Strain Energy Density Distribution Research Articles

The Outstanding Excellences of Interactive Energy Density Topology Change Method

It has not been solved that the fruits, vegetables, strawberries, cells, blood and a bottle of liquor are damaged, broken during transportation. It is the greatest factor in this situation that there is a danger vibration frequency band where these are easy to scratch and are prone to death. If there are eigen frequencies within this danger frequency band, it needs that those eigen frequencies within this danger frequency band are moved out of the band. In the former paper, it was shown difficult to apply the existing topology optimization methods using homogenization method or density method to control plural eigen frequencies for solving this problem. Therefore in the former paper, it was proposed the so called “Interactive Energy Density Topology (IEDT) change method” that is a new high precision and high efficiency method for controlling plural eigen frequencies simultaneously referring to the kinetic and the strain energy density distributions. Here it is discussed more about the IEDT change method and show that it is sometimes difficult for the traditional methods to get solutions because especially in the dynamic problem, eigen frequencies may go up or down depending on its size even if reinforcement or hall for change topology is applied at the same location. But with the proposed IEDT change method, always it can be realized because the proposed method has wider solution spaces than the traditional one. Lastly, it is shown that this IEDT method is also very effective to reduce the integral value of response curve by frequency.

Cold rolling pass design and process based on controlling the distribution of material strain energy density

To address the problem of excessive grain boundaries and difficulty in eliminating defects on the inner surface of a pipe under a cold rolling pass, a method to control the distribution of material strain energy density is proposed, and a reasonable reduction in wall thickness is designed. An analysis model for inner surface defects was established using the inner wall adhesion value as the main criterion to quantitatively analyze the degree of defects on the pipe's inner surface. A systematic analysis of the rolling process of three different pass-opening shapes revealed that the opening shape dramatically affects the deformation and flow of the metal. During the rolling process, because of the unreasonable distribution of metal deformation caused by the existing pass design, the pipe is prone to high or concentrated stress, resulting in a low adhesion value for the inner wall and a greater degree of defects on the inner surface. Therefore, we propose a new opening shape and reasonably allocate the wall thickness reduction of the pipe to obtain a new pass. Numerical simulation reveals that, after rolling with the new pass, the inner wall adhesion value becomes relatively high, and the mechanics parameters have also been improved. The experimental results show that, under the new pass design, grain boundaries on the inner surface of the pipe are reduced and the smoothness is significantly improved, proving the feasibility of a cold rolling pass design based on controlling the distribution of material strain energy density.

Fatigue life prediction of notched components under size effect based on strain energy density gradient

AbstractIn practical mechanical components, the characteristic of geometric discontinuous generates non‐uniform energy field near the notch, which affects the local damage evolution process of the notched specimens. Therefore, it is particularly crucial to formulate a parameter that can characterize such inhomogeneous state distribution. First, a strain energy density gradient factor is introduced to describe the complexity of strain energy density distribution near the notch. Meanwhile, a size effect factor is given considering the strain energy gradient factor and notch geometry parameter. Then, taking into account the impact of phase difference, a direct damage parameter considering non‐proportionality of different multiaxial load levels is developed. Furtherly, the modified strain energy density parameter is determined. Finally, the comparison between the proposed model and the other three models emphasizes the advantages of the model.

An equivalent anisotropy orientation tensor algorithm for integrated material-structure design

This study presents the Equivalent Anisotropy Orientation Tensor (EAOT) algorithm as a novel approach for the integrated design of thin-walled structures composed of composite materials. The EAOT algorithm is designed to enhance the buckling load of structures by efficiently optimizing the distribution of strain energy density. In addition, the algorithm considers the fiber orientation and stiffener structure as orthogonal isotropic material, providing an innovative design methodology that surpasses conventional approaches. The EAOT algorithm incorporates a weighted function to account for the effects of different buckling modes, which solves the problem of mode overlap and exchange that affects the optimization effect. Utilizing the optimization results from the EAOT algorithm enables the realization of effective material-structure integrated design. Two case studies demonstrate the superior performance of designs based on this algorithm compared to conventional methods, resulting in a substantial increase in the critical buckling load by 15.3 % to 34.7 %.

