FRACTURE LECTURE OF ABAQUS_甜梦文库
FRACTURE LECTURE OF ABAQUS
Basic Concepts of Fracture MechanicsLecture 1L1.2Overview? Introduction? Fracture Mechanisms? Linear Elastic Fracture Mechanics? Small Scale Yielding? Energy Considerations? The J-integral? Nonlinear Fracture Mechanics ? Mixed-Mode Fracture ? Interfacial Fracture ? Creep Fracture? FatigueModeling Fracture and Failure with AbaqusL1.3Overview? This lecture is optional.? It aims to introduce the necessary fracture mechanics concepts and quantities that are relevant to the Abaqus functionality that is presented in the subsequent lectures.? If you are already familiar with these concepts, this lecture may be omitted.Modeling Fracture and Failure with AbaqusIntroductionL1.5Introduction? Fracture mechanics is the field of solid mechanics that deals with the behavior of cracked bodies subjected to stresses and strains. ? These can arise from primary applied loads or secondary selfequilibrating stress fields (e.g., residual stresses).Modeling Fracture and Failure with AbaqusL1.6Introduction? Objective of fracture mechanics? The objective of fracture mechanics is to characterize the local deformation around a crack tip in terms of the asymptotic field around the crack tip scaled by parameters that are a function of the loading and global geometry.Modeling Fracture and Failure with AbaqusFracture MechanismsL1.8Fracture Mechanisms? For engineering materials, such as metals, there are two primary modes of fracture: brittle and ductile. ? Brittle fracture ? Cracks spread very rapidly with little or no plastic deformation.? Cracks that initiate in a brittle material tend to continue to grow and increase in size provided the loading will cause crack growth.? Ductile fracture? Three stages: void nucleation, growth, and coalescence. ? The crack moves slowly and is accompanied by a large amount of plastic deformation. ? The crack typically will not grow unless the applied load is increased.Modeling Fracture and Failure with AbaqusL1.9Fracture Mechanisms? Brittle fracture in polycrystalline materials displays either cleavage (transgranular) or intergranular fracture. ? This depends upon whether the grain boundaries are stronger or weaker than the grains .Cleavage fractureModeling Fracture and Failure with AbaqusL1.10Fracture Mechanisms? Ductile fracture has a dimpled, cup-and-cone fracture appearance .? Ductile fracture surfaces have larger necking regions and an overall rougher appearance than a brittle fracture surface.Modeling Fracture and Failure with AbaqusL1.11Fracture Mechanisms? Fracture process zone? The fracture process zone is the region around the crack tip where dislocation motions, material damage, etc. occur.? It is a region of nonlinear deformation. ? The fracture process zone size is characterized by? a number of grain sizes for brittle fracture or? either inclusion or second phase particle spacings for ductile fracture. ? Different theories have been advanced to describe the fracture process in order to develop predictive capabilities ? LEFM? Cohesive zone models? EPFM? Etc.Modeling Fracture and Failure with AbaqusLinear Elastic Fracture MechanicsL1.13Linear Elastic Fracture Mechanics? Fracture modes? Linear Elastic Fracture Mechanics (LEFM) considers three distinct fracture modes: Modes I, II, and III? These encompass all possible ways a crack tip can deform. ? Mode I: ? The forces are perpendicular to the crack, pulling the crack open. ? This is referred to as the opening mode.Modeling Fracture and Failure with AbaqusL1.14Linear Elastic Fracture Mechanics? Mode II:? The forces are parallel to the crack.? One force pushes the top half of the crack back and the other pulls the bottom half of the crack forward, both along the same line. ? This creates a shear crack: the crack slides along itself.? This is referred to as the in-plane shear mode. ? The forces do not cause out-ofplane deformation.Modeling Fracture and Failure with AbaqusL1.15Linear Elastic Fracture Mechanics? Mode III:? The forces are transverse to the crack.? This causes the material to separate and slide along itself, moving out of its original plane ? This is referred to as the out-of-plane shear mode. ? The objective of LEFM is to predict the critical loads that will cause a crack to grow in a brittle material.Modeling Fracture and Failure with AbaqusL1.16Linear Elastic Fracture Mechanics? Stress intensity factor? For isotropic, linear elastic materials, LEFM characterizes the local crack-tip stress field in the linear elastic (i.e., brittle) material using a single parameter called the stress intensity factor K.? K depends upon the applied stress, the size and placement of thecrack, as well as the geometry of the specimen.? K is defined from the elastic stresses near the tip of a sharp crackunder remote loading (or residual stresses).? K is used to predict the stress state (&stress intensity&) near the tipof a crack.? When this stress state (i.e., K) becomes critical, a small crack grows (&extends&) and the material fails. ? This critical value is denoted KC and is known as the fracture toughness (it is discussed further later).Modeling Fracture and Failure with AbaqusL1.17Linear Elastic Fracture Mechanics? Asymptotic crack tip solutions? The stress and strain fields in the vicinity of the crack tip are expressed in terms of asymptotic series of solutions around the crack tip.? They are valid only is a small region near the crack tip. ? This size of this region is quantified by small scale yielding assumptions (discussed later). ? The stress intensity factor is the parameter that relates the local crack-tip fields with the global aspects of the problem.Modeling Fracture and Failure with AbaqusL1.18Linear Elastic Fracture Mechanics? The leading-order terms of the asymptotic solution are:? ij (r ,? ) ?whereKI K K fijI (? ) ? II fijII (? ) ? III fijIII (? ), 2? r 2? r 2? rx2r?x1r is the distance from the crack tip,? = atan(x2/x1),KI is the Mode I (opening) stress intensity factor,KII is the Mode II (in-plane shear) stress intensity factor, KIII is the Mode III (transverse shear) stress intensity factor, and thefija define the angular variation of the stress for mode a.Modeling Fracture and Failure with AbaqusL1.19Linear Elastic Fracture Mechanics? Crack-tip singularity? The predicted stress state at the crack tip in a linear elastic (brittle) material possesses a square-root singularity:??1 . r? In reality, the crack tip is surrounded by the fracture process zone where plastic deformation and material damage occur. ? Inside this zone, the LEFM solution is not valid.? Outside of this zone (i.e., sufficiently &far& from the fracture process zone), the LEFM is accurate provided the plastic/damage zone is “small enough.”? This is called small-scale yielding (discussed further later).Modeling Fracture and Failure with AbaqusL1.20Linear Elastic Fracture Mechanics? Some comments on fracture toughness? Fracture toughness is strongly dependent on temperature.Fracture toughnessTemperature? The brittle-ductile transition temperature range depends on the material.? For many common metals it may lie within the reasonable operating temperature range for the design, so the temperature dependence of the fracture toughness must be considered.Modeling Fracture and Failure with AbaqusL1.21Linear Elastic Fracture Mechanics? Experimentally, the fracture toughness KC is a function of specimen thickness. ? Since plane strain gives the practical minimum value of KC …? The plane strain value is usually the value that is determined experimentally. ? However, if the application is fracture of thin sheets of material, KC values somewhere between the plane stress and plane strain values may be appropriate.Fracture toughnessKCThickness→Modeling Fracture and Failure with AbaqusL1.22Linear Elastic Fracture Mechanics? Aside from temperature and thickness, the fracture toughness is also a function of the crack extension. ? The fracture toughness as a function of crack extension is called the resistance curve (shown below).ductileVariation in fracture toughness with crack growth is Kr(Da):Kr(0)= KCbrittle? The