A total strain energy density model of metal fatigue

In this paper a total cyclic strain energy density equal to the sum of plastic strain energy and tensile elastic strain energy densities is used as a damage parameter for metal fatigue. It is shown that the total cyclic strain energy density is a consistent damage parameter for low- and high-cycle fatigue in the conditions of both uniaxial and multiaxial cyclic loading. This parameter is also consistent with the concept of crack initiation and subsequent propagation. The approach described here is applicable for both ideal Masing and non-Masing material response. The predictions of the proposed criterion are compared with the experimental data for medium carbon steel St5. The comparison has shown good agreement.Published inProblemy Prochnosti, Nos. 1–2, pp. 53–64, January–February, 1995.     

Energy criteria of multiaxial fatigue failure

This paper contains a review of energy-based criteria of multiaxial fatigue. The criteria have been divided into three groups, depending on the kind of strain energy density per cycle which is assumed as a damage parameter. They are: (i) criteria based on elastic strain energy for high-cycle fatigue; (ii) criteria based on plastic strain energy for low-cycle fatigue; and (iii) criteria based on the sum of plastic and elastic strain energies for both low- and high-cycle fatigue. The criteria which take into account strain energy density in the critical plane seem to be the most promising. In the energy approach to multiaxial fatigue there is an important unsolved problem, i.e. the evaluation of energy, especially plastic strain energy density, from closed stress–strain hysteresis loops under random loadings.     

Fatigue life of welded joints according to energy criteria in the critical plane

We have determined theoretical notch coefficients with use of the fictitious radius in tube-tube and flange-tube welds. The parameter of normal and shear strain energy density in critical planes is applied for estimation of fatigue life under cyclic conditions of pure bending, pure torsion and combined proportional bending with torsion. The critical planes were determined with use of two methods based on the maximum parameters of, respectively, normal and shear strain energy density. __________ Translated from Problemy Prochnosti, No. 4, pp. 118–124, July–August, 2006.     

An energy‐based damage parameter for the life prediction of AISI 304 stainless steel subjected to mean strain

Abstract

This study extends the plastic strain energy approach to predict the fatigue life of AISI 304 stainless steel. A modified energy parameter based on the stable plastic strain energy density under tension conditions is proposed to account for the mean strain and stress effects in a low cycle fatigue regime. The fatigue life curve based on the proposed energy parameter can be obtained directly by modifying the parameters in the fatigue life curve based on the stable plastic strain energy pertaining to fully reversed cyclic loading. Hence, the proposed damage parameter provides a convenient means of evaluating fatigue life on the mean strain or stress effect. The modified energy parameter can also be used to explain the combined effect of alternating and mean strain/stress on the fatigue life. In this study, the mean strain effects on the fatigue life of AISI 304 stainless steel are examined by performing fatigue tests at different mean strain levels. The experimental results indicate that the combination of an alternating strain and a mean strain strongly influences the fatigue life. Meanwhile, it is found that the change in fatigue life is sensitive to changes in the proposed damage parameter under the condition of a constant strain amplitude at various mean strain levels. A good agreement is observed between the experimental fatigue life and the fatigue life predicted by the proposed damage parameter. The damage parameter proposed by Smith et al. (1970) is also employed to quantify the mean strain effect. The results indicate that this parameter also provides a reasonable estimate of the fatigue life of AISI 304 stainless steel. However, a simple statistical analysis confirms that the proposed damage parameter provides a better prediction of the fatigue life of AISI 304 stainless steel than the SWT parameter.     

Fracture analysis of functionally graded materials with U‐ and V‐notches under mode I loading using the averaged strain‐energy density criterion

One of the most powerful criteria to predict the critical fracture load in plates with notches is the strain‐energy density averaged over a well‐defined control volume ahead of the notch tip. Although the averaged strain‐energy density (ASED) criterion has been proposed for homogeneous materials, it has been shown in this paper that this criterion can also be applied for non‐homogeneous materials, especially for functionally graded materials (FGMs). A numerical method has been used to evaluate the control volume boundary, the averaged strain‐energy density over the control volume, and also the critical fracture load in FGMs under mode I loading. A new set of experimental results on fracture of blunt V‐notched samples made of austenitic–martensitic functionally graded steel under mode I loading have been provided.     

