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

Figure 21 (a) Distribution of the bending stress sxx in a 100  20  10 mm3 specimen with a notch radius or 6 mm (b) FE computation results for the variation of local strain components at notch root Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 22 Representative strain range as function of the number of cycle to failure (From Ref 54.) 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 Detot ¼ Depl þ Deel with Depl according to the Manson– Coffin equation and Deel following a similar one: Detot ¼ aNÀb þ cNÀd i i ð53Þ The results of the notched bending specimens, obtained by the combination of FE analysis, strength hypothesis, and experimental life determination, 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 23b shows 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 _ e ¼ 0:0017 secÀ1 [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 qualitative 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 in Fig 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 displacements 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 completely 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 characteristic phenomenon is the initiation of an inclined shear band accompanied by a reduction in the specimen deformation resistance, deformation localization in the matrix, and hence a reduction in the fatigue life Figure 25 shows a comparison between the 2D-FEM results for the deformed mesh with the fiber configuration in longitudinal section of Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 coagulation of particles, oxidation, high-temperature corrosion or even phase transformation 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 _ e¼C ! s À si n sgnðs À si Þ sF ð1 À DÞ ð54Þ where in case of high-temperature creep   Q _ C ¼ e0 exp À 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 timedependent creep and the time-independent plastic behavior As discussed above, the low cycle fatigue can be described by s À si ¼ 1 sF ð1 À DÞ ð56Þ With decreasing strain rate, a transition takes place towards a timedependent creep behavior described by Eq (54), which can be rewritten in the form Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 26 Transition between the ranges of time-dependent (creep) and timeindependent (plasticity) ranges (From Ref 40.) s À si ¼ sF ð1 À DÞ  À1=n _ e C A continuous transition function may be given by À1=mà   à =n ! _ s À si e Àm ¼ þ1 _ sF ð1 À DÞ e0 ð57Þ ð58Þ as represented in Fig 26 In contrast to low cycle fatigue behavior, with its relative short life, the parameter sF is considered in the case of long-time creep exposure as a constant 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]: sd þ sS þ sp The first term sd accounts for the variation of the dislocation density An additional material resistance sS that accounts for subgrain formation can be taken into consideration, considering the material as a composite consisting of hard subgrain boundaries and soft subgrain interior [65] The particle stress sp accounts for the interaction between mobile dislocaCopyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 si ¼ sd þ sS with a unique function of strain The particle is then treated separately and eq (54) is rewritten as ! s À si À s p n _ e¼C sgnðs À si Þ ð59Þ sF ð1 À DÞ A Damage Function Different formulations can be applied for the damage function D Kachanov and Rabotnov [67,68] introduced the relation dD a ¼ ; dt ð1 À DÞp   t 1=ðrþ1Þ D¼1À 1À tf ð60aÞ where tf is the fracture time (Fig 28) Other applicable functions are dD ¼ bDm ; dt dD ¼ c þ gD; dt D ¼ ðt=tf ÞM D¼ expðgtÞ À expðgt0 Þ expðgtf Þ À expðgt0 Þ ð60bÞ ð60cÞ 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: D ¼ 1 À ½1 À ðe=ef ފ1=ðmþ1Þ or further function such as   D ¼ ðe ef Þ1 ð1ÀnÞ  D ¼ ½expðbe=ef Þ À 1Š ½expðbÞ À 1Š can be applied Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð61aÞ ð61bÞ ð61cÞ Figure 27 Main factors affecting internal back stress (a) Dislocations (b) Subgrain boundary (c) Particles (From Ref 66.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 29 Ref 69.) Creep strain after partial unloading as a function of time (From Similar to Eq (9), the internal stress component related to dislocation pffiffiffi density is defined as sd ¼ aGb r With the evolution Eq (10) of the dislocation density [5], the variation of the internal back stress is described by the relation dsi C1 ¼ ðsis À si Þ de e1 ð62Þ which is validated experimentally, e.g in Ref [69] In this equation,sis is the quasi-stationary value of the internal back stress and e1 is the corresponding creep strain Under constant stress and temperature, the internal back stress increases in the primary stage according to ! si e ¼ 1 À exp ÀC1 ð63Þ sis e1 with e1 as the strain at the end of the primary creep stage This relationship is shown in Fig 30 for 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 es ¼ AsN [71] On the other hand, _ _ es should follow the relation es ¼ C ðs À sis Þn It can be shown that Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 