FATIGUE LIMIT OF METALS UNDER MULTIAXIAL STRESS

CONDITIONS: THE MESOSCOPIC APPROACH

I.V. PAPADOPOULOS

Joint Research Centre, Commission of the European Communities, I-21020 Ispra(VA) Italy

e-mail: Ioannis_papadopoulos@msn.com

Abstract -------------

Among the various research areas in the field of multiaxial high-cycle fatigue

of metals, the problem of the fatigue limit is a research domain of high interest. The

knowledge of the fatigue limit of a metal component allows the distinction of a

damaging cyclic loading from a non damaging one. The latter can be sustained by the

component for a very high number of load cycles (theoretically infinite) without the

development of a catastrophic fatigue crack. The aim of this paper is to present a new

methodology concerning the evaluation of the fatigue endurance of metal components

subjected to multiaxial loading. This methodology is based on the mesoscopic

approach, first introduced in metal fatigue by Orowan in the late thirties. Within the

framework of the mesoscopic approach, the local stresses and strains at the scale of

the metal grain have to be evaluated. This can be done with the help of the Lin-Taylor

assumption. In the approach presented here, the fatigue limit is interpreted mainly as a

restraint applied on a measure (upper bound estimation) of the plastic microstrain

accumulated by a cyclic loading in some plastically deforming crystals of the metallic

aggregate. This bound depends on the mesoscopic hydrostatic stress. The general

theory of the mesoscopic approach to the problem of the fatigue criterion is presented

in this paper. In a companion paper the proposed theory has been applied on a

considerable amount of experimental data found in the scientific literature.

Nomenclature ---------------------

Σ =macroscopic stress tensor

Ε =macroscopic strain tensor macroscopic stiffness fourth-rank tensor C=

macroscopic compliance fourth-rank tensor L =

λ μ, =macroscopic Lamé coefficients of an isotropic material

σ =mesoscopic stress tensor

ε =mesoscopic strain tensor

ε εe p, =elastic and plastic parts of the mesoscopic strain tensor respectively mesoscopic stiffness fourth-rank tensor c =

mesoscopic compliance fourth-rank tensor l=

localisation tensor, B C= :l B=

ΣH =macroscopic hydrostatic stress, i.e. Σ Σ Σ Σ ΣH xx yyt zzr= = + +( ) ( )3 3

σH = mesoscopic hydrostatic stress, i.e. σ σ σ σ σH xx yyt zzr= = + +( ) ( )3 3

V =

Δ =

elementary material volume surrounding a point O of a body

elementary material plane, that is intersection of volume V and of a plane

passing through O

O xyz. = Orthogonal Cartesian system of axes attached to the specimen at point O

n =unit normal vector at an easy glide plane of a crystal, also unit normal vector at

an elementary material plane Δ

ϕ θ, = spherical co-ordinates of n in O.xyz

m =unit vector of an easy glide direction lying on a glide plane, also unit vector

along a given direction of an elementary material plane Δ

ψ =angle formed between m and an axis ξ fixed on Δ

a =orientation tensor of an easy glide system of a crystal, that is symmetric part of

the tensorial product of n by m , a sym n m= ( )

Σ =n macroscopic stress vector acting on an elementary material plane Δ

macroscopic normal stress vector acting on Δ, i.e. projection of Σ on N = n n

1

C = macroscopic shear stress vector, i.e. orthogonal projection of Σ on Δ n

T = macroscopic resolved shear stress acting on an easy glide direction m ,

i.e. projection of C on m

T =a amplitude of macroscopic resolved shear stress

σ =n mesoscopic stress vector acting on Δ

mesoscopic normal stress vector acting on Δ, i.e. projection of ν= σ on n n

mesoscopic shear stress vector, i.e. orthogonal projection of σ on Δ c= n

mesoscopic resolved shear stress acting along m τ =

shear yield limit of a crystal τy =

γ γp p, & =mesoscopic shear plastic strain and plastic strain rate vectors

b =mesoscopic kinematical hardening vector

&λ =mesoscopic plastic multiplier &Γ = rate of mesoscopic accumulated plastic shear strain, i.e. & & &Γ = ⋅γ γp p

δΓ( )i =shear plastic micro-strain accumulated over the i-th loading cycle

ΓN =accumulated shear plastic micro-strain after N loading cycles Γ∞ =asymptotic value of ΓN , i.e. Γ Γ∞ =

→∞limN N

crystal isotropic hardening modulus g =

c = crystal kinematical hardening modulus

p q, , ,α β = material constants

t− = fatigue limit under fully reversed pure shear 1

s− =fatigue limit under fully reversed pure normal stress 1

2

Introduction -------------------

Fatigue is one of the major considerations in the design of metallic structures.

It is now well recognised that fatigue is the principal cause of failure of mechanical

components subjected to varying loads. Despite over 100 years of research, the basic

mechanisms of damage accumulation under fatigue loading are still not well

understood. The progress already achieved concerns primarily the experimental part

of the research on fatigue of metals. Modelling of the phenomenon is still quite poor

and in many cases mostly empirical.

The general subject of fatigue of metals can be subdivided into two fields,

high-cycle fatigue and low-cycle fatigue. The main difference between these two

fields is that macroscopic plastic strain is predominant in low-cycle fatigue while

negligible in high-cycle fatigue. Regarding the field of uniaxial high-cycle fatigue,

experiments have shown that many metals possess a fatigue limit or endurance limit.

That is, a uniaxial loading that fluctuates within the stress bounds defined by the

corresponding fatigue limit, will be carried by the testing specimen for a very high

number of cycles. In theory this number of cycles is infinite, but in practice the

uniaxial endurance limit is the stress amplitude corresponding at a fatigue life of 106-

107 cycles. The uniaxial fatigue limit defines thus a range of safe operation for the

applied stress.

The case of uniaxial loading of a component of a structure is rather

uncommon. Very often many parts in a machine undergo multiaxial states of cyclic

stress. Examples of structures and components suffering multiaxial fatigue are,

pressure vessels, offshore structures, pipings, aircraft structures, as well as turbine and

jet engines, axles, crank shafts, propeller shafts and so forth. Often several

components in a structure are designed for an operation of many hundred thousands

of loading cycles. Especially in the automotive industry designing for high fatigue life

is the rule. The problem of how to extend the uniaxial safe stress range, that

corresponds to the endurance limit, in the case of multiaxial loading, is then a

research area of high industrial interest. The generalisation of the fatigue limit in

3

multiaxial conditions invokes the idea of the separation of the whole stress space into

two domains, the unsafe and the safe one. The safe domain of the stress space

contains the origin and is bounded by a closed surface. The equation of this bounding

surface will be called fatigue criterion. The fatigue criterion is thus the extension of

the uniaxial endurance limit in the case of multiaxial stress conditions. In spite of

many fatigue criteria proposed over decades of research [1-8], there is not yet a

universally accepted approach. This is because multiaxial stress conditions make the

task, of appraising how fatigue damage accumulates during the life of a component

very difficult.

It is known long that fatigue is a phenomenon that has its origin at the

mesoscopic scale of the metal, that is at the scale of the grains (crystals) of the

material. Because of this, research concerning the behaviour of single crystals under

fatigue loading has been undertaken as early as 1920 (for a review of these pioneer

experiments see H.J.Gough [9]). It was believed that a deep knowledge of the fatigue

behaviour of the crystals would permit, by a suitable assembling, to build appropriate

predictive models of the behaviour of the metallic polycrystalline aggregate. The

mesoscopic approach that will be used in this study is somewhat different in the sense

that we will start from establishing a link between macroscopic stresses and strains

and the corresponding mesoscopic quantities and after that a model will be built at the

mesoscopic scale. However, with the help of the relationships established before

between mesoscopic and macroscopic quantities, the proposed model will finally be

expressed in terms of the usual macroscopic stresses and strains. A quantitative

fatigue theory based on a similar mesoscopic approach has first been proposed by

E.Orowan in the late thirties [10]. Due to the limited knowledge of the fatigue

phenomenon at that time, Orowan's theory led to some erroneous conclusions. In

particular the fatigue limit, expressed as a limiting shear stress amplitude, was found

to be independent of the kind of loading. Consequently, according to Orowan the

endurance limits in fully reversed torsion and fully reversed tension-compression are

identical when expressed in terms of the maximum shear stress amplitude. Moreover

4

Orowan's theory cannot account for the well-known mean stress effect in asymmetric

tension-compression fatigue tests. A definite progress in the study of the fatigue

phenomenon within the framework of the mesoscopic approach has been achieved

with the work of K.Dang Van in the early seventies. More precisely this author

studied mainly the problem of the fatigue limit under multiaxial loading conditions

[11]. He proposed a formula (fatigue criterion) which has been successfully used in

industrial applications [12-13]. The mesoscopic stresses and strains have been

obtained with the help of the Lin-Taylor hypothesis. It was assumed that the yield of

the constituent crystals of the polycrystalline metallic material follows Schmid's law.

