The concept of Cyclic Normalised Equivalent Strain Functionals (CNESnFs) suitable for lowcyclic fatigue is developed in this paper. It reduces any fatigue strength and durability model to a unique form what facilitates the models comparison. Generalisation of some known fatigue durability models and introduction of new linear and nonlinear ones including those sensitive to the load sequence, are made. The functional concept allows to unite the strength and durability description under static, fatigue, creep and dynamic loading and their combinations. Examples of such new complex functional strength conditions are presented. The non-local versions of the CNESnFs applicable to strain fields sharply varying in space coordinates and to crack initiation and propagation analysis are also presented

Mikhailov, SE

Namestnikova, IV

English

Brunel University Research Archive

Proceedings of the Seventh International Conference on Biaxial/Multiaxial Fatigue & Fracture, DVM, Berlin, 2004, 409-414LOCAL AND NON-LOCAL NORMALISED EQUIVALENT STRAIN FUNCTIONALSFOR CYCLIC FATIGUES.E. Mikhailov and I.V. NamestnikovaDept. of Mathematics, Glasgow Caledonian Univ., Glasgow, G4 0BA, UKs.mikhailov@gcal.ac.uk, i.namestnikova@gcal.ac.ukABSTRACTThe concept of Cyclic Normalised Equivalent Strain Functionals (CNESnFs) suitable for low-cyclic fatigue is developed in this paper. It reduces any fatigue strength and durability model toa unique form what facilitates the models comparison. Generalisation of some known fatiguedurability models and introduction of new linear and nonlinear ones including those sensitive tothe load sequence, are made. The functional concept allows to unite the strength and durabilitydescription under static, fatigue, creep and dynamic loading and their combinations. Examplesof such new complex functional strength conditions are presented. The non-local versionsof the CNESnFs applicable to strain fields sharply varying in space coordinates and to crackinitiation and propagation analysis are also presented.KEYWORDSDurability; low-cyclic fatigue; multiaxial non-regularly oscillating loading; damage accumulation;local and non-local cyclic strength conditions; stress/strain concentration.INTRODUCTIONThere exist a number of cyclic fatigue theories accounting for damage accumulation undervariable load intensity, multiaxiality and non-proportional loading. However their forms are oftenincomparable analytically and their refinements to include some additional observed fatigueeffects are not evident. When one tries to develop a general approach to description of fatiguestrength and durability for different classes of materials under complex multiaxial loading, onehas to reduce all known fatigue theories to a unique comparable form and make sure the modelrefinements to include, e.g., overloading and sensitivity to the load sequence, are sufficientlysimple. For a material under uniaxial or multiaxial non-regularly oscillating stress, such a theorywas given in [6] by phenomenological strength conditions presented in terms of the CyclicNormalised Equivalent Stress Functionals (CNESFs). Since the strain history (strain process)is more representative for low-cyclic fatigue, the concept of Cyclic Normalised Equivalent StrainFunctionals (CNESnFs) will be developed in this paper.To overcome the well-known shortcomings of the the traditional local approaches, such asinability to describe adequately the high stress concentration, short crack effects and otherscale effects caused by the material pronounced micro-structure, non-local strength conditionsand fracture criteria may be employed for time and history independent materials under mono-tone (”static”) loading (see e.g. [8, 1, 2, 3, 4]). The non-local conditions use stresses or strainsobtained on the assumption of solid macro-homogeneity by solving usual local boundary valueproblems (e.g. of elasticity). All micro-heterogeneity is then implicitly taken into account by acorresponding non-locality of the strength condition. Non-local condition parameters should beobtained from macro-tests, although they could in principle be estimated also by developing ad-equate micro-mechanical models where local strength conditions operate. Some modificationsof the ”static” non-local approaches to fatigue crack initiation stage were used in [9, 10, 11].409S.E.Mikhailov & I.V.NamestnikovaIn this paper, we will describe the local and the non-local versions of the functional approachto fatigue, applicable to both the crack initiation and propagation.LOCAL CYCLIC NORMALISED EQUIVALENT STRAIN FUNCTIONAL AND STRENGTH CON-DITIONA general functional approach to the strength analysis of time dependent materials was de-scribed in [5]. It may still be applied to fatigue analysis using time as a natural parameter,however a special cyclic