resistance curve is used to predict crack growth stability.Modeling Fracture and Failure with AbaqusL1.23Linear Elastic Fracture Mechanics? Crack growth and stability? The condition for continued crack growth for a crack length a + Da isKapplied ? K R (Da).? The condition for stable continued crack growth is?K applied ?aload?dK R . d DaModeling Fracture and Failure with AbaqusSmall-Scale YieldingL1.25Small-Scale Yielding? Small-scale yielding (SSY) means the region of inelastic deformation at the crack tip is contained well within the zone dominated by the LEFM asymptotic solution. ? For LEFM to be valid, there must be an annular region around the crack tip in which the asymptotic solution to the linear elasticity problem gives a good approximation to the complete stress field.Plastic zoneK-dominated zoneTransition zoneModeling Fracture and Failure with AbaqusL1.26Small-Scale Yielding? The size of the process zone and the plastic region must be sufficiently small so that this is true. Typical shapes of plastic zones follow:plane strainplane stress (diffuse)plane stress (Dugdale)Modeling Fracture and Failure with AbaqusL1.27Small-Scale Yielding? We can estimate the plastic zone size, rp, by setting ?22 = ?0 in the LEFM asymptotic solution, where ?0 is the yield stress. This gives (for Mode I)rp ?1 2?? KI ? 1 ? KI ? ? ? ? ? ? . 6 ? ?0 ? ? ?0 ?22? Since the tractions across the boundary of the plastic zone have no net force or moments (St. Venant’s principle), the effect on the elastic field surrounding the plastic zone decays rapidly with distance from the boundary, becoming negligible at ~3rp.? LEFM predicts infinite stress at the crack tip ―obviously this is unrealistic. ? But we can use LEFM results if the region of inelastic deformation near the crack tip is small enough that there is a finite zone outside this region where the LEFM asymptotic solution is accurate.Modeling Fracture and Failure with AbaqusL1.28Small-Scale Yielding? If a is a characteristic dimension in the problem, such as remaining ligament size or thickness or crack length, then, to have a finite zone rK in which the K-field dominates, we need1 ? K IC ? a / 5 ? rK ? 3rp ? ? ? 2 ? ?0 ?or2? K IC ? a ? 2.5 ? ? . ? ?0 ?2ASTM Standard for validity of LEFM? This is the limit on specimen size in ASTM Standard E-399 for a valid KIC test.? KIC is KC (the fracture toughness) in Mode I.? The fracture toughness represents the critical value of K required to initiate crack growth.Modeling Fracture and Failure with AbaqusL1.29Small-Scale Yielding? For some typical metal materials rp is calculated by matching the yield stress to the Mises stress of the K field and the minimum characteristic length is calculated using the ASTM standard limit. ? For materials with high fracture toughness the size of the specimen for a valid fracture test is very large. T(? C)MaterialA061-T651 (Al) A075-T651 (Al)?0(MPa)KIC(MN/m3/2)rp(mm)Characteristic dimension (mm)20 20269 62033 365 0.3538 8.4AISI 4340 (Steel)A533-B (Steel)0931500620332000.05111.2260Modeling Fracture and Failure with AbaqusEnergy ConsiderationsL1.31Energy Considerations? Energy principles play an important role in studying crack problems.? This is motivated by the fact that crack propagation always involves dissipation of energy. Sources of energy dissipation include:? Surface energy, plastic dissipation, etc. ? By considering fracture from an energetic point of view, crack growth criteria can be postulated in terms of energy release rates. ? This approach offers an alternative to the K-based fracture criteria discussed earlier and reinforces the connection between global and local fields in fracture problems.? The energy release rate is a global parameter while the stress intensity factor is a local crack-tip parameter.Modeling Fracture and Failure with AbaqusL1.32Energy Considerations? The energy available to grow a crack is defined asG?-? ( PE ) , ?a Loadswhere PE is the potential energy and G is the Energy Release Rate. ? We consider the difference in the energy for two essentially identical specimens, one with crack length a, the other with crack length a + Da. ? The area under the loaddisplacement curve gives -PE for elastic materials.Modeling Fracture and Failure with AbaqusL1.33Energy Considerations? For isotropic linear elastic materials, one can show that1 - v2 2 G? K for plane strain EandK2 G? for plane stress. E? In a three-dimensional body under general loading that contains a crack with a smoothly changing crack-tip line, the energy release rate (assuming linear elasticity) per unit crack front length is? Thus, we see the stress intensity factors are directly related to the energy release rate associated with infinitesimal crack growth in an isotropic linear elastic material.Modeling Fracture and Failure with Abaqus1 - v2 2 1 2 2 G? ( K I ? K II )? K III . E 2GL1.34Energy Considerations? Initiation of crack growth in SSY? The necessary condition for crack growth expressed in terms of the energy release rate is G ? GC.? GC is a material property and represents the energy per unit crackadvance going into:? the formation of new surfaces,? the fracture process, and? plastic deformation.? As noted earlier, for linear elastic materials, G and K are related.? This leads to an alternative condition for K ? KC.? Recall KC is the fracture toughness of the material.Modeling Fracture and Failure with AbaqusThe J-integralL1.36The J-integral? The J-integral is used in rate-independent quasi-static fracture analysis to characterize the energy release associated with crack growth. ? It can be related to the stress intensity factor if the material response is linear. ? As will become apparent in the next section, it also has the advantage that it provides a method for analyzing fracture in nonlinear materials.Modeling Fracture and Failure with AbaqusL1.37The J-integral ? J is defined as follows:? ? ?u J ? ? Wn1 - i ? ij n j ? ds ?x1 ? ???x2x1? It is path independent when contours are taken around a crack tip.? The definition of J assumes: ? The material is homogeneous in the crack direction.? The material is elastic.? For linear elastic materials, the value of J is equal to the energy release rate associated with crack advance:J ?GModeling Fracture and Failure with AbaqusL1.38The J-integral ? J in small-scale yielding? Choose ?, the contour for J, to fall entirely within the annular region in which the K fields dominate.3rp? The integrand for J can be evaluated directly in terms of the (known) K fields. Direct calculation for Mode I in a linear elastic material gives1 - v2 2 J ?G ? K I for plane strain and E 1 J ? G ? K I2 for plane stress. EModeling Fracture and Failure with AbaqusNonlinear Fracture MechanicsL1.40Nonlinear Fracture Mechanics? LEFM applies when the nonlinear deformation of the material is confined to a small region near the crack tip. ? For brittle materials, it accurately establishes the criteria for failure. ? However, severe limitations arise when the region of the material subject to plastic deformation before a crack propagates is not negligible. ? Nonlinear fracture mechanics attempts to extend LEFM to consider inelastic effects.? The theory is sometimes called Elastic-Plastic Fracture Mechanics (EPFM). ? However, the theory is not based on an elastic-plastic material model, but rather a nonlinear elastic material.Modeling Fracture and Failure with AbaqusL1.41Nonlinear Fracture Mechanicsn? Consider a material that has a power-law hardening form,?? ? e ?a? ? , e0 ? ?0 ?where ?0 is the effective yield stress, e0 = ?0 / E is the associated yield strain, E is Young's modulus, and a and n are chosen to fit the stressstrain data for the material.Modeling Fracture and Failure with AbaqusnL1.42Nonlinear Fracture Mechanics? For such a material, Hutchinson, Rice, and Rosengren (extended to mixed mode loading by Shih) showed that the near-tip fields have the formLoading parameter is J? ij ? ? 0 ? e ij ? e 0 ???? a? e I r ? 0 0 n ? JJ1 ? n?1? ij (? ), ?? a? e I r ? 0 0 n ?n ? n?1?ij (? ), en ? n?1? J ?i ? ae 0 r ? ui - u ? a? e I r ? 