Moreover the cyclic strain hardening of the crystals was supposed to develop

according to a purely isotropic hardening rule. Later on, Papadopoulos [14] and

Papadopoulos and Dang Van [15], proposed a fatigue model covering both the

unlimited (fatigue criterion) as well as the limited (Wöhler curve) fatigue life regions.

The model is suitable for multiaxial loading conditions and a closed form expression

has been obtained for the Wöhler curve under multiaxial proportional loading.

Furthermore, the predictions of this model, concerning the damage accumulation

under two-stage fatigue loading qualitatively agreed with the general trends of the

experimental results in this field. A drawback of this model is that the modelisation of

the plastic behaviour of the crystal relied on von Mises criterion, although it is well-

known that the yield of metal crystals is governed by Schmid's law. Probably due to

this last hypothesis the results obtained by Papadopoulos fatigue criterion, in the case

of highly out-of-phase cycling loading, were not satisfactory enough [14]. An attempt

to improve the Papadopoulos approach has been provided by A.Deperrois [16]. This

researcher has included in his approach the concept of case A and case B cracks, first

introduced in metal fatigue studies by Brown and Miller [17].

The purpose of this work is to study the problem of the fatigue limit of metals

under multiaxial loading conditions with special emphasis to non-proportional

loading. The study will be performed within the framework of the mesoscopic

approach as established in the previous works of Papadopoulos and Dang Van, [14-

5

15]. However the unrealistic assumption of plastic flow of metal crystals according to

the von Mises criterion will be abandoned in the profit of the more natural Schmid's

law. On the other hand the hypothesis of pure isotropic strain hardening of the metal

grains will be replaced by the more pragmatic rule of combined kinematical and

isotropic hardening. Finally two new measures (upper bounds) of the accumulated

plastic microstrain will be introduced: one over an elementary material plane, and the

other within an elementary material volume. The terms elementary material plane and

elementary material volume will be clarified in the next section. These measures, lead

to two new fatigue criterion formulas, one for mild metals and one for hard metals. In

a companion paper the predictive capabilities of these fatigue criteria have been

tested. It is found that the new formulas provide very satisfactory results, especially

concerning out-of-phase multiaxial stress systems. Finally, it has to be mentioned that

the fatigue behaviour of brittle metals is not covered by this work.

Macroscopic and mesoscopic scales of material description ---------------------------------------------------------------------------------------

Relations between mesoscopic and macroscopic stress and strain fields

The mechanical state of a body considered as a continuous medium, is usually

described with the help of macroscopic quantities (stresses and strains), defined at

every point x of the body. For real materials this means that at the vicinity of any

point x an elementary volume V x( ) is defined, fig.(1). The size of the elementary

volume is chosen in such a way, that in a statistical sense, all the different constituent

microelements (grains) of the material and hence their properties, are equally

represented in V. Therefore the volume V x( ) is the smallest sample of the material

that can be considered as homogeneous. The elementary volume V x( ) defines thus

the macroscopic scale of the material. Let us denote as Δ( )x the intersection of the

volume V x( ) with a plane, that passes from the point x under consideration. This

intersection Δ defines an elementary material plane. Within the volume ( )x V x the ( )

6

usual macroscopic stress Σ( )x and strain Ε( )x are by definition constant

(homogeneous). For metals, the elementary volume contains a high number of

crystals of different orientations, sizes and forms. Accordingly a mesoscopic scale can

be defined as a partition of V x( ). The mechanical state of a crystal (grain) of the

metallic aggregate is described by the mesoscopic stress σ( ; )y x and strain ε( ; )y x ,

where y describes the position of the metal grain in a local frame attached to V x( ),

fig.(1). The quantities σ( ; )y x and ε( ; )y x clearly are not homogeneous in V x( );

consequently they differ from Σ( )x and Ε( )x . In what follows, the indication of the

dependence of the different quantities on x and y will be dropped, in order to make

the equations less cumbersome.

As it was pointed out before, the nucleation of fatigue cracks is a mesoscopic

phenomenon. The scale of this phenomenon is the scale of the grains of the metal.

Therefore to study the fatigue crack initiation, one has to evaluate the local stress and

strain fields developing at the mesoscopic scale of the material. The engineer has

access only to macroscopic quantities, that can be measured (with the use of strain

gauges, for instance) or calculated (by finite elements or other numerical or analytical

methods). It is then necessary to establish some theoretical relationships between

macroscopic and mesoscopic quantities in order to be able to evaluate the local stress

and strain fields σ and ε . This is a rather difficult problem and its rigorous treatment

leads to the well-known self-consistent scheme [18-19]. This problem will be treated

here in a simplified manner, in order to obtain closed form relationships between

mesoscopic and macroscopic quantities. However the simplifying assumptions to be

introduced, preserve these essential features of the micro-scale to macro-scale

passage, which are of interest for the description of the fatigue crack nucleation

phenomenon.

To start let us mention first, that a component cyclically loaded at the level of

its fatigue limit, behaves macroscopically in a purely elastic manner. The applied

loading is low enough to preclude the development of any plastic strain at the

7

macroscopic scale. The macroscopic stress Σ and strain Ε fields are thus related by

Hooke's law: Σ Ε Ε Σ= ⇔ =C: L: (1)

where C and L are respectively, the macroscopic stiffness and compliance fourth-

rank tensors of the material. Materials that are isotropic at the macroscopic scale will

be considered here. Then the corresponding stiffness and compliance tensors depend

only on two material constants, the Lamé coefficients λ and μ ,

C

L

ijkl ij kl ik jl jk il

ijkl ij kl ik jl jk il

= + +

=−+

+ +

λδ δ μ δ δ δ δ

λμ μ λ

δ δμ

δ δ δ δ

( )

( )( )

2 2 31

4

(2)

where is the Kronecker delta. Even if at the macroscopic scale the component

behaves in a purely elastic manner, at the microscale there are always some plastically

less resistant grains that undergo plastic glide. From now and in what follows our

attention will be focused on these plastically deforming grains, because it is from

there that the fatigue phenomenon originates. The mesoscopic strain of such a grain is

composed by an elastic

δij

εe and a plastic part εp :

ε ε ε= +e p (3)

The mesoscopic stress σ is related to the elastic part εe of the mesoscopic strain by

Hooke's law: σ ε ε= ⇔ =c : e e l σ: (4)

The mesoscopic stiffness c and compliance l tensors are not isotropic. For cubic

crystals, for instance, they depend on three material constants (elastic moduli).

To establish a link between the macroscopic fields Σ , Ε and the stress σ and

strain ε of a gliding crystal, an additional hypothesis will be introduced here. For the

quite low loading applied on a metal component at the fatigue limit level, the number

of the grains of the material that undergo plastic glide, is rather low. It seems then

natural to assume that any plastically deforming crystal is surrounded by grains that

are still elastic. Moreover one can reasonably make the hypothesis that the elastic

surrounding crystals impose their strain on the gliding crystal. That is, any plastically

8

deforming crystal is subjected to a strain-controlled loading. The average strain of

these surrounding elastic crystals is equal to the macroscopic strain Ε . This reasoning

leads to the scheme of an elastoplastic inclusion imbedded in an elastic matrix. It

leads as well to the Lin-Taylor assumption that can be stated as follows:

Ε = +ε εe p (5)

i.e., the total (elastic plus plastic) strain of a plastically deforming crystal (the

inclusion) is equal to the macroscopic strain (i.e. the strain of the elastic matrix).

Let us take the second order inner product of both sides of eq.(5) by the macroscopic stiffness tensor C.

C C Ce: : :Ε = +ε pε (6)

The left hand side of the above relationship is the macroscopic stress Σ as given by

eq.(1). The mesoscopic plastic strain εp is a deviatoric tensor. Then taking into

account eq.(2) one can easily show that: C p:ε με= 2 p (7)

Furthermore the mesoscopic elastic strain εe is related to the mesoscopic stress σ by

Hooke's law, eq.(4). After these manipulations, eq.(6) becomes: Σ = +C p: :l σ μ ε2 (8)

The second order inner product of the macroscopic stiffness C and the mesoscopic

compliance l , is the fourth-rank localisation tensor B, [20]. Then:

Σ = +B p:σ με2 (9)

In what follows our attention will be focused on polycrystalline aggregates of cubic

crystals as the majority of the metals, that are of interest in mechanical engineering,

belongs to this category. For isotropic aggregates of cubic crystals, it has been demonstrated that the macroscopic hydrostatic stress Σ ΣH tr=( )/3 is transmitted at

the crystals unchanged, [21]. That is: Σ ΣH H tr tr= ⇔ =σ 1

313( ) ( )σ (10)

The above result is exact, it is obtained without any simplification. The localisation

process thus operates only on the deviatoric part of the macroscopic stress.

