Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications

doi:10.1016/S0142-1123(01)00181-5

International Journal of Fatigue

Volume 24, Issue 6, June 2002, Pages 649-657

https://doi.org/10.1016/S0142-1123(01)00181-5 Get rights and content

Abstract

Criteria based on both critical plane and stress invariant approaches require routines for the calculation of the amplitude of critical stress quantities. The definition of the shear stress amplitude on a material plane as well as the evaluation of the amplitude of the second invariant of the stress deviator can be led back to an optimization problem which can be quickly solved with already existing routines. In this paper a new method is presented and results are compared with those obtained with other previously proposed algorithms.

Introduction

Many of the existing high cycle fatigue criteria can be divided into two groups [1]. The first group includes the criteria based on the critical plane concept. The second group comprises the criteria based on the stress invariants. Critical plane type criteria usually require the evaluation of the amplitude of the shear stress τ_a acting tangentially to the critical plane. Stress invariants based criteria typically call for the determination of the amplitude of the second invariant of the stress deviator J_2,a. From this, $J_{2,a}$ can be found, which has dimension of stress.

With the exception of proportional paths, the evaluation of τ_a and $J_{2,a}$ is not a trivial problem. A coherent definition of the shear stress amplitude has been given by Papadopoulos [2]. It is based on the construction of the minimum circle circumscribed to the path described by the shear stress vector on the critical plane. The radius of the minimum circumscribed circle is defined to be the shear stress amplitude. This definition can be extended to the case of $J_{2,a}$ . It is then required to construct the minimum hyper-sphere which encompasses the load path in the stress deviator space. Its radius is equal to $J_{2,a}$ .

Algorithms for constructing the minimum circumscribed circle or hyper-sphere have already been proposed by Papadopoulos [2] and Dang Van et al. [3]. However, these methods often necessitate a large amount of computational time. The purpose of the present paper is to introduce an efficient algorithm for the rapid calculation of τ_a and of $J_{2,a}$ . The proposed method makes use of numerical optimization routines currently available as free Fortran or C source codes or as part of commercial mathematical software.

Section snippets

The critical plane approach

One of the first fatigue criteria based on the critical plane concept first appeared in the paper of Findley [4]. On a given elementary plane Δ, identified by its normal vector n, the stress tensor σ combined with n defines the normal stress vector σ_N $σ_{N} =(n · σ · n)· n$ and the shear stress vector τ $τ = σ · n −(n · σ · n)· n .$ Following Findley, the most damaged plane is the one experiencing the greatest sum of the amplitude of the shear stress vector and of the normal stress. Thus the criterion is written as $τ_{a} (ϕ^{∗},θ$

Criteria based on the stress invariants

Concerning the stress invariant based criteria, the critical quantity is usually the second invariant of the stress deviator, calculated taking into consideration only the alternating components. This quantity is thus defined as J_2,a. Among these criteria let us recall the Sines criterion $J_{2,a} +κp_{m} ≤λ.$ The term J_2,a was originally intended as J₂ calculated over the alternating components, i.e. over σ_ij,a if σ_ij(t)=σ_ij,m+σ_ij,af(t).

Later Fuchs, as reported by Garud [14] and Bannantine et al. [15],

Shear stress amplitude calculation

Supposing the shear stress acting on Δ being defined by its components in a Cartesian reference (u,v) fixed on Δ $τ_{u} = sin (ωt) τ_{v} = sin (2ωt−π/4)$ the shear stress tip follows the curve shown in Fig. 4.

The centre of the circumscribed circle and its radius have been calculated with the Points Combination Algorithm (being affected by roundoff errors only, its results can be considered the reference solution), the Incremental Method and fminimax, all implemented in a Matlab [10] script, and yielded the

Concluding remarks

Critical plane type criteria as well as second invariant based criteria applied to non proportional load paths require to find the centre of respectively the circle and the hyper-sphere circumscribing the load path.

The Points Combination Algorithm allows to solve this problem with high accuracy but it makes calculation time too long if the load path is defined with a very fine time step.

Optimization routines give the same results with a number of function evaluations almost independent of the

Acknowledgements

The author expresses his gratitude to Dr. K. Dang Van and Dr. H. Maitournam of LMS, Ecole Polytechnique, France, for their valuable advice and encouragement and for providing him with the stress data and the Point Combination Algorithm results shown in the second example. The author also wishes to thank Dr. I.V. Papadopoulos of JRC, Ispra, Italy, for useful suggestions and comments on the contents of this paper.

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Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications

Abstract

Introduction

Section snippets

The critical plane approach

Criteria based on the stress invariants

Shear stress amplitude calculation

Concluding remarks

Acknowledgements

Int. J. Fatigue

Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane

Fatigue Fract. Engng Mater. Struct.

On a new multiaxial fatigue limit criterion: theory and applications

A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending

J. Engng Ind. Trans. ASME

An explanation on fatigue limit under combined stress

Bull. JSME

A general criterion for high cycle multiaxial fatigue failure

Fatigue Fract. Engng Mater. Struct.

A theory for fatigue under multiaxial stress–strain conditions

Proc. Inst. Mech. Engrs