Lifetime-Oriented Structural Design Concepts- P11 pps

It takes the loading frequency into accountand distinguishes between high-cycle and low-cylce fatigue: 1.2 − 0.2 rfat + smax− 0.053 1 − 0.445 rfat log In this expression, tfat is the dur

[386] within the mean part of the yield condition is a scaling factor, that

leads to total material softening for f ∗ qA=1 Nonlinear isotropic hardening is

considered by the following relation

i=1 κˆi with the followingassumption for the material time derivative of the back stress tensor

κ4|| > for i=4, ˆDp = ˙γ F,(Σˆ− ˆ κ)= ˙γN is theˆ

symmetric plastic strain rate, ˆDp = ˆG−1DˆpGˆ−1,N describes the symmetricˆgradient of the yield surface, ||ˆ κ ||=3/2 tr(ˆ κˆ κ ) is the norm of the backstress tensor ˆκ and ˆκ = ˆGˆκG, ¯ˆ κ, δ kin are model parameters and β controls

the decomposition of isotropic and kinematic hardening

The evolution of the void volume fraction f is described by

in (3.95) represents a nucleation law according to [197] with

To describe the physical process of void nucleation adequately, the evolution of

˙ pis only defined for loading In case of unloading no nucleation of micropores

is considered For the consideration of the coalescence of the micropores thephenomenological law according to [797] is used,

Fig 3.126 Numerical and experimental data for (a) material softening and (b)

equation proposed by [258] It is implemented into the Gurson-model by a

modification of the material parameter q B [386]

The implementation of the model is based upon the return-map algorithmand a consistent linearization procedure [743] Because of the anisotropy in-duced by the kinematic hardening, the iterative solution involves 8 unknowns(the components of the symmetric gradientN, the plastic multiplier ˙γ andˆ

the void volume fraction f ).

The proposed macroscopic elasto-plastic damage model has the ability toreplicate all typical phenomena of cyclic plasticity such as the Bauschinger-effect, ratcheting or mean stress relaxation, cyclic hardening or softening [386]

In the following, a comparison of numerical and experimental data shows theefficiency in case of simulating the effects of material softening and ratcheting

To this end, a cyclically loaded hollow cylindrical specimen of CS 1026 [86]

is re-analysed numerically Using the isotropic hardening law according to(3.93) the stress amplitude of the first 25 load cycles can be simulated in goodagreement to the experimental results, as Figure 3.126(a) shows The use ofthe Bari-Hassan-type of kinematic hardening rule allows for the simulation

of the ratcheting effect, which is demonstrated by the evolution of the radialstrain in case of biaxial loading in Figure 3.126(b)

3.3.1.2.1.2 Model Validation

The following analysis are performed for 20MnMoNi55, a low alloy steeltypically used for structures such as pressure vessels For this special type ofmaterial a calibration leads to the model parameters presented in Table 3.20[386]

After the calibration procedure, the micropore damage model is validatedaccording to Figure 3.125 by means of fatigue tests Therefore, results from

Table 3.20 Parameter of the elasto-plastic micropore damage model for

20Mn-MoNi55

E=204 [GPa] β=0.5 σ Y0=220 [MPa] δ iso=25 [-]

b1=25000 [-] b2=500 [-] b3=5 [-] b4=5000 [-] δ kin=0.18 [-]

c1=500000 [MPa] c2=60000 [MPa] c3=3000 [MPa] c4=100000 [MPa] ˘κ=0 [MPa]

f0=0.01 [-] S0=0.0 [-] q A=1.85 [-] q B=0.48 [-] q C=1.4 [-]

f n=0.08 [-] n=3.0 [-] s n=1.0 [-] f krit=0.09 [-] f Bruch=0.14 [-]

Fig 3.127 Low Cycle Fatigue in metals: Numerical and experimental results for

cyclically loaded round notched bar with (a) 2mm notch radius and (b) 10mm notchradius

cyclically loaded round notched bars with two different notch radii (2mm and10mm) performed by [638] are re-analyzed To study the damage evolutiondue to low cycle fatigue, the results are compared in terms of the degradation

of the peak reaction force in Figure 3.127

The chosen set of material specific model parameters result in a good ment of experimental and numerical results In particular the strong change ofthe slope of the reaction force curve, when coalescence becomes the dominantdamage mechanism, is simulated by the micropore damage model in a goodmanner for both cases This final change state can be correlated to the lifetime of the structure, which is reasonably well predicted (see Table 3.21)

agree-It should be noted that he numerical simulations also allow a localization

of the position of damage accumulation in accordance with the experimentalobservations For the smaller notch radius the micropore damage initiates andstarts to accumulate from the notch root (Fig.3.128(a,b)) In contrast, for thespecimen with the larger notch radius a nearly homogeneous damage accumu-lation initiating from the interior of the specimen is observed(Fig.3.128(c,d))