The sensitivity of the stiffness and thickness of a titanium inlay in a cementless PEEK femoral component to the micromotions and bone strain energy density

Polyetheretherketone (PEEK) has been proposed as alternative material for total knee arthroplasty implants due to its low stiffness, which may reduce stress-shielding. In cementless fixation, a proper primary fixation is required for long-term fixation. Previous research showed that the lower stiffness of a cementless PEEK femoral component results in larger micromotions at the implant-bone interface compared to a cobalt-chrome femoral component. A titanium inlay on the PEEK implant surface may improve the primary fixation while maintaining the favourable stiffness properties. Therefore, the effect of thickness and stiffness of a titanium inlay on the primary fixation and stress-shielding was investigated. A finite element model of the femur and femoral component was created with five titanium inlay variants. The micromotions and strain energy density (SED) were quantified as outcome measures. The distal thin – proximal thick variant showed the largest resulting micromotions (51.2 µm). Relative to the all-PEEK femoral component, the addition of a titanium inlay reduced the micromotions with 30 % to 40 % without considerably affecting the stress-shielding capacity (strain energy difference of 6 % to 10 %). Differences in micromotions (43.0–51.2 µm) and SED between the variants were relatively small. In conclusion, the addition of a titanium inlay could lead to a reduction of the micromotions without substantially affecting the SED distribution.

Failure characteristics and energy distributions of surrounding rock in hard rock tunnels subjected to different three-dimensional stress conditions

To study the failure characteristics of surrounding rock near an excavated tunnel under different three-dimensional stress conditions, a TRW3000 rock true triaxial electro-hydraulic servo mutagenesis testing system was used to conduct triaxial loading experiments on granite specimens with a circular hole. A high-definition micro-camera system was used to record the failure process near the circular hole. The results show that spalling failure occurs near the holed specimen under low-stress conditions, and rockburst appears around the holed specimen under high-stress conditions. Increasing the horizontal stress can prevent the occurrence of rockburst. The evolution of strain energy density (SED) in the surrounding rock of the holed specimen under different stress conditions was simulated. The findings show that the distribution of SED around the circular hole decreases with an increase in horizontal stress. The SED distributions around the circular hole are basically consistent with the damage distributions in the experiments, namely, butterfly-shaped, ear-shaped, and ring-shaped.

Flow-induced reconfiguration and cross-flow vibrations of an elastic plate and implications to energy harvesting

We numerically study flow-induced vibrations (FIV) of an elastic cantilevered plate in a cross-flow configuration. The plate is kept upright in an open domain with a fixed bottom end. Such a configuration eliminates energy losses in the wall boundary. Due to allowed vortex shedding from its bottom end, the resulting novel dynamics of the fluid-structure interaction (FSI) system is explored. An in-house sharp-interface immersed boundary method-based flow solver, coupled with a Saint-Venant Kirchhoff material model-based open-source finite element solver, is employed. Two sets of simulations were carried out in two-dimensional Cartesian coordinates at Reynolds number of 100 for a wide range of reduced velocity (UR), by performing a parametric sweep of dimensionless Young’s modulus (E) and mass ratio (M). In general, FIV characteristics of the plate are similar to the transverse vortex-induced vibration of an elastically-mounted cylinder. The plate deforms to a curved mean position depending on E and M and exhibits angular oscillations. In particular, the following FIV regimes are obtained with increasing UR: initial excitation, lock-in with the first mode, desynchronization after the first lock-in, lock-in with the second mode, and desynchronization after the second lock-in. During lock-in, the plate oscillation and vortex shedding frequency are closer to the first or second mode natural frequency of plate. A larger angular amplitude and increased energy exchange between the structure and fluid characterize the lock-in. We quantify the modal contributions of Euler-Bernoulli modes in the plate’s vibration response with a curved mean position. The spatiotemporal variation of energy exchange between the fluid and plate is analyzed to understand the observed plate dynamics. The flow physics of the wake is discussed using wake patterns, vortex formation length, and vortex strength. Drag over the plate reduces due to flow-induced reconfiguration and strongly correlates with these parameters. Lastly, based on the spatial distribution of strain energy density over the plate, we present a potential design for an ambient fluid energy harvester to harness the plate’s vibration energy with optimized piezoelectric patching. The optimum location and height of a piezoelectric patch on the plate for different cases are reported.