0 0 n ??i (? ). u?i is the displacement relative to the displacement of the crack ? Here ui - u ?i. These fields are commonly referred to as the HRR crack-tip fields. tip, uModeling Fracture and Failure with AbaqusL1.43Nonlinear Fracture Mechanics? Why not elastic-plastic?? The HRR field assumes a nonlinear elastic power law material:?? ? e ?a ? ? e0 ??0 ? ? Under monotonic loading, this nonlinear elastic material can be matched to the behavior of an elastic-plastic material whose hardening behavior is accurately modeled by a power law.n? Thus, evaluating J allows us to characterize the strength of the singularity in the crack-tip region in an elastic-plastic material subjected to monotonic loading.Modeling Fracture and Failure with AbaqusL1.44Nonlinear Fracture Mechanics? In unloading situations, the HRR fields do not describe the state around the crack tip, and hence J does not characterize the strength of the stress state ahead of a crack tip for plastic materials. Use caution when: ? The loading is not monotonic and an incremental plasticity material is used ? Crack growth occurs under monotonic loading (individual material particles may unload even when the overall structure is being loaded).? The HRR solution:? Gives the leading term in an asymptotic expansion of the deformation around the crack tip for
and ? Does not take into account finite-strain effects.Modeling Fracture and Failure with AbaqusL1.45Nonlinear Fracture Mechanics? Some comments on the HRR fields? The HRR fields, thus, describe the near-tip crack fields in terms of J.? J gives the strength of the near-tip singularity in any power-law material(nonlinear elastic or plastic) solid? Recall that in LEFM K plays this role in linear elastic materials.? J-based fracture mechanics is applied in much the same way as LEFM.? Crack growth initiates when J reaches a critical value: J ? JC . ? To apply the theory, must ensure conditions for J-dominance are satisfied (discussed next).Modeling Fracture and Failure with AbaqusL1.46Nonlinear Fracture Mechanics ? J-dominance? J-dominance refers to situations when J can be used as a method ofpredicting fracture.? In general, J is an adequate characterization when there exists a state of high triaxial tension (high triaxiality) ahead of the crack tip. ? High triaxiality ahead of the crack tip leads to low fracture toughness. ? Examples: states of small-scale and well-contained yielding (where the plastic zone is surrounded by an elastic zone):? Deeply notched bend specimen c ?dd cModeling Fracture and Failure with AbaqusL1.47Nonlinear Fracture Mechanics? In some situations the crack-tip stress field does not exhibit high triaxiality. ? Example: large-scale yielding (the plastic zone extends to the free boundaries of the body): ? Fully plastic flow of single-edge cracked specimens under tension loading? Shallow cracks under bending? Center-cracked panel? A two-parameter approach can be used to extend the fracture characterization to such cases (discussed next).Modeling Fracture and Failure with AbaqusL1.48Nonlinear Fracture Mechanics? Two-parameter fracture mechanics ? The Williams’ expansion of the Mode I stress field about a sharp crack in a linear elastic body with respect to r, the distance from the crack tip, is? ij (r ,? ) ?KI fij (? ) ? T ?1i?1 j ? O(r1/2 ). 2? r? The T-stress thus represents a stress parallel to the crack faces. ? The magnitude of the T-stress affects the size and shape of the plastic zone and the region of tensile triaxiality ahead of the crack tip. ? For positive T-stress, J-dominance exists and a single parameter J can be used for a fracture criterion. ? For negative T-stress, a two-parameter approach (J, T) is required to characterize the stress fields.Modeling Fracture and Failure with AbaqusMixed-Mode FractureL1.50Mixed-Mode Fracture? Under general loading almost all theories for the direction of crack growth assume or predict that the continued crack growth will be with KII = 0. ? Can assume that macroscopic cracks growing with continuously turning tangents will advance straight ahead, presumably under Mode I conditions.? The crack curvature will evolve in such a way as to maintain this in response to the loading. ? If the loading changes such that the local crack-tip stress field experiences a large change in local stress intensities, mixed-mode fracture will occur.Modeling Fracture and Failure with AbaqusL1.51Mixed-Mode Fracture? Different criteria for homogeneous, isotropic linear elastic materials have been proposed, including:? The maximum tangential stress criterion. ? The maximum energy release rate criterion. ? The KII = 0 criterion.? Although all three imply that KII = 0 as the crack extends, they predict slightly different angles for crack initiation.Comparison of predictions of crack propagation direction for different ratios of KII / KIModeling Fracture and Failure with AbaqusInterfacial FractureL1.53Interfacial Fracture? Many engineering applications involve bonded materials.? Examples:???? etc.? Engineers must be able to predict the strength of the bond. ? Interfacial fracture mechanics provides a method by which to do this. ? It extends LEFM to predict the behavior of cracks between two linear elastic materials.Modeling Fracture and Failure with AbaqusL1.54Interfacial Fracture? Once a crack has started to grow in an isotropic, homogeneous material, it generally does s that is, in Mode I. ? A crack lying on an interface can kink off the interface and grow under Mode I conditions, or it can grow along the interface under mixed mode conditions. ? Whether the crack kinks off the interface or propagates along it is frequently determined through energy considerations.Modeling Fracture and Failure with AbaqusL1.55Interfacial Fracture? If the crack kinks off the interface, the fact that there is an interface is important only in how it influences the stress and strain fields. ? If the crack grows along the interface, it grows under mixed mode conditions due to material asymmetry and possibly (though not necessarily) under mixed remote loading conditions. ? In such situations the conditions for crack growth depend on the interface properties. It is not sufficient to define crack initiation and growth criterion based on the conventional fracture toughness, KC.? Specifically KC = KC (?).? Toughness depends strongly on the mode mixity ?.Modeling Fracture and Failure with AbaqusL1.56Interfacial Fracture? Asymptotic fields? The asymptotic stress field for an interfacial crack between linear elastic materials is given by? K * ie ? ? ij (? , e ) ? ? ij ? Re ? r ? ? 2? r ? where K* = K1 ? iK2 is the complex stress intensity factor (i.e., it has real ? ij ?? , e ? is a complex function of the angle and imaginary parts) and ?and material mismatch parameter e :e?? (? - 1) - ?2 (?1 - 1) 1 1- ? log , where ? ? 1 2 , and 2? 1? ? ?1 (? 2 ? 1) ? ?2 (?1 ? 1)for plane stress for plane strain, axi, 3D? 3 -? ? ? ? ? 1 ?? ? ?3 - 4?Modeling Fracture and Failure with AbaqusL1.57Interfacial Fracture? The complex exponent rie indicates that the stresses will oscillate near the crack tip:? Both the stresses and crack opening displacements will oscillate wildly as the crack tip is approached. ? At some distance ahead of the crack tip, the fields settle down.? The fracture criterion should be measured at this point. Provided the location of this point is the same in different specimens, a fracture criterion is valid.Modeling Fracture and Failure with AbaqusCreep FractureL1.59Creep Fracture? High-temperature fracture? For temperatures above 0.3?M (where ?M is the melting temperature on an absolute scale), metals will typically creep. ? In plastics creep can occur even at room temperature.? There are typically two mechanisms that are active in creep fracture:? Blunting of the crack tip due to a relaxing stress field.? This tends to retard crack growth.? Accumulation of creep damage (microcracks, void growth, and coalescence). ? This enhances crack growth.? Steady-state creep crack growth occurs when the two effects balance one another.Modeling Fracture and Failure with AbaqusL1.60Creep Fracture? The stress state around a crack tip in a material