Von Mises Strain Research Articles

Biomechanical investigation of positive reduction in the femoral neck fracture: a finite element analysis.

Gotfried positive reduction offers an alternative strategy for femoral neck fracture (FNF) when achieving anatomical reduction is challenging. However, the biomechanical consequences of positive reduction remain unclear. The purpose of this study was to investigate the biomechanical behavior of positive reduction across different Pauwels classification, providing a reference for quantifying positive reduction in clinical practice. Three-dimensional (3D) models of FNF were established and categorized according to the Pauwels classifications (Pauwels I, II, and III), each of them contained seven models with different reduction qualities, including an anatomical reduction model, two negative reduction models, and four positive reduction models, all of which were stabilized with dynamic hip screws (DHS) and cannulated screws (CS). We investigated the maximal von-Mises stress of internal fixation and proximal femoral, femoral fragment displacement, and maximal von-Mises strain at the proximal fragment fracture site when a 2100N load was applied to the femoral head. The maximum von-Mises stress on the internal fixators in each Pauwels group was lowest in the anatomical reduction model. In the Pauwels I group, positive reduction exceeding 3mm resulted in the maximum von-Mises stress on the internal fixators surpassing that of the negative reduction model. For the Pauwels II group, positive reduction beyond 2mm led to the maximum von-Mises stress on the internal fixators exceeding that of the negative reduction model. In the Pauwels III group, positive reduction beyond 1mm caused the maximum von-Mises stress on the internal fixators to be higher than that of the negative reduction model. The maximum von-Mises strain at the fracture site of proximal femur fragment increased with positive reduction. Varus displacement increased in positive reduction models as the Pauwels angle rose, potentially exacerbating rotation deformity in Pauwels III group. Excessive positive reduction may increase the risk of FNF failure after internal fixation. From a biomechanical stability perspective, positive reduction should be limited to 3mm or below in the Pauwels I group, restricted to not exceed 2mm in the Pauwels II group, and should not exceed 1mm in the Pauwels III group. Negative reduction should be avoided in all Pauwels groups.

Effect of second-phase precipitates on deformation microstructure in AA2024 (Al–Cu–Mg): dislocation substructures and stored energy

The importance of comprehensive multiscale characterisation in advancing our understanding of engineering materials is undeniable but remains a challenging pursuit. Combining complimentary microstructure characterisation techniques, including transmission electron microscopy, electron backscatter diffraction and dark-field X-ray microscopy (DFXM), the formation of deformation microstructures is investigated in presence of shearable and non-shearable hardening precipitates in an industrial aluminium alloy (AA) 2024 (Al–Cu–Mg family). The alloy was used in naturally aged T3 (with shearable co-clusters and Guinier–Preston–Bagaryatsky (GPB) zones) and peak-hardened T8 (with non-shearable S-phase precipitates) states. After cold rolling with thickness reductions varying from 25 to 60% (or corresponding von Mises strain from 0.33 to 1.06), the T8 state revealed a higher sub-boundary density with slightly smaller mean disorientation angle, as compared to those in the T3 state. At a von Mises strain of 0.33, the T8 state exhibited higher long-range orientation gradients, as compared to the T3 state, for higher strain orientation gradients in T3 surpass those in T8 state. With DFXM, distinct 3D substructures are shown, revealing ellipsoidal sub-grains in the T8 state and pancake-like sub-grains in the T3 state. Moreover, the stored energy induced by cold rolling is higher for the T8 state. These results indicate different deformation microstructures, formed in the same AA2024 but hardened by shearable and non-shearable precipitates.Graphical abstract

Effects of Attachment Design and Aligner Material on Mandibular Canine Distal Bodily Movement in Aligner Treatment

