Averaged Strain Energy Density Research Articles

Understanding strain controlled low cycle fatigue response of P91 steel through experiment and cyclic plasticity modeling

In this paper, constitutive models for cyclic plasticity, namely, Chaboche and the Ohno-Wang models, are used to simulate the low cycle fatigue response of tempered ferritic-martensitic P91steel. A series of uniaxial strain-controlled low cycle fatigue tests are conducted at room temperature to evaluate various fatigue parameters. P91 steel shows cyclic softening behavior at all strain amplitudes. The analysis of the shape of hysteresis loops reveals that P91 steel deviates from Masing behavior. Plastic strain energy is used as a parameter to characterize the fatigue damage. The Morrow energy model estimates experimental average plastic strain energy density accurately. Remnant tensile properties are evaluated to quantify the damage during low cycle fatigue loading. The evaluation of back stress components and material Jacobian are implemented through a user defined subroutine UMAT into the commercial finite element package ABAQUS. Ohno-Wang material model predicts the hysteresis loop shape, area, and cyclic softening nature better than the Chaboche model. The predicted fatigue lives by simulation correlates well with the experimental fatigue lives within ± 200 cycles. These results suggest that energy based parameters can provide a reliable alternative to strain based approach in evaluating the fatigue damage of P91 steel.

Evaluation of Biomechanical Health Degree of Peri-Implant Bone Through Finite Element Analysis: A First Approach

The biomechanical health degree of peri-implant bone plays a critical role during the service of implants. This paper presents a preliminary exploration of the quantitative evaluation of the biomechanical health degree for the bone tissues around dental implant through finite element method. The finite element model of a part of mandible with three molars is constructed based on computer tomography scan image as a control sample, which is supposed to represent a healthy state. The model of treated mandible is made by replacing the middle tooth in the healthy model with a commercial implant. A regional average strain energy density (RASED) is proposed as a more accurate index to describe the stress state of peri-implant bone tissues, compared with the widely used maximum equivalent von Mises stress. The simulation shows that the stress state in peri-implant bone, i.e., the distribution and level of stress, is highly dependent on the modulus of implant material. Among the implants made of materials with various moduli, including Ti, stainless steel, zirconia, porous Ti, dentin material and polyether-ether-ketone (PEEK), the ones with medium modulus (15–40[Formula: see text]GPa) are found to achieve relatively healthy stress states. This study provides an effective tool to assess the risk of overloading or stress shielding in peri-implant bone tissues. It demonstrates a great potential in the optimization of design, production and usage of implants.

Finite element model of load adaptive remodelling induced by orthodontic forces

Many hypotheses have been formulated to explain orthodontic tooth movement. However, none of them can satisfactorily explain how small static orthodontic forces can induce bone remodelling. Our hypothesis assumes that small orthodontic forces do not lead to bone remodelling response directly, but rather indirectly by offsetting the tooth, leading to changes in bone loading during chewing that far exceed the changes caused by the orthodontic force alone. We developed a finite element model of a tooth with the periodontal ligament and the surrounding bone, and calculated the changes in strain energy density caused by the chewing force alone, the orthodontic force alone, and the combined action of the chewing and orthodontic forces. The results (average strain energy density 0.0005–0.01 kPa) demonstrate that the orthodontic loading alone does not produce strain energy density values in a range that is expected to induce remodelling (0.016–1.6 kPa). However, when the chewing and orthodontic forces are applied together, the highest values for the average strain energy density (0.02–0.99 kPa) as well as large changes in the bone tissue strain energy density (31.19–166.65%) are obtained. We conclude that the proposed hypothesis can indeed explain the observed bone remodelling that agrees with Wolff's law.

