Damage analysis of tetragonal perovskite structure ceramics implicated by asymptotic field solutions and boundary conditions

Cracking of ceramics with tetragonal perovskite grain structure is known to appear at different sites and scale level. The multiscale character of damage depends on the combined effects of electromechanical coupling, prevailing physical parameters and boundary conditions. These detail features are exhibited by application of the energy density criterion with judicious use of the mode I asymptotic and full field solution in the range of r/a=10⁻⁴ to 10⁻² where r and a are, respectively, the distance to the crack tip and half crack length. Very close to the stationary crack tip, bifurcation is predicted resembling the dislocation emission behavior invoked in the molecular dynamics model. At the macroscopic scale, crack growth is predicted to occur straight ahead with two yield zones to the sides. A multiscale feature of crack tip damage is provided for the first time. Numerical values of the relative distances and bifurcation angles are reported for the PZT-4 ceramic subjected to different electric field to applied stress ratio and boundary conditions that consist of the specification of electric field/mechanical stress, electric displacement/mechanical strain, and mixed conditions. To be emphasized is that the multiscale character of damage in piezoceramics does not appear in general. It occurs only for specific combinations of the external and internal field parameters, elastic/piezoelectric/dielectric constants and specified boundary conditions.     

Crack size and speed interaction characteristics at micro-, meso- and macro-scale

The motivation to examine physical events at even smaller size scale arises from the development of use-specific materials where information transfer from one micro- or macro-element to another could be pre-assigned. There is the growing belief that the cumulated macroscopic experiences could be related to those at the lower size scales. Otherwise, there serves little purpose to examine material behavior at the different scale levels. Size scale, however, is intimately associated with time, not to mention temperature. As the size and time scales are shifted, different physical events may be identified. Dislocations with the movements of atoms, shear and rotation of clusters of molecules with inhomogeneity of polycrystals; and yielding/fracture with bulk properties of continuum specimens. Piecemeal results at the different scale levels are vulnerable to the possibility that they may be incompatible. The attention should therefore be focused on a single formulation that has the characteristics of multiscaling in size and time. The fact that the task may be overwhelmingly difficult cannot be used as an excuse for ignoring the fundamental aspects of the problem.Local nonlinearity is smeared into a small zone ahead of the crack. A “restrain stress” is introduced to also account for cracking at the meso-scale.The major emphasis is placed on developing a model that could exhibit the evolution characteristics of change in cracking behavior due to size and speed. Material inhomogeneity is assumed to favor self-similar crack growth although this may not always be the case. For relatively high restrain stress, the possible nucleation of micro-, meso- and macro-crack can be distinguished near the crack tip region. This distinction quickly disappears after a small distance after which scaling is no longer possible. This character prevails for Mode I and II cracking at different speeds. Special efforts are made to confine discussions within the framework of assumed conditions. To be kept in mind are the words of Isaac Newton in the Fourth Regula Philosophandi:

“Men are often led into error by the love of simplicity which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is … We may learn something of the way in which nature operates from fact and observation; but if we conclude that it operates in such a manner, only because to our understanding that operates to be the best and simplest manner, we shall always go wrong.”––Isaac Newton

