Average Strain Energy Research Articles

Rupture assessment of rubber/clay nanocomposites containing a crack by means of an energy-based fracture criterion

Due to the growing use of nanotechnology in the modern era, the application of nanomaterials in elastomers has also been increased. Although some limited experiments are available in the previous studies regarding the fracture behavior of rubber nanocomposites containing a crack, no criteria have been presented so far. To fill this gap, the current research is devoted to develop a criterion for rupture assessment of rubber nanocomposites weakened by a crack. First, some fracture tests are performed on cracked rubber/clay nanocomposites. The prepared nanocomposites are made of ethylene–propylene-diene monomer/styrene-butadiene rubber/CLOISITE 15. Afterwards, the averaged strain energy density (ASED) criterion, as one of the most used energy-based criteria, is extended and utilized in nano-reinforced hyperelastic materials. The nearly uniaxial state of stress field next to the crack tip in rubber nanocomposites, which is the main prerequisite for the criterion extension, is proved by means of non-linear finite element modelling. Finally, the estimations of the criterion is compared with the corresponding experimental data and good agreement is achieved which reveals the high performance of the ASED criterion in the case of cracked rubbers filled with nanoparticles.

Predicting weld root notch stress intensity factors for rib-to-deck welded joint under deck loading modes

Orthotropic steel deck (OSD) has been widely employed in the bridge engineering owing to its superiorities of light-weight and high strength. However, the fatigue damage of rib-to-deck welded joint in OSD is one of the most challenging issues. The notch stress intensity factors (NSIFs) that characterize the local mechanical properties of notch tips are major parameters for fatigue evaluation of rib-to-deck welded joint. This paper proposes formulae of the NSIFs of weld root for rib-to-deck welded joint under deck loading modes. First, parametric studies are conducted using a finite element model to establish a holistic understanding of the effects of the geometrical parameters on the NSIFs. The investigated parameters include the weld penetration rate, relative weld height, deck toe flank angle, and plate thickness ratio. Then, multi-parameter analysis is carried out based on the data generated from the parametric study. Finally, the proposed formulae of the NSIFs are evaluated and applied to evaluate the stress distribution and the averaged strain energy density of single-weld specimens for rib-to-deck welded joints. The maximum error of 9.5% indicates that the NSIFs predicted using the proposed formulae are in good agreement with the finite element analysis results. The proposed formulae provide useful tools for determining the stress distributions and the averaged strain energy density of single-weld rib-to-deck welded joint specimens.

Averaged strain energy density estimated rapidly from the nodal stresses by FEM for cracks under mixed mode loadings including the T-stress contribution

The present contribution reviews a recently proposed method to rapidly estimate the averaged SED at the tip of short as well as long cracks under in-plane I+II and long cracks under out-of-plane I+III mixed mode loadings. Short cracks are distinguished from long cracks by considering that the stress fields within the control volume of short cracks are no longer governed solely by the stress intensity factors (SIFs), but the contribution of higher order terms, and primarily the T-stress, becomes significant to estimate the averaged SED. According to the proposed method, the averaged SED is calculated using the linear elastic nodal stresses evaluated by FEM either at the crack tip, to account for the SIFs contribution, and at selected FE nodes of the crack free edges, to include the T-stress contribution. The advantage of the proposed approach is two-fold: coarse FE meshes can be adopted; moreover, geometrical modelling the control volume is no longer necessary. To validate the approach, cracked plates subjected to in-plane I+II mixed mode loading as well as bars weakened by circumferential outer cracks subjected to out-of-plane mixed mode I+III loading have been analysed. A comparison between approximate values of the averaged SED according to the nodal stress approach and those derived directly from the FE strain energy adopting very refined FE meshes has been successfully performed.

Notch size effect on in‐phase multiaxial fatigue life prediction of steel bar specimens

The current paper presents a comprehensive analysis including experimental and numerical works that elucidate the effect of notch behavior on in-phase multiaxial fatigue life. Multiaxial fatigue experiments of 45 steel and 45QT steel considering the notch radius and opening angles were performed. An investigation including analytical and computational frameworks for multiaxial fatigue analysis on the basis of average strain energy density (SED) theory was conducted to deal with the energy gradient. A bulk of new fatigue data were summarized by normal and shear stress firstly and then reanalyzed by means of the local SED by control volumes surrounding the notch tip. The notch effect on the multiaxial fatigue life is further clarified by SED theory.

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.

Energy-based ductile failure predictions in cracked friction-stir welded joints

This research aims to study the fracture behaviour in dissimilar aluminum alloys adjoined by friction stir welding (FSW). In this way, experimental data dealing with this topic was taken from the recent literature. In those experimental results, two metal sheets made of Al 7075-T6 and Al 6061-T6 were adjoined together by FSW in the form of well-known specimen, namely the cracked semi-circular bend (CSCB) and then they are tested under mixed mode I/II loading condition. Due to the fact that substantial plastic behavior exist in the welded material and consequently significant plastic deformations were observed around the crack tip, failure prediction of the mentioned specimens needs failure prediction models in the basis of the elastic-plastic fracture mechanics which can be realized as sophisticated operations inquiring long time. In this way, the Equivalent Material Concept (EMC) is utilized in this research and then coupled with two eligible energy-based criteria, namely the averaged strain energy density (ASED) and J-integral criteria. Thus, the critical failure load of the welded samples is predicted. Comparison between the empirical data and theoretical predictions from energy-based evaluations showed that this model has enough capability in estimating the critical failure load of the CSCB samples.