that can creep is more complicated than for the corresponding plasticity problem. ? Because of the time-dependent effects there is no one parameter that can characterize the stress state around the crack tip for all possibilities.? This makes measuring the relevant parameters more difficult.? Hence, creep fracture is not as well established as elastic-plastic fracture.Initially, the crack-tip field is the elastic field. ? cr ) ? O(e ? el ) around the Stationary crack: O(e ? el ) ? O(e ? cr ) crack tip (RR field); around this field O(e (K field). ? el ) ? O(e ? cr ) Growing crack: region develops where O(e (HR field), which is in turn surrounded by the RR field. Eventually the HR field envelops the RR field (which ultimately disappears).Modeling Fracture and Failure with AbaqusL1.61Creep Fracture? Contour integrals? The contour integral for creep fracture is called the C(t)-integral.? It plays an analogous role to the J-integral in the context of timedependent creep fracture. ? Its development assumes a power law creep material:? ?e ? el ? e ? cr e? The C(t)-integral is proportional to the rate of growth of the crack-tip creep zone for a stationary crack under small-scale creep conditions:?? ? ? ? ? e0 ? ? E ? ?0 ? ? ?n? Under steady-state creep conditions, when creep dominates throughout the specimen, C(t) becomes path independent and is known as C*.?j ? ?u ? n ?ij n1 - ni ? ij C (t ) ? ? ije ds. ? ? ? r ?0 ? n ? 1 ?x1 ??Modeling Fracture and Failure with AbaqusL1.62Creep Fracture? Asymptotic fields for stationary crack? The near tip stress and strain fields were obtained by Riedel and Rice in terms of C(t). They are known as the RR fields and are analogous to the HRR fields in power law hardening plasticity.C(t) acts like a time-dependentloading parameter? ij ? ? 0 ?cr ?ij e? C (t ) ? ? e ? I r ? 0 0 n ?1 ? n?1? ij (? , n) ? ?ij (? , n) e? C (t ) ?0 ? ?e ? ? e ? I r ? 0 0 n ?n ? n?1Crack tip fields are similar to those for an elastic-plastic material?ij (? , n) is approximately Here In is a function of n and the magnitude of ? 1.Modeling Fracture and Failure with AbaqusL1.63Creep Fracture? Small-scale vs. extensive creep? For the case of no crack growth the loading parameters that characterize the crack-tip fields are reasonably well understood.? Under small-scale creep conditions with secondary creep, K is the loading parameter characterizing the crack-tip field. ? For extensive secondary creep C* is a loading parameter characterizing the crack-tip field upon which a fracture criterion may be based. ? Suitable criteria for crack extension that will predict an initiation time for crack growth for general cases are not yet available.Modeling Fracture and Failure with Abaqus??K ? (? ) ? rSmall-scale creepcreep zoneExtensive creepFatigueL1.65Fatigue? Fatigue is a special kind of failure in which cracks gradually grow under a prolonged period of subcritical loading. ? It is the single most common cause of failure in metallic structures.Damage at the ball grid array (BGA) in a solder joint after 2700 thermal loading cycles? The Paris Law can be used to predict crack growth as a function of cycles (or time):da ? C (DK ) n , where dN DK ? K max - K minModeling Fracture and Failure with AbaqusL1.66Fatigue? Abaqus offers a direct cyclic low-cycle fatigue capability based on the Paris Law. ? Models progressive damage and failure both in bulk materials and at material interfaces for a structure subjected to a sub-critical cyclic loading. ? For more advanced fatigue analysis capabilities, consult . ? fe-safe is a suite of fatigue analysis software that has a direct interface to Abaqus.Modeling Fracture and Failure with AbaqusModeling CracksLecture 2L2.2Overview? Crack Modeling Overview ? Modeling Sharp Cracks in Two Dimensions ? Modeling Sharp Cracks in Three Dimensions ? Finite-Strain Analysis of Crack Tips ? Limitations Of 3D Swept Meshing For Fracture ? Modeling Cracks with Keyword OptionsModeling Fracture and Failure with AbaqusCrack Modeling OverviewL2.4Crack Modeling Overview? A crack can be modeled as either? Sharp? Small-strain analysis? Singular behavior at the crack tip? Requires special attention? In Abaqus, a sharp crack is modeled using seam geometry ? Blunted? Finite-strain analysis? Non-singular behavior at crack tip? In Abaqus, a blunted crack is modeled using open geometry ? For example, a notchModeling Fracture and Failure with AbaqusL2.5Crack Modeling Overview? Mesh refinement? Crack tips cause stress concentrations.? Stress and strain gradients are large as a crack tip is approached.? The finite element mesh must be refined in the vicinity of the crack tip to get accurate stresses and strains. ? The J-integral
for LEFM, accurate J values can generally be obtained with surprisingly coarse meshes, even though the local stress and strain fields are not very accurate.? For plasticity or rubber elasticity, the crack-tip region has to be modeled carefully to give accurate results.Modeling Fracture and Failure with AbaqusL2.6Crack Modeling Overview? The crack-tip singularity in small-strain analysis? For mesh convergence in a small-strain analysis, the singularity at the crack tip must be considered.? J values are more accurate if some singularity is included in themesh at the crack tip than if no singularity is included. ? The stress and strain fields local to the crack tip will be modeled more accurately if singularities are considered. ? In small-strain analysis, the strain singularity is:? Linear elasticity ? ? r -? ? Perfect plasticity ? ? r -1 ? Power-law hardening ? ? r -n/(n+1)Modeling Fracture and Failure with AbaqusModeling Sharp Cracks in Two DimensionsL2.8Modeling Sharp Cracks in Two Dimensions? In two dimensions…? The crack is modeled as an internal edge partition embedded (partially or wholly) inside a face.? This is called a seam crack ? The edge along the seam will have duplicate nodes such that the elements on the opposite sides of the edge will not share nodes. ? Typically, the entire 2D part is filled with a quad or quad-dominated mesh. ? At the crack tip, a ring of triangles are inserted along with concentric layers of structured quads. ? All triangles in the contour domains must be represented as degenerated quads.Modeling Fracture and Failure with AbaqusL2.9Modeling Sharp Cracks in Two Dimensions? Example: Slanted crack in a plate? In Abaqus/CAE a seam is defined by through the Crack option underneath the Special menu of the Interaction module.? The seam will generate duplicate nodes along the edge.SeamCreate face partition t assign a seam to the partition.Modeling Fracture and Failure with AbaqusL2.10Modeling Sharp Cracks in Two Dimensions? To define the crack, you must specify ? Crack front and the crack-tip ? Normal to the crack plane or the direction of crack advance ? The crack advance direction is called the q vector.Select the vertex at either end as the crack front. (Repeat for the other end.)Crack tip same as crack front in this caseThe crack extension direction (q vector) defines the direction in which the crack would extend if it were growing. It is used for contour integral calculations.Modeling Fracture and Failure with AbaqusL2.11Modeling Sharp Cracks in Two Dimensions? Other options for defining the crack front and crack tipCrack front for a geometric instanceCrack tip for an orphan meshCrack front may be:Vertex/Node Edges/Element edges Faces/Elements Geometric Instances Orphan Mesh Crack tip may be:Vertex/NodeGeometric InstancesOrphan MeshModeling Fracture and Failure with AbaqusL2.12Modeling Sharp Cracks in Two Dimensions? Example: crack on a symmetry plane? If the crack is on a symmetry plane, you do not need to define a seam.? This feature can be used only for Mode I fracture.Crack normalCrack tipModeling Fracture and Failure with AbaqusL2.13Modeling Sharp Cracks in Two Dimensions? Modeling the crack-tip singularity with second-order quad elements? To capture the singularity in an 8-node isoparametric element:? Collapse one side (e.g., the side made up by nodes a, b, and c) so that all three nodes have the same geometric location at the crack tip. ? Move the midside nodes on the sides connected to the crack tip to the ? point nearest the crack tip.Modeling Fracture and Failure with AbaqusL2.14Modeling Sharp Cracks in Two Dimensions? If nodes a, b, and c are free to move independently, thenA B ?? ? as r ? 