Fatigue evaluation of rib-to-deck welded joint using averaged strain energy density method

This paper investigates the feasibility of an averaged strain energy density (SED) method for fatigue evaluation of rib-to-deck weld joint in orthotropic steel deck. The effect of weld geometry on fatigue resistance of rib-to-deck joint is evaluated. The analysis results of the presented average SED method are validated against fatigue testing data and compared with the results of the conventional hot-spot stress and effective notch stress methods. A W-N curve is derived using the averaged SED method and used for evaluating the fatigue strength of rib-to-deck welded joints. The averaged SED method is also used to investigate the effect of weld geometrical variables on the fatigue failure mode transition, and the fatigue strength of full-scale orthotropic steel deck specimens. The results indicate that the averaged SED method provides superior ability in evaluating fatigue resistance and failure mode of rib-to-deck welded joint.

Energy-based approach for fracture assessment of several rocks containing U-shaped notches through the application of the SED criterion

This work presents an energetic continuum approach for the fracture assessment of rocks containing U-shaped notches and subjected to Mode I loading conditions. Three different methodologies are proposed, all based on the premise that brittle failure will occur when the average strain energy density over a certain control area reaches a critical value that only depends on the material, as stated by the Strain Energy Density (SED) criterion.The first method proposed (A) deals with the application of the SED criterion through an expression with a series of already tabulated parameters, which are particularised for the analysed rocks by rational extrapolation. By contrast, the second method (B) aims to obtain numerically the previously extrapolated parameters, and the third method (C) directly relates the strain energy density with the applied load, without the use of those parameters.The research is based on the results obtained from an exhaustive experimental programme comprising 300 fracture specimens tested in four-point bending conditions. These tests combine parallelepiped samples made of six different types of rocks (two marbles, two limestones, a sandstone and a granite) and containing eight different notch radii (varying from 0.15 mm up to 15 mm).Thus, this work aims to show the potential, capacity and limitations of the SED criterion in rock fracture analyses, by comparing the experimentally obtained fracture loads to those predicted by the three proposed methodologies.

Rupture analysis of rubber in the presence of a sharp V-shape notch under pure mode-I loading

The rupture behavior of styrene-butadiene rubbers (SBR) in the presence of a V-shape notch is investigated for the first time both experimentally and theoretically. In the experiments, V-notched samples of SBR are tested under tensile loading and their rupture displacements are determined. Afterwards and in the analytical field, the rupture loads of tested rubbers are predicted using the averaged strain energy density (ASED) criterion. The key idea of this criterion (i.e. the almost uniaxial state of stress field near the notch tip) is verified through non-linear finite element modeling. It is shown that good agreement exists between the predictions of the ASED criterion and the experimental results obtained for SBR. Moreover, the microscopic study of the ruptured surfaces of the notched SBR demonstrates its high roughness which can be attributed to the resistance of the rubber chains against the crack growth.

Notched Ti-6Al-4V titanium bars under multiaxial fatigue: Synthesis of crack initiation life based on the averaged strain energy density

The fatigue life of notched components subjected to multiaxial loading conditions may be influenced by extrinsic mechanisms operating during crack propagation phase, such as sliding contact, friction and meshing between crack surfaces. These extrinsic mechanisms may disrupt the applicability of local approaches, such as that based on the strain energy density (SED) averaged over a structural volume having size R0 and surrounding the crack initiation point. In the present contribution, the multiaxial fatigue behaviour of circumferentially notched specimens made of titanium grade 5 alloy, Ti-6Al-4V, has been analysed. Pure bending, pure torsion and in-phase as well as out-of-phase combined bending-torsion fatigue tests have been carried out on notched specimens characterized by two different root radii, namely 0.1 and 4 mm. Fatigue crack initiation and subsequent propagation have been monitored by adopting the direct current potential drop (DCPD) technique. In principle, crack initiation life has been defined when the initiated fatigue crack fractures the structural volume, which corresponds to a given potential drop increase calibrated by means of electrical finite element (FE) analyses. The structural volume size R0 has been determined by fatigue testing plain and sharp V-notched specimens under pure axial or pure torsion loadings. Finally, the averaged SED approach has been adopted to correlate the experimental fatigue results expressed in terms of crack initiation life, in order to reduce the effect of extrinsic mechanisms.