Rapid assessment of multiaxial fatigue lifetime in notched components using an averaged strain energy density approach

This paper proposes a rapid fatigue lifetime assessment approach to deal with notched components undergoing multiaxial loading. The modus operandi consists of calculating an effective quantity of the strain energy density nearby the crack initiation site of the notched component using an elastic-plastic numerical model. This quantity is then inserted into a fatigue master curve connecting the strain energy density with the number of cycles to failure to estimate the number of cycles to fatigue crack initiation. The proposed approach is applied to lateral U-shaped notched round bars subjected to multiaxial in-phase loading histories. Both the experimental results and the numerical predictions are in good agreement.

Application of a New, Energy-Based ΔS* Crack Driving Force for Fatigue Crack Growth Rate Description.

This paper presents the problem of the description of fatigue cracking development in metallic constructional materials. Fatigue crack growth models (mostly empirical) are usually constructed using a stress intensity factor ΔK in linear-elastic fracture mechanics. Contrary to the kinetic fatigue fracture diagrams (KFFDs) based on stress intensity factor K, new energy KFFDs show no sensitivity to mean stress effect expressed by the stress ratio R. However, in the literature there is a lack of analytical description and interpretation of this parameter in order to promote this approach in engineering practice. Therefore, based on a dimensional analysis approach, ΔH is replaced by elastic-plastic fracture mechanics parameter—the ΔJ-integral range. In this case, the invariance from stress is not clear. Hence, the main goal of this paper is the application of the new averaged (geometrically) strain energy density parameter ΔS* based on the relationship of the maximal value of J integral and its range ΔJ. The usefulness and invariance of this parameter have been confirmed for three different metallic materials, 10HNAP, 18G2A, and 19th century puddle iron from the Eiffel bridge.

Multi-parameter failure assessment of a bi-material V-notch – Crack initiation from a free-edge singularity

Multi-parameter fracture mechanics criteria are used to describe crack initiation conditions in general singular stress concentrators. In the cases of stress concentrators described by only one singular stress term, multi-parameter description provides a qualitative and quantitative increase of accuracy. In this paper multi-parameter criteria, based on an average tangential stress and average strain energy density factor, are described and used to determine a crack initiation direction and the critical load in the case of a free-edge singularity. The critical load expresses a failure/stability criterion for the studied general singular stress concentrator. Contrary to criteria employing singular stress terms only, a criterion of this kind is able to cover the influence of the direction of the external load.

Uniform scatter bands to analyse the fatigue strength of welded joints

This paper analyses the possibilities of a rational and uniform interpretation of the fatigue strength of welded joints, as a necessary basis for the development of sound and easily applicable design procedures. The re-analyses of experimental data here considered refer to as-welded joints, in steel and aluminium alloys, fabricated by using arc-welding, under axial or bending loads, under different stress ratios and modelled with the assumption of a null notch tip radius at the crack initiation location. The first part of the paper gives a synthetic presentation of the principal unifying parameters proposed in the literature and of the fatigue scatter bands derived with their application. The second part applies to welded joints a new parameter, recently introduced as an extension of the averaged Strain Energy Density approach. This parameter, which has been called the Strain Energy Density Intensity Factor, does not depend on the radius of the chosen control area, since it is not related to the fatigue strength of a specimen without geometrical notches. For this reason, it should make the averaged strain energy density concept more easily applicable to welded structures and give more general validity to the scatter bands derived with its application.

Fracture analysis of rounded-tip V-notched components made of rubber-like materials using averaged strain energy density criterion

Nowadays, hyperelastic materials such as rubbers and elastomers have become increasingly important in research and industry. However, the existence of any geometric discontinuities like a crack or notch will affect the behavior of rubbers. Thus, the prediction of rupture condition in these materials is of paramount importance. One of the energy-based criteria, which has attracted many attentions due to its convenience in use and high precision, is the averaged strain energy density (ASED) criterion. Based on this criterion, a limited amount of energy in a specified volume around the stress concentration region controls the fracture of components. This criterion, which was already utilized extensively for brittle materials, has been recently extended for being used in cracked components having non-linear behavior like rubbers. On the other hand, by examining the previous studies, it can be concluded that despite the importance of rounded V-shaped notch, no research has been carried out in the presence of this type of stress concentration in hyperelastic materials. As a consequence, this study is devoted to investigate the rupture of rubbers containing a round-tip V-shape notch using the ASED criterion. To this end, first, some experiments are performed for rounded-tip V-notched rubber samples. Afterwards and in order to utilize the ASED criterion for predicting the rupture loads of tested samples, some finite element modeling considering both geometric and material non-linearities are performed. The comparison between the estimations of the ASED criterion and the corresponding experimental data reveals the high efficacy of this criterion in hyperelastic materials weakened by round-tip V-shape notches.

