Averaged Strain Energy Density Research Articles

Rock Fracture Toughness Under Mode II Loading: A Theoretical Model Based on Local Strain Energy Density

The values of mode II fracture toughness reported in the literature for several rocks are studied theoretically by using a modified criterion based on strain energy density averaged over a control volume around the crack tip. The modified criterion takes into account the effect of T-stress in addition to the singular terms of stresses/strains. The experimental results are related to mode II fracture tests performed on the semicircular bend and Brazilian disk specimens. There are good agreements between theoretical predictions using the generalized averaged strain energy density criterion and the experimental results. The theoretical results reveal that the value of mode II fracture toughness is affected by the size of control volume around the crack tip and also the magnitude and sign of T-stress.

RETRACTED: Fatigue Strength of Hot-Dip Galvanized Welded Steel Connections

Retracted article: This paper investigates the effect of a galvanizing coating on the fatigue strength of S355 structural steel. The aim of the present paper is to partially fill this lack of knowledge. A comparison is carried out, between hot dip galvanized fillet welded cruciform joints made by S355 structural steel and not treated welded joints characterized by the same geometry, subjected to a load cycle R = 0. Thirty four new experimental data are summarized in the present contribution, in terms of stress range Δσ and averaged strain energy density range in a control volume of radius R0 = 0.28 mm.

On the applicability of ASED criterion for predicting mixed mode I+II fracture toughness results of a rock material

A mixed mode fracture criterion based on the strain energy density averaged over a control volume has been successfully used earlier for evaluating the brittle fracture behavior of notched or cracked components made of engineering materials like metals, polymers and graphite. However, the applicability of the average strain energy density (ASED) criterion has never been investigated till now for rock materials. To fill this lack, a set of mixed mode I+II rock fracture toughness data provided experimentally by the authors are recalled and then predicted theoretically using the ASED criterion. The investigated fracture results have been obtained for a brittle white marble by testing an inclined edge crack triangular shape specimen subjected to symmetric three point bend loading. It is shown that very good ASED predictions are obtained for the fracture loads of the tested rock material in the entire range of mode mixities from pure mode I to pure mode II.

The application of strain energy density criterion to fatigue crack growth behavior of cracked components

The aim of this paper is to study the effects of specimen geometry on fatigue crack growth rate and fatigue life in cracked components by using a modified form of averaged strain energy density criterion. The modified criterion takes into account the first non-singular term of the Williams series expansion in addition to the classical stress intensity factors. Different samples are used to conduct high cycle fatigue experiments on Al 7075-T6. The fatigue behavior of cracked Al 7075-T6 samples is investigated experimentally and theoretically for the compact tension (CT), double cantilever beam (DCB) and Brazilian disc (BD) specimens. It is shown that the fatigue crack growth in these specimens is significantly geometry-dependent. The modified criterion can predict the fatigue crack growth rate and the fatigue life of the pre-cracked specimens well.

Tensile failure prediction of U-notched plates under moderate-scale and large-scale yielding regimes

The main goal of the present research is to check if the recently published experimental results on tensile load-carrying capacity (LCC) of ductile U-notched Al 7075-T6 and Al 6061-T6 plates could successfully be predicted by means of combining the Equivalent Material Concept (EMC) with the Averaged Strain Energy Density (ASED) criterion, which is fundamentally a brittle fracture criterion. By using the two main material properties, namely the fracture toughness and the tensile strength of the equivalent material, which is extracted from the EMC, the maximum load that each notched plate could sustain is predicted by means of the combined EMC-ASED criterion. It is revealed that the EMC-ASED criterion is capable of predicting well the LCC of both aluminum plates, independent of the level of ductility of materials and the size of plastic region around the notch at crack initiation instance.