Kinetics of microcrack blunting ahead of macrocrack approaching shear wave speed

The fracturing of glass and tearing of rubber both involve the separation of material but their crack growth behavior can be quite different, particularly with reference to the distance of separation of the adjacent planes of material and the speed at which they separate. Relatively speaking, the former and the latter are recognized, respectively, to be fast and slow under normal conditions. Moreover, the crack tip radius of curvature in glass can be very sharp while that in the rubber can be very blunt. These changes in the geometric features of the crack or defect, however, have not been incorporated into the modeling of running cracks because the mathematical treatment makes use of the Galilean transformation where the crack opening distance or the change in the radius of curvature of the crack does not enter into the solution. Change in crack speed is accounted for only via the modulus of elasticity and mass density. For this simple reason, many of the dynamic features of the running crack have remained unexplained although speculations are not lacking. To begin with, the process of energy dissipation due to separation is affected by the microstructure of the material that distinguishes polycrystalline from amorphous form. Energy extracted from macroscopic reaches of a solid will travel to the atomic or smaller regions at different speeds at a given instance. It is not clear how many of the succeeding size scales should be included within a given time interval for an accurate prediction of the macroscopic dynamic crack characteristics. The minimum requirement would therefore necessitate the simultaneous treatment of two scales at the same time. This means that the analysis should capture the change in the macroscopic and microscopic features of a defect as it propagates. The discussion for a dual scale model has been invoked only very recently for a stationary crack. The objective of this work is to extend this effort to a crack running at constant speed beyond that of Rayleigh wave. Developed is a dual scale moving crack model containing microscopic damage ahead of a macroscopic crack with a gradual transition. This transitory region is referred to as the mesoscopic zone where the tractions prevail on the damaged portion of the material ahead of the original crack known as the restraining stresses, the magnitude of which depends on the geometry, material and loading. This damaged or restraining zone is not assumed arbitrarily nor assumed to be intrinsically a constant in the cohesive stress approach; it is determined for each step of crack advancement. For the range of micronotch bluntness with 0 < β < 30° and 0.2 σ_∞/σ₀ 0.5, there prevails a nearly constant restraining zone size as the crack approaches the shear wave speed. Note that β is the half micronotch angle and the applied stress ratio is σ_∞/σ₀ with σ₀ being the maximum of the restraining stress. For σ_∞/σ₀ equal to or less than 0.5, the macrocrack opening displacement COD is nearly constant and starts to decrease more quickly as the crack approaches the shear wave speed. For the present dual scale model where the normalized crack speed v/c_s increases with decreasing with the one-half microcrack tip angle β. There prevails a limit of crack tip bluntness that corresponds to β 36° and v/c_s 0.15. That is a crack cannot be maintained at a constant speed if the bluntness is increased beyond this limiting value. Such a feature is manifestation of the dependency of the restraining stress on crack velocity and the applied stress or the energy pumped into the system to maintain the crack at a constant velocity. More specifically, the transitory character from macro to micro is being determined as part of the unknown solution. Using the energy density function dW/dV as the indicator, plots are made in terms of the macrodistance ahead of the original crack while the microdefect bluntness can vary depending on the tip geometry. Such a generality has not been considered previously. The macro-dW/dV behavior with distance remains as the inverse r relation yielding a perfect hyperbola for the homogeneous material. This behavior is the same as the stationary crack. The micro-dW/dV relations are expressed in terms of a single undetermined parameter. Its evaluation is beyond the scope of this investigation although the qualitative behavior is expected to be similar to that for the stationary crack. To reiterate, what has been achieved as an objective is a model that accounts for the thickness of a running crack since the surface of separation representing damage at the macroscopic and microscopic scale is different. The transitory behavior from micro to macro is described by the state of affairs in the mesoscopic zone.     

Effect of grain size on slow fatigue crack propagation and plastic deformation near crack tip

Fatigue crack growth and its threshold are investigated at a stress ratio of 0.5 for the three-point bend specimen made of Austenitic stainless steel. The effect of grain size on the crack tip plastic deformation is investigated. The results show that the threshold value Δk_th increases linearly with the square root of grain size d and the growth rate is slower for materials with larger grain size. The plastic zone size and ratio for different grain sizes are different at the threshold. The maximum stress intensity factor is k_max and σ_ys is the yield strength. At the same time, the characteristics of the plastic deformation development is discontinuous and anti-symmetric as the growth rate is increased from 2·10^—8 to 10⁻⁷ mm/cycle.A dimensionless relation of the form for collating fatigue crack starting growth data is proposed in which Δk_th represents the stress intensity factor range at the threshold. Based on experimental results, this relation attains the value of 0.6 for a fatigue crack to start growth in the Austenitic stainless steel investigated in this work. Metallurgical examinations were also carried out to show a transgranular shear mode of cyclic cleavage and plastic shear.     