Energy-based assessment of brittle fracture in VO-notched polymer specimens under combined compression-shear loading conditions

This research presents some experimental, numerical, and theoretical results on brittle fracture of disk-type test specimens weakened by V-notches with end-holes under mixed mode I/II loading with negative mode I contributions. First, 54 fracture tests are conducted on VO-notched Brazilian disk specimens made of the general-purpose polystyrene under mixed mode I/II loading with negative mode I contributions. Then, two energy-based brittle fracture criteria, namely the averaged strain energy density and averaged strain energy density based on the equivalent factor concept are proposed to predict the experimentally obtained fracture loads of the tested general-purpose polystyrene specimens. Additionally, the fracture initiation angles of the tested VO-notched Brazilian disk specimens are predicted by using averaged strain energy density criterion. The finite element analyses, as well as the experimental observations, show that although brittle fracture in the specimens under mixed mode I/II loading takes place from the applied load side of the notch border by local tensile stresses, the notch bisector line and the other sides of the notch border sustain compressive stresses. In fact, this phenomenon states the concept of mixed mode I/II loading with negative mode I contributions. Finally, it is shown that good agreement exists between the experimental results and the theoretical predictions of the two energy-based fracture criteria.

Study on the influence of elastic modulus heterogeneity on in-situ stress and its damage to gas shale reservoirs

Shale heterogeneity has a significant effect on drilling and completions, hydraulic fracturing, as well as hydrocarbon development performance. However, it lacks representation of rock damage/failure caused by the mechanical heterogeneity and “stress shadow” effect during the hydraulic fracturing process. In this paper, the Galerkin finite element method was adopted to numerically simulate the hydro-mechanical coupled interaction based on the solver in the COMSOL Multiphysics software and Matlab scripting development. The Weibull probability density function was used to represent the mechanical heterogeneity of gas shale. Under the condition of fully fluid-solid coupling during the fracking process, the effect of mechanical heterogeneity on von – Mises stress, strain energy density, damage factor, and fluid pressure was numerically simulated in the gas shale wells. The curves of von – Mises stress, strain energy density, and damage factor along a certain straight line showed the obvious decreasing distribution in a completely homogeneous formation. As the strata are heterogeneously enhanced, their distribution curves showed fluctuations. Moreover, von – Mises stress and strain energy density had a good relationship with damage factors. Accordingly, the method of rock damage/fracture and "stress shadow" effect caused by mechanic heterogeneity was put forward under the circumstance of two–dimensional plane strain. That was to use, the von – Mises stress or strain energy density at the single point or line to characterize the degree of local rupture or the shadow effect of stress, and the average strain energy density per unit area to characterize the degree of rupture of the rock or the intensity of the shadow effect. The study is of great significance to further improve the SRV(Stimulated Reservoir Volume) fracturing design and the productivity of gas shale wells.

Static assessment of nanoscale notched silicon beams using the averaged strain energy density method

This paper extends the averaged Strain Energy Density (SED) method to the static assessment of notched components at the nanoscale. First, in situ micromechanical testing of notched nano-cantilever beams made of single-crystal silicon is briefly reviewed. Then, an alternative strategy based on the Theory of Critical Distances is employed to evaluate the control volume and the critical SED. The method is later verified against experiments and FE analyses. The SED method successfully estimates the load at fracture of nanoscale notched specimens with a maximum discrepancy of 4.7%. Moreover, the method is mesh-independent, and therefore very coarse meshes can be employed in numerical analyses. Finally, the results are discussed on the basis of the breakdown of continuum fracture mechanics at the nanoscale. The extension of the SED approach to the micro- and nanoscales provides a fast and simple tool for the design of micro- and nanodevices.