31 Relation between the quasi-stationary internal back stress and the applied stress 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 33a shows the creep rate curve of the austenitic 18=11 Cr–Nisteel 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Xt ¼L tf ð66Þ the value of L decreases to about 0.6 in case of cyclic stress at constant temperature and about 0.8 in the case of pulsating temperature under constant stress [72] C Influence of Particles 1 Behavior of Dispersion-Strengthened Materials Dispersion-strengthened materials are usually produced by powder metallurgical 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 dispersoids 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 Al2O3 and 4% of Al4C3 as dispersoids with a mean particle size of 150 nm _ The creep rate can be described by e ¼ f ðs À sp Þ where sp is the additional resistance to deformation caused by the particles (Fig 35a) The following simple relation can be used for the estimation of the minimum creep rate hs À s iN p _ expðÀQ=RTÞ ð67Þ es ¼ C E where E is the modulus of elasticity at the creep temperature, N is the Norton–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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved The particle resistance may be estimated by sp % sO where sO is the Orowan stress given by   MG 2b LÀd ln sO ¼ 0:84 ð68Þ 4pð1 À nÞ L À d 2b 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 sà , which p 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 =sà as a function of the normalized creep stress s=sà A continuous p p Figure 36 Increase of additional particle resistance with increasing creep stress (From Ref 79.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved transition from the range of low stresses to the range of higher once may be described by a function of the type " " #m !#1=m sp s ¼ 1 À exp À à ð69Þ sp sà p The upper limit of the particle resistance can be set equal to the Orowan stress: sà % sO p 2 Precipitation Hardening The high-temperature creep behavior of precipitation hardenable industrial alloys is influenced by the kinetics of the precipitation and the ripening processes 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.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved The increase of the size of an existing precipitate can be considered according to the Oswald-ripening mechanism However, the precipitation process under creep loads is rather complex and is expected to be dependent not only on the temperature but also on the change in the degree of supersaturation of the matrix as well as on the defect structure and dislocation density Therefore, the rate of precipitation is assumed to depend on creep conditions The particles strengthen the material by exerting an internal, or threshold, stress sp on the moving dislocations According to the theory of Brown and Ham, reported by Martin [83], it is assumed that the high-temperature yield stress of particle-hardened material is controlled by local climb of dislocations over the particles sp ¼ S L ð70Þ where S is a material constant and L is the planar interparticle spacing From geometrical considerations, it can be shown that the planar interparticle pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL varies with Vp and d according to the relationship spacing L ¼ p=ð6Vp Þd The particle stress can be rewritten as rffiffiffi pffiffiffiffiffiffi Vp 6 sp ¼ ð71Þ S d p An analytical model [84] that considers the influence of a continuous precipitation process accompanied by particle coarsening on the creep behavior of metallic materials will be discussed briefly in the following lines The volume of particles per unit volume V p that precipitate out of the supersaturated matrix can be written as p ð72Þ Vp ¼ Np d3 6 where Np is the number of nuclei or the number of carbide particles per unit volume and d is the average particle diameter The number of growing nuclei per unit volume, Np, that form during precipitation out of a supersaturated solid solution is known to depend on the degree of supersaturation, the temperature, and the defect structure, especially the dislocation density At a given temperature, the value of Np can be written as Np ¼ Np0 þ fðtÞgðtÞ ð73Þ where Np0 is the initial number of nuclei and can be neglected in case of solution annealed materials, f(t) is a function of the temperature and the defect structure, g(t) is a function of the remaining supersaturation, which Copyright 2004 by Marcel Dekker, Inc All Rights Reserved decreases during the precipitation process During the primary creep stage, the dislocation density depends on the creep strain and f(t) can be written as fðtÞ ¼ f½a1 f1 ðTÞ þ a2 ðsn2 eÀQ2 =RT ÞM2 ŠtgM ð74Þ where a1 , a2 , n2 , M2 , and M are constants, s, t, and T are stress, time, and temperature in Kelvin, respectively, Q2 is the creep activation energy and is close to the activation energy of self-diffusion The first part of f(t) including f1(T) represents the nucleation under static conditions when no stress is acting on the material while the