0 r reverywhere in the collapsed element.? If nodes a, b, and c are constrained to move together, A = 0: ? The strains and stresses are square-root singular (suitable for linear elasticity). ? If nodes a, b, and c are free to move independently and the midside nodes remain at the midsides, B = 0 : ? The singularity in strain is correct for the perfectly plastic case.? For materials in between linear elastic and perfectly plastic (most metals), it is better to have a stronger singularity than necessary.? The numerics will force the coefficient of this singularity to be small.Modeling Fracture and Failure with AbaqusL2.15Modeling Sharp Cracks in Two Dimensions? Usage:The crack tip nodes are independent: r -1 singularityQuarter-point midside nodes on the sides connected to the crack tip The crack tip nodes are constrained: r -? singularity341,2,3,41, 22 311,1,2,3Modeling Fracture and Failure with AbaqusL2.16Modeling Sharp Cracks in Two Dimensions? Aside: Controlling the position of midside nodes for orphan meshes ? Singularity controls cannot be applied to orphan meshes. ? Use the Mesh Edit tools to adjust their position.Modeling Fracture and Failure with AbaqusL2.17Modeling Sharp Cracks in Two Dimensions? If the side of the element is not collapsed but the midside nodes on the sides of the element connected to the crack tip are moved to the ? point: ? The strain is square root singular along the element edges but not in the interior of the element. ? This is better than no singularity but not as good as the collapsed element.nodes moved to ? pointsModeling Fracture and Failure with AbaqusL2.18Modeling Sharp Cracks in Two Dimensions? Angular resolution? We need enough elements to resolve the angular dependence of the strain field around the crack tip.? Reasonable results are obtained for LEFM if typical elements around the crack tip subtend angles in the range of 10? (accurate) to 22.5? (moderately accurate).? Nonlinear material response usually requires finer meshes.Modeling Fracture and Failure with AbaqusL2.19Modeling Sharp Cracks in Two Dimensions? Modeling the crack-tip singularity with first-order quad elements? Collapsing the side of a first-order quadrilateral element with independent nodes on the collapsed side givesA ? ? as r ? 0. rModeling Fracture and Failure with AbaqusL2.20Modeling Sharp Cracks in Two Dimensions? Example: Slanted crack in a plate? To enable the creation of degenerate quads, you must create swept meshable regions around the crack tips (using partitions) and specify a quad-dominated mesh.24 elements around crack tip: 15? anglesQuarterpoint nodesCPE8R typical nodal connectivity shows repeated node at crack tip: Quad-dominated mesh + swept technique for the circular regions surrounding the crack tipsQuadratic element type assigned to part8, 8, 583, 588, 8, , 1970All crack-tip elements repeat node 8 in this example (nodes are constrained).Modeling Fracture and Failure with AbaqusL2.21Modeling Sharp Cracks in Two Dimensions? Example (cont’d): Alternate meshes ? No degeneracy:? Degenerate with duplicate nodes:With swept meshable region: CPE6M elements at crack tip ― cannot be used for fracture studies in Abaqus.With arbitrary mesh, singularity only along edges connected to crack tip.CPE8R elements at crack tip but no repeated nodes:, 583, 588, 2016, ...Coincident nodes located at crack tipModeling Fracture and Failure with AbaqusL2.22Modeling Sharp Cracks in Two Dimensions? Example (cont’d): Deformed shapeF deformation scale factor = 100A deformation scale factor = 100Modeling Fracture and Failure with AbaqusModeling Sharp Cracks in Three DimensionsL2.24Modeling Sharp Cracks in Three Dimensions? In three dimensions…? The seam crack is modeled as a face partition that is either partially or totally embedded into a solid body.? This can be done by partitioning or using a cut (Boolean) operation.? The face along the seam will have duplicate nodes such that the elements on the opposite sides of the face will not share nodes.? Wedge elements must be created along the crack front. ? Generally, this will require partitioning.Penny-shaped seam crack: Full modelQuarter modelWedge elementsMeshed modelModeling Fracture and Failure with AbaqusL2.25Modeling Sharp Cracks in Three Dimensions? Options for defining the crack front and crack lineCrack front for a geometric instanceCrack line for an orphan meshCrack front may be:Edges/Element edges Faces/Element faces Cells/Elements Geometric Instances Orphan Mesh Geometric Instances Orphan MeshCrack line may be:Edges/Element edgesModeling Fracture and Failure with AbaqusL2.26Modeling Sharp Cracks in Three Dimensions? Specifying the crack growth direction in three dimensions? In 3D you can specify either the? normal to the crack plane (only when the crack is planar)or the? virtual crack extension direction (the q vector).? Only a single q vector can be defined for geometric instances.? The implications of this will be discussed shortly.Modeling Fracture and Failure with AbaqusL2.27Modeling Sharp Cracks in Three Dimensions? Modeling the crack-tip singularity in three dimensions? 20-node and 27-node bricks can be used with a collapsed face to create singular fields.C3D20(RH)midplane edge plane2 nodes collapsed to the same locationcrack line midside nodes moved to ? points3 nodes collapsed to the same locationModeling Fracture and Failure with AbaqusL2.28Modeling Sharp Cracks in Three Dimensions? On an edge plane (orthogonal to the crack line):Double-edge notch specimen (symmetry model)??A as r ? 0 r??A B ? as r ? 0 r r??B as r ? 0 rCrack lineEdge plane nodes displace independentlyEdge plane nodes displace togetherModeling Fracture and Failure with AbaqusL2.29Modeling Sharp Cracks in Three Dimensions? On a midplane for 20-node bricks:? If the two nodes on the collapsed face at the midplane can displace independently, ? ? r -1 at the midplane (i.e., element interior).? If on each plane there is only one node along the crack line, no singularity is represented within the element. ? In either case the interpolation is not the same on the midplane as on an edge plane. ? This generally causes local oscillations in the J-integral values along the crack line.Modeling Fracture and Failure with AbaqusL2.30Modeling Sharp Cracks in Three Dimensions? On a midplane for 27-node bricks with all the extra nodes on the element faces:midplaneC3D27(RH)edge plane3 nodes collapsed to same locationcentroidcrack line3 nodes collapsed to same locationModeling Fracture and Failure with AbaqusL2.31Modeling Sharp Cracks in Three Dimensions? If all midface nodes and the centroid node are included and moved with the midside nodes to the ? points, the singularity can be made the same on the edge planes and midplane. ? Abaqus does not allow the centroid node to be moved from the geometric centroid of the element. ? Therefore, the behavior at the midplane will never be the same as at the edge planes. ? This usually causes some small oscillation of the crack fields along the crack line. ? The midface node marked “A” is frequently omitted.? This creates differences in interpolation between the midplane and the edge planes and, hence, causes further oscillation in the cracktip fields.? These oscillations are minor in most cases.Modeling Fracture and Failure with AbaqusL2.32Modeling Sharp Cracks in Three Dimensions? Example: Conical crack in a halfspace ? A conical crack in an infinite halfspace is considered. ? Only the aspects related to the geometric modeling are considered here. ? The results of this analysis (J-integral values, etc) will be considered in the next lecture. ? The modeling procedure is outlined next.Modeling Fracture and Failure with AbaqusL2.33Modeling Sharp Cracks in Three Dimensions1 Example (cont’d): Create the basic geometry? Because of symmetry, only a quarter model is createda = 15r = 10q= 45?Large solid