Influence of fibre taper on the work of fibre pull-out in short fibre composite fracture

A model has been formulated to determine the work of pull-out, U, of an elastic fibre as it shear-slides out of a plastic matrix in a fractured composite. The fibres considered in the analysis have the following shapes: uniform cylinder and ellipsoidal, paraboloidal or conical tapers. Energy transfer at the fibre–matrix interface is described by an energy density parameter which is defined as the ratio of U to the fibre surface area. The model predicts that the energy required to pull out a tapered fibre is small because the energy transfer at the fibre–matrix interface to overcome friction is small. In contrast, the pull-out energy of a uniform cylindrical fibre is large because the energy transfer is large. The pull-out energies of the paraboloidal and ellipsoidal fibres lay between those for the uniform cylindrical and the conical fibres. With the exception of the uniform cylindrical fibre which yields a constant energy density, tapered fibres yield expressions for the energy density which depend on the fibre axial ratio, q. In particular, the energy density increases as q increases but converges at large q. By defining the critical axial ratio, q ₀, as the limit beyond which u is independent of the fibre slenderness, our model predicts the value of q ₀ to be about 10. These results are applied to explain the mechanisms regulating fibre composite fracture.     

Recrystallization-etch approach to study the plastic energy absorbtion at crack initiation and extension of pressure-vessel steel A533B-1

To evaluate the elastic-plastic fracture toughness parameter of nuclear pressure-vessel steel A533B-1, a newly developed technique (the recrystallization-etch technique) for plastic strain measurement was applied to different sizes of compact tension specimens with a crack length/specimen width of 0.6–0.5 that were tested to generate resistance curves for stable crack extensions. By means of the recrystallization-etch technique, the plastic energy dissipation or work done within an intense strain region at the crack tip during crack initiation and extension was measured experimentally. Furthermore, the thickness effects on this crack tip energy dissipation rate were examined in comparison with other fracture-parameter J integrals. Thickness effects on critical energy dissipation and energy dissipation rate during crack extension were obtained and the energy dissipation rate dW _p/da in the mid-section shows a constant value irrespective of specimen geometry and size, which can be used as a fracture parameter or crack resistance property.     

Energy scale of the pairing model with energy-dependent density of state

The critical temperature of a pairing model for HT _c S has been calculated using an energy-dependent density of states. We showed that the problem which is connected with the van Hove scenario is very sensitive to the energy scale and to the behavior of the density of states at low and high energy.     

Revisiting intrinsic brittleness and deformation behavior of B2 NiAl intermetallic compound: A first-principles study

For NiAl intermetallic compound with B2 structure, there is still no calculation combining models of single and multiple layers while using the same basic set. Furthermore, some recently proposed criteria for brittleness and toughness have not been used to analyze its deformation behavior. Thus, first-principles calculation was applied to comprehensively study the elastic properties, ideal strength, generalized stacking fault energy and surface energy of B2-NiAl intermetallic compound. The results suggest that calculations based on the current basic set give more accurate lattice parameters and elastic moduli. The Pugh criterion and Cauchy pressure cannot be used to interpret the intrinsic brittleness of NiAl. In comparison, the ductility parameter based on the strain energy under elastic instability and ZCT and Rice criteria based on generalized stacking fault energy and surface energy successfully identify the intrinsic brittleness of the NiAl intermetallic compound. The reason why [111] slip always occurs in the deformation along [100] direction was clarified by examining the critical value for brittle-ductile transition. The results of density of state indicate shear deformation has less impact on structural stability, and the change of charge density difference implies that <001> shear induces more intensive redistribution of charge density, which is well correlated to the brittleness and deformation behavior of NiAl intermetallic compound.     

