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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.

The objective of this work is to develop an element technology to recover the plane stress response without any plane stress specific modifications in the large strain regime. Therefore, the essential feature of the proposed element formulation is an interface to arbitrary three‐dimensional constitutive laws. The easily implemented and computational cheap four‐noded element is characterized by coarse mesh accuracy and the satisfaction of the plane stress constraint in a weak sense. A number of example problems involving arbitrary small and large strain constitutive models demonstrate the excellent performance of the concept pursued in this work.

Engineering components/structures are usually subjected to complex and variable loads, which result in random multiaxial stress/strain states. However, fatigue analysis methods under constant loads cannot be directly applied to fatigue life prediction analysis under random loads. Therefore, the purpose of this study is how to effectively evaluate fatigue life under multiaxial random loading.

First, the average phase difference is characterized as the ratio of the number of shear strain cycles to the number of normal strain cycles, and the new non-proportional additional hardening factor is proposed. Then, the determined random typical load spectrum is processed into a simple variable amplitude load spectrum, and the damage in each plane is calculated according to the multiaxial fatigue life prediction model and Miner theory. Meanwhile, the cumulative damage can be calculated separately by projection method. Finally, the maximum projected cumulative damage plane is defined as the critical plane of multiaxial random fatigue.

The fatigue life prediction capability of the method is verified based on test data of TC4 titanium alloy under random multiaxial loading. Most of the predicting results are within double scatter bands.

The objective of this study is to provide a reference for the determination of critical plane and non-proportional additional hardening factor under multiaxial random loading, and to promote the development of multiaxial fatigue from experimental studies to practical engineering applications.

Crack tip stresses are used to relate the ability of structures to perform under the influence of cracks and defects. One of the methods to determine three-dimensional crack tip stresses is through the J-T_z method. The J-T_z method has been used extensively to characterize the stresses of cracked geometries that demonstrate positive T-stress but limited in characterizing negative T-stresses. The purpose of this paper is to apply the J-T_z method to characterize a three-dimensional crack tip stress field in a changing crack length from positive to negative T-stress geometries.

Elastic-plastic crack border fields of deep and shallow cracks in tension and bending loads were investigated through a series of three-dimensional finite element (FE) and analytical J-T_z solutions for a range of crack lengths ranging from 0.1⩽a/W⩽0.5 for two thickness extremes of B/(W − a)=1 and 0.05.

Both the FE and the J-T_z approaches showed that the combined in-plane and the out-of-plane constraint loss were differently affected by the T-stress and the out-of-plane size effects when the crack length changed from deep to shallow cracks. The conditions of the J-T_z dominance on the three-dimensional crack front tip were shown to be limited to positive T-stress geometries, and the J-T_z-Q^2D approach can extend the crack border dominance of the three-dimensional deep and shallow bend models along the crack front tip until perturbed by an elastic-plastic corner field.

The paper reports the limitation of the J-T_z approach, which is used to calculate the state of three-dimensional crack tip stresses in power law hardening materials. The results from this paper suggest that the characterization of the three-dimensional crack tip stress in power law hardening materials is still an open issue and requires other suitable solutions to solve the problem.

This paper demonstrates a thorough analysis of a three-dimensional elastic-plastic crack tip fields for geometries that are initially either fully constrained (positive T-stress) or unconstrained (negative T-stress) crack tip fields but, subsequently, the T-stress sign changes due to crack length reduction and specimen thickness increase. The J-T_z stress-based method has been tested and its dominance over the crack tip field is shown to be affected by the combined in-plane and the out-of-plane constraints and the corner field effects.

Stress state has a major influence on different phenomena, namely those involving diffusion and plastic deformation (like crack closure and high‐temperature fatigue crack growth, void formation or ductile fracture). The isolation of plane stress and plane strain states is crucial in fundamental studies of material behavior. The isolation of plane stress state is achieved with thin specimens, whilst the isolation of plane strain state is usually done increasing the thickness or introducing lateral grooves. The purpose of this paper is to propose a specimen geometry able to isolate the plane strain state, based on the standard M(T) geometry.

A numerical study was carried out aiming at obtaining a stress triaxiality parameter, h, as a function of different geometrical features of the specimen, such as the notch radius, notch depth and specimen thickness.

Results show that a pure plane strain state is achievable (i.e. 97 percent of specimen thickness has h>0.97) if a specimen with optimized geometrical features is used, which corresponds to a notch radius of 0.5 mm, a notch depth of 1 mm and a total specimen thickness of 12.56 mm.

This type of specimen geometry is a simple and efficient alternative to other common approaches used to obtain pure plain strain conditions for experimental purposes.