block (300 × 300 × 300) used to represent the half-space.Conical shell of revolution (revolved 90? ); this will be used to cut the block.Modeling Fracture and Failure with AbaqusL2.34Modeling Sharp Cracks in Three Dimensions2 Example (cont’d): Merge the block and cone? This will create the edges and surface necessary to define the seam and the crack.Instance and merge the two parts to create a new part. The instance must be independent.Modeling Fracture and Failure with AbaqusL2.35Modeling Sharp Cracks in Three Dimensions3 Example (cont’d): Define the seam and the crack front/lineOnly one q vector can be defined for geometry. The q vectors will be adjusted at the end of the modeling process by editing an orphan mesh.Modeling Fracture and Failure with AbaqusL2.36Modeling Sharp Cracks in Three Dimensions4 Example (cont’d): Partition the block for meshingThe regions surrounding the crack front are partitioned to permit structured meshing.A small curved tube is cente this region is meshed with a single layer of wedge elements. This mesh is swept along the length of the tube.Modeling Fracture and Failure with AbaqusL2.37Modeling Sharp Cracks in Three Dimensions? Aside: Why is the small curved tube needed?The swept meshing technique sweeps a mesh through a cross section. For the curved tube, this implies the sweep direction is along its length. In order for Abaqus to automatically create a focused mesh at the crack tip, however, it would need to sweep around the circumference. To overcome this, two conce the smaller one is meshed with a single layer of wedge elements (which is then swept along the length of the tube). If only a single curved tube was created (shown at right), the mesh around the crack tip would be arbitrary―not focused (wedge elements not created).Modeling Fracture and Failure with AbaqusL2.38Modeling Sharp Cracks in Three Dimensions? Aside: What about the seam?? After all the partitions are created for meshing purposes, the definition of the seam remains intact.Mesh seamModeling Fracture and Failure with AbaqusL2.39Modeling Sharp Cracks in Three Dimensions5 Example (cont’d): Mesh the part? Specify appropriate edge seeds to create a focused mesh around the crack front with minimal mesh distortion.Modeling Fracture and Failure with AbaqusL2.40Modeling Sharp Cracks in Three Dimensions6 Example (cont’d): Adjust theq vectors? As noted earlier, only a single q vector can be defined for geometry. As seen in the figure, the vector that was defined is only accurate at the left end of the crack line. ? Individual q vectors can be defined on an orphan mesh, however. Thus, either…? Create a mesh part (Mesh module)To take advantage of the input file approach, define a set that contains the conical region before writing the input file. Then you will be able to easily create a display group based on this set when manipulating the orphan mesh.or? Write an input file and import the model ? This approach has the advantage that it preserves attributes (sets, loads, etc).Modeling Fracture and Failure with AbaqusL2.41Modeling Sharp Cracks in Three Dimensions? For the orphan mesh, adjust each vector individuallyTo redefine this particular vector, select these nodes as the start and end points of the vector.Modeling Fracture and Failure with AbaqusL2.42Modeling Sharp Cracks in Three Dimensions? For all elements, the singularities are modeled best if the element edges are straight. ? In three dimensions the planes of the element perpendicular to the crack line should be flat. ? If they are not, when the midside nodes are moved to the ? points, the Jacobian of the element at some integration points may be negative. ? One way to correct this is to move the midside nodes slightly away from the ? points toward the midpoint.Modeling Fracture and Failure with AbaqusL2.43Modeling Sharp Cracks in Three Dimensions? Example: Conical crack modelModeling Fracture and Failure with AbaqusFinite-Strain Analysis of Crack TipsL2.45Finite-Strain Analysis of Crack Tips? Finite-strain analyses:? Singular elements should not be used (normally).? The mesh must be sufficiently refined to model the very high strain gradients around the crack tip if details in this region are required. ? Even if only the J-integral is required, the deformation around the crack tip may dominate the solution and the crack-tip region will have to be modeled with sufficient detail to avoid numerical problems.Modeling Fracture and Failure with AbaqusL2.46Finite-Strain Analysis of Crack Tips? Physically, the crack tip is not perfectly sharp, and such modeling makes it difficult to obtain results. ? Instead, we model the tip as a blunted notch, with a suggested radius ? 10-3rp. ? Here, rp is the size of the plastic zone (discussed in Lecture 1).? The notch must be small enough that under the applied loads, the deformed shape of the notch no longer depends on the original geometry.? Typically, the notch must blunt out to more than four times its original radius for this to be true.Modeling Fracture and Failure with AbaqusL2.47Finite-Strain Analysis of Crack Tips? Geometric modeling of blunt cracks? In 2D, the geometry of a blunted (or open) crack is modeled as a cut having a significant thickness.? Meshing is done in the usual way. ? A very fine mesh is required at the crack tip. ? This can be achieved by simply assigning small element sizes to the notch.Modeling Fracture and Failure with AbaqusL2.48Finite-Strain Analysis of Crack Tips? 3D open cracks can be created in Abaqus/CAE in one of two ways: ? Adding a Cut feature in the Part module. ? Subtracting a flaw from the original part with a Boolean operation in the Assembly module. ? Hex meshing more difficult due to irregular geometry. ? Creating a fine mesh at the crack front generally requires many partitions.Quarter model Penny shaped open crack: Full modelMeshed modelPartitions to control meshModeling Fracture and Failure with AbaqusRefined meshL2.49Finite-Strain Analysis of Crack Tips? The size of the elements around the notch must be about 1/10 th the notch-tip radius. Biased edge seeds canreduce the size of the mesh by focusing small elements towards the crack tip. SEN specimencrack-tip meshrnotch10% of rnotchModeling Fracture and Failure with AbaqusL2.50Finite-Strain Analysis of Crack Tips? For J-integral evaluation, the region on the surface of the blunted notch should be used to define the crack front.Crack tip regionq vectorCrack surface is detected automaticallyThe blunted notch surface is the crack front regionSymmetry plane? For the J- and Ct-integrals to be path independent, the crack surfaces must be parallel to one another (or parallel to the symmetry plane). ? If this is not the case, Abaqus automatically generates normals on the crack surface. ? If the notch radius shrinks to zero, all nodes that would be at the crack tip should be included in the crack-tip node set.Modeling Fracture and Failure with AbaqusL2.51Finite-Strain Analysis of Crack Tips? If the mesh is so coarse that the integration points nearest the crack tip are far from the tip, most of the details (accurate stresses and strains) of the finite-strain region around the crack tip will be lost. ? However, accurate J values may still be obtained if cracks are modeled as sharp.Modeling Fracture and Failure with AbaqusL2.52Finite-Strain Analysis of Crack Tips? Example: SEN specimenDeformed shapeModerate bluntingUndeformed shapeSevere bluntingDeformed vs Undeformed ShapesModeling Fracture and Failure with AbaqusContours of PEEQL2.53Finite-Strain Analysis of Crack Tips? In situations involving finite rotations but small strains, such as the bending of slender structures, a small keyhole around the crack tip should be modeled.crack-front region? The region defining the crack front for the contour integral consists of the region on the keyhole.? The elements should not be singular.Modeling Fracture and Failure with AbaqusLimitations Of 3D Swept Meshing For FractureL2.55Limitations Of 3D Swept Meshing For Fracture? For curved regions cannot generate wedges at the center using a hexdominated approach and then sweep along the length of