9

An important approximation regarding the localisation tensor will be introduced now. It will be assumed that the tensor B is equal to the fourth-rank unit

symmetric tensor. Of course this is only a first order approximation to the localisation

problem, but it has the advantage of greatly simplifying the equations of the problem.

Moreover, this approximation is found to provide satisfactory results within the

engineering framework of the fatigue limit question, as addressed in this work. With

this last hypothesis the macroscopic and mesoscopic fields are simply related by the

equation:

Σ = + ⇒ = −σ με σ με2 p Σ 2 p (11)

Because εp is deviatoric, the above equation preserves the transfer of the

macroscopic hydrostatic stress at the grains of the material without any change, in

accordance with the previously cited exact result, eq.(10).

Slip systems of a crystal and related macroscopic and mesoscopic quantities

The mesoscopic plastic strain εp of a plastically deforming crystal, is the

result of some plastic glide γ that takes place along some easy glide systems. An

easy glide system of a crystal is composed by a plane and a direction on this plane.

Easy glide planes and directions are those with the closest atomic packing. In general

there are more than one easy glide system in a crystal. For instance, there are twelve

easy glide systems in a face centred cubic crystal. Not all of the possible glide

systems take part in the plastic deformation of a grain. For rather low loads often only

one glide system operates. As the loads applied on a component at the vicinity of the

fatigue limit level are quite low, it seems legitimate to admit that only one glide

system is active per every plastically deforming grain of the metal.

p

To define an easy glide system, let us consider a point O of a body and a frame

O.xyz attached to the body. The easy glide plane of a plastically deforming crystal is

defined by its unit normal vector n . The easy glide direction is defined by a unit

vector m lying on the gliding plane. Then the unique active system of the crystal is

10

completely determined by the orientation tensor a , which is defined as the symmetric

part of the (tensorial) product of n by m . That is:

)(21)( ijjiij mnmnamnsyma +=⇒= (12)

It is easy to show that the orientation tensor has the property:

a a: = 12

(13)

Under the assumption of the single active glide system, the plastic strain tensor in the

crystal is given by,

ε γp p a= (14)

where γ is the magnitude of the plastic shear strain due to slip in the active system.

Then eq.(11) can be written as:

p

σ μγ= −Σ 2 pa (15)

Let us examine the mesoscopic stresses acting on the gliding plane. The

mesoscopic stress vector acting on the slip plane, is the vector σn given by the inner

product of σ by the unit normal n :

σ σ n= ⋅n (16)

The projection of the stress vector σn on the normal n is the mesoscopic normal

stress ν acting on the slip plane, that is:

ν σ= ⋅ ⋅( )n n n (17)

Substituting σ from eq.(15) in the above equation yields,

ν μγ ν μγ ν= ⋅ − ⋅ ⇒ = ⋅ ⋅ − ⋅ ⋅ ⇒ = ⋅ ⋅[ ( ) ] ( ) ( ) ( )n a n n n n n n a n n n n np pΣ Σ Σ2 2 (18)

because the product ( is equal to zero, as can be easily demonstrated using the

definition of

)n a n⋅ ⋅

a , eq.(12) and the fact that n and m are orthogonal. Clearly the right

hand side of eq.(18) above is nothing else but the macroscopic normal stress N acting

on the slip plane:

N = ⋅ ⋅( )n n nΣ (19)

Thus, it has been proved, that like the hydrostatic stress Σ , the macroscopic normal

stress

H

N acting on a material plane is transmitted unaltered at the scale of the crystal:

ν = N (20)

11

Clearly the algebraic value, positive if tension negative if compression, of the normal

stress is given by the scalar product N . = ⋅ ⋅( )n nΣ

The mesoscopic shear stress τ acting along the slip direction m (i.e. the

mesoscopic resolved shear stress), is the projection of the stress vector σ on n m :

τ σ= ⋅ ⋅( )n m m (21)

Because σ is symmetric the product (n )m⋅ ⋅σ is equal to ( : . The above

relationship can then be written as:

)σ a

τ σ= ( : )a m (22)

Upon substituting σ from eq.(15), the above formula becomes,

τ μ γ τ μ γ τ μ γ= − ⇒ = − ⇒ = −[( ) : ] [ : ( : )] ( : )Σ Σ Σ2 2p p p

a a m a a a m a m m (23)

where the property, eq.(13), of the orientation tensor a has also been taken into

account. The first term on the right hand side of eq.(23) is the macroscopic shear

stress T acting along m (i.e. the macroscopic resolved shear stress):

T = ( : )Σ a m (24)

It follows that

τ μ= −T pγ (25)

where the notation γ γp pm= has also been introduced.

An explanation is perhaps needed regarding the calculation of the macroscopic

and mesoscopic resolved shear stresses T and τ . Let us examine first the

macroscopic resolved shear stress T that has been defined by eq.(24). An

interpretation of the definition of T is the following. The macroscopic stress vector

Σ Σn n= ⋅ acting on an elementary material plane can be resolved into two

components. The normal stress N , eq.(19), and the shear stress C , which is the

orthogonal projection of Σ on the plane under consideration. Therefore n C is given

by:

C C= − ⇒ = ⋅ − ⋅ ⋅Σ Σn N n n( Σ n n) (26)

The macroscopic resolved shear stress T is the projection of C onto the unit vector m

defining the easy glide direction. With the help of eq.(26), it is easy to prove that this

12

projection ( )C ⋅ m is equal to the second order inner product ( : )Σ a . Similar

explanations hold regarding the definition of the mesoscopic resolved shear stress τ .

The relationship of eq.(25), established before, coupled to the model of

elastoplastic behaviour of the crystal that will be introduced in the next section, will

permit us to evaluate the plastic strain accumulated in a gliding crystal during a cyclic

loading.

Behaviour of the constituent grains of a polycrystalline aggregate ------------------------------------------------------------------------------------------------

Cyclic elastoplastic behaviour of cubic metal crystals

Many and extensive fatigue studies have been performed for polycrystalline

metals made of cubic crystals. Often single crystal specimens were used most prone to

single slip and materials such as copper, silver, nickel, α-iron and others have been

studied. There are several excellent articles that emphasise different aspects of the

subject, (for a recent review see Basinski and Basinski [22]). We will not go into the

details of the cyclic behaviour of metal crystals here. It is sufficient for our purposes

to focus our attention to some general properties of the plastic behaviour of cubic

crystals. Most of the experimental work performed, concerns copper crystals.

Nevertheless as was pointed out by Laird [23] almost all cubic metals and alloys show

a similar behaviour.

In fig.(2) typical fatigue loops with the corresponding number of cycles are

represented, showing resolved shear stress as a function of shear strain within

individual cycles. This figure could correspond to a test of a copper crystal oriented

for single slip and cyclically loaded between constant total strain amplitude limits.

Examination of an individual loop (for example the loop corresponding to 250 cycles)

shows the existence of a Baushinger effect as the crystal after reversal of the direction

of loading at point A, starts to yield again at point C, at a stress lower in absolute

value than the peak stress τ reached before. The Baushinger effect reveals the A

13

operation of a kinematical hardening mechanism within the crystal. But the

kinematical hardening is not the only hardening mechanism in action. Inspection of

the fatigue loops represented in fig.(2) clearly shows a sustained increase of the peak

stress reached at the extreme points of any half cycle (for example τ ). This

behaviour corresponds to what is known as isotropic hardening. Therefore the plastic

behaviour of a cubic crystal can be adequately modelled according to a combined,

kinematical and isotropic, hardening rule.

τB > A

Another interesting feature, that can be observed in fig.(2), is the decrease of

the width of the fatigue loops as the number of cycles increases. The accumulated

plastic strain after N cycles is equal to twice the sum of the widths of the N fatigue

loops. As this width decreases with the number of cycles it can be concluded that the

accumulated plastic strain is an increasing function of N with a decreasing rate of

increase. However the capabilities of the crystal to strain harden in an isotropic

manner are limited. It is experimentally observed that the isotropic hardening

mechanism operates only during a first stage of the loading, after which the peak

stress, reached at any half cycle, ceases to increase if the amplitude of the applied

strain is high enough. During this second stage the peak stress, also called saturation

stress, remains constant. The fatigue loops keep recurring in an identical manner.

Therefore the width of the loops remains constant. Consequently the plastic strain

carries on to accumulate, now with a constant rate. Many experimental results

reported in the literature confirm the existence of a saturation stress, independent of

the applied plastic strain amplitude (see for example [24]). From this it can be finally

concluded that the end of operation of the isotropic hardening mechanism,

corresponds to a particular material (crystal) dependent value of the accumulated

plastic strain.