Piezomagnetic and piezoelectric poling effects on mode I and II crack initiation behavior of magnetoelectroelastic materials

The ferrite and ferroelectric phase of magnetoelectroelastic (MEE) material can be selected and processed to control the macroscopic behavior of electron devices using continuum mechanics models. Once macro- and/or microdefects appear, the highly intensified magnetic and electric energy localization could alter the response significantly to change the design performance. Alignment of poling directions of piezomagnetic and piezoelectric materials can add to the complexity of the MEE material behavior to which this study will be concerned with.Appropriate balance of distortional and dilatational energy density is no longer obvious when a material possesses anisotropy and/or nonhomogeneity. An excess of the former could result in unwanted geometric change while the latter may lead to unexpected fracture initiation. Such information can be evaluated quantitatively from the stationary values of the energy density function dW/dV. The maxima and minima have been known to coincide, respectively, with possible locations of permanent shape change and crack initiation regardless of material and loading type. The direction of poling with respect to a line crack and the material microstructure described by the constitutive coefficients will be specified explicitly with reference to the applied magnetic field, electric field and mechanical stress, both normal and shear. The crack initiation load and direction could be predicted by finding the direction for which the volume change is the largest. In contrast to intuition, change in poling directions can influence the cracking behavior of MEE dramatically. This will be demonstrated by the numerical results for the BaTiO₃–CoFe₂O₄ composite having different volume fractions where BaTiO₃ and CoFe₂O₄ are, respectively, the inclusion and matrix.To be emphasized is that mode I and II crack behavior will not have the same definition as that in classical fracture mechanics where load and crack extension symmetry would coincide. A striking result is found for a mode II crack. By keeping the magnetic poling fixed, a reversal of electric poling changed the crack initiation angle from θ₀=+80° to θ₀=−80° using the line extending ahead of the crack as the reference. This effect is also sensitive to the distance from the crack tip. Displayed and discussed are results for r/a=10⁻⁴ and 10⁻¹. Because the theory of magnetoelectroelasticity used in the analysis is based on the assumption of equilibrium where the influence of material microstructure is homogenized, the local space and temporal effects must be interpreted accordingly. Among them are the maximum values of (dW/dV)_max and (dW/dV)_min which refer to as possible sites of yielding and fracture. Since time and size are homogenized, it is implicitly understood that there is more time for yielding as compared to fracture being a more sudden process. This renders a higher dW/dV in contrast to that for fracture. Put it differently, a lower dW/dV with a shorter time for release could be more detrimental.     

Micro/macro-crack growth due to creep–fatigue dependency on time–temperature material behavior

The implicit character of micro-structural degradation is determined by specifying the time history of crack growth caused by creep–fatigue interaction at high temperature. A dual scale micro/macro-equivalent crack growth model is used to illustrate the underlying principle of multiscaling which can be applied equally well to nano/micro. A series of dual scale models can be connected to formulate triple or quadruple scale models. Temperature and time-dependent thermo-mechanical material properties are developed to dictate the design time history of creep–fatigue cracking that can serve as the master curve for health monitoring.In contrast to the conventional procedure of problem/solution approach by specifying the time- and temperature-dependent material properties as a priori, the desired solution is then defined for a class of anticipated loadings. A scheme for matching the loading history with the damage evolution is then obtained. The results depend on the initial crack size and the extent of creep in proportion to fatigue damage. The path dependent nature of damage is demonstrated by showing the range of the pertinent parameters that control the final destruction of the material. A possible scenario of 20 yr of life span for the 38Cr2Mo2VA ultra-high strength steel is used to develop the evolution of the micro-structural degradation. Three micro/macro-parameters μ^*, d^* and σ^* are used to exhibit the time-dependent variation of the material, geometry and load effects. They are necessary to reflect the scale transitory behavior of creep–fatigue damage. Once the algorithm is developed, the material can be tailor made to match the behavior. That is a different life span of the same material would alter the time behavior of μ^*, d^* and σ^* and hence the micro-structural degradation history. The one-to-one correspondence of the material micro-structure degradation history with that of damage by cracking is the essence of path dependency. Numerical results and graphs are obtained to demonstrate how the inherently implicit material micro-structure parameters can be evaluated from the uniaxial bulk material properties at the macroscopic scale.The combined behavior of creep and fatigue can be exhibited by specifying the parameter ξ with reference to the initial defect size a₀. Large ξ (0.90 and 0.85) gives critical crack size a_cr = 11–14 mm (at t < 20 yr) for a₀ about 1.3 mm. For small ξ (0.05 and 0.15), there results critical a_cr = 6–7 mm (at t < 20 yr) for a₀ about 0.7–0.8 mm. The initial crack is estimated to increase its length by an order of magnitude before triggering global to the instability. This also applies ξ ≈ 0.5 where creep interacts severely with fatigue. Fine tuning of a_cr and a₀ can be made to meet the condition oft = 20 yr.Trade off among load, material and geometric parameters are quantified such that the optimum conditions can be determined for the desired life qualified by the initial–final defect sizes. The scenario assumed in this work is indicative of the capability of the methodology. The initial–final defect sizes can be varied by re-designing the time–temperature material specifications. To reiterate, the uniqueness of solution requires the end result to match with the initial conditions for a given problem. This basic requirement has been accomplished by the dual scale micro/macro-crack growth model for creep and fatigue.     