The peak stress method applied to the fatigue assessment of tube-tube steel joints with weld ends under multiaxial loadings

The Peak Stress Method (PSM) is an approximate, FE-oriented application of the notch stress intensity factor (NSIF) approach to fatigue design of welded joints, which is based on the singular linear elastic peak stresses calculated from FE analyses performed by using coarse mesh patterns. By adopting the averaged strain energy density (SED) as a fatigue strength criterion, a design stress (the equivalent peak stress) can be defined; in conjunction with a reference design curve previously defined, the fatigue strength assessment of welded joints subjected to multiaxial fatigue loadings can be performed. In the present contribution, the PSM has been applied to the fatigue assessment of tube-tube steel joints with weld ends, which have been fatigue tested in a previous contribution under combined loadings: namely pure axial, pure torsion and in-phase as well as out-of-phase axial-torsion loadings, all of which with two load ratios, i.e. R = 0 and R = -1. The experimental fatigue results have been re-converted in terms of equivalent peak stress by adopting a 3D FE model including an idealised weld end geometry. The equivalent peak stress has proved to assess the fatigue crack initiation location in agreement with experimental observations, moreover a quite good agreement has been obtained between the experimental results and the PSM-based design scatter band.

Mixed mode (I+II, I+III) fatigue crack growth description in S355/P355NL1 steel

Fatigue crack growth models (mostly empirical) are usually constructed using stress intensity factor – ΔK in linear-elastic fracture mechanics. Contrary to the kinetic fatigue fracture diagrams (KFFDs) based on ΔK, Kmax new energy diagrams based on strain energy density parameter ΔH, ΔS* demonstrates no sensitivity to mean stress effect expressed by the stress ratio R. According to the dimensional analysis approach ΔH, ΔS* is replaced by elastic-plastic fracture mechanics parameter based on the energy parameter associated with ΔJ energy parameter range. Hence, the main goal of this paper is the application of the new averaged strain energy density parameter ΔS* based on hysteresis loop curves for cracked components under uniaxial as well as multiaxial stress state. The presented approach assumes that the fatigue crack growth increment in the elastic-plastic range is caused by the combination of the maximum value of the energy parameter and its positive part of the range in each cycle of loading. The usefulness and invariance of this parameter have been confirmed in many works.

Notch-defect interaction in additively manufactured Inconel 718

Powder bed fusion based additively manufactured components are known to have poor surface quality, especially when building downward facing surfaces. These surfaces can contain defects, from which fatigue cracks can be initiated. In this work the notched fatigue behaviour of Inconel 718 specimens produced by selective laser melting is investigated. The main focus is set on the interaction between notch geometries and local defects due to the amount of overhang in the notch region. Four different geometries are considered, with different amount of notch acuities and degree of downward facing surfaces. A variation in failure sites, with respect to the notch bisector line, was fond in the specimens, and the position was found to be dependent on the amount of overhang and notch acuity. The fatigue life was found to be dependent on the size of surface defects measured in fracture surfaces. Further, the use of average strain energy density as a failure criteria in additively manufactured metals is discussed.

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.

Dynamic deformation behavior and microstructural evolution during high-speed rolling of Mg alloy having non-basal texture

The dynamic deformation behaviors and resultant microstructural variations during high-speed rolling (HSR) of a Mg alloy with a non-basal texture are investigated. To this end, AZ31 alloy samples in which the basal poles of most grains are predominantly aligned parallel to the transverse direction (TD) are subjected to hot rolling with different reductions at a rolling speed of 470 m/min. The initial grains with a TD texture are favorable for {10–12} twinning under compression along the normal direction (ND); as a result, {10–12} twins are extensively formed in the material during HSR, and this consequently results in a drastic evolution of texture from the TD texture to the ND texture and a reduction in the grain size. After the initial grains are completely twinned by the {10–12} twinning mechanism, {10–11} contraction twins and {10–11}-{10–12} double twins are formed in the {10–12} twinned grains by further deformation. Since the contraction twins and double twins have crystallographic orientations that are favorable for basal slip during HSR, dislocations easily accumulate in these twins and fine recrystallized grains nucleate in the twins to reduce the increased internal strain energy. Until a rolling reduction of 20%, {10–12} twinning is the main mechanism governing the microstructural change during HSR, and subsequently, the microstructural evolution is dominated by the formation of contraction twins and double twins and the dynamic recrystallization in these twins. With an increase in the rolling reduction, the average grain size and internal strain energy of the high-speed-rolled (HSRed) samples decrease and the basal texture evolves from the TD texture to the ND texture more effectively. As a result, the 80% HSRed sample, which is subjected to a large strain at a high strain rate in a single rolling pass, exhibits a fully recrystallized microstructure consisting of equiaxed fine grains and has an ND basal texture without a TD texture component.

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.