The fatigue criterion within the framework of the mesoscopic approach

14

After the brief description of the cyclic behaviour of cubic metals given in the

previous section, let us turn our attention to some general results of the mathematical

theory of plasticity. It has been proved that an elastoplastic material the hardening of

which has an isotropic component, tends to an elastic shake-down state, when

subjected to a cyclic loading applied a sufficiently high number of cycles, [25-27].

This proof has been obtained with the help of Melan's shake-down theorem as

extended by Mandel, [25], for hardening materials. Consequently, a cubic crystal,

having as shown before, an isotropic hardening at least in a first stage, will tend

asymptotically to an elastic shake-down state, when subjected to cyclic loading. The

term "asymptotically" means that the elastic shake-down state will be reached after a

theoretically infinite number of load cycles. But if a crystal after some plastic

straining tends to an elastic response, it will never break, because any process of

irreversible dissipation within the crystal tends to cease. In particular, the

accumulated plastic strain in the crystal tends to a finite asymptotic value, that

depends on the external load. Of course the capabilities of the crystal to harden in an

isotropic manner are not unlimited. As mentioned before, there is a certain (critical)

amount of accumulated plastic strain that the crystal can absorb within the range of

isotropic hardening behaviour. If during the first stage of a cyclic loading, the

accumulated plastic microstrain induced, reaches the above mentioned critical value

after a finite number of load cycles, then the yield limit of the crystal ceases to

increase with every loading cycle. As was explained before the plastic strain

continues to accumulate with a constant rate and becomes therefore an unbounded

increasing function of the number of loading cycles. Under these conditions, after a

finite number of cycles the accumulated plastic micro-strain will exhaust the ductility

of the crystal and the crystal will break, creating thus a fatigue microcrack.

Accordingly the methodology of the mesoscopic approach to the problem of

the fatigue limit can be stated as follows. Let us accept that the crystal has an

elastoplastic behaviour with a combined isotropic and kinematical hardening. Let us

assume furthermore that the capabilities of the crystal to harden in an isotropic

15

manner are unlimited; even after an infinite number of load cycles the crystal keeps

an isotropic hardening component. Therefore, under a cyclic load it will tend to an

elastic shake-down state. As will be shown later, it is possible to calculate the

asymptotic value of the cumulative plastic strain, which will be accumulated in the

crystal at the hypothetical elastic shake-down state. In reality, as it has been shown

before, the isotropic hardening stage of the crystal is limited by a certain critical

amount of accumulated plastic strain. If the calculated asymptotic value of

accumulated plastic strain is higher than this critical amount, then the crystal cannot

reach the elastic shake-down state. In fact after the end of the isotropic hardening

behaviour, plastic strain will start to accumulate in the crystal with a constant rate as

shown before and the crystal will ultimately break. On the contrary, if the calculated

asymptotic value of the accumulated plastic strain in the crystal is lower than the

critical amount, then the crystal will really reach an elastic shake-down state,

precluding any subsequent dissipation process and fracture. Hence, seen from the

mesoscopic approach point of view, the fatigue criterion mainly implies the

application of a bound on the plastic strain that has been accumulated in a gliding

crystal after an infinite number of load cycles. Clearly this condition precludes the

fracture of a crystal. Nevertheless, from an engineering point of view, the breaking of

the first plastically deforming crystal is not the most critical event, since crack

initiation in engineering terms, means the development of a small crack of the same

order of size as the elementary volume V of the material. Of course this corresponds

to the rupture of a certain number of metal grains and a subsequent coalescence of the

created sub-microcracks. It must be noticed that even in the case where the gliding

crystals in an elementary volume of the material shake-down elastically, the plastic

strain accumulated in the meantime produces some significant changes in the surface

morphology and in the bulk structure of the polycrystalline material. Due to the

accumulation of plastic strain some hills and embryo-crack like valleys appear in the

specimen surface. Concerning the bulk of the material, always due to the plastic

microstrain accumulation in some grains and the constraint imposed by neighbouring

16

elastic crystals, some microcracks could possibly develop in the grain interfaces

according to a mechanism suggested by Mughrabi and co-workers [28]. In fact, it has

been observed on metallic specimens cyclically loaded at the level of their endurance

limit, that very small cracks were present in the (unfractured) specimens at the end of

the loading. The existence of these embryo cracks introduces new parameters in the

problem of the fatigue limit. Thus the limitation of the accumulated plastic

microstrain alone can ensure that the gliding crystals reach an elastic shake-down

state, but regarding the avoidance of the creation of a fatigue crack, this limitation

appears to be only a necessary but not a sufficient condition. Therefore, to establish a

fatigue criterion useful in engineering, it is not enough to try just to preclude the

breaking of the first plastically deforming crystal. It seems better to seek to establish a

condition encompassing the whole elementary volume of the material or one section

(elementary material plane) of this volume. Fulfilment of this condition will prevent

the development of a small crack of the order of magnitude of the elementary volume.

This means that the existence of some very small crack like discontinuities of the

material could be tolerated. Of course in that case the mesoscopic stresses acting

normal to these embryo cracks will have a certain influence on the fatigue endurance

of polycrystalline metals. In a mean (average) sense, the influence of the mesoscopic

normal stresses can be adequately modelled with the use of the mesoscopic

hydrostatic stress. To conclude, within the framework of the mesoscopic approach, an

engineering fatigue criterion can be built based mainly upon the limitation of the

accumulated plastic microstrain; the effect of the mesoscopic hydrostatic stress, on

the bound applied on the accumulated plastic microstrain, has to be taken into

account.

Mathematical modelling of the plastic behaviour of metal crystals

The initiation of slip in the crystal is determined by Schmid's law. According

to it a crystal commences to deform plastically when the shear stress, acting on the

17

slip plane in the slip direction, reaches a critical value denoted as τ . To make clear

the calculations that follow, it is judicious to adopt a mathematical formalism similar

to the one of standard plasticity theory. Within such a framework, the plasticity

criterion of the crystal corresponding to Schmid's law, can be written in the form:

y

f b b by( , , ) ( ) ( )τ τ τ τ τ= − ⋅ − − ≤2y 0 (27)

Clearly, τ is the mesoscopic shear stress acting on the glide system under consideration, is the aforementioned critical value (yield limit) of the crystal and τy

b is the kinematical hardening parameter. The function f b y( , , )τ τ is the plastic

potential. The crystal is in plastic loading if,

f b b by( , , ) ( ) ( )τ τ τ τ τ= ⇒ − ⋅ − =0 2y (28)

and

∂∂τ

τf ⋅ >& 0 (29)

When the crystal is in plastic loading, the plastic shear strain rate &γ p is given by the

normality rule,

& & & & (γ λ ∂∂τ

γ λ τp pf b= ⇒ = −2 ) (30)

&λ being the plastic multiplier. For the sake of simplicity it will be assumed that the

isotropic strain hardening of the crystal follows a linear hardening rule. Then the rate

of increase of the yield limit is given by: & & &τ γy

p pg= ⋅ γ

τy

(31)

where g is a material constant with positive value. Taking into account eq.(30) and the

fact that during a plastic loading the equality, eq.(28), holds, the above hardening rule

becomes: (32) & &τ λy g= 2

A linear kinematical hardening rule will also be used,

b c b cp p= ⇒ =γ & &γ (33)

where c is another positive material constant. To calculate the plastic shear strain rate &γ p , one has to evaluate the multiplier . This can be done with the help of the

consistency condition:

&λ

18

& & & &f f fb

b f

yy= ⇒ ⋅ + ⋅ + =0 0∂

∂ττ ∂

∂∂

∂ττ (34)

Evaluating the derivatives of the plastic potential f b y( , , )τ τ appearing in the above

formula and taking into account the hardening rules eqs(32) and (33), the consistency

condition leads to the following value of the plastic multiplier : &λ

& ( ) &

( )λ τ

τ= − ⋅

+

τb

g c y22 (35)

Substituting the above value of into eq.(30) the plastic shear strain rate becomes: &λ

&[( ) & ] ( )

( )γ τ τ τ

τp

y

b bg c

= − ⋅ −+ 2 (36)

The mesoscopic shear stress τ , the kinematic hardening vector b and the shear stress

rate &τ are all vectors acting along the same line defined by m . In other words they are

parallel (collinear) vectors and an equality of the form & (τ )ρ τ= − b holds, where ρ is

an appropriate scalar. Therefore the product [( ) & ] ( )τ τ τ− ⋅ −b b appearing in the

numerator on the right hand side of eq.(36) can then be transformed as follows:

[( ) & ] ( ) [( ) ( )] ( )τ τ τ τ ρ τ τ− ⋅ − = − ⋅ − − =b b b b b = − ⋅ − − = − ⋅ − =[( ) ( )] ( ) [( ) ( )] & &τ τ ρ τ τ τ τ τ τb b b b b y

2 (37)

The equality, eq.(28), applying when the crystal is in plastic loading, has been taken

into account in the last step of the above transformation. From eqs(37) and (36) the

plastic shear strain rate is given by:

&&

( )γ τp

c g=

+ (38)

The relationship of eq.(25), established in a previous section, holds also in a rate

form:

& & &τ μ= −T pγ (39)

Introducing the above equation in eq.(38) and rearranging terms, leads to the

following fundamental relationship:

&&

γμ

pc g

=+ +T (40)

Hence, it has been demonstrated that the mesoscopic plastic shear strain rate is

proportional to the rate of the macroscopic shear stress applied on the gliding system

19

of the crystal. A direct link is therefore established between mesoscopic and

macroscopic quantities.