Table 3.21 Low Cycle Fatigue in metals: Number of load cycles until failure

ob-tained from numerical simulations and experiments

2 mm notch radius 10 mm notch radiusExperiment Num Model Experiment Num ModelFailure initiation 17 cycles 24 cycles 31 cycles 35 cyclesLife-time 23 cycles 28 cycles 43 cycles 40 cycles

Fig 3.128 Low Cycle Fatigue in metals: Damage accumulation and numerically

predicted damage in a cyclically loaded round notched bar: (a,b) 2 mm notch radius,(c,d) 10 mm notch radius

3.3.1.2.2 Quasi-Brittle Damage in Materials

moder-ing on the applied increment of load-cycles ΔN For a standardised evaluation

and formulation, the load cycles are related to the ultimate number of

load-cycles Nfaccording to the S-N -approach This leads to the standardised time scale n ∈ [0, 1] with increments Δn = ΔN

N This standardised time scale is

related to the time scale via Nf and the frequency f by:

Thus, one basic quantity for the fatigue model is the fatigue lifetime Nf

of concrete Generally, any S-N -curve can be applied for its evaluation In

the following the approach presented in [392] will be re-used It has beencompared to other approaches and to a large number of experiments in [627]and proved to be well suitable It takes the loading frequency into accountand distinguishes between high-cycle and low-cylce fatigue:

1.2 − 0.2 rfat + smax− 0.053 (1 − 0.445 rfat) log

In this expression, tfat is the duration of one load cycle, which is the inverse

of the frequency: tfat = 1/f , tref is a reference time of the same unit as tfat:

tref = 1[tfat], smax and smin are the related upper and lower stress limits,

respectively: smax = σmax/fc, smin = σmin/fc and rfat is the relationship of

lower to upper fatigue stress: rfat = σmin/σmax The approach, together with alarge number of experiments from the literature, is illustrated in Figure 3.129

The evaluation of the 0.05- and of the 0.95-quantile of Nf, which is also shown

in this diagram, will be introduced later in this section

Degradation of the Compressive Strength

The degradation of the compressive strength is formulated empirically with

a direct approach in the time scale of the related number of load cycles n, as introduced above To quantify the degradation, the variable d fc is introducedand the resulting compressive strength reads as follows:

According to the experimental results presented in [70, 374], the degradationprocess starts very slowly Nevertheless, fatigue failure is associated with adrop of the compressive strength onto the level of the upper fatigue stress

Thus, the value of d fc results in

100 102 104 106 108 1010

Fig 3.129.S-N approach (0.05-, 0.50-, and 0.95-quantiles) with experimental results

for the state of fatigue failure, n = 1 Based on an exponential approach, suggested in [370] for the description of sequence effects, d fcis sub-structuredinto

d fc= n a fc · dfc,fail with a fc= 26.5 − 25.0 |σmax|

Fig 3.131 Evaluation of the approach for sequence effects an comparison with

single simulation results from [383]

Sequence Effects

This direct formulation of the degradation of the compressive strength cludes indirectly the formulation of sequence effects As introduced, the degra-dation process depends on the applied upper fatigue-load level When this loadlevel is changed, the degradation curve changes, too As illustrated in the rightdiagram of Figure 3.130, this requires a modification of the related number

in-of load cycles n = N/Nf Keeping n constant would imply a sudden drop or increase of the compressive strength, which is physically nonsensical Thus, n has to be changed The updated value of n can be evaluated from the approach

for the description of the degradation process:

Strain Evolution

As introduced in Section 3.1.1.2.2.1, the strain evolution in concrete der fatigue loading can be interpreted as the sum of creep and cyclic strainevolution This interpretation is picked up for the modelling approach Inthe following, the approaches for the creep strain evolution as well as for theevolution of cyclic strains is introduced

un-spring frictional element

dashpot

σσ

Fig 3.132 Rheological element for the description of nonlinear creep processes

Creep Strain Evolution

The creep strain evolution is modelled with rheological elements The model

is based on an approach presented in [735], which has been enhanced in[133, 134] To account for the nonlinear relation of stress and creep strain

rate, especially for concrete under higher stresses than approximately 0.4 fc,nonlinear rheological elements are used They consist of a nonlinear springwith a friction element (to describe plastic deformations) and a nonlineardashpot Fig 3.132 shows such elements

The nonlinear behaviour of the spring is described with the stress-strainrelation for concrete under compression given in [182], with the compressive

strength fc replaced by f c,T = 0.8 fc, taking long-term effects into account.Thus, the stress-strain relation for the spring reads:

retardation time τ with the stiffness: η = τ Ec This relation is now enhancedand is reformulated dependent on time and on the applied stress:

Herein, σdis the stress in the dashpot and ncra material parameter According

to [133, 134], for concrete it takes values between 1.5 and 2.0

The classical relation between stress and strain rate,

Cyclic Strain Evolution

The rate of cyclic fatigue strains, in the time scale of related load cycles n:

results from [383] The borders between these domains are assumed at n = 0.1 and n = 0.9.

