Modeling and Simulation for Material Selection and Mechanical Design Part 10 potx

The estimation of the rate of growth of such microcracks can beachieved mainly by two methods: a the determination of the statistical dis-tribution of the microcrack length and its varia

Figure 42 Continued

The estimation of the rate of growth of such microcracks can beachieved mainly by two methods: (a) the determination of the statistical dis-tribution of the microcrack length and its variation with the creep time, and(b) finite-element simulation of the creep behavior regarding the material as

a composite consisting of grains separated by thin grain boundary layer withdifferent properties

1 Statistical Model

The fundamental concept of the statistical model is that a small fraction ofshort cracks and a high fraction of long cracks are expected when thegrowth rate is high, and vice versa [95–98] In order to achieve reliableresults using statistics, a great number of cracks have to be classified Over

a period of several years, about 50,000 cracks were classified in the steel X6CrNi18-11 and more than 60,000 cracks in the steel X8CrNiMoNb16-16 fordifferent temperatures and stresses

Based on the results of metallographic investigations, the followingassumptions are introduced: (a) A crack grows quickly along the grainboundary from one triple point to the next, where it rests for a longer timebefore it grows again to the next triple point, (b) The crack length is always

an integral multiple n of grain boundary facets and (c) every crack isinitiated in the length class n ¼ 1 and grows step by step to next higherlength classes

Let Z be the total number of cracks per unit area and Ynthe number ofcracks having a length n In a time unit, Vn,n þ 1 cracks grow out of thelength class n into the next higher length class (n þ 1) In the same time,

Vn  1,ncracks grow from the lower length class (n 1) into the consideredclass n The mean rate of growth is given by dn=dt ¼ Vn;nþ1=Ynand the rate

of change of dYn=dt ¼ Vn;nþ1 Vn1;n As all cracks initiate with the length

n¼ 1, the rate V0;1represents the rate of crack initiation and must be equal

to rate dZ=dt of the increase of the total crack number Therefore, followingrelation can be deduced:

Vn ;nþ1¼dZ

dt Xn i¼1

dYi

Denoting the fraction of cracks with the length of n by Xn, so that

dYn¼ XndZ þ ZdXn, the mean rate of growth ðdn=dtÞn of the cracks ofthe length class n can be written as

dZdt

!

¼ FðnÞGðtÞ  Hðn; tÞ

ð87Þ

The first term denoted F(n) is determined by the statistical distribution ofthe microcrack length This distribution can be described by

as represented in Fig 43 for two different austenitic steels Deviations aremainly observed in the range of long microcracks and low population Thesedeviations can be avoided by adding a second term including a Leibnitzseries, but it will be neglected here

The functionF ðnÞ is then given by

The parameter q depends on stress, temperature, and the constitution

of the material, but not on crack length Therefore, approximately no ence of the crack length on the growth rate arises from this term

influ-The function G, which is the relative rate of crack initiation, depends

on the material and the creep conditions In order to determine this Figure 43 Statistical distribution of the inter-crystalline creep microcracks

func-tion, several creep tests are to be carried out until different stages of damage

in tertiary creep stage are reached Using the results of metallographic tigations and digital image analyzing systems, the function Z(t) is found to

inves-be descriinves-bed adequately by the Kachanov–Rabotnov-relation given in

eq (60a), as well as by the empirical relation

Hence, H can be written as

Figure 46 Idealization of the grain=grain boundary combination.

Figure 47 (a) Regular and (b) randomly modified idealization

grains At higher temperatures, the grain boundary behaves as a viscouslayer with much higher strain rate sensitivity than the grains In the FEManalysis, two different material elements are used for the idealization ofgrains and grain boundaries with different material parameters (98), asshown inFig 46.

The creep behavior of the grain interior and the grain boundary layers

is described by the Norton–Bailey creep law

with sjust equal to the stress unit The parameters C and N are set mately equal to the values determined for the entire in the secondary creepstage, neglecting the influence of the grain boundaries in this stage due totheir small volume fraction

approxi-The grain boundary zone can be considered as a linear viscous ton solid Its stress exponent is set equal to unity as first approximation Asuitable thickness and the parameter C of the grain boundary layers aredetermined iteratively Their values are varied till the fracture time com-puted for different creep stresses coincides with the experimentallydetermined values

New-In order to avoid all grain boundaries having the same orientationfracture simultaneously, the size of each individual element in the network

is stochastically changed by adding random values to the grain nodecoordinates(Fig 47).The network determined in this manner has to be con-sidered as a quarter of the idealized body and to be symmetrically mirrored,

so that no additional anisotropy is induced The whole network can also berotated by an angle between 0 and 608 to exclude preferred orientations forcrack initiation