Abstract Purpose Dental crowding is a result of a mismatch between tooth size and arch dimensions, which leads to malocclusion; treatment often involves premolar extraction before orthodontic alignment. Clear aligners are limited in their ability to achieve canine distal bodily movement, a common orthodontic maneuver. This study investigated the impacts of attachment design and aligner material on the efficacy of canine distal bodily movement. Methods A finite element analysis was conducted to examine the impact of various attachment designs and two aligner materials, thermoplastic polyurethanes/polycarbonate (TPU/PC) and polyethylene terephthalate glycol-modified (PETG), on mandibular canine distal bodily movement. The investigation focused on the biomechanical responses in the periodontal ligament (PDL) and surrounding alveolar bone. Results Attachment configuration exerted a strong influence on mandibular canine movement. Vertically oriented attachment pairs positioned mesially (mesial occlusal–mesial cervical) resulted in the most effective canine distal bodily movement, followed by a rectangular attachment. TPU/PC aligners induced slightly higher principal stresses in the PDL and von Mises stress and strain in the surrounding alveolar bone compared with PETG aligners; however, the difference was negligible, amounting to less than 6%. Conclusion Attachment design, specifically vertically oriented pairs positioned mesially (mesial occlusal–mesial cervical), was determined to be the crucial factor influencing the efficacy of canine distal bodily movement. The choice of aligner material (TPU/PC or PETG) has minimal impact on this orthodontic procedure.

Finite element analysis of non-ultraviolet and ultraviolet-irradiated titanium implants.

The purpose of this study is to calculate von Mises stresses, von Mises strains, deformation, principal stresses and principal elastic strains of non-UV and UV-irradiated hybrid SLA (sandblasted, large-grit, acid-etched)-coated titanium implants. A cross-sectional analytical study was conducted at the Institute of Dentistry, CMH Lahore Medical College. Cone beam computed tomography (CBCT) data of One Hundred and Thirty Eight Dio Hybrid sandblasted and acid-etched implants of identical dimensions (10mm in length and 4.5mm in diameter) were allocated in the three groups. Control group A samples were not given UV irradiation, while groups B and C were given UVA (382nm, 25 mWcm-2) and UVC (260nm, 15 mWcm-2) irradiation, respectively. The CBCT data were analyzed using FEA (ANSYS software). CBCT images were taken before functional loading (8thweek) and after functional loading (26thweek). A 3-way ANOVA test was employed to see the difference between the three groups. Tukey test was utilized for multiple comparisons. p ≤ 0.05 was considered significant. The control group exhibited the highest average values for maximum von Mises stress, von Mises strain, deformation, principal stress, and principal elastic strain in both the maxilla and mandible compared to the UV-irradiated groups. Additionally, these measures consistently displayed higher averages in the maxilla across all groups compared to the mandible. Particularly, the UVC-irradiated group demonstrated the lowest von Mises stresses around the implants compared to the UVA group. Insignificant differences were observed between UVA- and UVC-irradiated implants in terms of principal stress, deformation, von Mises strain, and principal elastic strain. The only notable distinction was in von Mises stress, where the UVC-irradiated group exhibited lower von Mises stress around SLA-coated titanium implants.

Structural integrity assessment of a solid propellant grain considering confining pressure effect

Confining pressure has a significant effect on the mechanical properties of solid propellant, and it is crucial to develop a refined structural integrity analysis and assessment method considering confining pressure effect and nonlinearity of propellant material for the design of propellant grain of a solid rocket motor (SRM), especially for the design of high charge modulus. In this work, an existing viscoelastic-viscodamage constitutive model considering confining pressure effect was implemented as a user-defined material subroutine and verified using a few experiments under various confining pressures. The mechanical responses and safety factor of a typical variable cross-section propellant charge SRM with various charge modulus under ignition pressurization load and room temperature were analyzed and evaluated. The results show that the user-defined material subroutine can well describe nonlinear mechanical responses of solid propellant under different experimental conditions. The Von Mises stress and minimum safety factor considering the confining pressure effect are higher than the value without considering the confining pressure effect, while the Von Mises strain is opposite. In addition, regardless of whether the confining pressure effect is considered or not, the maximum Von Mises stress and strain show a nonlinear increasing relationship with the charge modulus, and the minimum safety factor decreases with charge modulus. However, when analyzing the structural integrity of high charge modulus, using two evaluation methods will yield completely different evaluation results. It is believed that this research can support structural integrity analysis and design of high charge modulus SRM.

