A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

Two-Parameter Model of a Mode I Crack in an Elastoplastic Body under Plane-Strain conditions

The Dugdale crack model is generalized to the case of plane strain. The governing equations are set up to determine the stresses in the plastic zone. Numerical results from specific problems are analyzed and compared with those for plane stress state and other cases. A relationship between the crack model and K _I-T theory is established in the case of small-scale yielding at the crack tip __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 44–55, June 2005.     

单纤维与裂纹交互微力学:完整粘接纤维

使用微拉曼光谱技术研究裂纹前喙Kevlar49芳纶纤维交互微力学行为.在建立Kevlar49芳纶纤维轴向变形与拉曼频移线性关系的基础上,通过逐点测量裂尖前喙完整粘接纤维上的应力分布,探讨了裂纹与完整粘接纤维的交互作用,建立了完整粘接纤维上的应力传递模型.Ⅰ型裂纹切口前方的高应力区造成粘接完好纤维上出现应力集中,应力强度因子K1影响着纤维应力集中长度.     

Characterization of crack tip stresses in plane-strain fracture specimens having weld center crack

Fracture toughness of metals depends strongly on the state of stress near the crack tip. The existing standards (like R-6, SINTAP) are being modified to account for the influence of stress triaxiality in the flaw assessment procedures. These modifications are based on the ability of so-called ‘constraint parameters’ to describe near tip stresses. Crack tip stresses in homogeneous fracture specimens are successfully described in terms of two parameters like J–Q or J–T. For fracture specimens having a weld center crack, strength mismatch ratio between base and weld material and weld width are the additional variables, along with the magnitude of applied loading, type of loading, and geometry of specimen that affect the crack tip stresses. In this work, a novel three-parameter scheme was proposed to estimate the crack tip opening stress accounting for the above-mentioned variables. The first and second parameters represent the crack tip opening stress in a homogeneous fracture specimen under small-scale yielding and are well known. The third parameter accounts for the effect of constraint developed due to weld strength mismatch. It comprises of weld strength mismatch ratio (M, i.e. ratio of yield strength of weld material to that of base material), and a plastic interaction factor (I_p) that scales the size of the plastic zone with the width of the weld material. The plastic interaction factor represents the degree of influence of weld strength mismatch on crack tip constraint for a given mismatch ratio. The proposed scheme was validated with detailed FE analysis using the Modified Boundary Layer formulation.     