A Review of Critical Plane Orientations in Multiaxial Fatigue Failure Criteria of Metallic Materials

The paper presents a review of multiaxial fatigue failure criteria based on the critical plane concept. The criteria have been divided into three groups, according to the fatigue damage parameter used in the criterion, i.e. (i) stress, (ii) strain and (iii) strain energy density criteria. Each criterion was described mainly by the critical plane orientation. Multiaxial fatigue criteria based on the critical plane concept usually apply different loading parameters in the critical plane whose orientation is determined by (a) only shear loading parameters (crack Mode II or III), (b) only normal loading parameters (crack Mode I) or sometimes (c) mixed loading parameters (mixed crack Mode). There are also criteria based on few critical plane orientations and criteria based on critical plane orientations determined by a weighted averaging process of rotating principal stress axes.     

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

The 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.     

Use of an energy‐based/critical plane model to assess fatigue life under low‐cycle multiaxial cycles

For engineering components subjected to multiaxial loading, fatigue life prediction is crucial for guaranteeing their structural security and economic feasibility. In this respect, energy‐based models, integrating the stress and strain components, are widely used because of their availability in fatigue prediction. Through employing the plastic strain energy concept and critical plane approach, a new energy‐based model is proposed in this paper to evaluate the low‐cycle fatigue life, in which the critical plane is defined as the maximum damage plane. In the proposed model, a newly defined NP factor κ^*  is used to quantify the nonproportional (NP) effect so that the damage parameter can be conveniently calculated. Moreover, a simple estimation method of weight coefficient is developed, which can reflect different contributions of shear and normal plastic strain energy on total fatigue damage. Experimental data of 10 kinds of materials are employed to assess the effectiveness of this model as well as three other energy‐based models.     

Criteria for brittle fracture in biaxial tension

The criteria of maximum tangential stress, maximum tangential principal stress, maximum tangential strain and strain energy density are applied to the problems of slit and elliptical cracks under remote uniform biaxial tension. The predicted direction of crack extension and the critical load are compared with experimental results reported by other investigators. The unstable crack paths are determined. The four criteria differ in the case of unequal tension; the strain energy density criterion is the least satisfactory. The criteria of maximum tangential strain and strain energy density can be modified to give a good prediction of critical load.     

A novel multiscale approach to brittle fracture of nano/micro‐sized components

Principles and advantages of a new concept based on the ab initio aided strain gradient elasticity theory are shown in comparison with the classical Barenblatt cohesive model. The method is applied to the theoretical prediction of the critical energy release rate and the crack tip opening displacement at the crack instability in nanopanels made of germanium and molybdenum crystals. The necessary length scale parameter l₁ is determined for germanium and molybdenum by the best gradient elasticity fits of ab initio computed screw dislocation displacements and phonon dispersions. Values of ab initio computed critical energy release rates and crack opening profiles revealed that the length l₁ is related to inflexion points of profiles. A novel ab initio method in combination with continuum mechanics was successfully tested to replace molecular statics dependent of availability of interatomic potentials. The asymptotic strain gradient elasticity solution for displacement components near the crack tip in materials with cubic lattice was also derived.     

Low‐cycle fatigue of IN 718: Effect of waveform

The influence of various strain waveforms on the low‐cycle fatigue of IN 718 tested at 650°C has been investigated. The straining paths are accompanied by dwell‐induced creep component(s) or unequal strain distribution in different portions of cycles reducing strength of material. The investigation intends to clarify mainly mechanistic aspects of relaxation‐fatigue interaction. Features of time‐dependent effect induced by nonpeak dwell and the same accompanied by peak dwell, slow unloading from the peak to a lower strain, and different loading and unloading rates are compared in terms of stress amplitude responses, mean stress relaxation, hysteresis loops, life, and damage parameter D_C‐F. Softening is common in all the cases, and degree of softening varies linearly with life. The energy‐based life prediction model has been found to work well for the data, and we have introduced energy fraction–based approach to observe simultaneous contribution from both creep and fatigue on life.     