Molecular dynamics study on nanodust removal strategies from nanotrench structures

The process of removing nanodust adhering to nanostructured surfaces is important for improving yield and sustainability in manufacturing fields that utilize the structural characteristics of patterns. In this study, we computationally proposed a method to remove nanodust from a trench-shaped Si substrate using an atomic force microscopy tip. A molecular system is thus constructed in which a Si substrate with a trench structure, a cylindrical C tip, and a spherical Cu nanoparticle coexist. By moving the C tip in a specific direction and pushing it into direct contact with the Cu nanoparticle, the interaction force between the Cu and the sidewall of the Si substrate can be substantially reduced. In particular, moving the C tip in a direction parallel to the pattern can effectively push nanodust away while minimizing damage to the Si substrate. The damage experienced by the substrate during the process was comprehensively evaluated using the von Mises stress, adhesion energy, contact area, and strain energy density developed by the axial stress components. The suitability of this method was systematically evaluated based on the diameter of the Cu nanoparticle and its position relative to the C tip. For each case, the maximum strain energy experienced at the edge of the trench structure was quantified, which suggests appropriate nanoparticle removal process conditions from a structural perspective. The strategy can be potentially used as a repair method for semiconductor production components, such as pellicles and hard masks.

Structural response analysis of a solid rocket motor charge under temperature and internal pressure loads

To study the structural integrity of a solid rocket motor under low temperature load, internal pressure load, and two combined loads, based on the linear viscoelastic model, the structural response of a solid rocket motor under low temperature load, internal pressure load and two combined loads were analyzed by ANSYS finite element software, and the temperature field and strain field of charge loading were obtained. The Von Mises strain distribution of the solid rocket motor along the axial characteristic line is obtained. The results are: Under the combined action of low temperature load, internal pressure load, and two kinds of loads, the maximum Von Mises strain appears at the transition position between the blade joint and the barrel section, which is the most dangerous position for charging.

Full-Field Strain Measurements of the Muscle-Tendon Junction Using X-ray Computed Tomography and Digital Volume Correlation.

We report, for the first time, the full-field 3D strain distribution of the muscle-tendon junction (MTJ). Understanding the strain distribution at the junction is crucial for the treatment of injuries and to predict tear formation at this location. Three-dimensional full-field strain distribution of mouse MTJ was measured using X-ray computer tomography (XCT) combined with digital volume correlation (DVC) with the aim of understanding the mechanical behavior of the junction under tensile loading. The interface between the Achilles tendon and the gastrocnemius muscle was harvested from adult mice and stained using 1% phosphotungstic acid in 70% ethanol. In situ XCT combined with DVC was used to image and compute strain distribution at the MTJ under a tensile load (2.4 N). High strain measuring 120,000 µε, 160,000 µε, and 120,000 µε for the first principal stain (εp1), shear strain (γ), and von Mises strain (εVM), respectively, was measured at the MTJ and these values reduced into the body of the muscle or into the tendon. Strain is concentrated at the MTJ, which is at risk of being damaged in activities associated with excessive physical activity.

Patient-specific mechanical analysis of PCL periodontal membrane: Modeling and simulation

This research fills a knowledge gap in bone tissue engineering by examining the mechanical characteristics of scaffolds at bone-tissue interfaces utilizing a cutting-edge technique involving the creation of 3D scaffolds from Polycaprolactone (PCL). The work employs Finite element analysis to measure the scaffolds' maximum principal and Von Mises stresses and strains. CT scans of the Maxilla and Mandible were used to apply load conditions to 3D models of the upper central incisor. In the derived computational model, four different load situations considered were: the masticatory load (70–100 N at 45°), two parafunctional habits (100–130 N) and 500–550 N at the incisal edge, both at 45°), and a trauma case (800–850 N applied perpendicularly from the inwards direction at 90°). The findings revealed that the central tooth region experiences the highest stress concentration, while the Maxilla and Mandible regions show the least stress. These results provide critical insights into the mechanical behavior of scaffolds at bone-tissue interfaces, suggesting a research direction for developing scaffolds that closely mimic real bone characteristics. The results of this study are particularly significant for using bone replacement materials, providing an approach to more effective healing options for bone traumas and degenerative bone disorders.

