Value Of Strain Energy Density Research Articles

Prediction of Bone Remodeling in Rat Caudal Vertebrae Based on Fluid-Solid Coupling Simulation.

Some previous researches have demonstrated that appropriate mechanical stimulation can enhance bone formation. However, most studies have employed the strain energy density (SED) method for predicting bone remodeling, with only a few considering the potential impact of wall fluid shear stress (FSS) on this process. To bridge this gap, the current study compared the prediction of bone formation and resorption via SED and wall FSS by using fluid-solid coupling numerical simulation. Specifically, 8-week-old female Sprague-Dawley rats were subjected to stretching of the eighth caudal vertebra using a custom-made device. Based on micro-computed tomography images, a three-dimensional model integrating fluid-solid coupling was created to represent compact bone, cancellous bone, and bone marrow. The animals were grouped into control, 1Hz, and 10Hz categories, wherein a tensile displacement load of 1000 με was applied to the loading end. The results revealed that SED values tended to increase with elevated porosity, whereas wall FSS values decreased it. Notably, wall FSS demonstrated the higher predictive accuracy for cancellous bone resorption than SED. These findings support the notion that fluid flow within cancellous bone spaces can significantly impact bone resorption. Therefore, the findings of this study contribute to a more comprehensive understanding of the role of wall FSS in bone remodeling, providing a theoretical support for the dynamic evolution of bone structures under mechanical stimulation.

Effect of Creep, Fatigue and Random Vibration on the Integrity of Solder Joints in BGA Package

This study employs Finite Element Analysis (FEA) to investigate the effects of creep, fatigue, and random vibration on the integrity of solder joints in BGA packages in electronic modules. Evaluating the response of lead-based Sn63Pb37 and lead-free SnAgCu alloy solders (SAC305, SAC387, SAC396, and SAC405) to induced loads - stress, strain, strain energy density, and displacement in the joints is obtained and studied better to understand the mechanism of the joints' degradation. SAC305 and SAC405 are modelled with linear stress-temperature relationships σ = 3.152T–68.167 and σ = 1.543T–34.983, respectively. The magnitude of the strain energy density in the joints is a key failure driver. SAC387 and SAC396 solder joints display lower values of strain energy density and thus have higher durability. Displacement analysis indicates that SAC305 and Sn63Pb37 are prone to deformation-induced failure. SAC387 exhibits the highest fatigue yield stress at 58 MPa, while SAC405 displays the lowest stress at 22 MPa. Analysis of the results of random vibration shows that Sn63Pb37 developed the highest stress at 34.62 MPa and is thus susceptible to stress-induced failure. The robust stress, strain, and strain energy responses of SAC405 and SAC396 provide key insights into improving the mechanical reliability of future electronic devices towards better sustainability.

A finite element study on the risk of bone loss around posterior short implants in an atrophic mandible.

This study aimed to evaluate the risk of bone loss around single short molar crown-supporting implants in an atrophic mandible. Implants of different lengths (L = 4 or 6 mm) and diameters (Ø = 4.1 or 4.8 mm) were placed in the molar area of an atrophic mandible. Additional control mandible models were simulated for 4.1 mm diameter implants (L = 4, 6, 8, and 10 mm). A vertical masticatory load of 200 N was applied to three or six occlusal contact areas (3ca or 6ca) of the prosthetic crown. The bone strain energy density (SED) of 109.6 µJ/mm3 was assumed to be the pathological threshold for cortical bone. The peri-implant bone resorption risk index (PIBRri) was calculated by dividing the maximum SED of the crestal cortical bone by the SED pathological threshold. Increasing the implant length from 4 to 6 mm, implant diameter from 4.1 to 4.8 mm, and number of contact areas from 3 to 6 reduced the SED and PIBRri values by approximately 20%, 35%, and 40%, respectively, when comparing pairs of models that isolated a specific variable. All models with 6ca had a low bone resorption risk (PIBRri<0.8), while the Ø4.1 short implant with 3ca had a medium (0.8≤PIBRri≤1.0) or high (PIBRri>1.0) resorption risk. Increasing the diameter or occlusal contact area of a 4 mm short implant in an atrophic mandible resulted in reduced bone resorption risks, similar to or lower than those observed in a regular mandible with standard-length implants.