Averaged strain energy density to assess mixed mode I/III fracture of U-notched GPPS samples

In the present contribution, fracture resistance of U-notched GPPS members under mixed mode I/III loading conditions is assessed by using the Averaged Strain Energy Density (ASED) criterion. This criterion has been founded based on the ASED parameter averaged over a well-defined control volume embracing the notch edge. The validation of the theoretical criterion predictions is evaluated through comparing with the results of a series of mixed mode I/III fracture tests conducted on rectangular-shaped GPPS specimens weakened by a single edge U-notch. A recently developed apparatus for mixed mode I/III fracture experiments is employed for measuring the fracture loads of the specimens. The test samples are fabricated with different notch tip radii with the aim of evaluating the influence of this major feature of the U-notched components on the mixed mode I/III fracture behavior. It is shown that the onset of brittle fracture in U-notched GPPS specimens under various combinations of tension and out-of-plane shear can well be predicted by means of the ASED criterion.

The Peak Stress Method to assess the fatigue strength of welded joints using linear elastic finite element analyses

In fatigue design of welded joints according to the notch stress intensity factor (NSIF) approach, the weld toe profile is assumed to be a sharp V-notch having tip radius equal to zero, while the root side is assumed to be a pre-crack in the structure. The Peak Stress Method (PSM) is an engineering, FE-oriented method to estimate the NSIFs starting from the singular linear elastic peak stresses calculated at the V-notch or crack tips by using a coarse FE mesh. The element type is kept constant and the average element size can be chosen arbitrarily within a given range. The method is used in conjunction with Ansys software. The FE meshes are claimed to be coarse in comparison to those necessary to evaluate the NSIFs from the local stress distributions. Two-dimensional as well as three dimensional FE analyses can be adopted to apply the method. By using the averaged Strain Energy Density (SED, which can be expressed as a function of the relevant NSIFs) as a fatigue strength criterion, a so-called equivalent peak stress is defined to assess either weld toe or weld root fatigue failures in conjunction with a properly calibrated design curve. After presenting the theoretical background of the method, the paper presents a review of applications of the PSM relevant to steel welded joints under uniaxial as well as multiaxial fatigue loadings. Because of the relatively coarse FE analyses required and simplicity of post-processing the calculated peak stresses, the PSM might be useful in the everyday design practice.

Multi-parameter average strain energy density factor criterion applied on the bi-material notch problem

The stress field near a bi-material notch tip has the singular character. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress term exponent. The terms can be either singular or non-singular depending on the value of stress term exponent. The exponents are determined as solution of eigenvalue problem and depend on material properties and local geometry. The factors are calculated by analytical-numerical approaches and depend on global geometry and applied loading. The singular terms become unbounded when approaching the bi-material notch tip while the non-singular terms vanish. The non-singular terms increase precision in stress description on larger distances from the notch tip. The crack initiation conditions from the notch tip can be assessed by the average strain energy density factor (SEDF) criterion. In the case of bi-material notch problem, the crack can initiate in the direction of global minimum, the local minimum or the interface. The actual crack initiation load is calculated for each of abovementioned directions considering the individual fracture toughnesses. Contrary to the case of a crack, the direction of minimum of SEDF changes in dependence on the distance from the tip, therefore the average value over specific distance is used. The multi-parameter criterion considers singular and non-singular stress terms. Therefore in cases, when the specific distance is large, the non-singular terms provide means to improve precision of prediction of crack initiation parameters.

Multi-parameter average strain energy density factor criterion applied on the sharp material inclusion problem

The tip of a sharp material inclusion (SMI) embedded in a parent material is considered as the singular stress concentrator. The SMI tip is modelled as a special case of a multi-material junction. The singular stress behaviour is caused by the material mismatch and geometrical discontinuity. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress terms exponent. The exponents are determined as a solution of eigenvalue problem. The factors are calculated by combination of analytical and numerical approaches. The terms can be either singular or non-singular depending on the stress term exponent. When approaching the concentrator, the singular terms become unbounded while the non-singular terms vanish. The non-singular terms increase precision of stress description on larger distances from the point of singularity. In some cases they provide the only means to describe the stress field well. Because of the singular stress behaviour near SMI tip, this location is prone to crack initiation. The crack initiation conditions are calculated by the average strain energy density factor (SEDF) criterion. Contrary to the case of a crack, the direction of minimum of SEDF changes with distance from the singular point. Therefore, an averaged value over specific distance is used. If the specific distance is relatively large, the employment of non-singular terms in multi-parameter criterion can significantly improve the critical parameters prediction. Thanks to that e.g. the particle composite design can be optimized.