Effect of material nonhomogeneities on the HRR dominance

This work studies the asymptotic stress and displacement fields near the tip of a stationary crack in an elastic–plastic nonhomogeneous material with the emphasis on the effect of material nonhomogeneities on the dominance of the crack tip field. While the HRR singular field still prevails near the crack tip if the material properties are continuous and piecewise continuously differentiable, a simple asymptotic analysis shows that the size of the HRR dominance zone decreases with increasing magnitude of material property gradients. The HRR field dominates at points that satisfy |α⁻¹ ∂α/∂x_δ|1/r, |α⁻¹ ∂²α/(∂x_δ ∂x_γ)|1/r², |n⁻¹ ∂n/∂x_δ|1/[r|ln(r/A)|] and |n⁻¹ ∂²n/(∂x_δ ∂x_γ)|1/[r²|ln(r/A)|], in addition to other general requirements for asymptotic solutions, where α is a material property in the Ramberg–Osgood model, n is the strain hardening exponent, r is the distance from the crack tip, x_δ are Cartesian coordinates, and A is a length parameter. For linear hardening materials, the crack tip field dominates at points that satisfy |E_tan⁻¹ ∂E_tan/∂x_δ|1/r, |E_tan⁻¹ ∂²E_tan/(∂x_δ ∂x_γ)|1/r², |E⁻¹ ∂E/∂x_δ|1/r, and |E⁻¹ ∂²E/(∂x_δ ∂x_γ)|1/r², where E_tan is the tangent modulus and E is Young’s modulus.     

Near tip field for pseudo Mode II crack growth: Power law hardening material

The asymptotic behavior of stress and strain near the tip of a Mode II crack growing in power law hardening material is analyzed by assuming that the crack grows straight ahead even though tests show otherwise. The results show that the stress and strain possess the singularities of (ln r)^2/(n−1) and (ln r)^2n/(n−1) respectively. The distance from the crack tip is r, and n is the hardening exponent, i.e. σⁿ. The amplitudes of the stress and strain near the crack tip are determined by the asymptotic analysis.     

A centrally cracked thin circular disk, Part I: 3D Elastic–plastic finite element analysis

A full field solution, based on small deformation, three-dimensional elastic–plastic finite element analysis of the centrally cracked thin disk under mode I loading has been performed. The solution for the stresses under small-scale yielding and lo!cally fully plastic state has been compared with the HRR plane stress solution. At the outside of the 3D zone, within a distance of rσ_o/J=18, HRR dominance is maintained in the presence of a significant amount of compressive stress along the crack flanks. Ahead of this region, the HRR field overestimate the stresses. These results demonstrate a completely reversed state of stress in the near crack front compared to that in the plane strain case. The combined effect of geometry and finite thickness of the specimen on elastic–plastic crack tip stress field has been explored. To the best of our knowledge, such an attempt in the published literature has not been made yet. For the qualitative assessment of the results some of the field parameters have been compared to the available experimental results of K, gives a fair estimate of the crack opening stress near the crack front at a distance of order 10⁻² in. On the basis of this analysis, the Linear Elastic Fracture Mechanics approach has been adopted in analyzing the fatigue crack extension experiments performed in the disk (Part II).     