parameterisations seem to be more relevant for the cyclic fatigue. Thecorresponding cyclic approach, based on the stress history analysis was developed in [6, 7].Here we present its counterpart based on the strain history, applicable also to low-cyclic fatigue.Let {ε} = {εc(m)}m=1,2,... be a cyclic, generally non-periodic, multiaxial strain processindependent on the space coordinates, where the tensor function εc(m) = {εij(τ ); τm−1 ≤τ ≤ τm} is an m−th cycle and τ is time. Denoting by n∗{ε} the durability (number ofcycles to rupture) under a process {ε}, the notion of the generalised strain–durability Sn–Ndiagram, λ 7→ n∗{λε}, (similar to the Coffin-Manson plot for plastic strain amplitude) may beintroduced, where λ is a non-negative number by which the process {ε} is multiplied. Then thecyclic functional safety factor, the cyclic normalised equivalent strain functional and the cyclicendurance normalised equivalent strain can be defined:The cyclic strain functional safety factor λN({ε};n) is supremum of λ ≥ 0 such thatn∗{λ′′²} > n for any λ′′ ∈ [0, λ]; if there is no such λ, we take λ({ε};n) = 0. The cyclicnormalised equivalent strain functional, CNESnF, is defined as ΛN({ε};n) := 1/λN({ε};n)if λN({ε};n) 6= 0, and ΛN({ε};n) := ∞ otherwise. The endurance normalised equivalentstrain functional is ΛNth({ε}) = ΛN({ε};∞).Note that the Cyclic Normalised Equivalent Strain Functional is a counterpart of the CyclicNormalised Equivalent Stress Functional introduced in [6, 7] and coincides with it for linearelastic materials. As follows from the definition, ΛN({ε};n) is positively homogeneous in {ε}and monotonously non-decreasing in n. The cyclic stable strength condition for a cycle n canbe written in the formΛN({ε};n) < 1.The CNESnF ΛN is a material characteristic which is not necessary connected with ageometrical, stiffness-related or abstract damage measure and can be identified from the cyclicstrain durability tests under homogeneous strain process fields. Using the above definition, anycyclic strength condition written in terms of a damage measure can be expressed in terms ofa corresponding ΛN , although not always analytically, and easily generalised to include e.g.instant overloading, creep or dynamic effects.For periodic multiaxial fatigue processes (i.e. with the same strain loop εcij on all cy-cles), one can easily determine the CNESnFs ΛN({εc};n) = ΛN∗(εc;n) from the cor-responding Sn–N diagrams. Suppose, e.g., a material strength under a periodic multiaxialin-phase loading is described by a power Sn–N diagram (Basquin type relation) n∗{ε} =(|||εc|||/ε∗(R))−b, where |||εc||| = supε∈εc |ε|; |ε| is a matrix norm of the tensor functionε, e.g., |ε| =√∑3i,j=1 ε2ij ; ε∗(R) and b = b(R) are positive material characteristics de-pending generally on the cycle εc shape in the strain space, i.e. on the tensor counterpartRij := εcij/|||ε||| of the asymmetry ratio but not on the cycle norm |||εc|||. Then we have thefollowing CNESnF and fatigue strength condition,ΛN∗(εc;n) =|||εc|||ε∗(R)n1/b < 1. (1)410S.E.Mikhailov & I.V.NamestnikovaThe Sines type fatigue strength condition can be rewritten in terms of the associated CNESnF,ΛN∗(εc;n) =σa,eqσ∗−1(n)+ 3σh,m[1σ∗0(n)− 1σ∗−1(n)]< 1, (2)whereas the Crossland type fatigue strength condition can be rewritten asΛN∗(εc;n) = σeq,a[2σ∗−1(n)− 1σ∗0(n)]+ 3σh,max[1σ∗0(n)− 1σ∗−1(n)]< 1, (3)Here σa,eq = µεa,eq is the von Mises form of the stress amplitude; σh,m = Kεh,m =Kεii,m/3 is the mean hydrostatic stress, and σh,max = Kεh,max = Kεii,max/3 is themaximum hydrostatic stress during the cycle; µ and K are the shear and volume expansionelastic moduli; σ∗−1(n) and σ∗0(n) are the (amplitude) S–N diagrams for the same materialunder uniaxial periodic loading with the asymmetry ratios R = −1 and R = 0, respectively.Palmgren-Miner linear accumulation ruleFor a Basquin type material under a multiaxial cyclic (non-periodic) in-phase self–similar load-ing or for b independent on R, the Palmgren-Miner linear accumulation rule leads to the follow-ing CNESnF and strain strength condition, c.f. [7],ΛN0 ({ε};n) = n∑m=1( |||εc(m)|||ε∗(R(m)))b1/b =∫ n0( |||εc(m)|||ε∗(R(m)))bdm1/b < 1, (4)where |||εc(m)||| is a piece-wise constant function of the cycle number m, ε∗(m) = ε∗(R(m))and b = b(R) should be taken as positive material characteristics in the power Sn–N diagramfor the corresponding multiaxial periodic process.If one would like to take into account an influence of both the instantaneous overloads ofmaterial and creep durability, one can add a term with εeq(ε) (e.g. von Mises, Tresca or other in-stantaneous equivalent strain) and a termΛT0 (ε; t′) =[1ε∗T∫ t′0 |ε(τ )|bT dτ]1/bTconnected withthe linear Robinson rule of time-dependent damage accumulation and power time-durability di-agram [5], and arrive at another, complex NESnF and local strength