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

Figure 42 Continued Copyright 2004 by Marcel Dekker, Inc All Rights Reserved The estimation of the rate of growth of such microcracks can be achieved mainly by two methods: (a) the determination of the statistical distribution 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 with different properties 1 Statistical Model The fundamental concept of the statistical model is that a small fraction of short cracks and a high fraction of long cracks are expected when the growth rate is high, and vice versa [95–98] In order to achieve reliable results 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 X6 CrNi18-11 and more than 60,000 cracks in the steel X8CrNiMoNb16-16 for different temperatures and stresses Based on the results of metallographic investigations, the following assumptions are introduced: (a) A crack grows quickly along the grain boundary from one triple point to the next, where it rests for a longer time before 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 is initiated in the length class n ¼ 1 and grows step by step to next higher length classes Let Z be the total number of cracks per unit area and Yn the number of cracks having a length n In a time unit, Vn,n þ 1 cracks grow out of the length class n into the next higher length class (n þ 1) In the same time, Vn À 1,n cracks grow from the lower length class (n À 1) into the considered class n The mean rate of growth is given by dn=dt ¼ Vn;nþ1 =Yn and the rate of change of dYn =dt ¼ Vn;nþ1 À VnÀ1;n As all cracks initiate with the length n ¼ 1, the rate V0;1 represents the rate of crack initiation and must be equal to rate dZ=dt of the increase of the total crack number Therefore, following relation can be deduced: n dZ X dYi ð86Þ À Vn;nþ1 ¼ dt dt i¼1 Denoting the fraction of cracks with the length of n by Xn, so that dYn ¼ Xn dZ þ ZdXn , the mean rate of growth ðdn=dtÞn of the cracks of the length class n can be written as ! P    ! n dn 1 À n Xi 1 dZ 1 X dXi i¼1 ¼ À ¼ FðnÞGðtÞ À Hðn; tÞ dt n Z dt Xn i¼1 dt Xn ð87Þ Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 43 Statistical distribution of the inter-crystalline creep microcracks The first term denoted F(n) is determined by the statistical distribution of the microcrack length This distribution can be described by Xn ¼ ð1 À qÞqnÀ1 ð88Þ as represented in Fig 43 for two different austenitic steels Deviations are mainly observed in the range of long microcracks and low population These deviations can be avoided by adding a second term including a Leibnitz series, but it will be neglected here The function F ðnÞ is then given by F¼ q 1Àq ð89Þ The parameter q depends on stress, temperature, and the constitution of the material, but not on crack length Therefore, approximately no influence of the crack length on the growth rate arises from this term The function G, which is the relative rate of crack initiation, depends on the material and the creep conditions In order to determine this func- Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 investigations and digital image analyzing systems, the function Z(t) is found to be described adequately by the Kachanov–Rabotnov-relation given in eq (60a), as well as by the empirical relation ðZ=Zf Þ ¼ exp½Àgðtf À tÞ=tf Š ð90Þ where the index f indicates the value at fracture (Fig 44) According to this relation g G¼ ð91Þ tf with g depending on material constitution, stress, and temperature The quantity G remains constant during the creep test, as long as Eq (90) is valid The function H is determined by the rate of change of the statistical dis_ tribution (Fig 45) and can be written as HðtÞ ¼ Ànq=ð1 À qÞ, where q(t) is described by a power law according to q ¼ qf ðt=tf Þm Hence, H can be written as nm q ð92Þ H¼ t 1þq Figure 44 Number of cracks Z per unit area, related to its value at fracture as a function of the life fraction Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 46 Idealization of the grain=grain boundary combination Figure 47 (a) Regular and (b) randomly modified idealization Copyright 2004 by Marcel Dekker, Inc All Rights Reserved grains At higher temperatures, the grain boundary behaves as a viscous layer with much higher strain rate sensitivity than the grains In the FEM analysis, two different material elements are used for the idealization of grains and grain boundaries with different material parameters (98), as shown in Fig 46 The creep behavior of the grain interior and the grain boundary layers is described by the Norton–Bailey creep law _ e ¼ Cðs=sà ÞN ð95Þ with sà just equal to the stress unit The parameters C and N are set approximately equal to the values determined