Application of an average strain energy density criterion to obtain the mixed mode fracture load of granite rock tested with the cracked asymmetric four-point bend specimens

Mixed mode brittle fracture behaviour of granite rock is studied experimentally and theoretically using Asymmetric Four Point Bend (AFPB) specimens containing pre-cracks subjected to different mixed mode loading conditions, ranging from pure mode I to pure mode II. The main aim of this paper is twofold. First, to present a complete set of experimental results on fracture of pre-cracked granite samples under various in-plane loading mixities, and second, to predict the fracture loads of the tested rock samples under mixed mode I/II conditions using an energy-based criterion, namely the Average Strain Energy Density (ASED) criterion. Good agreement is found between the experimentally obtained fracture loads and the theoretical predictions based on the constancy of the mean strain energy density over the material volume. It is shown that the ASED criterion is able to provide well predictions for the fracture loads of the investigated rock material containing a pre-crack.

40CrMoV13.9 notched specimens under multiaxial fatigue: an overview of recent results

The multiaxial fatigue strength of circumferentially V-notched and semicircular notched specimens made of 40CrMoV13.9 has been investigated in this paper. The multiaxial load were applied using combined tension and torsion loading, both in-phase and out-of-phase. The axissymmetric V-notched specimens were characterized by a blunt notch tip (radius: 1 mm) and notch opening angle of 90o and the semicircular specimens were characterized by a constant notch tip radius. The net sectional area of both cases was equal with a diameter of 12 mm. The notched specimens were tested under pure mode I (tension), mixed mode I/III and pure mode III (torsion). More than 120 fatigue test data are used in this paper to evaluate the fatigue behaviour of 40CrMoV13.9 alloy. All fatigue data are analysed using the Average Strain Energy Density (ASED) criterion which employs the mean value of the strain energy density evaluated over a finite size semicircular sector surrounding the tip of the notch to predict the fatigue life of the tested specimens.

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Fatigue assessment of steel rollers by means of the local energy

This paper aims to analyses the fatigue behavior of steel rollers using the average strain energy density (SED) criterion. Considering the variability of the V-notch opening angle, a simple scalar quantity, i.e. the value of the strain energy density averaged in a control volume surrounding the notch tip, has been introduced to overcome the complexities in failure assessment of this component. The strain energy is obtained using close form solutions based on the relevant Notch Stress Intensity Factors (NSIF) for modes I, II and III. Referring to the conventional arc welding processes, the radius of the control volume is carefully identified with reference to conventional arc welding processes being equal to 0.28 mm for welded joints made of steel. In this paper firstly the employed methodology for the fatigue assessment is described and then the first synthesis of fatigue data by means of local SED for a specific geometry is shown.

Crack initiation life in notched Ti-6Al-4V titanium bars under uniaxial and multiaxial fatigue: synthesis based on the averaged strain energy density approach

The fatigue behaviour of circumferentially notched specimens made of titanium alloy, Ti-6Al-4V, has been analysed. To investigate the notch effect on the fatigue strength, pure bending, pure torsion and multiaxial bending-torsion fatigue tests have been carried out on specimens characterized by two different root radii, namely 0.1 and 4 mm. Crack nucleation and subsequent propagation have been accurately monitored by using the direct current potential drop (DCPD) technique. Based on the results obtained from the potential drop technique, the crack initiation life has been defined in correspondence of a relative potential drop increase ?V/?V0 equal to 1%, and it has been used as failure criterion. Doing so, the effect of extrinsic mechanisms operating during crack propagation phase, such as sliding contact, friction and meshing between fracture surfaces, is expected to be reduced. The experimental fatigue test results have been re-analysed by using the local strain energy density (SED) averaged over a structural volume having radius R0 and surrounding the notch tip. Finally, the use of the local strain energy density parameter allowed us to properly correlate the crack initiation life of Ti-6Al-4V notched specimens, despite the different notch geometries and loading conditions involved in the tests.