20

Plastic microstrain accumulated in a flowing crystal under cyclic loading

In what follows our attention will be focused on the quantitative evaluation of

the accumulated plastic strain in a gliding crystal of a metal component subjected to a

cyclic loading. Before that, it would be useful to make clear how the macroscopic

resolved shear stress, applied on an active slip system, changes during the cyclic

loading. To do this, let us consider a point O of the loaded component and a frame

O.xyz attached to the body. Let us consider furthermore an elementary material plane

Δ, that passes through O. The plane Δ is defined by its unit normal n . On this plane

the macroscopic stress vector Σn is acting, with components the normal stress N ,

eq.(19), and the shear stress C , eq.(26). For a complex periodic loading the tip of the

macroscopic shear stress vector C describes on the plane Δ a closed curve Ψ, fig.(3).

The choice of the plane Δ is arbitrary. Because the metal is isotropic at the

macroscopic scale, there are always some plastically less resistant grains so oriented

that their easy glide planes coincide with the plane Δ. Let us study one of these

crystals. The easy glide direction of such a crystal is defined by the unit vector m that

obviously belongs to Δ, fig.(3). The macroscopic shear stress T acting on the gliding

system defined by the orientation tensor a , is given by eq.(24). As explained in a

preceding section the stress T is also the projection of C on the gliding direction m :

T = ⋅( )m m (41) C

Using this last definition it is easy to observe that for a cyclic loading the macroscopic

shear stress T , acting on the slip system of the crystal under consideration, oscillates

between the values T A and T B, where A and B are the extreme points of the

projection of the curve Ψ onto the gliding direction m , fig.(3). Although not

necessary, it will be assumed that the macroscopic shear stress T varies

monotonously from T A to T B and from T B back to T A . The main reason of this

hypothesis is to make the equations of the problem less cumbersome.

It was explained before, that the quantity of interest is the accumulated plastic

microstrain Γ . The rate of accumulation of Γ , is equal to,

21

& & & && &

Γ Γ= ⋅ ⇒ =⋅

+γ γ

μp p

c gT T+

(42)

where the eq.(40) has been taken into account. Our task is to calculate the

accumulated plastic microstrain for an infinite number of loading cycles,

dtgc

PN

N ∫ ++⋅

=Γ∞→∞

0

limμTT &&

(43)

P being the load period.

Before starting with the analytical evaluation of the above integral an interesting remark will be given, that will be of help for the calculation of Γ∞. Taking

into consideration eq.(25), the plasticity criterion of the crystal, eq.(27), can be

written in the form: ( ) ( )T T− − ⋅ − − − ≤μ γ μ γ τp p

yb b 2 0 (44)

Taking further into account the kinematical hardening rule, eq.(33), the criterion

becomes: [ ( ) ] [ ( ) ]T T− + ⋅ − + − ≤μ γ μ γ τc cp p

y2 0 (45)

The vectors T and γ p act along the line defined by m . Therefore geometrically the

above equation represents a segment of this line centred at the point ( )μ + c γ p and

of length 2τ . Before the beginning of the loading, y γ p = 0 , and this segment is

centred at the origin, see fig.(3). Its half-length is equal to the initial yield limit of the crystal τ . y

( )0

Without loss of generality, let us assume that the macroscopic shear stress T

moves initially from the origin toward the point A, in the direction of m , which is

considered as positive, fig.(3). The plastic microstrain that will be accumulated along

the path from O to A, will be given by the integral,

tdgc

A

AO

t

t∫ ⋅

++=Γ →

0

1TT &&

μ (46)

or in an equivalent form,

∫ ⋅++

=Γ →

A

y

AO

m

ddgc

T

TT)0(

1

τμ (47)

22

where the substitution dT T= & dt has been used. The lower limit ( ( )τy m0 ) of the

above integral, corresponds to the fact that the crystal enter plastic loading at the

moment at which the macroscopic shear stress T , moving in the positive direction (i.e. the m direction), will cross the frontier of the segment of radius τ that is, when y

( )0

T =τy m( )0 . The crystal remains in plastic loading until T reaches the value T A . It is

then clear, that the integral in eq.(47) is equal to the length of the vector ( ( )T A

y m− τ 0 ) . Consequently the plastic microstrain accumulated over the path

is given by: O A→

ΓO AA

c gmy→ =

+ +−1 0

μτ|| ||( )T (48)

T A acts along the positive m and it can be written as T TA A m= || || . Eq.(48) thus

takes the form: ΓO A

Ac g y→ =

+ +−1 0

μτ( || || )( )T (49)

During this path the yield limit of the crystal has been increased according to the

isotropic hardening rule given by eq.(31). When T T= A, the yield limit, denoted as , is equal to: τy

A( )

τ τ τ τμ

τy y y y yA

O AA Ag

gc g

( ) ( ) ( ) ( ) ( )( || || )= + ⇒ = ++ +

−→0 0Γ T 0 (50)

Now, at the moment at which T reaches the value T A , the crystal is in plastic

loading. Then the plasticity criterion, as given by eq.(45), holds as equality.

Therefore, when T T= A, the segment representing the plasticity criterion, is centred at a point O' lying at a distance τ from point A, fig.(3). y

A( )

Let us consider now the path from A to B. After the reversal of the loading at

A the crystal will be first elastic; it will be in plastic loading again from the moment at

which the macroscopic shear stress T , moving from A towards B, will reach the

frontier of the plasticity criterion. As the segment representing the plasticity criterion is centred at a distance τ from A, this will happen when y

A( ) T reaches the point C in

fig.(3), and the corresponding value of T will be: T TC A A

y m= − 2 τ( ) (51)

23

The crystal will remain next in plastic loading until T reaches the value T B, i.e. point

B in fig.(3). The accumulated shear plastic microstrain during this first half-cycle is

then given as:

||||11 CB

BA

B

CBA gc

ddgc

TTTT −++

=Γ⇒⋅++

=Γ →→ ∫ μμ

T

T

(52)

Substituting T C from eq.(51) into eq.(52) one obtains:

ΓA BB A

c gmA

y→ =+ +

− +1 2μ

τ|| ||( )T T (53)

Introducing the notation ΔT T T= −A B and after some elementary algebra the above

equation becomes: Γ ΔA B

Ac g y→ =

+ +−1 2

μτ( || || )( )T (54)

During the path A the yield limit of the crystal increased. At the moment at which

B→T becomes equal to T B the yield limit reaches the value τ given as: y

B( )

τ τ τ τμ

τy y y y yB A

A BB A Ag

gc g

= + ⇒ = ++ +

−→Γ Δ( || || )T 2 (55)

The shear plastic microstrain accumulated over the return half-cycle from B to A can

be calculated in the same way. It results: Γ ΔB A

Bc g y→ =

+ +−1 2

μτ( || || )( )T (56)

Let as denote as δΓ( )1 the plastic microstrain accumulated over the first cycle (path A and B ). Clearly δB→ A→ Γ( )1 is the sum of Γ

A B→ and Γ

B A→:

δ δμ

τ τΓ Γ Γ Γ Δ( ) ( ) || || ( )1 1 2= + ⇒ =+ +

− +→ →A B B AA B

c g y yT (57)

Introducing in this equation τ from eq.(55), one obtains: yB

δμ

μτΓ Δ( ) ( )( )

( )|| ||1 12

22=+

+ +−

c

c g yT (58)

The notation , has also been introduced in the above equation, where τ

has to be read as, yield limit of the crystal at the beginning of the first cycle.

