virtual mechanical testing using FEA, and the torsion constants measured from the scans. The results showed that virtual mechanical testing is the most reliable surrogate measure of postmortem torsional rigidity, having strong correlations and high absolute agreement. However, when FEA is not practical, the torsion constant is a viable alternative surrogate measure that is moderately correlated with postmortem torsional rigidity and can be readily calculated.

Longitudinal CT-based finite element analyses provide objective fracture healing measures in an ovine tibia model.

Measuring the healing status of a bone fracture is important to determine the clinical care a patient receives. Implantable devices can directly and continuously assess the healing status of fracture fixation constructs, while subject-specific virtual biomechanical tests can noninvasively determine callus structural integrity at single time points. Despite their potential for objectification, both methods are not yet integrated into clinical practice with further evidence of their benefits required. This study correlated continuous data from an implantable sensor assessing healing status through implant load monitoring with computer tomography (CT) based longitudinal finite element (FE) simulations in a large animal model. Eight sheep were part of a previous preclinical study utilizing a tibial osteotomy model and equipped with such a sensor. Sensor signal was collected over several months, and CT scans were acquired at six interim time points. For each scan, two FE analyses were performed: a virtual torsional rigidity test of the bone and a model of the bone-implant construct with the sensor. The longitudinal simulation results were compared to the sensor data at corresponding time points and a cohort-specific empirical healing rule was employed. Healing status predicted by both in silico simulations correlated significantly with the sensor data at corresponding time points and correctly identified a delayed and a nonunion in the cohort. The methodology is readily translatable with the potential to be applied to further preclinical or clinical cohorts to find generalizable healing criteria. Virtual mechanical tests can objectively measure fracture healing progressing using longitudinal CT scans.

The (Self-Similar, Variational) Rolling Stones

Abstract The interplay between variational functionals and the Brunn–Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems à la Firey’s worn stone problem.