Equivalent loads from the life-cycle of acetabular cages in relation to bone-graft transformation

Background and ObjectivesBone grafts placed behind acetabular cages change their structure in response to mechanical stimuli. The full consideration of lifestyle loads is extremely resource-intensive, so a method using substitutive loads was proposed to reduce the calculation cost. The aim of the study is to present and prove this method. MethodsBy means of mechanical equations and using the force vectors from the literature which have the same initial point and their relative frequency, while applying a linear model, the average strain energy density distribution for all load cases can be calculated, compiling a matrix from the external loads. From the elements of this matrix, three substitutive load vectors can be calculated, which can be proven to produce the same strain energy density distribution by averaging their effects. The feasibility of using this to model the transformation of bone grafts placed behind acetabular cages is demonstrated with a finite element model, along with a reference calculation. ResultsThe substitutive load vectors could be calculated in closed form and the simulations showed that they produced a similar density distribution to the reference model with a numerical calculation error range. Accordingly, the density distribution calculated from bone graft transformation is almost the same. ConclusionsIn addition to the aforementioned linearity and the same initial point limitations, the applied method is able to produce the substitutive load vectors with which the calculation of the strain energy density distribution and the bone graft's new density distributions can be carried out faster.

Accuracy of osseointegrated screw-bone construct stiffness and peri-implant loading predicted by homogenized FE models relative to micro-FE models

Computational predictions of stiffness and peri-implant loading of screw-bone constructs are highly relevant to investigate and improve bone fracture fixations. Homogenized finite element (hFE) models have been used for this purpose in the past, but their accuracy has been questioned given the numerous simplifications, such as neglecting screw threads and modelling the trabecular bone structure as a continuum. This study aimed to investigate the accuracy of hFE models of an osseointegrated screw-bone construct when compared to micro-FE models considering the simplified screw geometry and different trabecular bone material models.Micro-FE and hFE models were created from 15 cylindrical bone samples with a virtually inserted, osseointegrated screw (fully bonded interface). Micro-FE models were created including the screw with threads (=reference models) and without threads to quantify the error due to screw geometry simplification. In the hFE models, the screws were modelled without threads and four different trabecular bone material models were used, including orthotropic and isotropic material derived from homogenization with kinematic uniform boundary conditions (KUBC), as well as from periodicity-compatible mixed uniform boundary conditions (PMUBC). Three load cases were simulated (pullout, shear in two directions) and errors in the construct stiffness and the volume average strain energy density (SED) in the peri-implant region were evaluated relative to the micro-FE model with a threaded screw.The pooled error caused by only omitting screw threads was low (max: 8.0%) compared to the pooled error additionally including homogenized trabecular bone material (max: 92.2%). Stiffness was predicted most accurately using PMUBC-derived orthotropic material (error: -0.7 ± 8.0%) and least accurately using KUBC-derived isotropic material (error: +23.1 ± 24.4%). Peri-implant SED averages were generally well correlated (R2 ≥ 0.76), but slightly over- or underestimated by the hFE models and SED distributions were qualitatively different between hFE and micro-FE models.This study suggests that osseointegrated screw-bone construct stiffness can be predicted accurately using hFE models when compared to micro-FE models and that volume average peri-implant SEDs are well correlated. However, the hFE models are highly sensitive to the choice of trabecular bone material properties. PMUBC-derived isotropic material properties represented the best trade-off between model accuracy and complexity in this study.

Quantitative Load Dependency Analysis of Local Trabecular Bone Microstructure to Understand the Spatial Characteristics in the Synthetic Proximal Femur.

Analysis of the dependency of the trabecular structure on loading conditions is essential for understanding and predicting bone structure formation. Although previous studies have investigated the relationship between loads and structural adaptations, there is a need for an in-depth analysis of this relationship based on the bone region and load specifics. In this study, the load dependency of the trabecular bone microstructure for twelve regions of interest (ROIs) in the synthetic proximal femur was quantitatively analyzed to understand the spatial characteristics under seven different loading conditions. To investigate the load dependency, a quantitative measure, called the load dependency score (LDS), was established based on the statistics of the strain energy density (SED) distribution. The results showed that for the global model and epiphysis ROIs, bone microstructures relied on the multiple-loading condition, whereas the structures in the metaphysis depended on single or double loads. These results demonstrate that a given ROI is predominantly dependent on a particular loading condition. The results confirm that the dependency analysis of the load effects for ROIs should be performed both qualitatively and quantitatively.