τ τyA( ) ( )= 1

y

B A→ →

y( )1

In a merely similar way one can establish that the accumulated plastic

microstrain over the second cycle A is given as,

δμ

μτΓ Δ( ) ( )( )

( )|| ||2 22

22=+

+ +−

c

c g yT (59)

24

where is the yield limit at the beginning of the second cycle. According to the

linear isotropic hardening rule adopted, eq.(31), τ is equal to:

τy( )2

y( )2

(60) τ τ δy y g( ) ( ) ( )2 1= + Γ 1

Introducing this value of τ into eq.(59) the accumulated plastic shear strain over the

second cycle δ

y( )2

Γ( )2 becomes:

δ

μ

μτ δΓ Δ Γ( ) ( ) ( )( )

( )|| || ( )2 1 12

22=+

+ +− + ⇒

c

c ggyT

[ ] )1()1()2(

22 )()(42||||

)()(2 Γ

+++−−Δ

+++=Γ⇒ δ

μμτ

μμδ

gccg

gcc

yT (61)

But the first term on the right hand side of this equation is just equal to δΓ( )1 , as given

by eq.(58). Therefore eq.(61) becomes:

⇒Γ⎟⎟⎠

⎞⎜⎜⎝

⎛++−+=Γ⇒Γ

+++−Γ=Γ )1()2()1()1()2(

2

2)()(4 δ

μμδδ

μμδδ

gcgc

gccg

⇒ =δ η δΓ ( ) ( )2 Γ 1 (62)

The symbol η is equal to:

2

⎟⎟⎠

⎞⎜⎜⎝

⎛++−+=

gcgc

μμη (63)

The constants, μ (Lamé coefficient), c and g, are positive material constants.

Therefore η is positive and less than one, i.e. 0 1< <η .

Eq.(62) can be established in exactly the same way as before for any two

consecutive terms δΓ( )i and δΓ(i+1) (i.e. δ ). Consequently one has: ηδΓ( ) (i+ =1 Γ )i

Γ 1

(64)

δ δδ ηδδ ηδ η δ

δ ηδ η δ

Γ ΓΓ ΓΓ Γ Γ

Γ Γ

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 1

2 1

3 2 2 1

1 1

=== =

= =− −M

N N N

It is clear that the terms δΓ( )i form a geometrical progression with ratio η. Moreover

as , the above progression is convergent. The sum of the first N terms of this

progression is given as:

0< <η 1

25

⎟⎟⎠

⎞⎜⎜⎝

⎛−−Γ=Γ⇒+⋅⋅⋅+++Γ=Γ ∑∑

=

−

= ηηδδηηηδδ

11)1( )1(

1

)(1)1(

1

)( 2NN

i

iNN

i

i

(65)

The plastic shear strain, that has been accumulated in the gliding crystal after N loading cycles, denoted as Γ

N, is obviously equal to the above quantity, plus the

plastic strain ΓO A→

that has been produced during the initial part of the loading path

from O to A.

⎟⎟⎠

⎞⎜⎜⎝

⎛−−Γ+Γ=Γ → η

ηδ11)1(

N

AON (66)

It was already explained that our task is to calculate the accumulated plastic microstrain Γ∞ corresponding to an infinite number of cycles, see eq.(43). According

to the analysis performed above Γ∞ is given as,

ηδ

ηηδ

−Γ+Γ=Γ⇒

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

Γ+Γ=Γ →→∞→

∞∞ 11

11 )1()1(

AO

N

AON

mil (67)

where the fact that η is positive and less than one has been taken into account. Substituting Γ

O A→ from eq.(49), δΓ( )1 from eq.(58) and η from eq.(63) into eq.(67)

above, one obtains: Γ Δ∞ = − +

+ +−1

22 11 0

g c gy yA( || || ) ( || || )( ) ( )T τ

μτT (68)

Let us note that τ according to the notation introduced before. Finally,

substituting in eq.(68) τ by the value of τ given in eq.(50), leads to the following

fundamental relationship:

τy yA( ) ( )1 =

y( )1

yA( )

⎟⎠

⎞⎜⎝

⎛ −Δ

=Γ∞)0(

2||||1

ygτT

(69)

To make the equations in the succeeding sections less cumbersome, the following

notation will be adopted:

TT

a = || ||Δ2

(70)

Clearly Ta is the amplitude of the macroscopic resolved shear stress. Eq.(69) takes

then the simpler form:

Γ∞ = −1 0g a y( ( )T τ ) (71)

26

The above result is very important. It proves that the plastic microstrain, accumulated

over a very large number of cycles (i.e. infinite) on one gliding system of a plastically

deforming crystal, is proportional to the difference between, the amplitude of the

macroscopic resolved shear stress and the initial yield limit of the crystal. Of course

this is true under the hypothesis that the plastically deforming crystal reaches an

elastic shake-down state. It is interesting to notice that Γ∞ does not depend on the

kinematical hardening properties of the crystal, as the constant c does not appear in

the above formula. However Γ∞ still depends on two microstructural parameters, the isotropic hardening modulus g and the initial yield limit τ . In the next section, two

upper bound estimations of the value of the plastic microstrain, accumulated over an

elementary material plane on one hand and within an elementary material volume on

the other hand, will be introduced.

y( )0

The fatigue criterion -----------------------------

Measures of the plastic microstrain accumulated over an elementary material

plane and inside an elementary material volume: The Tσ and Mσ integrals

Let us consider a point O of a loaded component and the surrounding

elementary volume V. The volume V contains a high number of metal grains. Let us

consider further, a plane passing through O. The intersection of V and this plane

defines an elementary material plane Δ. The plane Δ is defined by its unit normal n in

a O.xyz frame attached to the specimen in the point O under consideration. The

components of n in O.xyz are given by,

n n nx y z= = =sin cos sin sin cosθ θ (72) ϕ θ ϕ

where the angles ϕ and θ are the spherical co-ordinates of n fig.(4). Let us consider

these crystals of V so oriented, that their easy glide planes coincide with the plane Δ.

The easy glide directions, m , of these slip planes are directions of Δ. Any m direction

will be located with the help of the angle ψ , formed between m and an arbitrary, but

27

fixed on Δ, axis denoted as ξ fig.(4). Among these crystals there are some with a

lower value of initial yield limit (i.e. plastically less resistant grains). The

corresponding gliding directions will be denoted as m∗ . The material is isotropic at

the macroscopic scale. Consequently, on any plane Δ the m∗ directions of the

plastically less resistant grains must have a regular angular distribution. The (discrete)

sum of the plastic strain accumulated over these m∗ slip directions will give the

plastic strain accumulated in the crystals of V with their easy glide planes parallel to

Δ. From a mathematical point of view it is more suitable to work with continuous and

not discrete functions. It will be assumed therefore that the m∗ directions have a

continuous regular angular distribution on Δ. That is, the m∗ directions fill in a

continuous manner the angle 2π around the origin on the plane Δ. It must be pointed

out that this assumption does not mean that all crystals with their glide planes parallel

to Δ flow plastically. It only means that the plastically less resistant grains with their

glide planes parallel to Δ are numerous enough to provide an approximately

continuous angular distribution of their m∗ directions. It must be pointed out also that

this assumption leads to an upper bound estimation of the accumulated plastic

microstrain on Δ, as compared to the value corresponding to the (real) discrete

distribution of the m∗ directions.

The amplitude of the macroscopic resolved shear stress T , acting along a slip

direction

a

m∗ , of the plane Δ of unit normal n , is obviously a function of n and m∗ .

The unit normal n is uniquely defined by its spherical co-ordinates ϕ and θ ; the unit

vector m∗ is uniquely defined by the angle ψ . Therefore T is a function of a ϕ , θ and

ψ , i.e. T Ta a= ( , , )ϕ θ ψ . Consequently Γ∞ is also a function of ϕ , and θ ψ . As it is

assumed that the m∗ directions have a continuous regular angular distribution on Δ, it

follows that the plastic microstrain accumulated by the plastically less resistant grains

of V, with their easy glide planes parallel to Δ, is given by the integral:

(73) ∫=

∞Γπ

ψ

ψθϕ2

0

,,( ψ)d

28

Let us mention that the very important problem of the high dispersion of the

fatigue resistance of metals, did not receive any consideration in our work until now.

