Modeling and Simulation for Material Selection and Mechanical Design Part 9 pptx

The current value of the strain rate depends on the current values effective stress, which is the difference between the applied stress s and the internal back stress si, the particle de

Figure 21 (a) Distribution of the bending stress sxx in a 100 20  10 mm specimen with a notch radius or 6 mm (b) FE computation results for the variation

of local strain components at notch root

The equivalent strains, determined according to Ref [48] for different notch radii vs the number cycles to fracture are plotted in Fig 22 The experimental results obtained for alternating tension–compression loading on smooth bars are also represented in Fig 22 and are described by the usual function Detot¼ Deplþ Deel with Depl according to the Manson– Coffin equation and Deel following a similar one:

Detot¼ aNb

i þ cNd

The results of the notched bending specimens, obtained by the combi-nation of FE analysis, strength hypothesis, and experimental life determina-tion, lie in a scatter band around the uniaxial data

D LCF of Metal Matrix Composite Materials

The fatigue life of metal matrix fiber composites is found to be strongly reduced in the range of low cycle fatigue due to the formation of kink bands,

at which fatigue cracks initiate

A cross-section of one of these materials is represented in Fig 23a

This composite consists of a pure copper matrix and continuous fibers of

Figure 22 Representative strain range as function of the number of cycle to failure (From Ref 54.)

tion takes place at two points of each fiber, and a kink band is observed, which is inclined to the load direction [59–59] Within the kink band, the matrix suffers high cyclic shear deformation, which may cause crack initiation If the fibers are brittle, fiber fracture occurs at the kink band boundaries

Figure 23bshows the initiation fatigue cracks in a specimen subjected

to alternating low cycle fatigue loading The specimens were tested strain controlled with an alternating total strain value by a strain rate _ee ¼ 0:0017 sec1 [60].

A two-dimensional idealization is chosen in order to get a clear idea about the deformation process even when the results are only of a qualita-tive character The material is supposed to consist of plain layers of cupper and austenitic steel To simulate buckling, an imperfection must be introduced to allow for mechanical instability This is done by bringing in

an inclination with a small angle b (for example 28) which may represent

a deviation between the load and the fiber directions resulting from a non-accuracy of the specimen geometry or a non-alignment of the testing machine axis

The computed distribution of the equivalent stress is shown inFig 24a

for a small elastic tensile strain of 0.001 The volume fraction of the harder material component equals 40% Due to the difference in the modulus of elasticity of the material components, higher stresses arise in the elements

of the stiffer materials and they appear brighter in the plot The gradient

of the stress in the lateral (horizontal) direction is related to bending caused

by stretching of the inclined network Fig 24b shows the stress distribution after reaching the maximum strain of 0.024 in the first cycle The displace-ments are exaggerated in the plot Both the material components are plastically deformed The harder material appears in the mean brighter than the softer one

Fourteen cycles later, the mesh and the stress distribution look com-pletely different, as shown in Fig 24c and d for the time instants of reaching the maximum compressive strain of the 14th cycle and the maximum tensile strain of the next cycle The originally softer materials show higher isotropic hardening due to the greater amount of accumulated strain An inclined kink band can be easily recognized with high stresses in the matrix The successive fiber buckling with increasing number of cycles can be more obviously observed under pulsating compressive stresses A character-istic phenomenon is the initiation of an inclined shear band accompanied by

a reduction in the specimen deformation resistance, deformation localiza-tion in the matrix, and hence a reduclocaliza-tion in the fatigue life

Figure 25shows a comparison between the 2D-FEM results for the deformed mesh with the fiber configuration in longitudinal section of

III CREEP BEHAVIOR

In contrast to plasticity, a long-time high-temperature creep exposure causes

a continuous change in the constitution of the materials Beside hardening

by the increasing dislocation density, several microstructural events take place such as initiation of subgrains, precipitation, ripening, and coagula-tion of particles, oxidacoagula-tion, high-temperature corrosion or even phase trans-formation The creep strain is accompanied by a slowly increasing damage process that covers a great fraction of the creep life

Constitutive equations based on a combination of overstress concept [62,62] and threshold stress concept [5,63] allow an adequate description

of the materials behavior if successive damage is taken into consideration The current value of the strain rate depends on the current values effective stress, which is the difference between the applied stress s and the internal back stress si, the particle deformation resistance sp, the material creep resistance sF and the degree of damage D For long-time creep under low stresses, the creep rate can be represented by