Changes in strain energy density in the temporomandibular joint disk after sagittal split ramus osteotomy using acomputed tomography-based finite element model.

We evaluated the changes in the strain energy density (SED) in the temporomandibular joint (TMJ) disk after sagittal split ramus osteotomy (SSRO) at three time points. Afinite element model (FEM) based on real patient-based computed tomography (CT) data was used to examine the effect of SSRO on the TMJ. Measurements of the condylar position and angulation in CT images and FEM analyses were performed for 17patients scheduled to undergo SSROs at the following time points: before surgery, immediately after surgery, and 1year after surgery. SED on the entire disk was calculated at each of the three time points using FEM. Furthermore, the relationship between individual SED values and the corresponding condylar position was also evaluated. No significant change was observed in the condylar position at the three time points. The FEM analysis showed that SED was the highest and lowest immediately after and 1year after surgery, respectively. Apossible SED distribution imbalance between the left and right joints was improved 1year after SSRO. Concerning the effect of fossa morphometry and condylar position, wide and deep glenoid fossae and amore posterior condylar position tended to show lower SED. SED in the articular disk temporarily increased after surgery and significantly decreased 1year after surgery compared with that before surgery. SSRO generally improved the imbalance between the left and right joints. Thus, SSRO, which improves maxillofacial morphology, may also improve components of temporomandibular disorders.

Distribution law of strain energy density for stiffness design of GFRP, UT/GFRP, and VRB/GFRP hybrid hat‐shaped beams

AbstractThe structural optimization for stiffness performances with hundreds or thousands of design parameters is a complex and time‐consuming process. This is because, on the one hand, the direct relationship between the stiffness and the many design parameters is unknown, and on the other hand, the sensitivity of a single design parameter to the global stiffness is hardly obtained. Therefore, transforming the global stiffness performance to the strain energy density (SED) distribution over the whole structure can significantly simplify the optimization problem. In this work, the relationships among the design parameters, SED distribution and stiffnesses of glass fiber reinforced plastic (GFRP), uniform thickness metal (UT)/GFRP, and variable‐thickness rolled blanks (VRB)/GFRP hat‐shaped beams are investigated through numerical simulations. It is found that the higher SED of the all hat‐shaped beam structure indicates a worse stiffness performance under both bending and torsional conditions. When the SED of the GFRP hat‐shaped beam increases 6.65%, its bending stiffness decreases 10.8%. It is also found that the smaller SED distribution variance of the all hat‐shaped beam structures indicates a better stiffness performance. When the SED distribution variances of the VRB and GFRP in the VRB/GFRP hybrid beam are, respectively, reduced by 70.78% and 85.56%, the bending stiffness of the hybrid beam is increased by 50.83%. The above relations and conclusions indicate that the structural stiffness optimization problem with many design parameters can be replaced with the structural SED distribution variance minimization problem, which would not need the unavailable sensitivities of the design parameters to the global stiffness. Besides, compared to the GFRP hat‐shaped beam, with the same mass, volume or cost, the bending stiffnesses of the VRB/GFRP hat‐shaped beam, respectively, increases by 196.1%, 476.5%, and 676.4%, and its torsional stiffnesses, respectively, increases by 70.9%, 233.0%, and 348.4%. This means the combination of VRB and GFRP in the beam structure can reach a better stiffness performance and a higher lightweight level.

The mechanical influence of bone spicules in the osteochondral junction: A finite element modelling study.

While much has been done to study how cartilage responds to mechanical loading, as well as modelling such responses, arguably less has been accomplished around the mechanics of the cartilage-bone junction. Previously, it has been reported that the presence of bony spicules invading the zone of calcified cartilage, preceded the formation of new subchondral bone and the advancing of the cement line (Thambyah and Broom in Osteoarthr Cartil 17:456-463, 2009). In this study, the morphology and frequency of bone spicules in the cartilage-bone interface of osteochondral beams subjected to three-point bending were modelled, and the results are discussed within the context of biomechanical theories on bone formation. It was found that the stress and strain magnitudes, and their distribution were sensitive to the presence and number of spicules. Spicule numbers and shape were shown to affect the strain energy density (SED) distribution in the areas of the cement line adjacent to spicules. Stresses, strains and SED analyses thus provided evidence that the mechanical environment with the addition of spicules promotes bone formation in the cartilage-bone junction.