the region. ? This was discussed earlier in the context of the conical crack problem. ? To create a focused mesh in this case, embed a small tube within a larger concentric tube. Mesh the smaller tube with a single lay the surrounding regions are meshed with hex elements.Sweep directionModeling Fracture and Failure with AbaqusL2.56Limitations Of 3D Swept Meshing For Fracture? Partition for a penny-shaped crack? Illustrates the limitation that the path for the partition must be perpendicular to i thus, cannot properly partition along the arc of a circle as shown in this example:Tangent direction of arcarc (not a semi-circle as in previous example)Cross-sectional view of blockPartition by sweeping circular edge along arcModeling Fracture and Failure with AbaqusL2.57Limitations Of 3D Swept Meshing For Fracture? The workaround is to partition the face with circular arcs, and then partition the cell using the n-sided patch technique.n-sided patchFace partitionNote that the cross-sectional area of the swept region is not constant along its length because the tangents at the ends are not perpendicular to the block (generalized sweep meshing)Resulting mesh around the crack front using wedge elementsModeling Fracture and Failure with AbaqusModeling Cracks with Keyword OptionsL2.59Modeling Cracks with Keyword Options? Defining a crack with keyword options:? The *CONTOUR INTEGRAL option is used to define both, the crack itself and the fracture output, in an Abaqus input ( .inp) file.? In this section, we focus solely on the crack-specific parameters of this option. ? These include:*CONTOUR INTEGRAL, SYMM, NORMAL? In the next lecture, we discuss the output-specific parameters of this option. ? As noted earlier, the main requirements in defining a crack are: ? Defining the crack front? Defining the crack extension directionModeling Fracture and Failure with AbaqusL2.60Modeling Cracks with Keyword Options? Crack symmetry*CONTOUR INTEGRAL, SYMM? The crack lies on a plane of symmetry and only half the structure is being modeled ? This feature should only be used for Mode I problems.Modeling Fracture and Failure with AbaqusL2.61Modeling Cracks with Keyword Options? Crack extension*CONTOUR INTEGRAL, NORMAL? The NORMAL parameter is used to define the normal to the crack plane when the crack is planar. ? Usage:*contour integral, normal nx, ny, nz nodeSet1, nodeSet2, ...? In this case, give a list of the node set names defining the crack front from one end to the other end, in sequential order, without missing any points on the crack line. ? In two-dimensional cases, only one node set is needed.These sets de the first node in each set defines the crack tip node for that set.(An optional CRACK TIP NODES parameter is available to specify the crack tip nodes directly).Modeling Fracture and Failure with AbaqusL2.62Modeling Cracks with Keyword Options? Example: Penny-shaped crack in an infinite space*Contour integral, symm, normal, ... 0.0, 1.0, 0.0 Crack-Front-1, Crack-Front-2, Crack-Front-3, ...Crack-Front-1Modeling Fracture and Failure with AbaqusL2.63Modeling Cracks with Keyword Options? If the NORMAL parameter is omitted, we must give the crack-tip node set name, and the crack propagation direction q, at each node set defining the crack front. ? Usage:*contour integral, ... nodeSet1, (qx)1, (qy)1, (qz)1 nodeSet2, (qx)2, (qy)2, (qz)2 :? Data must start with the node set at one end and be given for each node set defining the crack line sequentially until the other end of the crack is reached.? The first node in each set is the crack tip node for that set unless the CRACK TIP NODES parameter is used.? This format allows nonplanar cracks to be analyzed.Modeling Fracture and Failure with AbaqusL2.64Modeling Cracks with Keyword Options? Example: conical crack in an infinite half-space*Contour integral, ... Crack-Front-1, 0.707107, -0.. Crack-Front-2, 0.705994, -0..0396478 Crack-Front-3, 0.702661, -0..0791708Crack-Front-1Modeling Fracture and Failure with AbaqusL2.65Modeling Cracks with Keyword Options? Generating a focused mesh with keyword options? Example: DEN specimen? The focused mesh shown in the figure will be generated with the use of keyword options. ? The options include*NODE*NGEN*NFILL*ELEMENT*ELGENModeling Fracture and Failure with AbaqusL2.66Modeling Cracks with Keyword Options? Node definitions*node 1, 1, , 16101, *ngen, 1, *ngen, 101, , 0.0 0.0 0.0 0.5 0.5 0.0 nset=tip 1 nset=outer
1 11210114101210116101101tip *NGEN generates nodes incrementally between any two previously defined nodes. In this example, 17 crack-tip nodes are created (contained in the set tip); the 17 nodes on the outer boundary are contained in set outer.Start nodeEnd nodeIncrement in node numberModeling Fracture and Failure with AbaqusL2.67Modeling Cracks with Keyword Options? Quarter-point nodes*nfill, singular=1 tip, outer, 10, 10This parameter generate the 1 indicates the first bound represents the crack tipStart set: first boundEnd set: second boundNode number increment21 1021 21Number of intervals between bounding nodes4011 1131*NFILL generate nodes for a region of a mesh by filling in nodes between two bounds. In this example, 10 rows of nodes are generated between each tip node and its corresponding outer node.Modeling Fracture and Failure with AbaqusL2.68Modeling Cracks with Keyword Options? Element definitions*element, type=cps8r 1, 1, 21, , 11, , 1001 *elgen, elset=plate 1, 5, 20, 10, 8, First row of elements Total number of rows Nodes 1, 1001, and 2001 are coincident1*ELGEN generates elements incrementally.21111In this example, 5 elements form the first row (extending radially outward from the tip); a total of 8 rows of elements (based on the first row) are created around the crack tip.Modeling Fracture and Failure with AbaqusL2.69Modeling Cracks with Keyword Options? Crack-tip nodes? If the crack-tip nodes are permitted to behave independently, the strength of the strain-field singularity is ? ? r -1. ? The crack-tip nodes can be constrained using equations, multi-point constraints, using repeated nodes in the element definition, etc. For example, to constrain the crack-tip nodes with a multi-point constraint:*nset, nset=constrain, generate 1, 1 *mpc tie, constrain, 16001? Only node 16001 is independent in this case.? The strain-field singularity is ? ? r -?.Modeling Fracture and Failure with AbaqusFracture AnalysisLecture 3L3.2Overview? Calculation of Contour Integrals ? Examples ? Nodal Normals in Contour Integral Calculations? J-Integrals at Multiple Crack Tips? Through Cracks in Shells? Mixed-Mode Fracture? Material Discontinuities? Numerical Calculations with Elastic-Plastic Materials? Workshop 1? Workshop 2Modeling Fracture and Failure with AbaqusCalculation of Contour IntegralsL3.4Calculation of Contour Integrals? Abaqus offers the evaluation of J-integral values, as well as several other parameters for fracture mechanics studies. These include: ? The KI, KII, and KIII stress intensity factors, which are used mainly in linear elastic fracture mechanics to measure the strength of lo ? The T-stress in linear
? The crack propagation direction: an angle at which a preexisting
