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Engineering components/structures with geometric discontinuities normally bear complex and variable loads, which lead to a multiaxial and random/variable amplitude stress/strain state. Hence, this study aims how to effectively evaluate the multiaxial random/variable amplitude fatigue life.

Recent studies on critical plane method under multiaxial random/variable amplitude loading are reviewed, and the computational framework is clearly presented in this paper.

Some basic concepts and latest achievements in multiaxial random/variable amplitude fatigue analysis are introduced. This review summarizes the research status of four main aspects of multiaxial fatigue under random/variable amplitude loadings, namely multiaxial fatigue criterion, method for critical plane determination, cycle counting method and damage accumulation criterion. Particularly, the latest achievements of multiaxial random/variable amplitude fatigue using critical plane methods are classified and highlighted.

This review attempts to provide references for further research on multiaxial random/variable amplitude fatigue and to promote the development of multiaxial fatigue from experimental research to practical engineering application.

Continuum‐atomistic modeling denotes a mixed approach combining the usual framework of continuum mechanics with atomistic features like e.g. interaction potentials. Thereby, the kinematics are typically characterized by the so called Cauchy‐Born rule representing atomic distance vectors in the spatial configuration as an affine mapping of the atomic distance vectors in the material configuration in terms of the local deformation gradient. The application of the Cauchy‐Born rule requires sufficiently homogeneous deformations of the underlying crystal. The model is no more valid if the deformation becomes inhomogeneous. By virtue of the Cauchy‐Born hypothesis, a localization criterion has been derived in terms of the loss of infinitesimal rank‐1 convexity of the strain energy density. According to this criterion, a numerical yield condition has been computed for two different interatomic energy functions. Therewith, the range of the Cauchy‐Born rule validity has been defined, since the strain energy density remains quasiconvex only within the computed yield surface. To provide a possibility to continue the simulation of material response after the loss of quasiconvexity, a relaxation procedure proposed by Tadmor et al. [1] leading necessarily to the development of microstructures has been used. Alternatively to the above mentioned criterion, a stability criterion has been applied to detect the critical deformation. For the study in the postcritical region, the path‐change procedure proposed by Wagner and Wriggers [2] has been adapted for the continuum‐atomistics and modified.

The purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity.

The new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level.

Two-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure.

This paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

In this paper, the peak kinetic energy density (KED) of soil particles during earthquake excitation is used as an intensity measure for the evaluation of liquefaction potential under field conditions. The paper seeks to discuss this measure.

Using centrifuge tests data, it is shown that seismic pore water pressure buildup is proportional to cumulative KED at a particular soil depth. Linear relationships are found between cumulative kinetic energy and corresponding cumulative strain energy. To consider the effect of soil amplification, several equivalent linear ground response analyses are performed and the results are used to derive an equation for depth reduction factor of peak kinetic energy density. Two separate databases of liquefaction case histories are used in order to validate the proposed model. The performance of the proposed model is compared with a number of commonly used shear stress‐based liquefaction assessment methods. Finally, the logistic regression method is employed to obtain probabilistic boundary curves based on the present model. Parametric study of the proposed probabilistic model is carried out to verify its agreement with the previous methods.

It has been shown that the kinetic energy model works satisfactorily in classifying liquefied and non‐liquefied cases compared with the existing recommendations of shear stress‐based criterion. The results of the probabilistic kinetic energy model are in good agreement with those of previous studies and show a reasonable trend with respect to the variations of fines content and effective overburden pressure. The proposed model can be as used an alternative approach for assessment of liquefaction potential.

These findings make a sound basis for the development of a kinetic energy‐based method for assessment of liquefaction potential.

This paper demonstrates the application of a new multiaxial creep damage model developed by authors using stress traixiality to predict the failure time of a component made of 0.5% Cr-0.5% Mo-0.25% V low alloy steel. The model employs strain energy density and assumes that the uniaxial strain energy density of a component can be easily calculated and can be converted to multi-axial strain energy density by multiplying it to a function of stress trixiality which is a ratio of mean stress to equivalent stress. For comparison, an elastic-creep and elastic-plastic-creep FEA analysis is performed to get multi-axial strain energy density of the component which is compared with the calculated strain energy density for both cases. The verification and application of the model are demonstrated by applying it to thin tube for which the experimental data are available. The predicted failure times by the model are compared with the experimental results. The results show that the proposed model is capable of predicting failure times of the component made of the above-mentioned material with an accuracy of 4.0%.