Effect of the notch opening angle on the quasi-static behavior of PLA and carbon fibers reinforced PLA realized through Fused Deposition Modelling

Additive manufacturing provides significant advantages over conventional manufacturing. Among the others, the almost unconstrained freedom in the geometrical design for this technology can be pointed out. However, the geometrical complexness of such components requires for adequate tools to assess both their fracture behavior and fatigue life. A suitable solution for such a design challenge is to rely on the so-called local approaches whose main advantage is to consider a local parameter to evaluate the behavior of the entire component; besides, such methods have the advantage that their critical value can be assumed to be independent on both the overall geometry of the component and the loading conditions. With this purpose, the present work investigates the fracture behavior of notched specimens made of PLA and carbon fiber reinforced PLA realized through additive manufactured technique. The specimen's geometry considered is smooth and double notched while the notch opening angles varies between 30 and 120 degrees. The results of the experimental campaign have been summarized through the averaged strain energy density (SED) method, an energy-based local approach, widely proved to be a valid tool to investigate both fracture in static condition and fatigue failure. The critical value of SED has been obtained through the stress-strain curve of smooth specimens for the two studied materials. After the determination of the control volume characteristic length, R0, the data have been summarized in terms of averaged SED values. The critical loads for the different geometries and the different materials considered are predicted by the method with an average error of ±7%.

Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification

The literature in the field of higher-order homogenization is mainly focused on 2-D models aimed at composite materials, while it lacks a comprehensive model targeting 3-D lattice materials (with void being the inclusion) with complex cell topologies. For that, a computational homogenization scheme based on Mindlin (type II) strain gradient elasticity theory is developed here. The model is based on variational formulation with periodic boundary conditions, implemented in the open-source software FreeFEM to fully characterize the effective classical elastic, coupling, and gradient elastic matrices in lattice materials. Rigorous mathematical derivations based on equilibrium equations and Hill–Mandel lemma are provided, resulting in the introduction of macroscopic body forces and modifications in gradient elasticity tensors which eliminate the spurious gradient effects in the homogeneous material. The obtained homogenized classical and strain gradient elasticity matrices are positive definite, leading to a positive macroscopic strain energy density value—an important criterion that sometimes is overlooked. The model is employed to study the size effects in 2-D square and 3-D cubic lattice materials. For the case of 3-D cubic material, the model is verified using full-field simulations, isogeometric analysis, and experimental three-point bending tests. The results of computational homogenization scheme implemented through isogeometric simulations show a good agreement with full-field simulations and mechanical tests. The developed model is generic and can be used to derive the effective second-grade continuum for any 3-D architectured material with arbitrary geometry. However, the identification of the proper type of generalized continua for the mechanical analysis of different cell architectures is yet an open question.

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.

The Effect of the Stress State, Testing Temperature, and Hardener Composition on the Strength of an AlMg5/Epoxy Metal-Polymer Joint.

The regularities of the effect of a complex stress state on the strength of an AlMg5/epoxy adhesive joint are experimentally studied at −50 and +23 °C in tension+shear and compression+shear tests with different normal-to-shear stress ratios. The tests use modified Arcan specimens and Brazil-nut-sandwich specimens, with the lateral faces of the adhesive layer having a shape of a mushroom-like “ridge” aimed at reducing stress concentration at the specimen edges. An original computational model of a selected microvolume including the interface together with the adjacent substrate and adhesive layers is used to process the experimental results. The attainment of the threshold value of strain energy density in the selected microvolume, W*, is used as the failure criterion. The effect of the hardener composition, the testing temperature, and the value of the phase angle β determining the proportion of normal and shear stresses at the adhesive interface on the threshold value W* is detected. W*(β) diagrams (fracture loci) are plotted and analytically described logarithmic functions. They can be used to make strength calculations for adhesive joints in structures and metal-polymer composites.

Reliability and Lifetime Prediction Model of Sintered Silver Under High-Temperature Cycling

Although excellent reliability has been reported for sintered silver as a die-attach material under both thermal and power cycling loads in power electronics applications, the promise of this material as a large-area attachment at temperatures beyond 200 °C needs to be investigated. This article presents insights into the thermomechanical behavior and reliability of sintered silver under extreme thermal cycling conditions. In this study, we bonded sintered silver samples and subjected it to a thermal cycling profile of −40 °C to 200 °C with high ramp rates. We periodically monitored samples under thermal cycling to detect the presence of any failure mechanisms using a scanning acoustic microscope. We also included 95Pb5Sn solder in the study to obtain reference data. Results show the occurrence of cracks in sintered silver followed by a rapid rate of crack growth that exceeded the failure criterion in just 50 cycles. The predominant failure mechanism we observed was adhesive failure. As a large-area attachment, solder exhibited a higher reliability than sintered silver but failed within 100 cycles. Finally, we performed thermomechanical modeling to compute strain energy density values and correlated these with the experimentally observed crack growth rates to formulate a lifetime prediction model for sintered silver.

