Averaged Strain Energy Density Research Articles

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.

Fatigue failure investigation of thick plate butt‐welded joints with incomplete penetration effect

AbstractThe objective of this paper was to investigate the fatigue performance of the thick plate structural steel butt‐welded joints influenced by its geometrical factors, such as the notch angles, incomplete penetration length, and plate thickness. Utilizing experimental tests, finite element calculations, and analytical models, the fatigue behavior of various weld joint shapes and dimensions was analyzed, and the discrepancy of fatigue strength for different incomplete penetration lengths was evaluated using the nominal stress method. Meanwhile, the local approaches, including notch stress intensity factor (NSIF), averaged strain energy density (SED), effective notch stress (ENS) method, peak stress method (PSM), and implicit gradient effective stress (IGES) method, are employed to assess the fatigue life of thick plate welded joints. According to the results, incomplete penetration ratios in butt‐welded joints led to a decrease in fatigue strength. The analytical solutions of butt‐welded joints in high cycle regime based on SED results were obtained to estimate the fatigue characteristic values for different plate thickness and incomplete penetration ratios. The proposed equations could illustrate the relationship between the nominal stress and local SED dealing with the variations of incomplete penetration length and geometrical dimensions. Moreover, the results obtained by the analysis of ENS, PSM, and IGES could be deduced by the analytical models to determine the fatigue life of welded joints.

An innovative experimental energy-based approach for the life prediction of notched details

Fatigue assessment of mechanical components can be very challenging due to the presence of macroscopic defects and notches. However, local approaches as the Stain Energy Density, have shown how the fatigue resistance is strictly related to the local stress field near the notch tip and not on the overall dimensions of the component. It is necessary to know some peculiar material’s parameters to apply those local approaches. A short way to obtain such parameters can be to apply Thermographic Methods, such as the Risitano’s Thermographic Method and the Static Thermographic Method, monitoring the energy release during stepwise fatigue and static tensile tests. In this work, experimental tests were performed on plain and double edge steel notched specimens. The SED method has been applied through finite element models, with the easy application of the volume-free procedure, to evaluate its trend at several stress levels. The analysis of the energy release and of the local averaged strain energy density during a static tensile test may be adopted as a novel approach to relate the local stress state with a macroscopic stress value that introduce in the material the first micro plasticization leading to the failure of the component.

On the combined use of averaged strain energy density criteria (ASED) and equivalent material concept (ECC) for the fracture load prediction of additively manufactured PLA v-notched specimens

On the combined use of averaged strain energy density criteria (ASED) and equivalent material concept (ECC) for the fracture load prediction of additively manufactured PLA v-notched specimens

Heat treatment effects on fracture resistance of additively manufactured PLA specimens under mode I loading

AbstractThe current paper deals with the influence of heat treatment on the tensile, flexural, and fracture strength of polylactic acid (PLA) specimens made by the fused deposition modeling (FDM) technique. Three different temperatures of 80°C, 100°C, and 120°C were used to anneal the dog bone and enlarged compact tension (ECT) specimens to examine the mechanical and fracture performance of the FDM‐PLA parts, respectively. The fracture behavior was assessed using the equivalent material concept (EMC) combined with the J‐integral, average strain energy density (ASED), and maximum tangential stress (MTS) criteria, and the theoretical estimates were compared with the experimental results. Finally, the effectiveness of these criteria was discussed comprehensively, and the failure mechanisms in the broken tensile and flexural specimens were investigated using scanning electron microscopy (SEM). Overall, applying heat treatment to the FDM specimens could improve their structural integrity by around 57% from the fracture resistance point of view.

Length scale parameters to estimate fatigue lifetime of 3D-printed titanium alloy Ti6Al4V containing notches in the as-manufactured condition

The accuracy of the Theory of Critical Distances, Gradient Elasticity and the Averaged Strain Energy Density criterion in estimating fatigue lifetime of notched additively manufactured Ti6Al4V is assessed against numerous experimental results generated under load ratios equal to −1 and 0.1. The 3D-printed Ti-alloy under investigation was tested by keeping the notches in the as-manufactured condition. The common feature of the considered design approaches is that they all make use of a material length scale. The validation exercise based on the generated experimental results demonstrates that the length scale concept can be extended successfully also to the fatigue assessment of notched 3D-printed metallic components.

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.