condition,ΛITN0 ({ε};n(t)) = sup0≤t′≤t{εeq(ε(t′))εr+ΛT0 (ε; t′) + ΛN0 ({ε};n(t′))}< 1 (5)where εr is a material strain strength under uniaxial monotone tensile loading; n(t) is aninteger-valued function of time t. The term ΛT0 (ε; t′) can be also considered as describingdynamic strength, c.f. [5].Note that strength condition (5) lead to an accumulation rule sensitive to the loading se-quence. One can create not only sums but also other homogeneous combinations of thosethree terms to get other possible simple forms of the NESnF describing instant, time-dependentand cycle-dependent effects on the durability.Linear and non-linear accumulation rules for CNESnF partial incrementsLet us consider another type of accumulation rules sensitive to loading sequence. In contrastto the Palmgren-Miner linear accumulation rule for partial life–times, we can write a linear (c.f.[6]) or non-linear accumulation rule for partial increments of CNESnF.411S.E.Mikhailov & I.V.NamestnikovaLet a CNESnF ΛN∗(εc;n) for a periodic multiaxial fatigue process, i.e. with equal strainloop εcij on all cycles, be known, for example be given by one of expressions (1)-(3). Then aCNESnF and corresponding strength condition for a non-periodic oscillating process {ε} canbe taken in the following linear formΛN({ε};n) = max1≤n′≤nΛN∗(εc(n′); 1)+n′−1∑m=1[ΛN∗(εc(m);n′ −m+ 1)− ΛN∗(εc(m);n′ −m)] < 1. (6)Non-linear versions of the accumulation law can be also introduced. For example, one canconsider the following one with the power-type nonlinearity,ΛN({ε};n) = max1≤n′≤n(ΛN∗(εc(n′); 1))β+n′−1∑m=1[(ΛN∗(εc(m);n′ −m+ 1))β − (ΛN∗(εc(m);n′ −m))β]1β< 1. (7)Here the constant β > 0 is a material parameter.To obtain the fatigue strength condition for the Basquin, the Sines, or the Crossland typematerial under non-periodic cyclic loading, one has to substitute, respectively, expressions (1),(2) or (3) in (6) or (7). Similarly, one can use the linear (6) or non-linear (7) accumulation lawsto extend the Dang Van, Kakuno, Mucha and other periodic fatigue strength conditions to thenon-periodic cycling. One can check that (6) and (7) interpolate the Sn–N diagram, that is,degenerate into the corresponding strength conditions for a periodic processes if εc(n) = εc.Note that the strength conditions (6) and (7) are sensitive to the loading sequence in contrastto the Palmgren–Miner rule. They account for instant overloading and can be also combinedwith the creep/dynamic strength conditions similar to (5).Local quasi-ductile and brittle fatigue strength conditionsThe above fatigue strength conditions can be employed not only to the strain fields independenton space coordinates but also to ones moderately varying in the coordinates, leading to the so-called local cyclic quasi-ductile strength conditions at any material point y.To describe cyclic fracture, i.e. crack initiation and propagation under cycling loading, onehave to analyse the brittle strength, that is strength at a particular point y along a particularinfinitesimal plane with a normal vector ~ζ at that point. The local brittle cyclic strength conditionfor a plane ~ζ at a point y can be taken in the formΛ({εc(y)};n, y, ~ζ) < 1. (8)Here Λ({εc(y)};n, y, ~ζ) is a local brittle cyclic normalised equivalent strain functional, whichis defined similar to the above and is a material characteristics positively homogeneous in {εcij}and non-decreasing in n. Particular forms of the brittle NESnFs can be obtained from (4)-(7),where the strain tensor loop sequence {εcij(m; y)}m=1,2,... is replaced by the strain vectorloop sequence {~εc(m; y, ~ζ)}m=1,2,... on the plane ~ζ.There is no fracture in the body D if inequality (8) is satisfied on all planes ~ζ at all pointsy ∈ D but a crack nucleates or initiates on the plane ~ζ∗(y∗) at the point y∗ during the cyclen∗(D), where and when the inequality violates. During the next cycles, the crack set Y ∗412S.E.Mikhailov & I.V.Namestnikovapropagates and one can track it taking into account the strain field redistribution on each cycledue to the body shape change when a new portion of Y ∗ becomes a part of the body boundary.This allows to describe the growth of the CNESnFΛN({ε(D)};n, y∗0, ~ζ) on each plane at eachpoint before and after the fracture initiation as well as the cyclic crack propagation through thedamaged material points and planes.Particularly, local strength condition (8) and CNESnF associated with (4) can be used forcrack initiation as well as crack propagation analysis (for the CNESF it was shown in [7]). How-ever, it works only for the power b < 2 in the Basquin relation and the crack start delay is notpredicted. Due to the strain singularity at the crack tip, the CNESnFs like (6) with instant termsare not applicable to fatigue crack analysis unless their corresponding non-local counterpartsare used.NON-LOCAL CYCLIC