Deep Learning-Based Prediction of Stress and Strain Maps in Arterial Walls for Improved Cardiovascular Risk Assessment

Conducting computational stress-strain analysis using finite element methods (FEM) is a common approach when dealing with the complex geometries of atherosclerosis, which is a leading cause of global mortality and complex cardiovascular disease. The considerable expense linked to FEM analysis encourages the substitution of FEM with a considerably faster data-driven machine learning (ML) approach. This study investigated the potential of end-to-end deep learning tools as a more effective substitute for FEM in predicting stress-strain fields within 2D cross sections of arterial walls. We first proposed a U-Net-based fully convolutional neural network (CNN) to predict the von Mises stress and strain distribution based on the spatial arrangement of calcification within arterial wall cross-sections. Further, we developed a conditional generative adversarial network (cGAN) to enhance, particularly from the perceptual perspective, the prediction accuracy of stress and strain field maps for arterial walls with various calcification quantities and spatial configurations. On top of U-Net and cGAN, we also proposed their ensemble approaches to improve the prediction accuracy of field maps further. Our dataset, consisting of input and output images, was generated by implementing boundary conditions and extracting stress-strain field maps. The trained U-Net models can accurately predict von Mises stress and strain fields, with structural similarity index scores (SSIM) of 0.854 and 0.830 and mean squared errors of 0.017 and 0.018 for stress and strain, respectively, on a reserved test set. Meanwhile, the cGAN models in a combination of ensemble and transfer learning techniques demonstrate high accuracy in predicting von Mises stress and strain fields, as evidenced by SSIM scores of 0.890 for stress and 0.803 for strain. Additionally, mean squared errors of 0.008 for stress and 0.017 for strain further support the model’s performance on a designated test set. Overall, this study developed a surrogate model for finite element analysis, which can accurately and efficiently predict stress-strain fields of arterial walls regardless of complex geometries and boundary conditions.

Quantifying 3D crack propagation in nodular graphite cast iron using advanced digital volume correlation and X-ray computed tomography

Advanced digital volume correlation (DVC) combined with X-ray computed tomography (XCT) was utilized to investigate the 3D crack propagation in nodular graphite cast iron under tensile loading. The objective of this work was to quantify crack growth at the voxel-scale and extract fracture mechanics parameters such as the 3D crack front, crack opening displacements (CODs), and stress intensity factors (SIFs). By employing a crack growth updating strategy and an adaptive multiscale mesh, the 3D crack shape and propagation were accurately captured throughout the entire process. Von Mises strain and COD fields were characterized to provide insights into plastic zones and variations in crack opening behavior. Estimation of SIFs revealed a mode I dominant regime and limited influence from modes II and III. This study provides comprehensive insights into crack propagation and crack opening behavior, providing valuable information to fracture mechanics.

Effects of open and minimally invasive Transforaminal Lumbar Interbody Fusion (TLIF) surgical techniques on mechanical behaviour of fused L3-L4 FSU: A comparative finite element study

For predicting the biomechanical effects of the fusion procedure, finite element (FE) analysis is widely used as a preclinical tool. Although several FE studies examined the efficacies of various fusion surgical techniques, comparative studies on Open and minimally invasive (MIS) transforaminal lumbar interbody fusion (TLIF) procedures incorporating a follower coordinate system have not been investigated yet. The current FE study evaluates the ranges of motion (ROM) and load distributions of Open-TLIF and MIS-TLIF implanted models, under physiological loading such as compression, flexion, extension and lateral bending. The most noteworthy finding from the investigation is that both the fusion procedures significantly reduced the ROMs of the implanted segment (L3-L4) and full model (L1-L5) by at least 89 % and 44 %, respectively, compared to the intact model. For all loading situations, over 95 % of the implanted models' cancellous bone volume was subjected to von Mises strains ranging from 0.0003 to 0.005. The maximum von Mises strain was observed to be localized on a small amount of cancellous bone volume (<5 %). The likelihood of adjacent segment degeneration is higher in the case of MIS-TLIF due to the higher stress (22–53 MPa) and strain (0.018–0.087) noticed on the upper facet of the L3 vertebra.

Assessment of biomechanical properties of the new two-hole miniplate internal fixation for bony mallet finger treatment by finite element analysis

Surgery is highly recommended for a bony mallet finger when the fracture fragment involves greater than one-third of the articular surface. K-wire based and plated-based internal fixation are widely used for mallet fracture. However, the outcomes of different surgical treatment options make the treatment of the bony mallet finger controversial due to frequent complications. The two-hole miniplate is a new and promising plate-based internal fixation treatment for the bony mallet finger with low complication rates in recent years. The aim of this study was to evaluate the biomechanical parameters (von Mises stress, strain and deformation) of the two-hole miniplate fixation compared to the traditional K-wire-based fixation using finite element analysis (FEA). Further, the biomechanical parameters of each part of the two-hole miniplate internal fixation were also analyzed. The results indicated that the two-hole miniplate model had the minimum von Mises stress value and the displacement of fracture fragment was less than 30 µm. The two-hole miniplate had an apparent compression effect on the avulsion fracture and inhibited the fracture displacement. This study would provide further guidance for clinical application in using the two-hole miniplate internal fixation.