The purpose of this paper is to perform an analytical study of non-linear elastic delamination fracture in the multilayered functionally graded split cantilever beam (SCB) configuration. The SCB studied may have an arbitrary number of vertical layers. The material in each layer is functionally graded along the layer thickness. Also, the material properties may be different in each layer. The analytical solution derived was applied for parametric investigations in order to evaluate the effects of material properties and delamination crack location on the non-linear fracture behaviour.

The delamination fracture was studied in terms of the strain energy release rate. The SCB mechanical response was described by using a power-law stress-strain relation. A non-linear analytical solution for the strain energy release rate was derived by considering the SCB complementary strain energy. In order to verify the solution, an additional analysis of the strain energy release rate was developed by considering the complementary strain energy in the beam cross-sections ahead and behind the crack front.

The effects of material gradient, crack location along the beam width and non-linear material behaviour on the delamination fracture were evaluated. The analytical solution derived is useful for parametric studies of non-linear fracture in multilayered functionally graded beams.

Delamination fracture in the multilayered functionally graded SCB configuration was analysed with considering the non-linear material behaviour.

The lead‐free solder joint reliability of several printed circuit board mounted high‐density packages, when subjected to temperature cycling was investigated by finite element modelling. The packages were a 256‐pin plastic ball grid array (PBGA), a 388‐pin PBGA, and a 1657‐pin ceramic column grid array. Emphasis was placed on the determination of the creep responses (e.g. stress, strain, and strain energy density) of the lead‐free solder joints of these packages.

The purpose of this paper is to carry out a delamination fracture analysis of the three-dimensional functionally graded split cantilever beam (SCB) configuration.

The fracture behaviour was studied analytically in terms of the strain energy release rate by applying methods of linear elastic fracture mechanics. It was assumed that the material is functionally graded along the beam width, height and length. The strain energy release rate was derived by analysing the stress and strain state in the beam cross-sections ahead and behind the crack front. An additional analysis of the strain energy release rate was performed by considering the beam strain energy for verification.

The influence of material gradient along the beam width, height and length on the delamination fracture behaviour was investigated. The effect of crack length was analysed too. The analytical approach developed is very useful for parametric fracture investigations. The results obtained in the present study can be applied for optimisation of three-dimensional functionally graded beam systems in their design with respect to delamination fracture behaviour.

An analytical approach for studying the delamination fracture in the SCB configuration that is functionally graded along the beam width, height and length was developed.

The purpose of this paper is to develop an analysis of longitudinal fracture behaviour of a functionally graded non-linear-elastic circular shaft loaded in torsion. It is assumed that the material is functionally graded in both radial and longitudinal directions of the shaft (i.e. the material is bi-directional functionally graded).

The Ramberg–Osgood stress-strain relation is used to describe the non-linear mechanical behaviour of the functionally graded material. The fracture is studied in terms of the strain energy release rate by analysing the balance of the energy. The strain energy release rate is obtained also by differentiating of the complementary strain energy with respect to the crack area for verification.

Parametric studies are carried out in order to evaluate the influence of material gradients in radial and longitudinal directions, the crack location in radial direction and the crack length on the fracture behaviour of the shaft. It is found that by using appropriate gradients in radial and longitudinal directions, one can tailor the variations of material properties in order to improve the fracture performance of the non-linear-elastic circular shafts to the externally applied torsion moments.

A longitudinal cylindrical crack in a bi-directional functionally graded non-linear-elastic circular shaft loaded in torsion is analysed by using the Ramberg–Osgood stress-strain relation.

Proposes an automatic adaptive meshing scheme. Error in strain energy is directly obtained through strain energy density function (SED). Versatility of this function, in comparison with that of others, is looked at in detail. Mesh enrichment method consists of a series of h‐refinement steps and concludes with a single p‐refinement step. Adds that an examination of the accuracy of the element used in the refinement procedure is made. This scheme has been implemented in ZATILAN, a FE code developed in the Department of the Mechanical Engineering of the University of the Basque Country.