Fracture assessment of V-notched specimens with end holes made of tungsten-copper functionally graded material under mode I loading

The present research is the first to assess fracture of V-notches with end holes (VO-notches) made of tungsten-copper functionally graded material (W-Cu FGM) under mode I both experimentally and theoretically. W-Cu FGM specimens were fabricated by powder metallurgy technique. A number of fracture tests were carried out on VO-notched W-Cu FGM specimens under mode I for various notch tip radii, notch depths, and notch opening angles.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. Besides, the effect of notch geometry (notch tip radius, notch depth, and notch opening angle) on fracture load was elaborated. This study demonstrates that SED works well on VO-notched FGM specimens under mode I loading.

An engineering approach to predict mixed mode fracture of PUR foams based on ASED and micromechanical modelling

The Averaged Strain Energy Density (ASED) criteria is applied herein to reinterpret the fracture data of PUR foams. Four type of specimens were used in fracture tests. The ASED parameters were determined based on micromechanical models. The volume control for cracked components is represented by a circle with the centre at the crack tip for all type of fracture modes. It was also demonstrated that the SED parameters obtained from pure mode I could be applied successfully for mixed modes and mode II. This approach represents an useful engineering tool for the assessment of brittle fracture of components made of cellular materials.

Mixed-mode (I/II) rupture assessment of rubber-like materials weakened by cracks using the averaged strain energy density criterion

In the present study, the application of the averaged strain energy density (ASED) criterion is extended from pure mode-I to mixed-mode (I/II) loading for hyperelastic materials. Indeed, the use of a recently proposed method by the present authors for determination of the critical values of strain energy density and radius of control volume in mode-I loading has been generalized to the mixed-mode loading conditions. The key point in the generalization is the almost uniaxial state of stress fields near the crack tip in rubber-like materials. To validate the accuracy of the criterion in mixed-mode (I/II) loading, a new set of experiments on rubbers weakened by cracks has been carried out. The experimental results confirm the accuracy of the ASED criterion in the case of cracked rubbers under mixed-mode (I/II) loading.

Static Strength of V-Notches With End Holes Under Combined Tension-Shear Loading: Experimental Measurement by the Disk Test and Theoretical Prediction by the Local Energy

Abstract The aim of the present research work was twofold. Firstly, to provide some new experimental results regarding brittle fracture in V-notches with end holes (VO-notches) under mixed mode I/II loading; secondly, to check the suitability of the well-established averaged strain energy density criterion (SED) in predicting the load-carrying capacity of notched specimens. A new test specimen, called Brazilian disk containing central VO-notches (VO-BD), made of PMMA was proposed and utilized to perform fracture tests at room temperature for different mode mixities. It was found that the average SED criterion seems to successfully evaluate the static strength of VO-BD specimens for different notch angles and various notch radii.

Mixed mode fracture analysis using generalized averaged strain energy density criterion for linear elastic materials

A modified energy-based model called the generalized averaged strain energy density (ASED) criterion is proposed to predict the mixed mode I/II fracture resistance of brittle and quasi-brittle materials. The proposed criterion takes into account the effect of T-stress in addition to the stress intensity factors KI and KII when calculating the strain energy density averaged in a control volume around the crack tip. The effect of T-stress on the mixed mode fracture resistance of materials is investigated by using the generalized ASED criterion and the generalized SED criterion. The results predicted by these criteria are compared with the experimental results reported in the literature from fracture tests conducted on the Brazilian disc specimens and the semi-circular bend specimens made of Harsin marble and Guiting limestone. It is shown that the theoretical estimates of the generalized ASED criterion are consistent very well with the experimental results. The theoretical results also show that the specimens having a negative T-stress require higher loads in order their averaged strain energy density around the crack tip attains its critical value.