Comparison of local, gradient-enhanced and integral form of continuum damage approaches to strain localization in fiber reinforced composites

Structures often have inserts and notches that cause stress concentration and consequent reduction in the load bearing capacity. The literature on numerical implementation of continuum damage mechanics (CDM) has generally focused on using local theories of damage. However, their use to analyze deformations of notched laminates that are prone to strain localization can cause numerical convergence issues that can be alleviated by employing a non-local theory. There are a few studies using a non-local damage theory for fiber-reinforced polymeric composites (FRPCs) and even fewer for strain localization in FRPCs. The non-local theories employ an equivalent (or effective or von Mises) strain often used in metal plasticity while most theories for studying failure of FRPCs use individual strain/stress components. Here, we compare predictions from local and nonlocal CDM approaches that are effective in studying strain localization in FRPCs. For the local approach, we employ a rate-dependent evolution relation. For the non-local approach, we either compute damage at a point as the weighted sum of damage at its neighbors that introduces a length scale into the problem, or employ a gradient-enhanced approach that also introduces a length scale. It is shown that numerical predictions from both the rate-dependent and the non-local theory alleviate to different degrees sensitivity of results to the finite element mesh used and the strain localization issues. Of these, a local rate-type approach is more convenient and cost effective.

Design of hexacopter and finite element analysis for material selection

PurposeThe purpose of this study is to offer a better and more effective hexacopter design for a 3 kg payload using finite element analysis (FEA), facilitating the use of different materials for different components that too without compromising strength.Design/methodology/approachA 3D computer-aided design (CAD) model of a hexacopter with a regular hexagonal frame is presented. Furthermore, a finite element model is developed to perform a structural analysis and determine Von Mises stress and strain values along with deformations of different components of the proposed hexacopter design.FindingsThe results establish that carbon fibre outperforms acrylonitrile butadiene (ABS) with respect to deformations. Within the permissible limits of the stress and strain values, both carbon fiber and ABS are suggested for different components. Thus, a proposed hexacopter offers lighter weight, high strength and low cost.Originality/valueThe use of different materials for different components is suggested by making use of static structural analysis. This encourages new research work and helps in developing new applications of hexacopter, and it has never been reported in literature. The suggested materials for the components of the hexacopter will prove to be suitable considering weight, strength and cost.

Comparative analysis of material for flat & dome top piston for SI engine using ansys

Compared to other areas of the IC engine, the working condition of piston attracts the most attention rather than any part in IC engine due to its complexity. Pistons have undergone many significant material and topographic changes since their first industrial introduction to overcome these working conditions. It is being said that electric vehicles will take over the world, but until the Lithium-ion battery remains, there will always be a shortage for batteries. Taking this as an assumption and the fact that carbon neutral fuel is currently in development, our goal is to make the piston lighter and more durable to make it future ready. In this research article, our objective is to substantiate and analyze the stress distribution of the piston with various [1] piston crown designs, namely, [1] flat top and dome top; integrated with various synthetic and alloy materials, namely, Titanium alloy, [2] Aluminum alloy, [3] Super magnesium, and [4] 3D Graphene. [5] Static Structural and Steady state thermal analysis is performed on these pistons using ANSYS R2 2022 software, while designing is performed on AutoCAD 2020 software. This article presents and analyses [5] thermal stress analysis and damages due to pressure application. Results are displayed comparing based on [5,6,7] total deformation, Von-mises strain, Von-mises stress, strain energy, normal elastic strain, normal stress, total heat flux and directional heat flux. Result is concluded by comparing to find the most preferable material for the piston in an engine. The preference is given based upon weight reduction, strength, and future scope of the material.