Within the framework of a deterministic approach as this-one, a conservative way to

face the dispersion problem can be provided by the so-called upper bound

methodology. Practically in our case this means that one has to abandon the exact

value of the accumulated plastic microstrain, eq.(73), in the profit of upper bound

estimations of this quantity. This has already be done in some extent, with the

assumption of the continuous angular distribution of the m∗ directions, introduced

before. Furthermore, a stronger upper bound estimation of the plastic microstrain,

accumulated on an elementary material plane Δ, will be introduced here by the

following formula:

∫=

∞Γπ

ψ

ψψθϕ2

0

2 ),,( d (74)

It can be shown easily that the above expression has always a value higher than, or at

least equal to, the value of the integral of eq.(73). Taking into account eq.(71), one

has:

∫∫==

∞ >−<=Γπ

ψ

π

ψ

ψτψθϕψψθϕ2

0

22

0

2 )0(),,(1),,( dg

d yaT (75)

The notation . used in the formula above, means: positive part of the quantity inside

the brackets. Use of this notation is necessary because it is possible that some m*

directions exist, along which the amplitude of the macroscopic resolved shear stress , is lower than the corresponding initial yield limit τ . Therefore to calculate the

integral of eq.(75) one has before to find the range of the angle

Ta y

( )0

ψ over which the

quantity is positive. Obviously this creates important complications

concerning the evaluation of this integral. Accordingly a simplification will be

introduced to confront this problem. In particular it will be assumed that τ has a

value equal to zero. With this assumption the integral of eq.(75) extends over the

( )( )

Ta y− τ0

y( )0

29

whole range of the angle ψ, defined by the integration limits, as Ta is positive for

every ψ π∈[ , ]0 2 . It must be pointed out that this assumption is coherent with the

upper bound methodology introduced before, because the value of the expression of

eq.(75), is strengthened by the elimination of τ . Let us introduce the following

integral denoted as

y( )0

Τσ :

∫=

=π

ψ

ψψθϕ2

0

2 ),,(( daTθϕ ),Τσ (76)

Taking into account the assumption τ and with the help of the y( )0

0= Τσ integral as

given by eq.(76), the upper bound estimation, eq.(75), of the plastic strain

accumulated by the plastically less resistant grains of V with their easy glide planes

parallel to Δ, can be written simply as:

),(1),,(0

2Γ∞

2

θϕψψθϕ σ

π

ψ

Τ=∫= g

d

2

(77)

Let us consider now that the orientation of the elementary material plane Δ,

passing through point O, is varied in such a way that all the directions of the three-

dimensional space around O are covered. This can be done by making the angle ϕ

vary from 0 to 2π and the angle θ from 0 to π (i.e. 0 ≤ ≤ϕ π and 0 ). In a

similar way as before, an upper bound estimation of the accumulated plastic

microstrain now of all the gliding crystals inside V irrespective of their orientation

will be given by the expression:

≤ ≤θ π

ϕθθϕϕθθψψθπ

ϕ

π

θσ

π

ϕ

π

θ

π

ψ

ddTg

ddd ),(1sin),,2

0 0

22

0 0

2

0∫ ∫∫ ∫ ∫= == = =

=⎟⎟⎠

⎞⎜⎜⎝

⎛θsinϕ(2

∞Γ (78)

Upon introducing the following integral denoted as Mσ,

ϕθθθϕπ

ϕ

π

θσσ ddTM sin),(

2

0 0

2∫ ∫= =

= (79)

the plastic microstrain accumulated within the elementary volume V becomes:

σ

π

ϕ

π

θ

π

ψ

ϕθθψψθϕ Mg

ddd 1sin),,(2

0 0

2

0

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛Γ∫ ∫ ∫

= = =∞ (80)

30

In conclusion, at any point O, surrounded by an elementary volume V, of a cyclically

loaded body, the quantities T gσ and Mσ g are upper bound approximations of the

plastic microstrain accumulated respectively: by the gliding grains of V with their

easy glide planes parallel to a given elementary material plane Δ passing from O, and

by all the gliding crystals within the elementary volume V surrounding O. Clearly the

above quantities vary from one point to another; they are functions of the co-ordinates

of the point under consideration. Furthermore the quantity T gσ depends, for every

point O, on the orientation of the plane Δ. That is T gσ is a function of the spherical

co-ordinates ϕ and θ, see eq.(76), whereas Mσ g is by definition, eq.(79),

independent of these angles.

Mild and hard metals: Two new fatigue criterion formulas

The pioneer work of Gough, Pollard and Clenshaw [1] on the fatigue limit of

metals under in-phase bending-twisting, has shown that a single empirical formula

could not represent successfully the fatigue behaviour of all metals tested. Extensive

analysis of experimental data under non-proportional loading performed in a

companion paper, [29], has confirmed the above conclusion of Gough et al., that is,

the whole range of metals cannot be covered by only one fatigue criterion.

Accordingly the metals have been classified into three categories: mild, hard and

brittle metals. The third category (i.e. brittle metals) is not covered by the present

work. For any metal the above classification is based on the value of the ratio of the

fully reversed torsion endurance limit t-1, over the fully reversed tension-compression

endurance limit s-1. Mild metals are the metals with a t s− −1 1 value in the range of 0.5

to 0.6 (i.e. 0 5 0 61 1. ≤ ≤− −

t s . ). For hard metals this ratio has a value from 0.6 to

approximately 0.8 (i.e. 0 6 0 8. ≤ ≤− −t s1 1 . ). Finally brittle metals have a t s− −1 1 ratio

close to one. It must be pointed out that this classification is empirical; it is not a

theoretical result of the mesoscopic approach. More research is needed in that

31

direction to link this empirical classification to the theoretical development of the

mesoscopic approach.

Let us consider first the broader category of hard metals (i.e.

0 6 0 81 1. ≤ ≤− −t s . ). For these metals the best results have been obtained with the use

of a fatigue formula based on a limitation of the plastic microstrain accumulated by all

the gliding crystals within an elementary material volume V. This means that the

fatigue criterion can be expressed as a limitation of the quantity M gσ introduced

before. It has been explained in a previous section that an engineering fatigue

criterion seeks to prevent the creation of a small fatigue crack of the order of

magnitude of the elementary material volume. Therefore, the development, in the

elementary volume, of some very small crack like discontinuities, can be tolerated.

Consequently, the effect of the mesoscopic stresses, acting normal to these embryo

cracks, has to be taken into account. It has been shown that the mesoscopic and

macroscopic normal stresses acting on a material plane are equal, see eq.(20). The

normal stress N depends on the orientation of the plane on which it is acting, that is N

is a function of the angles ϕ and θ, N N= ( , )ϕ θ . The orientation of the embryo

cracks mentioned above it is not specified in any way. Therefore the best manner to

include the effect of the normal stresses is to take the integral of N over all the

directions of the three-dimensional space around a point of the body. It can be shown

that this integral is equal to 4πΣH, where ΣH is the hydrostatic stress, eq.(10):

(81) Hdd Σ=∫ ∫= =

πϕθθθϕπ

ϕ

π

θ

4sin),(2

0 0

N

Clearly for a cyclic loading the hydrostatic stress (and the integral of eq.81) is a

function of time. It will be assumed that the limit which will be applied on Mσ g is a

linear function of the maximum value that the above integral achieves, during a

loading cycle. The fatigue criterion for the hard metals (i.e. 0 6 0 8. .≤ ≤− −t s1 1 ) can

then be written as:

1 4g

M q p Hσ π≤ ′ − ′( ,maxΣ ) (82)

32

ΣH,max

′q ′p

denotes the maximum value of the hydrostatic stress in a loading cycle and

, are material constants. It is noteworthy that one microstructural parameter, the

isotropic hardening modulus g, appear in the above formula. Nevertheless this does

not constitute an important drawback. Upon multiplying both members of eq.(82)

with g and after introducing the notation,

(83) q g q p g p= ′ = ′(4π )

the fatigue criterion for hard metals becomes: M q p Hσ ≤ − Σ ,max (84)

This formula is independent of any microstructural parameter of the material. Furthermore the quantities M and σ ΣH,max , can be evaluated through the knowledge

of the macroscopic stresses alone. The constants p and q can be expressed via the

endurance limits t-1 and s-1, by applying the criterion, eq.(84), in the cases of fully

reversed torsion and fully reversed tension-compression, respectively.

In the case of mild metals (i.e.0 5 0 61 1. ≤ ≤− −t s . ) better results have been

obtained by employing the plastic microstrain accumulated by the gliding crystals of

an elementary volume V with their easy glide planes parallel to a plane Δ. That is, the

quantity T gσ has to be used in the place of Mσ g. Of course as T is a function of

the orientation of the plane Δ, (i.e.Tσ

Tσ σ ϕ θ= ( , ) ), one has to find beforehand the

couple ( , )ϕ θ that maximises T . Concerning the isotropic hardening modulus g,

similar manipulations as before can be performed in such a way that the fatigue

criterion for mild metals can finally be written as:

σ

max ( , ), ,maxϕ θ σ ϕ θ β αT H≤ − Σ (85)

As before, the quantities appearing in this formula are expressed via the macroscopic

stresses alone. The constants α and β can be easily linked to the endurance limits t

and s of the material. −1

−1

The two formulas, eqs(84) and (85), based on the T and M integrals,

conclude the development of the mesoscopic approach in the field of the fatigue limit

of mild and hard metals. More work is needed to extend this approach in the case of

σ σ

33

brittle metals where the plastic microstrain is not the main parameter of the fatigue

fracture process.