A numerical analysis of skin\u2013PPE interaction to prevent facial tissue injury

The use of close-fitting PPE is essential to prevent exposure to dispersed airborne matter, including the COVID-19 virus. The current pandemic has increased pressure on healthcare systems around the world, leading to medical professionals using high-grade PPE for prolonged durations, resulting in device-induced skin injuries. This study focuses on computationally improving the interaction between skin and PPE to reduce the likelihood of discomfort and tissue damage. A finite element model is developed to simulate the movement of PPE against the face during day-to-day tasks. Due to limited available data on skin characteristics and how these vary interpersonally between sexes, races and ages, the main objective of this study was to establish the effects and trends that mask modifications have on the resulting subsurface strain energy density distribution in the skin. These modifications include the material, geometric and interfacial properties. Overall, the results show that skin injury can be reduced by using softer mask materials, whilst friction against the skin should be minimised, e.g. through use of micro-textures, humidity control and topical creams. Furthermore, the contact area between the mask and skin should be maximised, whilst the use of soft materials with incompressible behaviour (e.g. many elastomers) should be avoided.

The Role of the Loading Condition in Predictions of Bone Adaptation in a Mouse Tibial Loading Model.

The in vivo mouse tibial loading model is used to evaluate the effectiveness of mechanical loading treatment against skeletal diseases. Although studies have correlated bone adaptation with the induced mechanical stimulus, predictions of bone remodeling remained poor, and the interaction between external and physiological loading in engendering bone changes have not been determined. The aim of this study was to determine the effect of passive mechanical loading on the strain distribution in the mouse tibia and its predictions of bone adaptation. Longitudinal micro-computed tomography (micro-CT) imaging was performed over 2 weeks of cyclic loading from weeks 18 to 22 of age, to quantify the shape change, remodeling, and changes in densitometric properties. Micro-CT based finite element analysis coupled with an optimization algorithm for bone remodeling was used to predict bone adaptation under physiological loads, nominal 12N axial load and combined nominal 12N axial load superimposed to the physiological load. The results showed that despite large differences in the strain energy density magnitudes and distributions across the tibial length, the overall accuracy of the model and the spatial match were similar for all evaluated loading conditions. Predictions of densitometric properties were most similar to the experimental data for combined loading, followed closely by physiological loading conditions, despite no significant difference between these two predicted groups. However, all predicted densitometric properties were significantly different for the 12N and the combined loading conditions. The results suggest that computational modeling of bone’s adaptive response to passive mechanical loading should include the contribution of daily physiological load.

Modelling extracellular matrix and cellular contributions to whole muscle mechanics.

Skeletal muscle tissue has a highly complex and heterogeneous structure comprising several physical length scales. In the simplest model of muscle tissue, it can be represented as a one dimensional nonlinear spring in the direction of muscle fibres. However, at the finest level, muscle tissue includes a complex network of collagen fibres, actin and myosin proteins, and other cellular materials. This study shall derive an intermediate physical model which encapsulates the major contributions of the muscle components to the elastic response apart from activation-related along-fibre responses. The micro-mechanical factors in skeletal muscle tissue (eg. connective tissue, fluid, and fibres) can be homogenized into one material aggregate that will capture the behaviour of the combination of material components. In order to do this, the corresponding volume fractions for each type of material need to be determined by comparing the stress-strain relationship for a volume containing each material. This results in a model that accounts for the micro-mechanical features found in muscle and can therefore be used to analyze effects of neuro-muscular diseases such as cerebral palsy or muscular dystrophies. The purpose of this study is to construct a model of muscle tissue that, through choosing the correct material parameters based on experimental data, will accurately capture the mechanical behaviour of whole muscle. This model is then used to look at the impacts of the bulk modulus and material parameters on muscle deformation and strain energy-density distributions.