DYNAMIC LOADING EFFECTS ON CAR SUSPENSION COMPONENTS DURING ROAD DRIVING

Suspension system plays a vital role in providing a pleasure and safe driving experience. Because of that, studying the behaviour of suspension system during road driving is crucial. This paper describes the work carried out to investigate the dynamic loading effects on suspension components during rough road driving and analyse based on safety factor and fatigue life. A multibody simulation using SolidWorks software is conducted. Von Mises stress, strain, and displacement distribution plots are generated to visualize stress and deformation patterns. Fatigue life distribution plots indicate infinite fatigue life strength for the majority of components. Kinematic and compliance test examines roll centre, vertical movement of the wheel centre, toe angle and camber angle. This comprehensive analysis contributes to suspension system design and optimization, improving vehicle safety and durability.

Numerical Investigation on Crack Propagation of Three-Dimensional Pipeline Under Seismic Wave Loading

Cracks in transport pipelines significantly impact their structural safety, making pipeline fracture damage a major concern. While experimental tests can study crack growth, their setup and costs are substantial. To address this, our work involves parametric analyses to investigate the effects of initial crack length, load types and sizes, and pipe wall thickness on the crack tip stress field. The finite element model is established according to the actual operation, and the crack defects of the pipe are established by the extended finite element method (XFEM). The crack tip variation is studied by extracting the von Mises stress and strain at the crack tip. When the crack propagates, the stress field at the crack tip will fluctuate in a certain range. The relationship between stress field fluctuation and crack propagation was studied by changing the XFEM crack growth for comparative analysis. Utilizing 429 numerical results obtained from a parametric numerical model implemented with Python and Abaqus, we establish a back propagation (BP) neural network prediction model to forecast the stress field at the crack tip. The relative errors between the numerical results and the predictions by our model remain below 5%. In conclusion, our proposed approach for analyzing pipeline fracture under multiple loads proves valuable for pipeline safety assessment, and it offers useful insights for evaluating the suitability of transportation pipelines.

A novel bio-inspired helmet with auxetic lattice liners for mitigating traumatic brain injury

The human head is most vulnerable to injury during activities such as road traffic and sports. To mitigate the risk of traumatic brain injury (TBI), helmets serve as an important protective device. This study proposes a hedgehog biomimetic helmet with auxetic lattice liners in the shape of a hemisphere. The helmeted head impact configuration is built based on a high bio-fidelity head-neck finite element model incorporated into our novel helmet model. Biomechanical responses including acceleration, intracranial pressure, and von Mises strain of head are extracted from the simulation model to assess TBI risks. The results indicate that the helmet featuring auxetic lattice liners outperforms those without liners or with other liner designs, offering superior protection. Compared to the threshold, the novel helmet design was found to reduce the head injury criterion value by 72.65%. Additionally, parametric studies of lattice’s bar radius for uniform and graded auxetic lattice liners are discussed. Finally, this study also carries out the optimization design of lattice strut radius and height, resulting in a lightweight auxetic lattice liner with superior protective performance. The outcomes of this study extend the application of auxetic materials and provide guidance for designing helmet liners that better mitigate TBI.

Estimation of Crack Formation By Digital-Twin Structure of a Single Secondary Particle

Most of the recently commercialized active materials for lithium-ion batteries (LiB) adopt secondary particles, and most of them are designed by evaluating cell-level electrochemical properties. Therefore, the relationship between particle design and battery performance is not properly understood because it is not possible to grasp the intrinsic properties of the electrode active material. Therefore, a single particle measurement technique has been developed in which a gold or platinum filament having a diameter of several tens of microns is directly contacted with a single particle to evaluate the single particle electrochemical properties. However, since it is a non-sealed battery system, only less-volatile electrolytes can be applied. Also, the experiment is difficult because the contact between particle and filament can easily break down with small vibration. Additionally, there are a lot of particle design parameters such as particle shape, size, and size distribution. Moreover, with operating conditions, the particle design needs to be changed. So, it is challenging to cover all cases by experiment.In this study, digital-twin structure of a single secondary particle was formed using focused ion beam / scanning electron microscope. Using the digital-twin structure, digital-twin electrochemo-mechanical model was developed and more than 99% of accuracy was secured by comparing the voltage profiles with result of single particle measurement technique. Using the well-matched model. Operando analysis of electrochemical properties (State of Lithiation, overpotential, potential, and concentration) and mechanical properties (von Mises stress and strain energy density) were performed. The strain energy density can be the criteria to determine the crack formation by comparing with the fracture toughness which is the strain energy density when the fracture occurs.