Averaged strain energy density criterion for rupture assessment of cracked rubbers: A novel method for determination of critical SED

In the present study, the application of averaged strain energy density (ASED) criterion has been extended to hyperelastic materials. Because of the material and geometry nonlinearities, commonly known for rubber-like materials, the use of conventional relations for determining the criterion parameters is no longer allowable. Therefore, by taking the advantage of a simple uniaxial state of stress field ahead of the crack tip in hyperelastic materials, a novel method has been proposed for determining the critical value of strain energy density. The sound agreement between the theoretical estimates based on the employed ASED criterion and the experimental data, taken from the literature, confirms the suitability of the proposed method.

Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids.

Understanding the mechanical behavior of multicellular monolayers and spheroids is fundamental to tissue culture, organism development, and the early stages of tumor growth. Proliferating cells in monolayers and spheroids experience mechanical forces as they grow and divide and local inhomogeneities in the mechanical microenvironment can cause individual cells within the multicellular system to grow and divide at different rates. This differential growth, combined with cell division and reorganization, leads to residual stress. Multiple different modeling approaches have been taken to understand and predict the residual stresses that arise in growing multicellular systems, particularly tumor spheroids. Here, we show that by using a mechanically robust agent-based model constructed with the peridynamic framework, we gain a better understanding of residual stresses in multicellular systems as they grow from a single cell. In particular, we focus on small populations of cells (1-100 s) where population behavior is highly stochastic and prior investigation has been limited. We compare the average strain energy density of cells in monolayers and spheroids using different growth and division rules and find that, on average, cells in spheroids have a higher strain energy density than cells in monolayers. We also find that cells in the interior of a growing spheroid are, on average, in compression. Finally, we demonstrate the importance of accounting for stochastic fluctuations in the mechanical environment, particularly when the cellular response to mechanical cues is nonlinear. The results presented here serve as a starting point for both further investigation with agent-based models, and for the incorporation of major findings from agent-based models into continuum scale models when explicit representation of individual cells is not computationally feasible.

A synthesis of geometry effect on brittle fracture

There are numerous analytical solutions in order to take into account the geometry effect on the fracture toughness and the fracture load of the cracked specimens. However, due to the complexities of the mentioned criteria it is practical to have an engineering method which can be used for fracture prediction in cracked samples of various shapes and loading conditions. In this paper, the fracture behaviors of five different testing samples made of various brittle and quasi-brittle materials have been studied using an energy based criterion namely the Average Strain Energy Density (ASED) criterion. According to the formulation of the ASED criterion, all the stress terms around the crack tip were taken into account and the brittle fracture of different samples with various geometry constraints were well predicted.

Fracture investigation of V-notch made of tungsten-copper functionally graded materials

The fracture of V-notches with end holes made of tungsten-copper functionally graded material under mode I has been studied in this paper. The averaged strain energy density over a well-defined control volume was employed to predict the fracture loads. A numerical approach was used to determine the outer boundary of the control volume. Mechanical properties such as elasticity modulus, Poisson’s ratio, fracture toughness KIc, and ultimate tensile stress have been considered to obey the power law function through the specimen width.

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Local strain energy density for the fatigue assessment of hot dip galvanized welded joints: some recent outcomes

Since in literature only data about the effect of the hot-dip galvanizing coating on fatigue behavior of unnotched specimens are available, whereas very few for notched components and none for welded joints, the aim of this paper is to partially fill this lack of knowledge comparing fatigue strength of uncoated and hot-dip galvanized fillet welded cruciform joints made of structural steel S355 welded joints, subjected to a load cycle R = 0. 34. The results are shown in terms of stress range ?? and of the averaged strain energy density range DeltaW in a control volume of radius R0 = 0.28 mm