Two different crack initiation criteria are tried out: a strain criterionand a stress one According to the strain criterion, a crack initiates as soon

as the equivalent creep strain reaches a critical value In this case, the grainboundary element is not totally eliminated but its thickness is reduced by afactor of 1=1000 Such a weakened element behaves during further deforma-tion like a crack The second criterion which is based on the maximum prin-ciple stress or the maximum shear stress instead of the equivalent strain isfound to be non-applicable because the experimentally determined stress-lifefunction could not be achieved with this criterion

With increasing extension of the whole mesh under constant loadforces, the crack opening criterion is fulfilled first at a single grain boundaryfacet The next crack opens at a different grain far from the first crack, but

at a place where the orientation of the grain boundary facet is favorable.After the initiation of several individual cracks having a length of one grain

boundary facet, the total creep extension of the mesh is high enough toinduce crack growth along the neighboring facets which are steeply inclined

to the load direction In this way, cracks of length class n¼ 2 initiate at ferent locations With further growth, the individual cracks start to coales-cence resulting in a great additional extension of the mesh Fracture isconsidered to take place as soon as the total creep extension of the meshreaches a predefined value, and the computation is stopped

dif-Figure 48 presents the ratio of the number of cracks Z to that of cracks

at fracture Zfas a function of relative strain e=ef determined by the element simulation and by the creep experiment The comparison showsthat most of the data from the finite-element simulation lie in the same scat-ter band as those of the experimental investigation

finite-Figure 49shows that the fraction X1of short cracks having a length ofone grain boundary facet slightly increases with increasing nominal stresses

as determined in experiments and by the finite-element simulation

With these results, the main reason for crack initiation and growthseems to be the relatively high local strains, and not the local stress, inthe neighborhood of the grain boundaries Metallographic investigationconfirms the existence of such deformations in the neighborhood of theFigure 48 Comparison between experimental data and computational results forthe increase of the number of cracks with increasing creep strain

front surface, one can assume that the stress F=A induced is uniformly tributed over the whole rod On the other hand, if the rod is impacted, e.g.

dis-by a hammer at the front surface, the mass inertia forces cannot beneglected The rod front is pushed forward by a velocity v An arbitrarycross-section at a distance x from the free end dose not start immediately

to move with the same velocity, before all masses between the front surfaceand the cross-section considered have been accelerated to the velocity v ThisFigure 50 Creep microcrack initiation and growth in a notched specimen

needs a certain time interval Dt The longer the distance x, the longer thetime interval This explains why displacements, strains, and stresses propa-gate throughout the material in the form of mechanical waves with the char-acteristic wave properties, such as reflection at surfaces.

At an arbitrary time point, the cross-section at the distance x is placed by u At a neighboring cross-section x þ dx, the displacement is

dis-u þ ð@dis-u=@xÞdx The strain in the material element dx is given by e ¼

@u=@x The forces acting on the element are As and A½s þ ð@s=@xÞ dx.The mass inertia force is rA dxð@2u=@x2Þ Therefore,

s

ð99Þ

A certain value of the displacementu¼ fðx0 ct0Þ that is observed

at the distance x0 at the time point t0 arises at the distance x0þ Dx afterFigure 51 Material element in an impacted bar

the time interval Dt, yielding fðx  ctÞ ¼ f½x þ Dx  cðt þ DtÞ and hence,

Dx ¼ cDt Therefore, c is the propagation velocity of the longitudinal wave

If the load is applied in the lateral direction or if the load is a torsionmoment, a transversal wave is induced that propagates with a velocity of

c¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi

@s=@er

s

; cT¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi

@t=@gr

s

ð100Þ

While these equations are essential for analytical modeling, they have not to

be necessarily externally considered in the numerical simulation when quate computation codes are used These codes must account for the mass

ade-of the material, for example, by considering point masses lumped at thenodes of the finite elements Beside the FEM, the finite difference methodand the method of characteristics are often applied

A Non-uniformity of Strain Distribution

If a tensile specimen is chosen too long or the impact energy input is tively low, the local strain at the impacted specimen end is found experimen-tally to be much lower than that measured at the far end of the specimen.Such phenomena can be explained by an FE-simulation using a code fortransient dynamic problems The loading time function and the idealization

rela-of the impact tensile test arrangement are shown in Fig 52 The material isconsidered as strain hardening and strain rate sensitive

Immediately after loading the specimen, an elastic and a plastic wavepropagate along the axial direction of the specimen The elastic wave ismuch faster than the plastic one An elastic deformation propagates alongthe specimen to the far specimen head, where the elastic wave reflects Itruns back towards the near specimen head, where it reflects again This

Figure 52 Input load time function and idealization of impact tension test

process is repeated many times during the propagation of the plastic wave,representing an elastic vibration superimposed plastic deformation process.The plastic wave propagates first throughout the specimen and is thenreflected from the far specimen head Due to superposition of the advancingand the reflected wave, high stresses and strains are induced at the far end ofthe specimen If the impact energy is completely consumed by the plasticdeformation, a permanent non-uniform strain distribution remains in thespecimen (Fig 53).