second part represents the effect of the applied creep stress As precipitation progresses, the supersaturation of the matrix decreases The function g(t) can be represented as a hyperbolic function of time gðtÞ ¼ ½1 þ ðBtÞn Š Àm ð75Þ where n and m are parameters that control the rate of precipitation, B is a function of stress and temperature The increase in particle diameter d with time due to Oswald ripening can be represented by the function [85]  Ã1=m d ¼ dm þ Ct % ðCtÞ1=m 0 ð76Þ where d0 is the original particle diameter and may be neglected, m is a constant whose value depends on the particle growth mechanisms and C is a function of temperature given by C ¼ C0 expðÀQ1 =RT Þ, where Q1 is the activation energy of the coarsening process, e.g for carbon in iron Based on the experimental results, the first part of the function f(t) in Eq (74) is expected to be small compared with the second part so that it may be neglected for creep loading With this approximation, the volume fraction Vp can be written as Vp ¼ ðp=6Þ½a2 ðsn2 eÀQ2 =RT ÞM2 ŠM C3=m tMþ3=m ½1 þ ðBtÞn Š m ð77Þ With increasing time, Vp approaches asymptotically a final value Vp1 that depends only on the initial supersaturation, but not on a function of time or stress, M þ ð3=mÞ must be equal to mn and the volume fraction Vp can be written as !m ðBtÞn ð78Þ Vp ¼ Vp1 1 þ ðBtÞn Copyright 2004 by Marcel Dekker, Inc All Rights Reserved As M must be positive, mn > 3=m Taking m ¼ 3, mn > 1 Based on the above assumptions, the parameter B can be expressed as a function of stress and temperature as follows: ! Qa na B ¼ B0 s exp À ð79Þ RT where Qa ¼ ð3Q1 =m þ M2 MQ2 Þ=ðmnÞ is an apparent activation energy without a specific physical interpretation and na ¼ n2 M2 M=ðmnÞ Substituting Eqs (76) and [78] in eq (71), sp can be written as rffiffiffiffiffiffiffiffiffiffiffi !m=2 S 6 ðBtÞn Vp1 sp ¼ 1=m tÀ1=m ð80Þ p 1 þ ðBtÞn C The condition for maximum strengthening due to precipitation, _ _ e ¼ emin , is achieved when sp reaches a maximum value,sp max , after time tà , thus   nmm À 2 1=n 1 ð81Þ tà ¼ 2 B Figure 38 Time to minimum creep rate of Alloy 800HT for different creep stresses and temperatures (From Ref 84.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 39 (a) Experimental data (markers) and (b) model results (curves) for the creep rate of Alloy 800HT as a function of time or creep strain (From Ref 66.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 40 Creep rate curves of AA2024 in two heat treatment conditions (a) Highly coarsened precipitates (b) Solution annealed (From Ref 86.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved sp max S ¼ 1=m C rffiffiffiffiffiffiffiffiffiffiffi !Àm=2 6 2 Vp1 1 þ tÃÀ1=m p mmn À 2 ð82Þ Considering the stress dependence of the parameter B, the following relationships are determined: tà ¼ A0 sÀna eQa =RT ð83Þ This relation is found to fit well experimental results for precipitation hardening austenitic steel (Fig 38) and Aluminum AA2024 The variation of sp with creep time is given by the relationship sp sp max !m=2   t À1=m ¼ Àn tà ðnmm=2Þ À 1 þ ðt=tÃ Þ nmm 2 ð84Þ Figure 39 shows a comparison between experimental results and model results for Alloy 800HT Similar results, obtained for the Aluminum Alloy AA2024, are presented in Fig 40 In order to determine the matrix behavior, the material is over-aged for a long span of time until the precipitates coagulate The particle diameter and the distance between them are so increased that the particle stress can be neglected This matrix behavior is described by Eq (54) For the solution annealed condition, the influence of the particle stress is considered using Eqs (59) and [84] For practical applications, some of the parameters can be set equal to certain values Assuming that ripening takes place on the basis of volume diffusion, the parameter m is to be set equal to 3 Furthermore, the variation of the volume fraction of the precipitates with time, according to Eq (78), can be well described with m ¼ 2 as far as the parameter n can be determined by best fit of the experimental results (Fig 41) With these values, the particle stress can be given by t À1=3 sp 3n ¼ ð85Þ Ã ÞÀn tà sp max 3n À 1 þ ðt=t D Simulation of Creep Microcrack Growth Under high-temperature creep exposure, failure takes place due to the initiation and growth of inter-crystalline cavities, in the form of voids or intercrystalline microcracks In the case of high temperatures, low stresses and small grain size of the material, the inter-crystalline void initiation, growth and coalescence is the dominant damage mechanism Several well-founded Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 42 (a) Microcracks in a longitudinal section of a creep specimen of steel X6CrNi18-11 after creep fracture (s ¼80 MPa, T ¼ 7008C) (b) The role of grain boundary sliding in crack initiation Copyright 2004 by Marcel Dekker, Inc All Rights Reserved  ... 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... 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... 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