for the entire in the secondary creep stage, neglecting the influence of the grain boundaries in this stage due to their small volume fraction The grain boundary zone can be considered as a linear viscous Newton solid Its stress exponent is set equal to unity as first approximation A suitable thickness and the parameter C of the grain boundary layers are determined iteratively Their values are varied till the fracture time computed for different creep stresses coincides with the experimentally determined values In order to avoid all grain boundaries having the same orientation fracture simultaneously, the size of each individual element in the network is stochastically changed by adding random values to the grain node coordinates (Fig 47) The network determined in this manner has to be considered as a quarter of the idealized body and to be symmetrically mirrored, so that no additional anisotropy is induced The whole network can also be rotated by an angle between 0 and 608 to exclude preferred orientations for crack initiation Two different crack initiation criteria are tried out: a strain criterion and 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 grain boundary element is not totally eliminated but its thickness is reduced by a factor of 1=1000 Such a weakened element behaves during further deformation like a crack The second criterion which is based on the maximum principle stress or the maximum shear stress instead of the equivalent strain is found to be non-applicable because the experimentally determined stress-life function could not be achieved with this criterion With increasing extension of the whole mesh under constant load forces, the crack opening criterion is fulfilled first at a single grain boundary facet 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 48 Comparison between experimental data and computational results for the increase of the number of cracks with increasing creep strain boundary facet, the total creep extension of the mesh is high enough to induce crack growth along the neighboring facets which are steeply inclined to the load direction In this way, cracks of length class n ¼ 2 initiate at different locations With further growth, the individual cracks start to coalescence resulting in a great additional extension of the mesh Fracture is considered to take place as soon as the total creep extension of the mesh reaches a predefined value, and the computation is stopped Figure 48 presents the ratio of the number of cracks Z to that of cracks at fracture Zf as a function of relative strain e=ef determined by the finiteelement simulation and by the creep experiment The comparison shows that most of the data from the finite-element simulation lie in the same scatter band as those of the experimental investigation Figure 49 shows that the fraction X1 of short cracks having a length of one 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 growth seems to be the relatively high local strains, and not the local stress, in the neighborhood of the grain boundaries Metallographic investigation confirms the existence of such deformations in the neighborhood of the Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 50 Creep microcrack initiation and growth in a notched specimen front surface, one can assume that the stress F=A induced is uniformly distributed over the whole rod On the other hand, if the rod is impacted, e.g by a hammer at the front surface, the mass inertia forces cannot be neglected The rod front is pushed forward by a velocity v An arbitrary cross-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 surface and the cross-section considered have been accelerated to the velocity v This Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 51 Material element in an impacted bar needs a certain time interval Dt The longer the distance x, the longer the time interval This explains why displacements, strains, and stresses propagate throughout the material in the form of mechanical waves with the characteristic wave properties, such as reflection at surfaces At an arbitrary time point, the cross-section at the distance x is displaced by u At a neighboring cross-section x þ dx, the displacement is u þ ð@u=@xÞdx The strain in the material element dx is given by e ¼ @u=@x The forces acting on the element are ÀAs and A½s þ ð@s=@xÞ dxŠ The mass inertia force is rA dxð@ 2 u=@x2 Þ Therefore, Að@s=@xÞ dx ¼ rA dxð@ 2 u=@x2 Þ ð96Þ In the case of elastic behavior, @s @s @2u ¼E ¼E 2 @x @x @x ð97Þ and the following differential equation is obtained for the local displacement: @2u E @2u ¼ @t2 r @x2 Any function fðx À ctÞ or fðx þ ctÞ fulfills this condition, when sffiffiffiffi E c¼ r