With increasing impact energy, the plastic wave can run several timesalong the specimen, reflecting at both ends, before the impact energy W iscompletely consumed by the plastic deformation of the material In thiscase, the strain distribution is approximately uniform over the whole gaugelength(Fig 54)

Under quasi-static compressive loading of a composite material, the slimfibers buckle within the softer matrix leading to a global plastic bending

of the work piece In order to simulate this behavior, an imperfection isFigure 53 Variation of the distribution of the plastic strain in a tensile specimen atdifferent time point after dynamic loading

Figure 55 Fiber buckling under quasi-static loading of copper reinforced by 45%volume fraction of austenitic steel fibers with 0.2 mm diameter (From Ref 101.)

stress increases as well, so that higher tensile forces are needed for the tinuation of extension Other specimen regions undergo additional deforma-tion, so that the uniform elongation increases with increasing strain ratesensitivity and extension rate On the other hand, the adiabatic characterFigure 56 Fiber buckling in a composite material under dynamic loading (FromRef 101.)

con-of the deformation process reduces the flow stress and promotes instability.Mass inertia in the lateral direction arises in connection with radial accelera-tion due to the reduction of area This causes the initiation of either lateraltensile or lateral compressive stresses depending on the time function of spe-cimen elongation.

In addition to these ductility considerations, an increased notch tivity is observed under dynamic loading One of the reasons is that the localfracture strain decreases with increasing strain rate This will be discussedlater on in this chapter The other reason lies in the interaction betweenFigure 57 Idealization of perforated plates

sensi-Figure 58 Stress distribution around voids at different time points after impact loading: (a) t¼10 ms, smax¼598 MPa, (b) t¼18 ms,

smax¼647 MPa, and (c) t¼24 ms, smax¼661 MPa

mechanical waves and notches.Figure 57shows the idealized part of a forated plate used in a study of the wave notch interaction [102,103] Theholes are chosen as circular or elliptical with different axes ratio and orien-tation Also, the distance between the holes is variable.

per-The variation of the stress distribution with increasing time, cally computed, shows the propagation of the mechanical wave throughthe material (Fig 58).Stress peaks are observed at the notch roots, beforethe maximum loading stress reaches this points High stress values remain

numeri-at the peaks, even when the maximum lading stress has passed through.Compared with the notch effect under quasi-static loading, the dynamicnotch effect is characterized by higher stress and strain concentrations,greater strain gradients, lower stress relief by neighboring voids and lowerinfluence of the orientation in the case of elliptical voids

Rice and Tracy [120] deduced a closed-form solution for the of-change of the mean radius of a void, in an ideal plastic material, as afunction of the current value of the radius and of the ratio between the meanstress and the effective stress

rate-
ss ¼ const:; 1

R

dR

Hancock and Mackenzie [121] showed that the failure strain is assumed to

be inversely proportional to the relative cavity growth rate (d ln R=de) Thestrain at fracture can be deduced from the Rice and Tracy criterion and beexpressed as

where enis the effective strain before void nucleation The Rice and Traceymodel has been used, e.g., in Ref [122] and was verified by Thomason[123–125] in numerical simulations Experimental results of Marini et al.[126] showed that the factor 0.28 of Eq (101) should be replaced by highervalues according to the volume fraction of inclusions In Ref [121], the localplastic strain which leads to coalescence of cavities was found to be highlyinfluenced by the volume fraction of inclusions fN Using special treatmentsfor ferritic steels, different residual sulfur-concentrations were realized byHolland et al [127] which were found to affect the fracture strain(Fig 60a).These results were described by the modified relation

where instead of the factor 3=2, a parameter b is introduced with values ging between 5 and 23 The degree of purity had a drastic influence on en,which was affirmed by the investigation of further materials and treatments(Fig 60b)

ran-Based on the models of McClintock [128] and of Rice and Tracey,Gurson [112] deduced a yield function for materials with randomly distrib-uted voids of a volume fraction f In this model, the flow rule according toMises is extended by two additional terms including the porosity f In moredetailed investigations carried out by Tvergaard and Needleman [129–131],the Gurson model is modified yielding a plastic potential in the form

2s2 Y

In this equation, Sijis the stress deviator given bySij¼ sij dijskk=3 where

dijis the second order unit tensor sYis the yield stress of the matrix and skk

is the sum of the normal stress components.f is a function of the volumefraction f of the voids according to