NORMALISED EQUIVALENT STRAINFUNCTIONAL AND STRENGTH CONDITIONTo analyse strength and durability under oscillating in time and highly inhomogeneous strainfields and predict both the crack initiation and propagation as a united process, we merge inthis section the above functional approach to the cyclic strength with the non-local approach of[3] and present a strain based counterpart of the corresponding stress-based non-local cyclicstrength condition [6, 7].In the non-local version of brittle fracture, we will suppose that cyclic strength of an infines-imal plane ~ζ at a point y ∈ D does depend not only on the cyclic strain history at the point,{εc(m; y)}m=1,2,..., but also on the stress history in its neighbourhood and generally, in thewhole of the body, {ε} = {εc(m;x)}m=1,2,..., x ∈ D.Then we can repeat for a plane ~ζ at a point y the reasonings of the previous section under-standing under {ε} = {εc(m)}m=1,2,... the cyclic process field in D and arrive at the notionsof non-local cyclic durability n∗¯({ε}; y, ~ζ), CNESnF ΛN¯({ε};n, y, ~ζ) and the non-localcyclic strength condition for a plane ~ζ at a point y ∈ D,ΛN¯({ε};n, y, ~ζ) < 1. (9)One can obtain the simplest examples of the non-local CNESnFs and strain strength condi-tions for a plane ~ζ by replacing the local strain εij(τ ; y) by its non-local counterpart ε¯ij(τ ; y, ~ζ)in the local CNESnFs, e.g. in ΛN0 ({ε(y)};n), ΛIN0 ({ε(y)};n(t)) or ΛITN0 ({ε(y)};n(t))above. Here ε¯ij(y, ~ζ) is a weighted average of εij(x) along some neighbourhood (non-localityzone) Ω(y, ~ζ) of y depending also on ~ζ. That is, one can take,ΛN¯({ε};n, y, ~ζ) = ΛN({ε¯(y, ~ζ)};n), ε¯ij(τ ; y, ~ζ) =∫Ω(y,~ζ)w(x, y, ~ζ)εij(τ ;x)dx,(10)where w(x, y, ~ζ) as well as Ω(y, ~ζ) ⊂ D are characteristics of material and (generally) of thebody D shape, such as∫Ω(y,~ζ)w(x, y,~ζ)dx = 1. Particularly, Ω(y, ~ζ) can be taken as a discof a diameter 2d (or a segment of a length 2d for a two-dimensional bodyD) in the plane ~ζ withthe center in y (or as intersection of the disc/segment with D for y near the boundary of D),where d is considered as a material parameter, and w(x, y, ~ζ) = 1/|Ω(y, ~ζ)| where |Ω(y, ~ζ)|is the area/length of Ω(y, ~ζ).As demonstrated in [7], the fatigue crack propagation numerical analysis using CNESnF isapplicable for the Basquin type material with any power b > 0 of the Sn–N diagram and canpredict the crack start delay and short–crack effect.413S.E.Mikhailov & I.V.NamestnikovaNote that one can similarly introduce the non-local cyclic quasi-ductile strength conditionsand normalised equivalent strain functionals, if one discards above in this section the infinites-imal plane ~ζ and considers Ω(y) in (10) as a 3D domain for the 3D body or, respectively, 2Ddomain for the 2D body.CONCLUSIONIntroduction of the notion of normalised equivalent strain functional and its use in the strengthconditions allowed to unite the strength and durability description under static, fatigue, creepand dynamic loading and their combinations often appearing in practice.Unlike some damage measures usually used for such purposes, the functionals are me-chanically meaningful material characteristics, which can be experimentally determined and/orverified. This makes a refinement of the CNESnF approximations to reflect e.g. influence ofthe load sequence, more straightforward and justified than for damage measures.The non-local versions of the CNESnFs allow to unite the cyclic strength, durability andfatigue crack propagation analysis including crack initiation and propagation through the dam-aged material. It makes also possible to analyse the fatigue strength and durability of structureelements with singular stress concentrators and describe the short–crack effects and crackstart delay.REFERENCES[1] S. Bavaglia and S. E. Mikhailov. Application of a non-local failure criterion to a crackin heterogeneous media. In: A. Carpentery, C. A. Brebbia (eds), Damage and FractureMechanics: Computer aided assessment and control. Southampton-Boston: Computa-tional Mechanics Publ., 1998, pp. 155–164.[2] L. P. Isupov and S. E. Mikhailov. Comparative analysis of several non-local fracture criteria.Archive of Applied Mechanics 68 (1998) 597–612.[3] S. 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Theory of notch stresses: Principles for exact stress calculation. Ann Arbor,Michigan: J.W.Edwards, 1946.[9] A. Seweryn and Z. Mro´z. On the criterion of damage evolution for variable multiaxial stressstates. Internat. J. Solids and Structures, 35 (1998) 1589–1616.[10] D. Taylor. A mechanistic approach to critical-distance methods in notch fatigue.Fatigue and Fracture of Eng. Mater. and Struct. 24 (2001) 215–224.[11] W. X. Yao, K. Xia, and Y. Gu. On the fatigue notch factor, kf . Internat. J. of Fatigue 17(1995) 245–251.Contact address: s.mikhailov@gcal.ac.uk414