ð98Þ ð99Þ A certain value of the displacement uà ¼ fðx0 À ct0 Þ that is observed at the distance x0 at the time point t0 arises at the distance x0 þ Dx after Copyright 2004 by Marcel Dekker, Inc All Rights Reserved the time interval Dt, yielding fðx À ctÞ ¼ f½x þ Dx À cðt þ Dtފ 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 torsion moment, a transversal wave is induced that propagates with a velocity of pffiffiffiffiffiffiffiffiffi cT ¼ G=r, where G is the shear modulus When plastic deformation takes place, the modulus of elasticity E and the shear modulus G are to be replaced by the tangent modules sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi @s=@e @t=@g c¼ ; cT ¼ ð100Þ r r While these equations are essential for analytical modeling, they have not to be necessarily externally considered in the numerical simulation when adequate computation codes are used These codes must account for the mass of the material, for example, by considering point masses lumped at the nodes of the finite elements Beside the FEM, the finite difference method and 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 relatively low, the local strain at the impacted specimen end is found experimentally 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 for transient dynamic problems The loading time function and the idealization of the impact tensile test arrangement are shown in Fig 52 The material is considered as strain hardening and strain rate sensitive Immediately after loading the specimen, an elastic and a plastic wave propagate along the axial direction of the specimen The elastic wave is much faster than the plastic one An elastic deformation propagates along the specimen to the far specimen head, where the elastic wave reflects It runs back towards the near specimen head, where it reflects again This Figure 52 Input load time function and idealization of impact tension test Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 53 Variation of the distribution of the plastic strain in a tensile specimen at different time point after dynamic loading 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 then reflected from the far specimen head Due to superposition of the advancing and the reflected wave, high stresses and strains are induced at the far end of the specimen If the impact energy is completely consumed by the plastic deformation, a permanent non-uniform strain distribution remains in the specimen (Fig 53) With increasing impact energy, the plastic wave can run several times along the specimen, reflecting at both ends, before the impact energy W is completely consumed by the plastic deformation of the material In this case, the strain distribution is approximately uniform over the whole gauge length (Fig 54) B Fiber Composites Under Dynamic Compression Under quasi-static compressive loading of a composite material, the slim fibers buckle within the softer matrix leading to a global plastic bending of the work piece In order to simulate this behavior, an imperfection is Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 56 Fiber buckling in a composite material under dynamic loading (From Ref 101.) stress increases as well, so that higher tensile forces are needed for the continuation of extension Other specimen regions undergo additional deformation, so that the uniform elongation increases with increasing strain rate sensitivity and extension rate On the other hand, the adiabatic character Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 57 Idealization of perforated plates of the deformation process reduces the flow stress and promotes instability Mass inertia in the lateral direction arises in connection with radial acceleration due to the reduction of area This causes the initiation of either lateral tensile or lateral compressive stresses depending on the time function of specimen elongation In addition to these ductility considerations, an increased notch sensitivity is observed under dynamic loading One of the reasons is that the local fracture strain decreases with increasing strain rate This will be discussed later on in this chapter The other reason lies in the interaction between Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved mechanical waves and notches Figure 57 shows the idealized part of a perforated plate used in a study of the wave notch interaction [102,103] The holes are chosen as circular or elliptical with different axes ratio and orientation Also, the distance between the holes is variable The variation of the stress distribution with increasing time, numerically computed, shows the propagation of the mechanical wave through the material (Fig 58) Stress peaks are observed