Some useful expressions and a proof of the validity of the volume free procedure for the SED method application

This work provides some useful expressions for the application of the strain energy density method to component having sharp V-notches under mixed loading conditions. The strain energy density method is characterized by showing a low sensitivity to the mech refinement; however, both the conventional procedure for the application of this method and the volume free procedure result in unavoidable approximation of the theoretical control volume integration domain. The analytical expressions for the parameters e1, e2 and e3, useful to evaluate the averaged strain energy density value in the control volume starting from the notch stress intensity factors values have been obtained. The theoretical difference, obtained by approximating the control volume shape according to both the conventional and the volume free procedure, has been assessed and the mesh size, below whose value the approximation of the control volume shape results in negligible difference with respect to the SED value that should be obtained theoretically, has been found as a function of R0. It has been found that for the conventional procedure, a mesh size equal to or lower than R0/3 ensures having an absolute difference lower than 4% while as regards the volume free procedure a mesh size equal to or lower than R0/7 is required regardless of the mode mixity and the Poisson ratio.

Bending working law of corroded ductile iron pipe flange connections revealed by structural stressing state theory

Applying structural stressing state theory, this study finds out the bending working law of corroded ductile iron pipe flange (CDIPF) connections from their tested strain data. Firstly, the strain data are transformed into the generalized strain energy density values to build the stressing state mode and the characteristic parameter. Then, the Mann-Kendall criterion is used to detect the mutation points in the evolution curves of the stressing state characteristic parameters with the load increase according to the natural law from quantitative change to qualitative change of a system. Meanwhile, it is verified that the evolutions of the stressing state modes also present the corresponding mutation features. The mutation points define the starting points of the bending CDIPF connections' failure processes. Furthermore, the Mann-Kendall criterion distinguishes the elastoplastic branch (EPB) points in the normal working processes of CDIPF connections. Both the failure starting point and the EPB point lay the new foundation to analyze the working performances of the CDIPF connections and to improve their design.

Misalignment effect on the fatigue failure behavior of load-carrying cruciform welded joints

This paper presents an analytical model based on the Strain Energy Density (SED) method to predict the high cycle fatigue failure behavior of load-carrying cruciform welded joints (LCWJ) weakened by axial and angular misalignments. Compared with the Hot Spot Stress (HSS) and Effective Notch Stress (ENS) methods recommended in standard codes, the SED values at weld toe and weld root were carried out systematically considering the welded details and the types of misalignments. Moreover, the analytical model was applied to estimate the magnitudes of fatigue indicators and illustrated the effects of misalignment on the fatigue failure transition region in LCWJ. It is validated effectively as a supplementary analysis method for fatigue life prediction and fatigue design of welded joints compared with the experimental data and published reports.

Predicting damage evolution in panel paintings with machine learning

Understanding the mechanical behavior and properties of works of art, such as panel paintings, is helpful for evaluating the failure mechanisms in play, especially during exhibitions in confined spaces. A panel painting is typically formed by a wooden panel and by layers of gesso, followed by paints and varnishes on the top. Induced by environmental changes, the mismatch in the moisture response of a stiff gesso layer and of the underlying wood panel produces risk of fracture in the pictorial layer. This happens because the gesso layer experiences tension which leads to cracking if the mechanical strain exceeds a critical level. The proposed contribution aims to develop a 3D simplified model for paintings, to detect the environmental conditions which may lead to exceeding the critical strain levels. To this aim, a penny-shaped crack has been simulated inside the gesso layer, centered along the symmetrical planes. To establish if the crack is in critical conditions, the failure criterion of the strain energy density (SED) method has been used: when the critical SED value is reached, the crack is assumed to be in unsafe conditions, otherwise it is in safe conditions. Several combinations of the geometric parameters describing the model have been checked, allowing to define whether the different conditions are likely to lead to a further damage development. Finally, the obtained results have been used to train a preliminary extreme gradient boosting machine (XGBoost) that may be able to classify and predict the two possible outcomes: safe and unsafe. This way, by exploiting the capabilities of machine learning, it could be possible to limit the need of numerical simulations and to introduce new rationales in the framework of work of arts conservation.