Energy TCD - robust and simple failure prediction unifying the TCD and ASED criterion

Robust and simple failure prediction is a relevant and active field of research. The present work aims at unifying the Theory of Critical Distances (TCD) and Averaged Strain Energy Density (ASED) criterion. The proposed Energy TCD is formulated along with analytical expressions for the characteristic length scales and validated using experimental data. On this data set, the accuracy of the Energy TCD is similar to that of the classical methods. Most importantly, the presented methods allow to transfer the point-based calibration technique from notched specimens, which is common for the TCD, to the Energy TCD ASED criterion.

Improving the Stability of a Hemipelvic Prosthesis Based on Bone Mineral Density Screw Channel and Prosthesis Optimization Design.

In pelvic reconstruction surgery, the hemipelvic prosthesis can cause significant changes in stress distribution due to its high stiffness, and its solid structure is not suitable for osseointegration. The purpose of this study was to identify a novel bone mineral density screw channel and design the structure of the prosthesis so as to improve the distribution of stress, promote bone growth, and enhance the biomechanical properties of the prosthesis. The mechanical characteristics of bone mineral density screw and traditional screw were compared by finite element analysis method, and redesigned by topology optimization. The direction of the newly proposed screw channel was the posterolateral entrance of the auricular surface, ending at the contralateral sacral cape. Compared to the original group, the maximum stress of the optimized prosthesis was decreased by 24.39%, the maximum stress of the sacrum in the optimized group was decreased by 27.23%, and the average strain energy density of the sacrum in the optimized group was increased by 8.43%. On the surface of screw and connecting plate, the area with micromotion more than 28 μm is reduced by 12.17%. On the screw surface, the area with micromotion more than 28 μm is reduced by 22.9%. The newly determined screw channel and optimized prosthesis design can effectively improve the biomechanical properties of a prosthesis and the microenvironment of osseointegration. This method can provide a reference for the fixation of prostheses in clinical pelvic reconstruction.

Effects of anisotropy and pre-strain on the shear fatigue behaviour of DP600 steel

The effects of anisotropy and pre-strain on the low-cycle-fatigue properties of DP600 steel sheet under shear path were investigated. Fully reversed strain-controlled fatigue tests were performed along the 0° and 90° directions with the initial strain direction. The cyclic-loading amplitude determined whether the pre-strained material had a hardening process. The high dislocation density in the crystals of pre-strained material can promote the formation of low-energy substructures, obviously accelerating the cyclic-softening rate and reducing the fatigue life. Average plastic strain energy density was affected by the pre-straining state and loading direction and had a linear relationship with the strain amplitude. Overall, these results can be used to verify the calculation model of structural parts.

Fracture Load Predictions in Additively Manufactured ABS U-Notched Specimens Using Average Strain Energy Density Criteria.

This paper provides a methodology for the prediction of fracture loads in additively manufactured ABS material containing U-notches. The approach is based on the Average Strain Energy Density (ASED) criterion, which assumes that the material being analysed develops fully linear-elastic behaviour. Thus, in those cases where the material has a certain (non-negligible) amount of non-linear behaviour, the ASED criterion needs to be corrected. In this sense, in this paper, the ASED criterion is also combined with the Equivalent Material Concept (EMC) and the Fictitious Material Concept (FMC), both being corrections in which the non-linear real material is substituted by a linear equivalent or fictitious material, respectively. The resulting methodologies have been applied to additively manufactured ABS U-notched single-edge-notched bending (SENB) specimens combining five different notch radii (0, 0.25, 0.5, 1 and 2 mm) and three different raster orientations (0/90, 45/−45 and 30/−60). The results obtained demonstrate that both the ASED-EMC and the ASED-FMC combined criteria provide more accurate predictions than those obtained directly through the ASED criterion, with the ASED-EMC criterion generally providing safe more accurate predictions, with an average deviation from the experimental fracture loads between +1.0% (predicted loads higher than experimental loads) and −7.6% (predicted loads lower than experimental loads).

U-notch fracture in additively manufactured ABS specimens under symmetric three-point bending

This paper aims to investigate the fracture behavior of U-notched semi-circular bend (USCB) specimens fabricated from the Acrylonitrile Butadiene Styrene (ABS) using the fused deposition modelling (FDM) method under opening-mode loading. The notched specimens are manufactured with different notch tip radii and various unidirectional layup configurations for which the fracture loads are experimentally acquired by conducting fracture tests under symmetric three-point bending. Some additional tests are also performed on cracked and plain tensile specimens to obtain the fracture toughness and tensile properties of the ABS material. With the goal of estimating the fracture loads theoretically, the recently proposed virtual isotropic material concept (VIMC) is linked to three different brittle fracture models, namely the point-stress (PS), mean-stress (MS), and averaged strain energy density (ASED) criteria. Comparing with the experimental results, it is shown that the VIMC-MS criterion is generally the best model among the three ones, while VIMC-PS criterion gives fair results and VIMC-ASED criterion is excellent only for larger notch tip radii. The crack growth path of the mentioned samples as well as their type of failure is also evaluated. It is revealed that the unidirectional USCB samples with 0, 30, and 45 degrees layup configurations experience inter/cross-laminar fracture, while those with 60 and 90 degrees layups experience inter-laminar fracture.