In order to approximate the fatigue strain rates as functions of the appliedload level, a scalar measure for the fatigue loading, taking upper and lowerfatigue stress into account, is introduced as the product of mean stress andstress difference:

s = smax+ smin

The evaluation of the experimental results of [383, 70] are shown in theright diagram of Figure 3.133 The experiments, that exhibit significant creep

1.0 2.0 3.0 4.0

approach

¯ Holmen

¯ Awad & Hilsdorf

Fig 3.133 Fatigue strain evolution (stress measure vs strain rate by [383, 70])

strains due the test duration are plotted in white These experiments have notbeen used for the evolution of cyclic strain evolution Those ones that exhibitpure cyclic strain are plotted in black By least square fitting, second orderpolynomials are evaluated to approximate the (pure cyclic) fatigue strain rate

as function of the stress measure s:

˙ε fat,∗ 1,3 =−113.189 s2+ 67.5492 s − 4.50913 , (3.115)

˙ε fat,∗2 =−6.54818 s2+ 4.55811 s − 0.268655 (3.116)The right diagram in Figure 3.133 shows the polynomial as well as experimen-tal results, exemplarily for domain 2 The results of [383] have been used forthe evaluation of the polynomial For additional proof, the results of [70] areplotted in the diagram These values have been evaluated graphically Theyare therefore regarded as too imprecise and where not taken into account forthe evaluation

Fatigue Damage

For the evaluation of fatigue damage, again the tests of [383] deliver the perimental basis In these experiments, not only the evolution of the maximum,but also of the minimum4 fatigue strains is reported That can be utilised forthe sub-division of the total fatigue strains into damaging and plastic parts Atfirst, the measured total strains are reduced by the initial ones Assuming linearunloading and reloading, according to the damage theory, the fatigue strains

ex-corresponding to σ = 0 can be extrapolated, like illustrated in Figure 3.134.

This yields reversible and irreversible parts of the total fatigue strains, whichare interpreted as damaging and plastic, respectively

4That means the strains, that correspond to the lower fatigue stress level

Fig 3.134 Split of total fatigue strains into reversible and irrversible parts

To distinguish the damaging part from the total fatigue strains, the variable

0.2 0.4 0.6

due to the convex curvature of the unloading path, to an overestimation of

the irreversible strains The series 0.675/0.05 exhibits significant creep strains,

which are regarded as plastic Thus, the reduction of the damaging part oftotal measured time-dependent strains proves the assumption, that the totalfatigue strains can be interpreted as the sum of time- and cycle-dependentparts, see Section 3.1.1.2.2.1

Multi-Axial Stress States

To enhance the uniaxial approach for fatigue and creep strain evolution, apotential and a flow rule, analogous to the classical time-independent descrip-tion of the behaviour of concrete according to the smeared crack approach,which has been introduced earlier in this section, is assumed:

˙

ε fat,∗=∂φc

∂ σ ˙λ fat,∗

In this equation, the derivative ∂φc/∂ σ describes the direction of the strain

evolution, ˙λ fat,∗c the norm The latter one is derived from the uniaxial strainrate and the assumption of energetic equality of uniaxial and multi-axial stressstates:

Fatigue damage is quantified by a fourth order compliance tensor, analogously

to the smeared crack model described earlier in this section The evolutionlaws of the two independent variables read

Dda,fat,∗= ˙D da,fat,∗s1 1⊗1+ ˙D da,fat,∗s2 III + ¯III, (3.122)

to evaluate increments of this tensor, it has to multiplied with the increment

Δn of the number of related load cycles:

Thus, the final compliance relation, which takes damage and plastic strainsdue to time-independent and time-dependent modelling of the material be-haviour of concrete into account, reads

Scatter of Basic Model Properties

In order to account for the scatter, which is inherent to fatigue processes, the

scatter of the basic quantity Nf is investigated According to the observations

in e.g [738, 821], the scatter of the fatigue lifetime is related to the scatter of

the compressive strength fcof concrete An approach presented in [738] takes

this into account; it describes the standard deviation of log Nf as a function

of the standard deviation of fc and of the applied fatigue load:

s(log Nf) =

5.714 s(fc)m(fc)

This is a reasonable suggestion that is hardly to proof experimentally: At

load levels below 0.5 fc, very high fatigue lifetimes Nfoccur, which makes therealisation of a large number of experiments, the basis for the evaluation ofthe scatter, difficult