Impact of surgical alignment, tray material, PCL condition, and patient anatomy on tibial strains after TKA

Bone remodeling after total knee arthroplasty is regulated by the changes in strain energy density (SED), however, the critical parameters influencing post-operative SED distributions are not fully understood. This study aimed to investigate the impact of surgical alignment, tray material properties, posterior cruciate ligament (PCL) balance, tray posterior slope, and patient anatomy on SED distributions in the proximal tibia.Finite element models of two tibiae (different anatomy) with configurations of two implant materials, two surgical alignments, two posterior slopes, and two PCL conditions were developed. The models were tested under the peak loading conditions during gait, deep knee bending, and stair descent. For each configuration, the contact forces and locations and soft-tissue loads of interest were taken into consideration. SED in the proximal tibia was predicted and the changes in strain distributions were compared for each of the factors studied.Tibial anatomy had the most impact on the proximal bone SED distributions, followed by PCL balancing, surgical alignment, and posterior slope. In addition, the thickness of the remaining cortical wall after implantation was also a significant consideration when evaluating tibial anatomy. The resulting SED changes for alignment, posterior slope, and PCL factors were mainly due to the differences in joint and soft-tissue loading conditions. A lower modulus tray material did result in changes in the post-operative strain state, however, these were almost negligible compared to that seen for the other factors.

Development of an Instrument to Assess the Stability of Cementless Femoral Implants Using Vibration Analysis During Total Hip Arthroplasty.

Objective: The level of primary implant fixation in cementless total hip arthroplasty is a key factor for the longevity of the implant. Vibration-based methods show promise for providing quantitative information to help surgeons monitor implant fixation intraoperatively. A thorough understanding of what is driving these changes in vibrational behavior is important for further development and improvement of these methods. Additionally, an instrument must be designed to enable surgeons to leverage these methods. This study addresses both of these issues. Method: An augmented system approach was used to develop an instrument that improves the sensitivity of the vibrational method and enables the implementation of the necessary excitation and measurement equipment. The augmented system approach took into account the dynamics of the existing bone-implant system and its interaction with the added instrument. Results: Two instrument designs are proposed, accompanied by a convergence-based method to determine the insertion endpoint. The modal strain energy density distribution was shown to affect the vibrational sensitivity to contact changes in certain areas. Conclusion: The augmented system approach led to an instrument design that improved the sensitivity to changes in the proximal region of the combined bone-implant-instrument system. This fact was confirmed both in silico and in vitro. Clinical Impact: The presented method and instruments address practical intraoperative challenges and provide perspective to objectively support the surgeon’s decision-making process, which will ensure optimal patient treatment.

Topology and fibre orientation simultaneous optimisation: A design methodology for fibre-reinforced composite components

Additive manufacturing for fibre-reinforced composite structures is rapidly diffusing, since it enables the production of lightweight structural parts characterized by complex geometries and tailored fibre orientations. Therefore, the development of design methodologies capable to simultaneously optimize the shape of the fibre-reinforced composite part and the fibre orientation in the additive manufacturing process is, at present, of utmost interest among industries and research centres. In this paper, a novel simultaneous optimisation method capable to optimise the topology and the local fibre orientation is proposed. The method is computationally cheap, fast convergent and permits to avoid stress peaks, working efficiently on 2D and on 3D models. The analytical formulation of the problem and the optimisation algorithm are at first described. The optimisation criteria are based on the uniform strain energy density distribution and the fibre alignement along the principal stress direction. The proposed method is then verified with several benchmarks from the literature and with a 3D illustrative example, confirming that it can be effectively and efficiently employed for the optimisation of composite components to be produced through additive manufacturing.

Compliant Hydrostatic Bearings Utilizing Functionally Graded Materials

Abstract Deformability of hydrostatic bearings could potentially increase their application range significantly. When a bearing is made compliant, the film pressure starts to influence the deformation of the support itself. This effect, called compliant-hydrostatic pre-loading, is especially crucial to take into account when designing highly deformable hydrostatic bearings. This work introduces the principle of pressure profile matching to minimize the effect of this pre-loading. A two-dimensional design model is introduced to determine the performance of such an elastic bearing utilizing stiffness profile matching. Additionally, an extension to the model is presented to analyze the basic performance of these type of bearings over small counter surface eccentricities. Finally, an embodiment of such a material distribution is presented utilizing functionally graded materials. These embodiments are analyzed with respect to their failure behavior, showing an improved shear stress and strain energy density distribution with the functionally graded supports when compared with conventional elastic supports.