Application of energy density theory on finite cracked bodies

Abstract

A new concept of the energy release rate of a finite cracked body is proposed. Considering the global view of the strain energy density field, the new fracture parameter presented here is different from the conventional energy release rate that only depends on the stress field around the crack tip but neglects the influences induced by the boundary conditions on the far field. Based on the hypothesis of the energy density theory, fracture initiation and termination, respectively can be predicted by the local and global relative minima of the strain energy density function. The new energy release rate is then defined as the integration of the strain energy density along the fracture trajectory from the initiation point to the destination point. The results show that the difference between the new and the conventional energy release rate becomes more pronounced if the material has a large core region (or the material is more ductile) and if the height‐width ratio of a finite cracked plate is comparatively small.     

Acoustic emission imaging of shear failure in large reinforced concrete structures

The averaged value of the strain energy density over a well-defined volume is used to predict the static strength of U-notched specimens under mixed-mode conditions due to combined bending and shear loads. The volume is centered in relation to the maximum principal stress present on the notch edge, by rigidly rotating the crescent-shaped volume already used in the literature to analyse U- and V-shaped notches subject to mode I loading. The volume size depends on the ultimate tensile strength σ _u and the fracture toughness K _IC of the material. In parallel, an experimental programme was performed. All specimens are made of polymethyl-metacrylate (PMMA), a material which exhibits quasi-brittle behaviour at -60°C. Good agreement is found between experimental data for the critical loads to failure and theoretical predictions based on the constancy of the mean strain energy density over the control volume.     

J-integral evaluation for U- and V-blunt notches under Mode I loading and materials obeying a power hardening law

The paper deals with calculations of the J-integral for a plate weakened by U- and V-blunt notches under mode I loading in the case of a linear and nonlinear elastic material. The main aim of the study is to suggest simple equations suitable for rapid calculations of the J-integral. The semicircular arc of the notch, which is traction free, is assumed as integration path and the J-integral is given as a function of the strain energy over the notch edge. For a numerical investigation of the strain energy density distribution on the notch edge the equation W(θ)=W_max cos^δ(θ) has been assumed, where δ has been determined from finite element analyses. In particular, the following values of the notch acuity a/ρ and the opening angle 2α have been analyzed: 4 ≤ a/ρ ≤ 400 and 0 ≤ 2α ≤ 3π /4. Considering plates weakened by lateral and central notches under symmetric mode I loading, the approximate relationships for the strain energy density, which require the presence of a non zero notch radius for their application, and the J-integral are discussed firstly considering a linear elastic material and then a material obeying a power hardening law during the loading phase. The predicted results of the J-integral are consistent with those directly obtained from finite element analyses.     

Investigation of the Temperature and Density Dependences of the Effective Pair Potential Parameters Using Variational Theory

A variational theory (VT), in which the potential energy of a real system is evaluated relative to the hard-sphere system, has been used to investigate the medium's effects on the pair potential parameters. By adding the medium's effects to the isolated pair potential, the concept of an effective pair potential (EPP) has been introduced. The advantage of such a potential (EPP) over the isolated pair potential is that the configurational energy can exactly be written as the sum of all EPP of all pairs available in the system. The parameters of such a pair potential will then show state dependence. The EPP parameters for different dense fluids at various temperatures have been obtained via the VT, and they have been shown to be density independent for densities greater than the Boyle density, _B 1.8 _c , (where _c is the critical density), while at lower densities the parameters depend on density as well as temperature. For any dense fluid, the depth of the EPP, , is found to be larger than its corresponding isolated pair. When the EPP parameters are used to reduce temperature and density, the cut-off parameter, C=d/ depends only on the reduced density, and this parameter shows a strong principle of corresponding states for different fluids. The resulting expression for the cut-off parameter has then been used to accurately predict the internal energy. Finally, the EPP parameters are compared with those of the average effective pair potential (AEPP) for Ar, to show the importance of the medium effects and the long-range interactions of the AEPP in dense fluids, individually. This comparison shows that the depth parameter of the AEPP is much larger than that of the EPP. Since the long-range interactions are mainly attractive, such a conclusion is reasonable.