A two-strain-gage technique for determining mode I stress-intensity factor

A finite element analysis and a experimental test were performed to show that the terms with r^−1/2 and ^1/2 in the eigenfunction expansion of the strains can describe the crack tip strain distribution with sufficient accuracy. A set of two linear equations can be obtained to determine the stress intensity factor K_I using only two strain-gages. Errors within 5% can be achieved provided that the two strain-gages are placed at the appropriate locations. The technique can be developed to treat crack bodies with irregular geometry and complex loading.     

Pseudo plane stress mode II crack growth in plastic materials

The near tip field of mode II crack that grows in thin bodies with power hardening or perfectly plastic behavior is analyzed. It is shown that for power hardening behavior, the pseudo plane stress field possesses the logarithm singularity, i.e. σ (ln r)²/(n−1), (ln r)^{2n/(n − 1)}, where r is the distance from the crack tip, n the hardening exponent is σⁿ. When n → ∞ the solution reduced to that for the perfectly plastic case.     

Strain strengthening effects in Witwatersrand quartzite

A series of uniaxial compression specimens were tested over a range of applied ram displacement rates of 8.9 × 10⁻⁴ to 8.9 mm/sec to elucidate the effects of loading rate on the uniaxial compressive fracture stress of Witwatersrand quartzite. It was demonstrated that even within standard loading rate ranges, considerable scatter in the fracture strength (under uniaxial compression) existed in this particular quartzite rock. Nevertheless, a definite trend of increasing fracture resistance with increasing monotonic loading rate was evident inasmuch that increasing the loading rate (strain rate) by four orders of magnitude increase the fracture strength by almost 2.8 times. Prior fatigue loading also produced a significant strain strengthening as the uniaxial compressive fracture stress tended to increase in a sigmoidal fashion with increasing number of fatigue cycles prior to testing. Indeed, the fracture strength of quartzite was almost doubled in value after 10⁻ cycles. Plane strain fracture toughness tests utilising three point bend specimens were conducted and an average of K_lc = 1.7 MPa√m was realized. In both the uniaxial compression tests and the fracture toughness tests, failure occurred by crack extension predominantly by a transgranular flat cleavage-like mode through pure quartzite (silica) regions. However, crack extension was also observed to occur in an intergranular “ductile-like” mode through areas associated with inclusions prevalent in the quartzite.     

Effect of stress ratio and specimen thickness on fatigue crack growth of CK45 steel

Presented are the effect of stress ratio and thickness on the fatigue crack growth rate of CK45 steel according to DIN 17200. Test results are obtained for constant amplitude load in tension with three stress ratios of R=0, 0.2 and 0.4 and three specimen thicknesses of B=6, 12 and 24 mm. Microgauge crack opening values were used to calculate ΔK_eff values from which the da/dN − ΔK_eff curves are obtained. Crack closure can be applied to explain the influence of mean stress and specimen thickness on the fatigue crack growth rate in the second regime of the two-parameter crack growth rate relation. An empirical model is chosen for calculating the normalized load ratio parameter U as a function of R, B and ΔK and, for correlating the test data.     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

An integral equation for dynamic elastic response of an isolated 3-D crack

By use of a steady state (e^−iωt) dynamic elastic representation theorem for fields created by relative motions ΔU_k on the faces of a crack, we reduce the problem of steady state response of an isolated three-dimensional planar crack, loaded by tractions on its surfaces, to an integral equation for ΔU_k.     