ON ONE POSSIBILITY OF THE ASSESSMENT OF PROGRESSIVE COLLAPSE OF STRUCTURES IN FIRE

. The paper proposes an algorithm for determining the responsible elements in the progressive collapse of a building in fire. The algorithm is based on the use of the most flexible calculation schemes and iterative procedure of energy portraits of the building according to the Vasilkov-Shmukler criterion. The implementation of the algorithm is demonstrated on the example of determining the responsible elements of a real residential complex in Lviv. The object of the study is the residential complex «America», which consists of two 17-storey paired blocks and a block of two-storey shopping centre, located between them. Structurally, each of two buildings is a monolithic reinforced concrete frameless structure with voided overlappings made according to «Monofant» technology. At the beginning, all the initial data related to the structural scheme of the building are collected and systematized, such as geometric parameters, physical and mechanical characteristics of materials, engineering and geological conditions, loads and impacts. Then the computational model of the building is created. Herein the «Lira» software based on finite element method is used. Based on the results of the generated initial data and the building model, a static calculation is performed. After verification of compliance with the requirements for structures of the 1st and 2nd groups of limit states, the analysis of possible fire scenarios and modes is performed. The next step is determination of temperature fields, which gives the possibility to generate temperature loads. In general, 16 separate downloads were formed. The first download included all vertical loads from static calculation, and the next 15 downloads - all vertical loads and 1 different temperature load. The main aspect in the proposed algorithm is the criterion for selecting the responsible elements based on the approach of building of energy portrait of structure. After Vasylkov-Shmukler criterion, it is proposed to determine the values of the strain energy density for each finite element and strain energy of the whole system. According to the results of compiling 15 scenarios of emergency fire on one of the buildings and determining the largest values of strain energy density, the most dangerous accident scenario for the whole complex was identified.

Mechano-Regulation of Trabecular Bone Adaptation Is Controlled by the Local in vivo Environment and Logarithmically Dependent on Loading Frequency.

It is well-established that cyclic, but not static, mechanical loading has anabolic effects on bone. However, the function describing the relationship between the loading frequency and the amount of bone adaptation remains unclear. Using a combined experimental and computational approach, this study aimed to investigate whether trabecular bone mechano-regulation is controlled by mechanical signals in the local in vivo environment and dependent on loading frequency. Specifically, by combining in vivo micro-computed tomography (micro-CT) imaging with micro-finite element (micro-FE) analysis, we monitored the changes in microstructural as well as the mechanical in vivo environment [strain energy density (SED) and SED gradient] of mouse caudal vertebrae over 4 weeks of either cyclic loading at varying frequencies of 2, 5, or 10 Hz, respectively, or static loading. Higher values of SED and SED gradient on the local tissue level led to an increased probability of trabecular bone formation and a decreased probability of trabecular bone resorption. In all loading groups, the SED gradient was superior in the determination of local bone formation and resorption events as compared to SED. Cyclic loading induced positive net (re)modeling rates when compared to sham and static loading, mainly due to an increase in mineralizing surface and a decrease in eroded surface. Consequently, bone volume fraction increased over time in 2, 5, and 10 Hz (+15%, +21% and +24%, p ≤ 0.0001), while static loading led to a decrease in bone volume fraction (−9%, p ≤ 0.001). Furthermore, regression analysis revealed a logarithmic relationship between loading frequency and the net change in bone volume fraction over the 4 week observation period (R2 = 0.74). In conclusion, these results suggest that trabecular bone adaptation is regulated by mechanical signals in the local in vivo environment and furthermore, that mechano-regulation is logarithmically dependent on loading frequency with frequencies below a certain threshold having catabolic effects, and those above anabolic effects. This study thereby provides valuable insights toward a better understanding of the mechanical signals influencing trabecular bone formation and resorption in the local in vivo environment.

Fatigue assessment of high strength welded joints through the strain energy density method

AbstractThe main aim of the present work is to investigate, through the strain energy density method, the fatigue behaviour of high strength welded joints realised employed in hydraulic runner blades. The geometries, found in literature, present the welding bead machined in order to have no geometrical discontinuities in the specimen realising a wide fitting radius between the two welded plates; the only critical geometrical discontinuity in the specimen is given by the lack of penetration that leads to an internal crack‐like defect. The specimens presented failure both from the weld toe and from the weld root depending on the amount of welding penetration. The results, summarised in this work with the strain energy density method, show clearly the possibility to consider a unique master curve for this kind of joints regardless of the failure initiation point. Acquiring, through a finite element model, the strain energy density value both at the weld toe and at the weld root and comparing them, the method is proved to be adequate to detect the most critical area of the joint. A first fatigue master curve in terms of cyclic averaged strain energy density value is provided for the fatigue design of high strength welded joints.