Conclusions ------------------

The general theory of the mesoscopic approach to the problem of the fatigue

limit of metallic materials has been presented in this paper. Two scales of material

description have been introduced, the usual macroscopic scale and a mesoscopic one.

The former is defined with the help of the elementary volume V determined at any

point O of a body as the smallest sample of the material surrounding O that can be

considered as homogeneous. The volume V contains a high number of metal grains

(crystals) and the mesoscopic scale is therefore defined as a partition of V. The

mesoscopic stresses and strains acting at the level of crystals are related to the

corresponding macroscopic quantities with the help of the Lin-Taylor assumption.

Under an external cyclic loading some plastically less resistant grains of V

suffer plastic glide. The possibility of these plastically deforming crystals to reach an

elastic shake-down state is investigated. The fatigue limit is related to some

characteristic quantities of this elastic shake-down state. In particular, at any point O

of a body surrounded by an elementary volume V two new measures (upper bounds)

of the accumulated plastic microstrain have been introduced. These two measures are

associated to the elastic shake-down state of the plastic crystals in V. The first

measure is an upper bound of the plastic microstrain accumulated in only these

crystals of V which are so oriented that their active glide planes are parallel to a given

plane Δ passing through O. The second is an upper bound of the accumulated plastic

microstrain of all the plastically deforming crystals in V irrespective of their

orientation. For mild metals the fatigue criterion is defined mainly as a limitation of

the first of the above measures, while for hard metals the second measure of the

accumulated plastic microstrain has to be limited in order to avoid fatigue fracture. It

34

has been shown that in both cases the limits to apply on the accumulated plastic

microstrain depend on the maximum value that the mesoscopic hydrostatic stress

reaches during a loading cycle. It is noteworthy that the two proposed criteria,

although built at the mesoscopic scale, are finally expressed through the usual

macroscopic stresses, with the help of the appropriate micro-scale to macro-scale

passage relationships established before.

References ----------------

1. Gough, H. J., Pollard, H. V. and Clenshaw, W. J. (1951) "Some experiments on

the resistance of metals to fatigue under combined stress", Aeronautical

Research Council, Reports and Memoranda No. 2522, His Majesty's Stationary

Office, London.

2. Stulen, F. B. and Cummings, H. N. (1954) "A failure criterion for multi-axial

fatigue stresses", Proceedings ASTM, Vol. 54, pp. 822-835.

3. Crossland, B. (1956) "Effect of large hydrostatic pressures on the torsional

fatigue strength of an alloy steel", in Proceedings of the International

Conference on Fatigue of Metals, London-New York, Published by the

Institution of Mechanical Engineers, London, pp. 138-149.

4. Findley, W. N. (1959) "A theory for the effect of mean stress on fatigue of

metals under combined torsion and axial load or bending", Transactions

ASME, Vol. 81, Series B, pp. 301-306.

5. Sines, G. (1959) "Behaviour of metals under complex static and alternating

stresses", in Metal Fatigue (Edited by Sines, G. and Waisman, J.L.), McGraw

Hill, New York, pp. 145-169.

35

6. Kakuno, H. and Kawada Y. (1979) "A new criterion of fatigue strength of a

round bar subjected to combined static and repeated bending and torsion",

Fatigue Engng Mater. Struct., Vol. 2, pp. 229-236.

7. McDiarmid, D. L. (1991) "A general criterion for high-cycle multiaxial fatigue

failure", Fatigue Fract. Engng Mater. Struct., Vol.14., No. 4, pp. 429-453.

8. Dietmann, H., Bhongbhibhat, T. and Schmid, A. (1991) "Multiaxial fatigue

behaviour of steels under in-phase and out-of-phase loading, including different

wave forms and frequencies", in Fatigue Under Biaxial and Multiaxial

Loading, ESIS10 (Edited by Kussmaul, K., McDiarmid, D. L. and Socie, D.),

Mechanical Engineering Publications, London, pp. 449-464.

9. Gough, H. J. (1933) "Crystalline structure in relation to failure of metals-

Especially by fatigue", Eighth Edgar Marburg Lecture 33, Proceedings ASTM.

10. Orowan, E. (1939) "Theory of the fatigue of metals", Proc. Roy. Soc. A 187,

pp. 79-106.

11. Dang Van, K., (1973) "Sur la résistance à la fatigue des métaux", Thèse de

doctorat ès sciences, Science et Techniques de l'Armement 47, 3e fascicule

1973, pp. 647-722.

12. Le Douaron, A., Lim, T. et Lafont R. (1983) "Méthode d'analyse en fatigue sous

sollicitations complexes", Revue de Métallurgie-Mémoires et Etudes

Scientifiques, pp. 105-109.

13. Dang Van, K., Papadopoulos, I. V., Griveau, B., et Message, O. (1987) "Sur le

calcul des structures soumises à la fatigue multiaxiale", in Matériaux et

Structures, Fatigue et Contraintes Résiduelles, Hermes, Paris, pp.79-97.

14. Papadopoulos, I. V. (1987) "Fatigue Polycyclique des métaux: Une nouvelle

approche", Thèse de doctorat, Ecole Nationale des Ponts et Chaussées, Paris.

15. Papadopoulos, I. V., Dang Van, K. (1988) "Sur la nucléation des fissures en

fatigue polycyclique sous chargement multiaxial", Arch. Mech., Vol. 40 (5-6),

pp. 759-774.

36

16. Deperrois, A. (1991) "Sur le calcul de limites d'endurance des aciers", Thèse de

doctorat, Ecole Polytechnique, Paris.

17. Brown, M. W. and Miller, K. J. (1973) "A theory for fatigue failure under

multiaxial stress strain conditions", Proc. Inst. Mech. Engrs, Vol. 187 (65/73),

pp. 745-755.

18. Hill, R. (1965) "Continuum micro-mechanics of elastoplastic polycrystals",

J. Mech. Phys. Solids, Vol. 13, pp. 89-101.

19. Berveiller, M. and Zaoui, A. (1979) "An extension of the self-consistent scheme

to plastically-flowing polycrystals", J. Mech. Phys. Solids, Vol. 26, pp. 325-344.

20. Mandel, J. (1976) "Plasticité classique et viscoplasticité", Courses and Lectures

of CISM, No. 97, Springer-Verlag, Wien.

21. Walpole, L. J. (1985) "The stress-strain law of a textured aggregate of cubic

crystals", J. Mech. Phys. Solids, Vol. 33, No. 4, pp. 363-370.

22. Basinski, Z. S. and Basinski, S. J. (1992) "Fundamental aspects of low

amplitude cyclic deformation in face-centred cubic crystals", Progress in

Material Science, Vol. 36, pp. 89-148.

23. Laird, C. (1976) "The fatigue limit of metals", Mater. Sci. Engng, Vol. 22,

pp. 231-236.

24. Winter, A. T. (1974) "A model for the fatigue of copper at low plastic strain

amplitudes", Phil. Mag., Vol. 30, pp. 719-738.

25. Mandel, J. (1976) "Adaptation d'une structure plastique écrouissable",

Mech. Res. Comm., Vol. 3, pp. 251-256.

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approximations", Mech. Res. Comm., Vol. 3, pp. 483-488.

27. Mandel, J., Zarka, J. et Halphen, B. (1977) "Adaptation d'une structure

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pp. 309-314.

28. Mughrabi, H., Renhui, W., Differt, K. and Essmann, U. (1983) "Fatigue crack

initiation by cyclic slip irreversibilities in high-cycle fatigue", in Fatigue

37

Mechanisms: Advances in Quantitative Measurements of Physical Damage,

ASTM STP 811, (Edided by Lankford, J., Davidson, D. L., Morris, W. L. and

Wei, R. P.), American Society for Testing and Materials, pp. 5-45.

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multiaxial stress conditions: Application of the mesoscopic approach",

(submitted for publication in Fatigue Fract. Engng Mater. Struct.)

38

Figures ----------

Δ

nV

~1mm

V

σ ε,

Σ Ε,

Σ Ε,

Fig.1 Macroscopic and mesoscopic scales of material description.

39

τ

γ

A

BC

100

250

1000

10000

Fig.2 Typical fatigue loops, showing resolved shear stress as a function of total

shear strain within individual cycles.

40

Δ

m

nA

B

C

Ψ

τy02

τy2 A

C

0

0'

m

Fig.3 Path Ψ of the macroscopic shear stress C acting on a material plane Δ and

corresponding path of the macroscopic resolved shear stress T acting on

an easy glide direction m .

41

z

x

y

ϕ

θ

Δξ

L

χ 0

n

m

42

Fig.4

Spherical co-ordinates (ϕ, θ) of unit vector n normal to a material plane

Δ. Location of an easy glide direction m belonging to Δ with the help of

angle ψ.