_ee ¼ C s si

sFð1 DÞ

where in case of high-temperature creep

C¼_ee0 exp  Q

RT

ð55Þ

This relation is applicable for true stress and true strain rate If the creep tests are carried out with a constant force, the applied engineering stress value is to be multiplied by the factor ð1 þ eÞ in order to account for the increase of true stress due to reduction of area The engineering strain rate is to be divided by the same factor

In many cases of the modeling of high-temperature material behavior,

it is highly recommended to use the same set of equation to describe time-dependent creep and the time-intime-dependent plastic behavior As discussed above, the low cycle fatigue can be described by

s si

With decreasing strain rate, a transition takes place towards a time-dependent creep behavior described by Eq (54), which can be rewritten in the form

s si

sFð1 DÞ¼

_ee C

 1=n

ð57Þ

A continuous transition function may be given by

s si

sFð1 DÞ¼

_ee _ee0

 m =n

þ1

!1=m

ð58Þ

as represented in Fig 26

In contrast to low cycle fatigue behavior, with its relative short life, the parameter sFis considered in the case of long-time creep exposure as a con-stant reference stress and can be set equal to 1 MPa In this case, only the overstress concept is considered

The kinematic hardening may include several components [64]:

sdþ sSþ sp The first term sdaccounts for the variation of the dislocation density An additional material resistance sSthat accounts for subgrain for-mation can be taken into consideration, considering the material as a com-posite consisting of hard subgrain boundaries and soft subgrain interior [65] The particle stress sp accounts for the interaction between mobile

disloca-Figure 26 Transition between the ranges of dependent (creep) and time-independent (plasticity) ranges (From Ref 40.)

tions and precipitates It depends on the mean distance between the particles which may change in the course of creep exposure (Figs 27)

For engineering construction and life assessment, it is required to reduce the number of parameters to the amount that is essential for the description of the mechanical behavior and that can be determined with a tolerable experimental effort Therefore, most engineering materials may

be described well using si¼ sdþ sS with a unique function of strain The particle is then treated separately and eq (54) is rewritten as

_ee ¼ C s si sp

sFð1 DÞ

Different formulations can be applied for the damage function D Kachanov and Rabotnov [67,68] introduced the relation

dD

ð1  DÞp; D ¼ 1  1  t

tf

 1 =ðrþ1Þ

ð60aÞ

wheretf is the fracture time (Fig 28).Other applicable functions are dD

dD

dt ¼ c þ gD; D ¼ expðgtÞ  expðgt0Þ

As the damage increases very rapidly with time in the late tertiary stage, a more accurate description can be achieved by formulating damage

as a function of strain instead of the time function used above In this case, a modified Kachanov and Rabotnov relation:

or further function such as

D¼ ðe

efÞ1

D¼ ½expðbe=efÞ  1

can be applied

Figure 27 Main factors affecting internal back stress (a) Dislocations (b) Subgrain boundary (c) Particles (From Ref 66.)

Similar to Eq (9), the internal stress component related to dislocation density is defined as sd¼ aGb ffiffiffirp With the evolution Eq (10) of the dislo-cation density [5], the variation of the internal back stress is described by the relation

dsi

de ¼C1

e1

sis si

which is validated experimentally, e.g in Ref [69] In this equation,sisis the quasi-stationary value of the internal back stress and e1is the corresponding creep strain Under constant stress and temperature, the internal back stress increases in the primary stage according to

si

sis

¼ 1  exp C1

e

e1

ð63Þ

with e1as the strain at the end of the primary creep stage This relationship

is shown inFig 30for different materials stresses and temperatures The quasi-stationary value sis depends on the applied stress (Fig 29)

With increasing creep stress, sis increases approaching a saturation value

siss The experimental data for the secondary creep rate are usually well described by the Norton–Bailey relation _ees¼ AsN [71] On the other hand, _ees should follow the relation _ees¼ C s  sð isÞn It can be shown that

Figure 29 Creep strain after partial unloading as a function of time (From Ref 69.)