Modeling and analysis of forced vibration of the thin-walled cylindrical shell with arbitrary multi-ring hard coating under elastic constraint

In this paper, a parameterized multi-partitioning method is presented for modeling arbitrary multi-ring hard coating treatment for the thin-walled cylindrical shell under different ring numbers and coating ratios, while the Rayleigh–Ritz method and the Chebyshev orthogonal polynomials of the second kind are employed to derive the governing equations of motion and the admissible displacement, respectively. Moreover, a unified formulation of external load based on the trigonometric and Chebyshev series is developed for forced vibration analysis of the shell with arbitrary axial half-wave number and circumferential wave number. Based on the experimental resonant frequencies, an improved inverse identification procedure is proposed in which the genetic algorithm and pattern search algorithm are sequentially employed to identify the stiffness coefficients of elastic constraint. Good agreement between the experimental and numerical resonant frequencies and responses exhibits the reliability and effectiveness of both the stiffness identification method and semi-analytical model. The identified results happen to find that the constraint stiffness may have the inherent property of frequency dependence due to the existence of micro sliding which would further result in soft nonlinearity in frequency response curve as shown in experimental results. Additionally, the influences of ring number and coating ratio on vibration and damping characteristics of the thin-walled cylindrical shell are investigated by introducing the average strain energy density and average modal loss factor.

J integral and local strain energy density approach to characterize the cracks in anisotropic hyperelastic skin type composite materials

It has been proposed that the response of soft tissues and skin type materials critically depends on the orientation and re-orientation of reinforcing fiber. Here, we quantify the effect of fiber orientation on the mechanical and fracture properties of anisotropic hyperelastic glass fiber (GF)/polydimethylsiloxane (PDMS) composites through a unified experimental and computational analysis. Experimental results show that the tensile strength and fracture toughness of skin type materials depends on the initial orientation of fiber families and is maximum for 0–90 fiber orientation. The re-orientation of fibers under tensile loading is experimentally demonstrated. The experimental findings are well complimented by the finite element (FE) simulations performed using bilinear strain stiffening fiber and matrix (BLFM) material model for soft tissues. The material model parameters are obtained by fitting the experimental data for a meaningful comparison. It is observed that fiber rotation along the loading direction indeed enhances the resistance to crack growth in skin type materials. The size of process zone (RC) ahead of the crack tip and relative contribution of anisotropic energy is characterized using J integral and average strain energy density (ASED) criteria. The RC and anisotropic energy contribution (ϕaniso-avg/ ϕavg) depends on the fiber orientation with higher values for 0–90 fiber orientation. The larger process zone enhances the fracture toughness for 0–90 orientation in comparison to other orientations. Moreover, RC and ϕaniso-avg/ ϕavg are found to be independent of initial crack length and therefore, they can be considered as material parameters for a given fiber orientation. The findings of the present investigations may be used to explore GF/PDMS composite as a possible candidate for soft actuator and bio-medical applications with enhanced toughness.

Automated implementation of the Peak Stress Method for the fatigue assessment of complex welded structures

Dealing with the fatigue design of complex welded structures, the Peak Stress Method (PSM) is a rapid approach to the fatigue strength assessment, which is based on the singular, linear-elastic opening, in-plane and out-of-plane shear peak stresses calculated at the weld toe or weld root adopting coarse FE meshes. By adopting the averaged Strain Energy Density (SED) as a fatigue strength criterion, a design stress, the so-called equivalent peak stress, has been defined as a proper combination of the local peak stresses and adopted in conjunction with a reference design curve to estimate the fatigue lifetime of welded structures. To furtherly enhance the applicability to complex welded structures, the PSM has recently been calibrated by using 10-node tetra elements, which allow to easily discretize complex geometries. Moreover, an interactive analysis tool has been developed within Ansys® Mechanical, in order to automate the application of the PSM to complex both aluminum and steel 3D welded structures subjected to multiaxial fatigue loads. The automated PSM tool has been developed taking advantage of the integrability between Ansys® Mechanical and modern programming languages, enforcing PSM compatibility requirements and automating the implementation tasks and phases of the method. In this perspective, the present work deals with the employment of the PSM tool for the fatigue strength assessment of some complex welded details adopted in modern roller-coaster structures. Due to the geometrical complexity of large-scale structures, private companies usually employ finite element beam models in order to evaluate nominal stresses, that must be compared with appropriate fatigue strength references (FAT classes) in design standards. The assessment procedure proposed in this work aims instead at defining appropriate stress-based FAT classes from the results of automated PSM analyses on the considered joint geometries. Moreover, the proposed approach allows to compensate the limited number of FAT classes and details available in International Standards and Recommendations, when dealing with complex joint geometries.