A functional approach to non–local strength conditions and fracture criteria.

A mechanistic approach to critical-distance methods in notch fatigue.

Application of a non-local failure criterion to a crack in heterogeneous media.

Comparative analysis of several non-local fracture criteria.

Fatigue strength and durability analysis by normalised equivalent stress functionals. In:

Local and non-local approaches to fatigue crack initiation and propagation.

Non-local strength conditions based on generalized ±c cohesive models.

Non-local strength conditions based on generalized δc cohesive models.

On the criterion of damage evolution for variable multiaxial stress states.

On the fatigue notch factor, kf .

On the fatigue notch factor, kf.

Theoretical backgrounds of durability analysis by normalized equivalent stress functionals.

Theory of notch stresses: Principles for exact stress calculation.

Local and non-local normalised equivalent strain functionals for cyclic fatigue

The concept of Cyclic Normalised Equivalent Strain Functionals (CNESnFs) suitable for lowcyclic fatigue is developed in this paper. It reduces any fatigue strength and durability model to a unique form what facilitates the models comparison. Generalisation of some known fatigue durability models and introduction of new linear and nonlinear ones including those sensitive to the load sequence, are made. The functional concept allows to unite the strength and durability description under static, fatigue, creep and dynamic loading and their combinations. Examples of such new complex functional strength conditions are presented. The non-local versions of the CNESnFs applicable to strain fields sharply varying in space coordinates and to crack initiation and propagation analysis are also presented. KEYWORDS Durability; low-cyclic fatigue; multiaxial non-regularly oscillating loading; damage accumulation; local and non-local cyclic strength conditions; stress/strain concentration

S. E. Mikhailov

I. V. Namestnikova

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Local and non-local normalised equivalent strain functionals for cyclic fatigue

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