After a sudden increase of the applied stress, the internal back stress starts to increase gradually approaching the quasi-stationary value (Fig 32a)

If the applied stress is then reduced to the original value, a gradual reduction

of the internal back stress towards the corresponding quasi-stationary value

is observed (Fig 32b)

As the strain rate depends on the difference between the applied load and the internal back stress, a load enlargement leads to a very high strain rate that reduces gradually to normal value In the same way, a load drop causes a severe strain rate reduction, even to negative strain rate values when

s declined to a value lower than si

Figure 33ashows the creep rate curve of the austenitic 18=11 Cr–Ni-steel under cyclic creep loading The stress is changed periodically between

150 and 125 MPa The period is equal to 96 hr The influence of fatigue can

be neglected and the material behavior can be described as a pure cyclic creep

Under cyclic stress, the influence of the creep strain transients is found

to reduce the creep life Compared with life values calculated by the linear damage accumulation rule

Figure 31 Relation between the quasi-stationary internal back stress and the applied stress

X t

tf

the value of L decreases to about 0.6 in case of cyclic stress at constant tem-perature and about 0.8 in the case of pulsating temtem-perature under constant stress [72]

C Influence of Particles

1 Behavior of Dispersion-Strengthened Materials

Dispersion-strengthened materials are usually produced by powder metal-lurgical techniques, especially mechanical alloying They include very fine oxide or carbide particles embedded within the grains The particles obstacle the dislocation motion and increase the resistance to deformation The strength depends mainly on the size and the volume fraction of the disper-soids as well as on the consolidation process and the matrix material [73,74] Contrary to precipitation hardening, the dispersoids are thermally stable and do not ripen or coagulate during long-time high-temperature exposure [75,76] Therefore, such materials are predestined for applications under high-temperature creep conditions Their behavior is studied under tensile and compressive loads, e.g in Refs [77–79].Fig 34a shows the creep rate curves of the Aluminum alloy AlSi20 which was produced from its powder without any additions by cold pressing and hot extrusion This material includes an oxygen content of less than 0.5 mass% The corresponding curves in Fig 34b are determined for a dispersion-strengthened version AlSi20C1O2 produced by mechanical alloying Carbon powder is added

to the matrix powder and the mixture undergoes intensive milling before cold pressing and hot extrusion The material includes a volume fraction

of 4% of Al2O3and 4% of Al4C3as dispersoids with a mean particle size

of 150 nm

The creep rate can be described by_ee ¼ f ðs  spÞ where spis the addi-tional resistance to deformation caused by the particles (Fig 35a)

The following simple relation can be used for the estimation of the minimum creep rate

_ees¼ C s sp

E

where E is the modulus of elasticity at the creep temperature, N is the Nor-ton–Bailey stress exponent of the matrix material, Q is the activation energy for self-diffusion of the base element of the matrix Fig 35b shows that the creep strength is increased by a constant value which depends only on the volume fraction of the particles and their morphology

The particle resistance may be estimated by sp sO where sO is the Orowan stress given by

sO¼ 0:844pð1  nÞMG 2b

L dln

L d 2b

ð68Þ

In this equation, M is the Tailor factor, G is the shear modulus depending on temperature, L is the mean distance between particles, and

b is the Burger vector

A precise description of the experimental data in the range of low creep stresses and high fracture times allows a model introduced by Reppich

et al [80] According to this model, the additional particle resistance sp

depends not only on the particle morphology but also on the applied stress

In the range of high creep stresses, sp approaches an upper limit sp, which can be set equal to the Orowan stress sO With decreasing creep stress, the dislocation can overcome the particle resistance by partial climb, and the particle resistance is assumed to be directly proportional to the applied stress Fig 36 shows the relation between the relative particle resistance

sp=s

p as a function of the normalized creep stress s=s

p A continuous

Figure 36 Increase of additional particle resistance with increasing creep stress (From Ref 79.)

transition from the range of low stresses to the range of higher once may be described by a function of the type

sp

sp¼ 1  exp  s

sp

" #m!

ð69Þ

The upper limit of the particle resistance can be set equal to the Oro-wan stress: sp sO

2 Precipitation Hardening

The high-temperature creep behavior of precipitation hardenable industrial alloys is influenced by the kinetics of the precipitation and the ripening pro-cesses Creep specimens are found to exhibit a longer creep life after solution treatment compared to those ones additionally aged before testing [81] This phenomenon is attributed to the precipitation of fine particles during the early stages of creep [82], which strengthen the material and reduce the creep rate (Fig 37) With increasing the creep time, the particle coarsening leads

to an increase of the interparticle spacing and to an acceleration of creep strain rate

Figure 37 Precipitates and dislocations in Alloy 800HT (From Ref 66.)