Energy dissipation mechanisms in ductile fracture

The objective is to investigate energy dissipation mechanisms that operate at different length scales during fracture in ductile materials. A dimensional analysis is performed to identify the sets of dimensionless parameters which contribute to energy dissipation via dislocation-mediated plastic deformation at a crack tip. However, rather than using phenomenological variables such as yield stress and hardening modulus in the analysis, physical variables such as dislocation density, Burgers vector and Peierls stress are used. It is then shown via elementary arguments that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms at different length scales from the crack tip; the energy dissipations mechanisms are cleavage, crack tip dislocation nucleation and also dislocation nucleation from a Frank-Read source. Therefore, the material behavior is classified into three groups. The first two groups are the well-known intrinsic brittle and intrinsic ductile behavior. The third group is designated to be extrinsic ductile behavior for which Frank-Read dislocation nucleation is the initial energy dissipation mechanism. It is shown that a material is predicted to exhibit extrinsic ductility if the dimensionless parameter bρ_disl^1/2 (b is Burgers vector, ρ_disl is dislocation density) is within a certain range defined by other dimensionless parameters, irrespective of the competition between cleavage and crack tip dislocation nucleation. The predictions compare favorably to the documented behavior of a number of different classes of materials.     

Fatigue crack growth of cable steel wires in a suspension bridge: Multiscaling and mesoscopic fracture mechanics

Intrinsically, fatigue failure problem is a typical multiscale problem because a fatigue failure process deals with the fatigue crack growth from microscale to macroscale that passes two different scales. Both the microscopic and macroscopic effects in geometry and material property would affect the fatigue behaviors of structural components. Classical continuum mechanics has inability to treat such a multiscale problem since it excludes the scale effect from the beginning by introducing the continuity and homogeneity assumptions which blot out the discontinuity and inhomogeneity of materials at the microscopic scale. The main obstacle here is the link between the microscopic and macroscopic scale. It has to divide a continuous fatigue process into two parts which are analyzed respectively by different approaches. The first is so called as the fatigue crack initiation period and the second as the fatigue crack propagation period. Now the problem can be solved by application of the mesoscopic fracture mechanics theories developed in the recent years which focus on the link between different scales such as nano-, micro- and macro-scale.On the physical background of the problem, a restraining stress zone that can describe the material damaging process from micro to macro is then introduced and a macro/micro dual scale edge crack model is thus established. The expression of the macro/micro dual scale strain energy density factor is obtained which serves as a governing quantity for the fatigue crack growth. A multiscaling formulation for the fatigue crack growth is systematically developed. This is a main contribution to the fundamental theories for fatigue problem in this work. There prevail three basic parameters μ^∗, σ^∗ and d^∗ in the proposed approach. They can take both the microscopic and macroscopic factors in geometry and material property into account. Note that μ^∗, σ^∗ and d^∗ stand respectively for the ratio of microscopic to macroscopic shear modulus, the ratio of restraining stress to applied stress and the ratio of microvoid size ahead of crack tip to the characteristic length of material microstructure.To illustrate the proposed multiscale approach, Hangzhou Jiangdong Bridge is selected to perform the numerical computations. The bridge locates at Hangzhou, the capital of Zhejiang Province of China. It is a self-anchored suspension bridge on the Qiantang River. The cables are made of 109 parallel steel wires in the diameter of 7 mm. Cable forces are calculated by finite element method in the service period with and without traffic load. Two parameters α and β are introduced to account for the additional tightening and loosening effects of cables in two different ways. The fatigue crack growth rate coefficient C₀ is determined from the fatigue experimental result. It can be concluded from numerical results that the size of initial microscopic defects is a dominant factor for the fatigue life of steel wires. In general, the tightening effect of cables would decrease the fatigue life while the loosening effect would impede the fatigue crack growth. However, the result can be reversed in some particular conditions. Moreover, the different evolution modes of three basic parameters μ^∗, σ^∗ and d^∗ actually have the different influences on the fatigue crack growth behavior of steel wires. Finally the methodology developed in this work can apply to all cracking-induced failure problems of polycrystal materials, not only fatigue, but also creep rupture and cracking under both static and dynamic load and so on.