at the notch roots, before the maximum loading stress reaches this points High stress values remain at the peaks, even when the maximum lading stress has passed through Compared with the notch effect under quasi-static loading, the dynamic notch effect is characterized by higher stress and strain concentrations, greater strain gradients, lower stress relief by neighboring voids and lower influence of the orientation in the case of elliptical voids V DUCTILE FRACTURE The ductile fracture results usually from nucleation, growth, and coalescence of microvoids, that initiate mostly around inclusions In accordance to its appearance of the fracture surface, ductile fracture can be classified into two cases [105] In the case of softer materials, void nucleation at inclusions followed by marked void growth with internal necking and shear fracture of the intervoid matrix The fracture surface shows a structured configuration of dimples often orientated perpendicular to the loading direction (Fig 59) In case of high strength materials, shear fracture takes place without distinctive void growth The matrix fails due to instabilities like shear bands forming between voids resulting in fracture with nearly no necking, promoted by low strain hardening material, high stress multiaxiality, and regions of high porosity [106] A Failure Criteria Beside macromechanical empirical failure criteria [107,108], several mesoscopic mechanical models are introduced The failure criterion is defined by the local failure strain f ðsm =Þ as a function of the ratio of the local e s  mean stress sm to the equivalent stress s For the nucleation of microvoids, different models have been deduced considering an energy criterion [109–111], critical stress [112–114], or critical strain [115–119] Rice and Tracy [120] deduced a closed-form solution for the rateof-change of the mean radius of a void, in an ideal plastic material, as a function of the current value of the radius and of the ratio between the mean stress and the effective stress Copyright 2004 by Marcel Dekker, Inc All Rights Reserved  s ¼ const:; 1 dR ¼ 0:28 expð3sm =2Þ s R de ð101Þ 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) The strain at fracture can be deduced from the Rice and Tracy criterion and be expressed as s ef ¼ en þ a exp½À3sm =ð2ފ ð102Þ where en is the effective strain before void nucleation The Rice and Tracey model 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 higher values according to the volume fraction of inclusions In Ref [121], the local plastic strain which leads to coalescence of cavities was found to be highly influenced by the volume fraction of inclusions fN Using special treatments for ferritic steels, different residual sulfur-concentrations were realized by Holland et al [127] which were found to affect the fracture strain (Fig 60a) These results were described by the modified relation s ef ¼ en þ a exp½Àbsm =Š ð103Þ where instead of the factor 3=2, a parameter b is introduced with values ranging 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) Based on the models of McClintock [128] and of Rice and Tracey, Gurson [112] deduced a yield function for materials with randomly distributed voids of a volume fraction f In this model, the flow rule according to Mises is extended by two additional terms including the porosity f In more detailed investigations carried out by Tvergaard and Needleman [129–131], the Gurson model is modified yielding a plastic potential in the form   h i 3 q2 skk à À 1 þ ðq1 f à Þ2 ¼ 0 f ¼ 2 Sij Sij þ 2q1 f cosh 2sY 2sY ð104Þ In this equation, Sij is the stress deviator given by Sij ¼ sij À dij skk =3 where dij is the second order unit tensor sY is the yield stress of the matrix and skk is the sum of the normal stress components f à is a function of the volume fraction f of the voids according to f fc ; fà ¼f Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð105aÞ f > fc ;  1 f À fc f ¼ fc þ À fc q1 fF À fc à  ð105bÞ where fc is the volume fraction at the beginning of void coalescence and fF is _ the volume fraction at fracture The rate-of-change f of the void volume fraction, is considered as the sum of three different contributions: (1) the growth rate of existing voids, which is proportional to (1Àf ) and to the local strain rate, (2) the nucleation rate of new voids depending on the effec_ tive strain rate  in the matrix, and (3) the nucleation rate of new voids which e is proportional to the rate of change of the mean stress sm ¼ dij skk =3 When the third contribution is neglected, the following relation is used for the evolution of fv : _ ij _ f_ ¼ f_growth þ f_nucleation ¼ ð1 À f Þdij epl þ Apl e ð106aÞ A non-zero