Structural state of stress analysis of confined concrete based on the normalized generalized strain energy density

This paper investigates the behavior of 32 high-strength concrete square columns confined by high-strength transverse reinforcement under concentric compression, by applying structural state of stress theory. First, the sum of generalized strain energy density values (Eij) of columns at every load value (Fj) is normalized as Ej,norm to describe the structural state of stress. Then, the Ej,norm-Fj curves reveal the characteristics of the state of stress and failure mechanism of confined concrete columns under load-bearing condition. The Ej,norm-Fj curves also reveal an updated failure load definition and failure criterion. Furthermore, the effects of unconfined concrete compressive strength, volumetric ratio, and transverse reinforcement yield strength on the state of stress are studied. Finally, a formula is proposed to predict the updated failure loads of confined concrete columns. In summary, a novel method to identify some working characteristics of structures is explored.

Averaged strain energy density estimated rapidly from nodal displacements by coarse FE analyses: Cracks under mixed mode loadings

AbstractThe strain energy density (SED) averaged over a structural volume allows to assess the fracture and fatigue behaviour of cracked components under mixed‐mode loadings. To rapidly estimate the averaged SED, two approaches have been previously proposed: (a) the direct approach; (b) the peak stress method (PSM) and nodal stress (NS) approach. In this paper, the nodal displacement (ND) approach is presented to rapidly estimate the averaged SED from the nodal displacements by FE analyses. This method combines the advantages of all previous approaches, ie, the use of coarse meshes able to capture the contributions of both stress intensity factors (SIFs) and higher order terms, without requiring geometrical modelling of the control volume. To validate the method, cracked plates and cracked bars under mixed‐mode loadings have been analysed. The averaged SED values estimated by the ND approach agree with those calculated by the direct approach, within a range of applicability.

Volume free strain energy density method for applications to blunt V-notches

This work investigates the application of the strain energy density method to rounded V-notches under mode I loading conditions and to components without stress concentrators through finite element models that do not involve the construction of the control volume in the preprocessing phase of the finite element analysis. The application of the strain energy density method to components without stress concentrators like sharp V-notches requires, according to the conventional procedure of the method, two different numerical simulations; the first simulation defines the point of maximum of the first principal stress along the notch fillet; the second simulation calculates the strain energy density value within the control volume built using the position of the first principal stress maximum evaluated in the first simulation. Several numerical analyses, with changing the notch angle, the notch radius, the control volume radius and the mesh refinement, were carried out to evaluate the error in calculating the strain energy density value through the new procedure presented in the present work whose main advantages are to simplify the method and to decrease the calculation time that, dealing with complex geometries, is reduced by 50% requiring only one simulation instead of the two requested by the conventional procedure

Francis-99: Evaluation of the strain energy density value for welded joints typical of turbine runner blades

The main aim of this work is to investigate the fatigue behaviour of welded joints through an energetic approach based on the Strain Energy Density failure criteria. The geometries, taken from the literature, are typical of turbine runner blades. The results of the fatigue tests on these details were summarised through the Strain Energy Density approach. The application of this method to these geometries is the first step of a wider research with the aim to provide a suitable tool in FEM code for the lifetime estimation of components characterised by complex geometries.

Ductile Failure Prediction of U-Notched Bainitic Functionally Graded Steel Specimens Using the Equivalent Material Concept Combined with the Averaged Strain Energy Density Criterion

In this paper, the ductile fracture of bainitic functionally graded steel has been studied. Fracture tests was performed on U-notched specimens made of bainitic functionally graded steel under mode I. The averaged strain energy density criterion combined with equivalent material concept was employed to predict the ductile fracture of bainitic functionally graded steel. For this purpose, first, based on equivalent material concept, the mechanical properties of virtual brittle functionally graded steel were obtained. Then the averaged value of strain energy density over a well-defined control volume was calculated by finite element analysis for U-notched virtual brittle functionally graded steel. After that, the fracture loads was obtained based on the averaged strain energy density criterion. The agreement between experimental fracture loads and theoretical predictions was good.