The Peak Stress Method applied to fatigue lifetime estimation of welded steel joints under variable amplitude multiaxial local stresses

Fatigue actions of most engineering welded structures involve variable amplitude (VA) loading cycles. The fatigue design of welded joints in such structures is based on the use of a fatigue strength criterion for constant amplitude (CA) loading in conjunction with a cumulative damage rule. In the present work, the Peak Stress Method (PSM), has been reformulated to estimate the lifetime of welded joints subjected to VA loadings. The PSM is an engineering finite element (FE)-oriented technique to rapidly estimate the mode I, II, and III notch stress intensity factors (NSIFs) at the weld toe or at the weld root, by taking advantage of the singular linear elastic, opening, sliding, and tearing peak stresses, respectively. Such stresses are computed at the V-notch tip by means of 2D or 3D FE models discretized using coarse meshes, provided that some conditions of applicability of the method are satisfied. The fatigue strength under CA loading is then evaluated by combining the simplicity and rapidity of the PSM in evaluating the NSIFs with a robust and validated fatigue strength criterion such as the one based on the averaged Strain Energy Density (SED), which can be written as a function of the relevant NSIFs. To preserve the simplicity of the method, its extension to VA loading conditions has been achieved by assuming the Miner’s Linear Damage Rule (LDR) as a cumulative damage rule. The proposed method has been validated against new experimental results generated by testing non-load-carrying (nlc) fillet-welded joints with double inclined attachment and made of structural steel under pure axial loading.

Numerical and Experimental Investigations of the Thermal Fatigue Lifetime of CBGA Packages

A thermal fatigue lifetime prediction model of ceramic ball grid array (CBGA) packages is proposed based on the Darveaux model. A finite element model of the CBGA packages is established, and the Anand model is used to describe the viscoplasticity of the CBGA solder. The average viscoplastic strain energy density increment ΔWave of the CBGA packages is obtained using a finite element simulation, and the influence of different structural parameters on the ΔWave is analyzed. A simplified analytical model of the ΔWave is established using the simulation data. The thermal fatigue lifetime of CBGA packages is obtained from a thermal cycling test. The Darveaux lifetime prediction model is modified based on the thermal fatigue lifetime obtained from the experiment and the corresponding ΔWave. A validation test is conducted to verify the accuracy of the thermal fatigue lifetime prediction model of the CBGA packages. This proposed model can be used in engineering to evaluate the lifetime of CBGA packages.

Experimental Verification of the Averaged Strain Energy Density Criterion for Brittle Fracture in Blunt V-Notches Under Pure Compression

Experimental Verification of the Averaged Strain Energy Density Criterion for Brittle Fracture in Blunt V-Notches Under Pure Compression

Applicability of strain energy density criterion for fracture prediction of notched PLA specimens produced via fused deposition modeling

The Averaged Strain Energy Density (ASED) criterion is validated for the failure prediction of notched Polylactide Acid specimens fabricated by Fused Deposition Modeling by means of experimental data and the results are compared to the Theory of Critical Distances. The common approach of estimating the ASED control volume radius based on the measured fracture toughness was shown to be suboptimal, arguably because of the difficulties of obtaining the fracture toughness with such complex materials. Therefore, a more robust approach is evaluated in analogy of the TCD and it is shown to successfully extend the range of applicability of the ASED criterion.

Critical Load Prediction in Notched E/Glass-Epoxy-Laminated Composites Using the Virtual Isotropic Material Concept Combined with the Average Strain Energy Density Criterion.

This paper attempts to validate the application of the Virtual Isotropic Material Concept (VIMC) in combination with the average strain energy density (ASED) criterion to predict the critical load in notched laminated composites. This methodology was applied to E/glass–epoxy-laminated composites containing U-notches. For this purpose, a series of fracture test data recently published in the literature on specimens with different notch tip radii, lay-up configurations, and a number of plies were employed. It was shown that the VIMC–ASED combined approach provided satisfactory predictions of the last-ply failure (LPF) loads (i.e., critical loads).