value of A is only used if pl exceeds its current maximum in the e time increment considered In this case  ! e fN 1 pl À eN ð106bÞ A ¼ pffiffiffiffiffiffi exp À sN 2 2psN where fN is volume fraction of particles that may nucleate voids, eN is the mean value strain for nucleation, and sN is the corresponding standard deviation B Influence of Strain Hardening and Strain Rate Sensitivity In an early study on the growth of cavities by plastic deformation of the surrounding material, McClintock [128] deduced a closed-form analytical solution for the rate-of-growth of cylindrical cavities of elliptical cross-section with the semi-axes a and b in a strain-hardening material which is "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # pffiffiffi 3ð1 À nÞ sa þ sb 1 dR 3 Àn ¼ sinh s ¼ Ce ; ð107Þ  2 s R de 2ð1 À nÞ Where R¼(aþb)=2 is the mean cross-sectional radius and sa and sb are the normal stresses in the direction of the ellipse axes Because of its simplicity, the Hancock–Mackenzie relation is also applied to the range of high strain rates after introducing correction factors considering the influences of strain rate Carroll and Holt [132] introduced a visco-plastic modification of the Hancock–Mackenzie model Johnson and Cook [133] considered the strain rate sensitivity as well as the influence of the temperature on the local fracture strain ef ¼ ½D1 þ D2 expðD3 sm =ފ½1 þ D4 lnð_ =_ 0 Š½1 þ D5 T=Tm Š s e e Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð108Þ _ with e0 ¼ 1 secÀ1 and Tm the absolute melting point As sm =; e and T s _ change during deformation, it is assumed that fracture takes place when a _ damage parameter D ¼ S½De=ef ðsm ; e; Tފ reaches the value of 1 A failure criterion for void growth considering non-linear visco-plastic behavior of a strain-hardening and rate-sensitive material can be obtained using an analytical solution [134] The void growth is to be determined by _  means of flow stress described by s ¼ Kn ð=_ à Þm , with equivalent stress e e e _ _  e s, equivalent strain rate , reference strain rate eà ¼ 1 secÀ1 , equivalent plastic strain  and the material constants K, n and m A spherical void of radius e R is considered to exist at the center of a metallic sphere (Fig 61) At the outer radius L of this hollow sphere, a radial stress component srL is acting which is set equal to the mean stress sm ¼ ðs1 þ s2 þ s3 Þ=3, which leads to a visco-plastic deformation of the material and hence to an increase in void volume For an arbitrary void radius r, the tangential strain rate is given _ _ by et ¼ r=r Under consideration of the plastic volume constancy, Figure 61 Spherical void growing in a hollow sphere matrix Copyright 2004 by Marcel Dekker, Inc All Rights Reserved  ¼ 2_ =r Regarding the continuity condition r2 r ¼ R2 R, the equivalent _ _ _ e r _ ¼ ðR=rÞ3R ¼ ðL=rÞ3L , and the correspond_ _ e e e strain rate can be rewritten as  _ _   e e e e ing equivalent stress s ¼ sL ðR =L Þn ðR =L Þm ðR=rÞ3ðmþnÞ follows from _ _   s ¼ sL for  ¼ L In order to determine the distribution of the radial stress, e e the condition of equilibrium @sr =@r ¼ À2ðsr À st Þ=r is taken into considera tion According to the von Mises yield criterion, st À sr ¼ s where st is the tangential and sr is the radial stress component With the boundary condition sr ¼ 0 for r ¼ R, a closed-form analytical solution is deduced for the rate of radius increase reading ! 1 dR 1 3ðm þ nÞ ðsr ÞL 1=ðmþnÞ ¼ ð109Þ  R dL 2 2ð1 À fmþn Þ sL e with f ¼ ðR=LÞ3 , which is approximately equal to the volume fraction of  voids At the outer radius of the sphere (r ¼ L), the values of sr L , sL and _ _ L can be regarded as equal to sm , s and , which are determined for the  e e construction element geometry considering the material as continuum if f 51, the failure criterion is given by ! 3ðm þ nÞ sm À1=ðmþnÞ n  à m  ð110Þ s ¼ K ðe=_ Þ ; ef ¼ en þ a e _ e  2 s In the cases of high temperatures or very high strain rate, this relation can be applied using n¼0 and m ¼ 1 as a special case _ ef % en ðe; TÞ þ C a ðsm =Þ s ð111Þ Growth of Microcracks In order to increase the strength of engineering materials, several strengthening mechanism are adopted Beside precipitation hardening, the strength of the matrix is increased by alloying elements During plastic deformation, microcavities initiate in two different ways (a) At low temperatures and high strain rates delamination takes place at the interface between matrix and particles leading to microcrack formation (Fig 62a) (b) At higher temperatures or lower strain rates, particles fracture causing a microcavity that elongates with further plastic deformation (Fig 62b) In order to consider damage by both cavitation