3.3.1.2.2.2 Metallic Materials

Authored by Henning Sch¨ utte

It is known from experiments that most materials, and in particularbrittle and quasi-brittle materials, under general loading conditions developanisotropic damage [441] For a given stress state, materials damaged by mi-crocracks in general accumulate additional damage through the growth ofthese microcracks Considering this and the mentioned points, the concern

of this paper is to provide a consistent, continuum damage model based onthe micromechanical framework and the local anisotropy induced by kinking

and growing elliptical and/or circular microcracks The reason for consideringelliptical and circular cracks is that these geometries are good approximationsfor the shape of a flaw in many engineering materials For clarity purposesand to explain the main issues of the proposed model in a more clear mathe-matical way, the complexity of the proposed damage model is reduced here byleaving out the thermal effects and other non-mechanical phenomena Strainsand rotations are assumed to be small, hence the framework of linear elasticfracture mechanics can be applied Furthermore, viscous effects and perma-nent deformations are neglected and the material behavior is assumed to belinear elastic in its pristine state The small strain assumption, and the lack

of permanent deformations in this model makes it suitable to show the tion of damage in structures with brittle and quasi-brittle fracture behaviorexperiencing high-cycle fatigue

evolu-Effective Continuum Elastic Properties of Damaged Media

The micromechanical models are commonly referred to a class of analyticalmodels which give the relation between the macroscopic state of a specimenand its micro-structure [164] One of the goals of the micromechanical models

is to provide relatively simple constitutive laws Within this approach, theeffective elastic properties are derived by using the pertinent results of micro-constituent analysis, such as that of a planar crack embedded in an infinitemedium Using the concept of micromechanics, continuum damage modelsbased on the framework of fracture mechanics and elasticity can give the lo-cal details of the damage response within a representative volume element.These class of models are based on the hypothesis of statistical homogeneityand weak interaction of defects, which are justifiable for reasonably modestconcentration of heterogeneities [565] In this respect, the first step in the for-mulation of the proposed continuum damage model requires the formulation

of the change of continuum elastic properties due to the presence, kinking andgrowth of elliptical and/or circular microcracks

Applying the approach of micromechanics, the components of the effectivecompliance tensor of an infinite, homogeneous, isotropic (in its pristine state)and elastic continuum damaged by a single internal circular crack is given

by [440] Here, applying the same method, the components of the tensor forthe change of compliance due to the presence of a single internal ellipticalcrack are derived, from which the results corresponding to a single circularcrack can be reproduced (see also [163]) Within the approach of microme-chanics, the effective elastic properties of a solid damaged by a planar internalelliptical crack are derived from the contribution to the complementary strainenergy corresponding to the quasi-static, selfsimilar growth of the crack Forthis, the stress intensity factors suffice to give the energy released during thequasi-static, selfsimilar growth of the crack However, for the formulation ofthe complementary strain energy corresponding to the kinking of a crack,the analytical expressions for the so called T-stresses are required as well.The complete set for the T-stresses for internal elliptical and circular cracks

embedded in a homogenous isotropic infinite solid have been addressed by[549] and [716], respectively.

Kinking of an Internal Elliptical Crack

To study the degradation of the elastic material properties due to the ing and growth of elliptical and/or circular microcracks, consider a singleelliptical crack in an infinite, homogeneous, isotropic and elastic continuumsubjected to mechanical loads applied at infinity For a stress state outside thedamage surface, the considered crack will kink and propagate to a new geom-etry, and the local kinking angle and the local crack extension length can becalculated from the considered fracture criterion coupled with a fatigue crackevolution law In an analogous manner to the previous section, this problemcan be decomposed into two sub-problems: that of the continuum without acrack subjected to the remote traction field, and that of the same continuum,where only the crack faces are subjected to the traction field

kink-In the framework of linear elasticity, the compliance tensor of a materialcontaining a kinked crack can be decomposed into three parts [715]

*SSS = SSSMatrix+ ΔSSSCrack+ ΔSSSKink, (3.127)whereSSSMatrix and ΔSSSCrack refer respectively to the compliance tensor of thematrix material in its pristine state and the change of compliance due to the

presence of the microcrack, and ΔSSSKink is the change of compliance due tothe kinking and growth of the microcrack

The analytical expression for the tensor of the change of compliance due tothe presence of a single active elliptical or circular microcrack was given by[163] In a similar way, the tensor of the change of compliance due to the kink-ing of a crack can be calculated from the contribution to the complementarystrain energy corresponding to the kinking of the crack, which is the energyreleased during the kinking growth of the crack

The rate of the change of the compliance tensor for a volume element V of elastic material, attributable to the extension rate ˙s, through which a point

on the perimeter of a single crack kinks to a new position and integrating thisalong the crack perimeter, the rate of the change of compliance due to thegrowth of an internal crack is resulting