and ? The Ct-integral, which is used with time-dependent creep behavior.? Output can be written to the output database ( .odb), data (.dat), and results (.fil) files.Modeling Fracture and Failure with AbaqusL3.5Calculation of Contour Integrals? Domain representation of J? For reasons of accuracy, J is evaluated using a domain integral. ? The domain integral is evaluated over an area/volume contained within a contour surrounding the crack tip/line.? In two dimensions, Abaqus defines the domain in terms of rings of elements surrounding the crack tip. ? In three dimensions, Abaqus defines a tubular surface around the crack line.Modeling Fracture and Failure with AbaqusL3.6Calculation of Contour Integrals? Different contours (domains) are created automatically by Abaqus. ? The first contour consists of the crack front and one layer of elements surrounding it.? Ring of elements from one crack surface to the other (or the symmetry plane).? The next contour consists of the ring of elements in contact with the first contour as well as the elements in the first contour.Contour 1Contour 2? Each subsequent contour is defined by adding the next ring of elements in contact with the previous contour.Contour 3Contour 4Modeling Fracture and Failure with AbaqusL3.7Calculation of Contour Integrals? The J-integral and the Ct-integral at steady-state creep should be path (domain) independent. ? The value for the first contour is generally ignored. ? Examples of contour domains:2nd contour1st contour crack-front nodesCrack-tip node2nd contour1st contourCrack-tip nodeModeling Fracture and Failure with AbaqusL3.8Calculation of Contour Integrals? Usage:*CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}Specifies the number of contours (domains) on which the contour integral will be calculatedThis is the output frequency in incrementsNote: In this lecture, we focus on the output-specific parameters of the *CONTOUR INTEGRAL option. The crack-specific parameters SYMM and NORMAL were discussed in the previous lecture.Modeling Fracture and Failure with AbaqusL3.9Calculation of Contour Integrals? Usage (cont’d):*CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}? J for J-integral output,? C for Ct-integral output. ? T STRESS to output T-stress calculations ? K FACTORS for stress intensity factor outputModeling Fracture and Failure with AbaqusL3.10Calculation of Contour Integrals? Usage (cont’d):*CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}Three criteria to calculate the crack propagation direction at initiation? Use with TYPE=K FACTORS to specify the criterion to be used for estimating the crack propagation direction in homogenous, isotropic, linear elastic materials:? Maximum tangential stress criterion (MTS) ? Maximum energy release rate criterion (MERR) ? KII = 0 criterion (KII0)Modeling Fracture and Failure with AbaqusL3.11Calculation of Contour Integrals? Output files*CONTOUR INTEGRAL, OUTPUT? Set OUTPUT=FILE to store the contour integral values in the results (.fil) file.? Set OUTPUT=BOTH to print the values in the data and results files. ? If the parameter is omitted, the contour integral values will be printed in the data (.dat) file but not stored in the results (.fil) file.Modeling Fracture and Failure with AbaqusL3.12Calculation of Contour Integrals? Loads? Loads included in contour integral calculations:? Thermal loads.? Crack-face pressure and traction loads on continuum elements as well as those applied using user subroutines DLOAD and UTRACLOAD. ? Surface traction and crack-face edge loads on shell elements as well as those applied using user subroutine UTRACLOAD. ? Uniform and nonuniform body forces.? Centrifugal loads on continuum and shell elements. ? Not all types of distributed loads (e.g., hydrostatic pressure and gravity loads) are included in the contour integral calculations. ? The presence of these loads will result in a warning message.Modeling Fracture and Failure with AbaqusL3.13Calculation of Contour Integrals? Other loads not included in contour integral calculations:? Contributions due to concentrated loads are not included.? If needed, modify the mesh to include a small element and apply a distributed load to the element. ? Contributions due to contact forces are not included.? Initial stresses are not considered in the definition of contour integrals.Modeling Fracture and Failure with AbaqusExamplesL3.15Examples? Penny-shaped crack in an infinite space? Model characteristics? The mesh is extended far enough from the crack tip so that the finite boundaries will not influence the crack-tip solution. ? The radius of the penny-shaped crack is 1.? Two types of loading are considered: ? Uniform far-field loading? Nonuniform loading on the crack face: p = Ar n.Modeling Fracture and Failure with AbaqusL3.16Examples? Different mesh characteristics:20? Axisymmetric or three-dimensional? Fine or coarse focused meshes? With or without ? point elements? Various element types used:20? First- and second-order? With and without reduced integrationAxisymmetric modelCrack tipFocused mesh around crack tipModeling Fracture and Failure with AbaqusL3.17Examples? Fine mesh vs. coarse mesh (axisymmetric and 3D models)0.00040.08The fine mesh
the coarse mesh above. The length perpendicular to crack line of the crack-tip elements are indicated.~0.08Modeling Fracture and Failure with AbaqusL3.18Examples? Axisymmetric model: geometrySymmetry planesClose up of crack tip region for coarse mesh model (identical for fine mesh model―only the inner semicircular region is smaller)Model geometryModeling Fracture and Failure with AbaqusL3.19Examples? Axisymmetric model: crack definitionCrack tip with extension directionSet to 0.5 to use midpoint rather than ? point elementsModeling Fracture and Failure with AbaqusL3.20Examples? 3D model: geometry and meshFine 3D mesh? A 90? sector is modeled because of symmetry.Additional partition required for swept meshSymmetry planesOn planes perpendicular to the crack front, the mesh is very similar to the axisymmetric mesh Partitions used for coarse mesh model (identical for fine mesh model―only the inner semicircular region is smaller) In the circumferential direction around the crack line, 12 elements are used.Modeling Fracture and Failure with AbaqusL3.21Examples? Why is the additional partition required?? Without the additional partition, the region shown below would require irregular elements at the vertex located on the axis of symmetry.? This is not supported by Abaqus.Irregular elements required here because revolving about a pointA 7-node element is an example of an irregular element.Modeling Fracture and Failure with AbaqusL3.22Examples? 3D model: crack definition? Orphan mesh created to edit q vectors.Modeling Fracture and Failure with AbaqusL3.23Examples? Contour integral output requests (axisymmetric and 3D)Separate output requests are required for J, K-factors, and the T-stress.Modeling Fracture and Failure with AbaqusL3.24Examples? Loads (axisymmetric and 3D)The far-field load is suppressed.Modeling Fracture and Failure with AbaqusL3.25Examples? Results? MISES stress shown below for the axisymmetric fine mesh.J analytical ? J numerical J analytical?100%Deformation scale factor = 250AnalyticalContour 1Contour 2Contour 3Contour 4Contour 55.796E-025.8169E-02Contour 6 5.8064E-025.8095E-02Contour 7 5.8044E-025.8121E-02Contour 8 5.8024E-025.8104E-02Contour 9 5.8005E-025.8084E-02Contour 10 5.7985E-02Modeling Fracture and Failure with AbaqusL3.26Examples J values from meshes with ? point elements (reduced integration)3-D Loading Uniform far field Uniform crack face Nonuniform crack face (n = 1) Nonuniform crack face (n = 2) Analytical result C3D20R Coarse Fine Axisymmetric CAX8R Coarse Fine.0580. ..0578. ..0580. ..0579. ..0581. .Nonuniform crack face (n = 3)? Abaqus values are based on the average of contours 3 ?5 in each mesh.Modeling Fracture and Failure with AbaqusL3.27Examples J values from meshes with ? point elements (full integration)3-D Loading Uniform far field Uniform crack face Nonuniform crack face (n = 1) Nonuniform crack face (n = 2) Analytical result C3D20 Coarse Fine Axisymmetric CAX8 Coarse Fine.0580. ..0577. ..0572. ..0578. ..0580. .Nonuniform crack face (n = 3)? Abaqus values are based on the average of contours 3 ?5 in each mesh.Modeling Fracture and Failure with AbaqusL3.28Examples J values from meshes without ? point elements (reduced integration)3-D AxisymmetricLoading Uniform far fieldUniform crack face Nonuniform crack face (n = 1) Nonuniform crack face (n = 2) Nonuniform crack face (n = 3)Analytical resultC3D20RCoarse FineC3D8RCoarseCAX8RCoarse FineCAX4RCoarse.0580.0574.0580.0563.0574.0581.0562..............? Abaqus values are based on the average of contours 3 ?5 in each mesh.Modeling Fracture and Failure with AbaqusL3.29Examples J values from meshes without ? point elements (full integration)3-D AxisymmetricLoading Uniform far fieldUniform crack face Nonuniform crack face (n = 1) Nonuniform crack face (n = 2) Nonuniform crack face (n = 3)Analytical resultC3D20Coarse FineC3D8CoarseCAX8Coars