Averaged strain energy density criterion to predict ductile failure of U-notched Al 6061-T6 plates under mixed mode loading

The well-known Averaged Strain Energy Density (ASED) criterion, which is basically a brittle fracture criterion, is employed in conjunction with the Equivalent Material Concept (EMC) to predict the experimental failure loads of some U-notched ductile Al 6061-T6 plates reported in the literature under mixed mode I/II loading. While the EMC-ASED criterion does not claim that it is able to predict the fracture plane, the growth path of the crack emanating from the notch border, etc.; it is shown to be successful in predicting the load-carrying capacity (LCC) of the U-notched Al 6061-T6 plates, that fail by the large-scale yielding (LSY) regime, without performing time-consuming and complex elastic-plastic analyses.

Retracted: Fracture investigation of U‐notch made of tungsten–copper functionally graded materials by means of strain energy density

AbstractThe averaged strain energy density over a well‐defined control volume was employed to assess the fracture of U‐notched specimens made of tungsten–copper functionally graded materials under prevalent mode II loading. The boundary of control volume was evaluated by using a numerical method. Power law function was employed to describe the mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness and ultimate tensile stress) through the specimen width. The effect of notch tip radius and notch depth on notch stress intensity factors and mode mixity parameter χ were assessed. In addition, a comparison based on fracture load between functionally graded and homogeneous W–Cu was made. Furthermore, in this research, it was shown that the mean value of the strain energy density over the control volume can be accurately determined using coarse meshes for functionally graded materials.

Multiaxial fatigue strength assessment of welded joints using the Peak Stress Method – Part II: Application to structural steel joints

In Part I of the present manuscript, the Peak Stress Method (PSM) has been extended for the first time to assess the fatigue strength of welded joints in aluminium alloys subjected to in-phase as well as out-of-phase multiaxial loadings. A so-called equivalent peak stress has been defined on the basis of the averaged Strain Energy Density (SED) fatigue strength criterion. The equivalent peak stress can be used in principle to assess either weld toe and weld root fatigue failures in conjunction with an appropriate design curve for multiaxial fatigue loadings. In the present Part II of the paper, the Peak Stress Method is applied to assess the multiaxial fatigue strength of welded joints made of structural steels. The method has been validated against a bulk of experimental data taken from the available literature. The equivalent peak stress has shown to correlate with good approximation most of the analysed experimental data.

Strain energy density based fatigue cracking assessment of load-carrying cruciform welded joints

This paper presents fatigue failure crack initial points and the fatigue failure mode transition relationship between the weld toe and root of load-carrying cruciform welded joints on the basis of LEFM and Williams’ notch stress intensity theory. The notch stress intensity factors (NSIFs) of different failure locations were calculated quantitatively. The averaged strain energy density (SED) in a control volume near the notches is used to unify the scalar quantity and rectify inconsistencies of NSIFs units for the weld toe and root. The failure mode transition relations of different geometrical parametric models (such as main plate thickness, attachment plate thickness, weld length, and penetration size) were compared and analyzed based on the SED values from weld toe and root locations, and the effect of unequal main plate thickness ratio in a cruciform joint on failure mode was also investigated. The results show that SED approach can be used to accurately predict the transition region of equal base plate thickness. The weld length and penetration size are related to significant changes in the fatigue failure location of cruciform joints. The SED approach effectively reflects failure characteristics and can be used to assess fatigue failure behaviors for load-carrying cruciform welded joints.

Mixed mode I/II fracture investigation of Perspex based on the averaged strain energy density criterion

In this work, some recent mixed mode I/II fracture toughness results obtained from Perspex (or polymethylmethacrylate (PMMA)) with four simple cracked specimens subjected to the conventional three-point bend loading are reanalysed based on local energy concept. Although all the mentioned samples have been tested under the same and similar mode mixities, different fracture toughness envelopes were obtained for mixed mode I/II fracture of PMMA. The averaged strain energy density (SED) criterion has been applied in the past for different types of notched specimens (including U, V, O and keyhole notches). It is shown that the mixed mode tensile-in plane shear fracture toughness data obtained from the semicircular and triangular crack type specimens are successfully predicted for sharp cracked PMMA samples using the SED criterion.