mechanisms, a new model is introduced in Refs [104,135] In analogy to the Avrami theory of the kinetics of phase change [136], the following assumptions are made for the initiation and growth of microcavities Precipitations and inhomogeneities embedded in a matrix can be interpreted as active nuclei for void Copyright 2004 by Marcel Dekker, Inc All Rights Reserved and crack initiation The total number of particles representing possible nuclei for damage decreases with increasing global strain due to cavitation initiation at some of them Around each cavitation, a region of reduced stresses and strains exists (hatched areas in Fig 63) in which no further cavitations can initiate This region is spherical with radius r in case of penny-shaped cracks and ellipsoid in case of microvoids which can be approximated by a cylinder with a constant radius a, which is equivalent to the mean particle diameter and a length of l It can be assumed that the number of new cracks initiated per unit strain is proportional to the number of remaining particles lying outside the relieved regions The size distribution of cracks in impacted specimens was determined by Curran et al [137] It was found that the linear crack growth rate dr=de is not a function of the current value of the radius r but only proportional to the relative nucleation rate of new small cavitations In the case of penny-shaped microcracks, the spherical region of relieved stresses and strain grows spherically with a constant radial rate dr=de In the case of microvoids, the cavitation radius remains constant, but its length changes with a constant rate (onedimensional growth) The degree of damage is proportional to the relieved volume fraction, so that the fraction of damaged area DðeÞ reads &   '! e k DðeÞ ¼ C 1 À exp À à ð112Þ e Figure 63 Particles (spots) acting as nuclei for cavity initiation (a) Penny-shaped microcracks with spherical regions of reduced stresses (hatched areas) growing spherical (b) Microvoids with cylindrical regions of reduced stresses growing one dimensional (From Ref 104.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 64 Experimental (marker) and computational (curves) results at different mean strain rates de=dt for tensile specimens of Aluminum Alloy AA7075 T7351 (From Ref 138.) with the material constant C and eà , which is proportional to that strain, at which first damage occurs The exponent k was found to be equal to 4 in the case of microcracks and 2 in the case of microvoids As an application, this damage model was used to describe the flow curves of Aluminum AA7075 [138] measured in impact tensile tests at room temperature (microcracks) and 1508C (microvoids) (Fig 64) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved D Starting Point of Ductile Fracture In order to determine the failure criterion, which is defined by the local failure strain f ðsm =Þ as a function of the ratio between local mean stress sm e s  and local equivalent stress s, tensile tests on differently notched specimens may be carried out The time functions of specimen elongation measured experimentally can be applied as a boundary condition to FE computations in order to determine the local values of stresses and strains along the narrowest cross-section, which is assumed to be critical for fracture initiation (Fig 65) As a result, the time-dependent distributions can be determined, as it is shown for two examples in Fig 66 [135] The analysis shows that in case of unnotched or smoothly notched bars, both the maximum equivalent plastic strain and degree of multiaxiality lie at the specimen axis (radius ¼ 0), whereas in case of a sharply notched specimen, the maximum equivalent plastic strain is reached in the notch root, where the degree of multiaxiality shows a minimum Therefore, it can be stated that, in case of unnotched bars, the starting point of fracture lies at the specimen axis On the other Figure 65 FE Simulation of dynamic tensile test on a notched bar of Aluminum AA7075 (explicit code) (From Ref 139.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved  ... hardening material, high stress multiaxiality, and regions of high porosity [106 ] A Failure Criteria Beside macromechanical empirical failure criteria [107 ,108 ], several mesoscopic mechanical. .. investigation of further materials and treatments (Fig 60b) Based on the models of McClintock [128] and of Rice and Tracey, Gurson [112] deduced a yield function for materials with randomly distributed... Rice and Tracy criterion and be expressed as s ef ẳ en ỵ a expẵ3sm =2ị 102 ị where en is the effective strain before void nucleation The Rice and Tracey model has been used, e.g., in Ref [122] and