This paper aims to propose a new robust membrane finite element for the analysis of plane problems. The suggested element has triangular geometry. Four nodes and 11 degrees of freedom (DOF) are considered for the element. Each of the three vertex nodes has three DOF, two displacements and one drilling. The fourth node that is located inside the element has only two translational DOF.

The suggested formulation is based on the assumed strain method and satisfies both compatibility and equilibrium conditions within each element. This establishment results in higher insensitivity to the mesh distortion. Enforcement of the equilibrium condition to the assumed strain field leads to considerably high accuracy of the developed formulation.

To show the merits of the suggested plane element, its different properties, including insensitivity to mesh distortion, particularly under transverse shear forces, immunities to the various locking phenomena and convergence of the element are studied. The obtained results demonstrate the superiority of the suggested element compared with many of the available robust membrane elements.

According to the attained results, the proposed element performs better than the well-known displacement-based elements such as linear strain triangular element, Q4 and Q8 and even is comparable with robust modified membrane elements.

Severe accuracy loss may occur when finite element comes to the distorted mesh model, and the calculation may fail when element mesh degenerates into concave quadrangle or the element boundary is curved. This is a valuable research topic, and many efforts have been made to develop new finite element models. This paper aims to propose two quasi-conforming membrane elements based on the assumed stress quasi-conforming method and fundamental analytical solutions to overcome the difficulties.

First, the fundamental analytical solutions which satisfied both the equilibrium and the compatibility relations of plane stress problem are used as the initial assumed stress of both elements. Then, the stress-function matrices are used as the weighted functions to weaken the strain-displacement equations, which makes only string-net functions on the boundary of the elements are needed in the process of strain integration. Finally, boundary interpolation functions expressed by unknown nodal displacement parameters are adopted to the process of strain integration.

The formulations of both elements are simple and concise, and the elements are immune to the distorted mesh, which can be used to the mesh shape degenerates into a triangle or concave quadrangle and curved-side element. The results of the numerical tests have proven that the new models possess high accuracy.

New formulations of quasi-conforming method are described is detail, and the new strategy exhibits advantages of both analytical and discrete methods.

A numerical study on the stretching of a Newtonian fluid filament is analysed. Stretching is performed between two retracting plates, moving under constant extension rate. A semi‐implicit Taylor‐Galerkin/pressure‐correction finite element formulation is employed on variable‐structure triangular meshes. Stability and accuracy of the scheme is maintained up to large Hencky‐strain levels. A non‐uniform radius profile, minimum at the filament mid‐plane, is observed along the filament‐length at all times. We have found maintenance of a suitable mesh aspect‐ratio around the mid‐plane region (maximum stretch zone) to restrict early filament break‐up and consequently solution divergence. As such, true transient flow evolution is traced and the numerical results bear close agreement with the literature.

The purpose of this paper is to present a numerical model of coupled heat, moisture transfer and their effects on the mechanical deformations of wood during the drying process.

Coupling among heat, moisture, and mechanical deformations is solved consecutively by use of sparse solver of MATLAB. The weighted residual of the equilibrium equations of drying process of wood, based on finite element method, is investigated. The stress and plastic strain increments can be solved with Newton's method.

The numerical model is applied to a plain strain problem of a long wood board taken from the outer region of the wood log. Numerical simulation reveals the stress reversal during the drying process. The mechanical deformations and the principle stresses of a three‐dimensional wood board in consideration of the orthotropic properties are presented.

Plane strain and plane stress are analysed. The tangential modulus is derived. The transformation of the stress and strain tensors between the local coordinate system resulting from the cylindrical properties of wood and the global one is evaluated. Selection of element type for temperature, moisture content and displacement is discussed.

The use of enhanced strains leads to an improved performance of low order finite elements. A modified Hu‐Washizu variational formulation with orthogonal stress and strain functions is considered. The use of orthogonal functions leads to a formulation with B (overline) ‐strain matrices which avoids numerical inversion of matrices. Depending on the choice of the stress and strain functions in Cartesian or natural element coordinates one can recover, for example, the hybrid stress element P‐S of Pian‐Sumihara or the Trefftz‐type element QE2 of Piltner and Taylor. With the mixed formulation discussed in this paper a simple extension of the high precision elements P‐S and QE2 to general non‐linear problems is possible, since the final computer implementation of the mixed element is very similar to the implementation of a displacement element. Instead of sparse B‐matrices, sparse B (overline) ‐matrices are used and the typical matrix inversions of hybrid and mixed methods can be avoided. The two most efficient four‐node B (overline) ‐elements for plane strain and plane stress in this study are denoted B (overline)(x, y)‐QE4 and B (overline)(ξ, η)‐QE4.