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      6. Characterization of crack-tip strain singularity in brittle-microcracking materials by means of the photoelastic-coating method      Toshiyuki Hashida 《Experimental Mechanics》1989,29(3):307-311 The strain fields ahead of crack tips in rock, mortar, and graphite were measured using a photoelastic coating method. A transparent ferroelectric ceramic was used as a coating material to observe the photoelastic fringe pattern. A coating plate of 110–150 μm thick was placed on single-edge-notch specimens, and photoelasticity experiments were conducted in three-point bending under a polarizing microscope. The results show that the principal-strain difference ahead of the crack tip is given by $$\Delta \in = \Delta \in _o [(J/\sigma _{ult} )/r]^m $$ whereσ ult is the ultimate tensile strength,r is the distance from the crack tip, and9? o andm are material constants. Based on this observation, the use of theJ Ic concept in determining the fracture toughness of brittle-microcracking materials is discussed.      7. <Emphasis Type="Italic">T</Emphasis>-stress analysis for a Griffith crack in a magnetoelectroelastic solid      Xian-Ci Zhong  Xian-Fang Li 《Archive of Applied Mechanics (Ingenieur Archiv)》2008,78(2):117-125 T-stress as an important parameter characterizing the stress field around a cracked tip has attracted much attention. This paper concerns the T-stress near a cracked tip in a magnetoelectroelastic solid. By applying the Fourier transform, we solve the associated mixed boundary-value problem. Adopting crack-faces electromagnetic boundary conditions nonlinearly dependent on the crack opening displacement, coupled dual integral equations are derived. Then, the closed-form solution for the T-stress is obtained. A comparison of the T stresses for a cracked magnetoelectroelastic solid and for a cracked purely elastic material is made. Obtained results reveal that in addition to applied mechanical loading, the T-stress is dependent on electric and magnetic loadings for a vacuum crack.      8. Numerical simulation of stress intensity factors via the thermoelastic technique      S. H. Ju  J. R. Lesniak  B. I. Sandor 《Experimental Mechanics》1997,37(3):278-284 The purpose of this study is to investigate the accuracy of the least squares method for finding the in-plane stress intensity factorsK I andK II using thermoelastic data from isotropic materials. To fully understand the idealized condition ofK I andK II calculated from thermoelastic experiments, the total stress field calculated from finite element analysis is used to take the place of data obtained from real thermoelastic experiments. In the finite element analysis, theJ-integral is also calculated to compare with (K I 2 +K II 2 )/E evaluated by the least squares method. The stress fields near the crack tip are dominated by the two stress intensity factors; however, the edge effect will cause inaccuracy of the thermoelastic data near the crack tip. Furthermore, the scan area of thermoelastic experiments cannot be too small. Therefore, we suggest that three or four terms of stress function be included in the least squares method for evaluating stress intensity factors via the thermoelastic technique. In the idealized condition, the error can be smaller than 3 percent from our numerical simulations. If only ther –1/2 term (K I andK II ) is included in the least squares method, even in the idealized case the error can be up to 20 percent.      9. An Experimental Study of Mixed Mode Crack Initiation and Growth in Functionally Graded Materials   总被引：2，自引：0，他引：2   J. Abanto-Bueno  J. Lambros 《Experimental Mechanics》2006,46(2):179-196 Quasi-static mixed mode crack initiation and growth in functionally graded materials (FGMs) was studied through fracture experiments on polymer-based FGMs manufactured by selective ultraviolet irradiation poly(ethylene carbon monoxide)—a photo-sensitive copolymer that becomes more brittle and stiffer under ultraviolet irradiation. The objective of the study was to determine whether crack kinking criteria for homogeneous materials, e.g., maximum hoop stress criterion, also hold for FGMs. Single edge notched tension specimens with different spatial variations of Young's modulus, failure stress and failure strain, were tested. Near tip mode mixity was introduced either by inclining the crack to the remote loading direction, as in the case of homogeneous materials, or to the direction of material gradient, or both. A full-field digital image correlation technique was used to measure in real-time the displacement field around the crack tip while it propagated through the graded material, and to extract the fracture parameters of stress intensity factor K I and K II , and the T-stress. It was found that the nonsingular T-stress term in the asymptotic expansion for stresses plays a very important role in accurately measuring fracture parameters. It was also found that the maximum tangential stress criterion can be applied to the case of FGMs to predict crack kinking provided that the effect of the T-stress is accounted for and the process zone size is small compared to the intrinsic material gradient length scale. However, for accurate crack path prediction at a length scale comparable to the material gradient, detailed material property information is required. In general, the crack will propagate towards a region that exhibits less fracture toughness, but, unlike the case of homogeneous materials, along a path where K II is not necessarily equal to zero.      10. Crack tip field andJ-integral with strain gradient effect      夏嵩  王自强  陈少华 《Acta Mechanica Sinica》2004,20(3):228-237 The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventionalJ 2 deformation theory. No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory. Two typical crack problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-IK-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory. The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, theJ-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases. The project supported by the National Natural Science Foundation of China (19704100 and 10202023) and the Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20)      11. Behaviour of the Reynolds stress on rough walls      A. Mazouz  L. Labraga  C. Tournier 《Experiments in fluids》1994,17(1-2):39-44 Quadrant analysis is used to study the contributions, associated to the four quadrants of the (u,v) plane, to the production of the turbulent shear stress on rough walls. The measurements are described for a fully developed turbulent flow between two rough plates with varying the parameterλ z(span/height ratio of roughness elements). The application of this technique indicates that the (Q 2) events (ejections) and (Q 4) events (sweeps) cause an intense production of the Reynolds stress —ρ as compared with the (Q 1) and (Q 3) quadrants. The (Q 2) contribution to the Reynolds stress depends on the geometry factorλ z. Variation of the parameterλ zaffects the distributions of and forH⩽3, whereH is a particular threshold. Comparison with boundary layer flow shows that the region 0.2⩽y/h⩽0.7 is characterized by a Reynolds stress production, independent of the flow nature. The third moment of the longitudinal velocity fluctuations is found to be sensitive to the surface roughness.      12. Effect of material nonhomogeneities on the HRR dominance      Z. -H. Jin   《Mechanics Research Communications》2004,31(2):203-211 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/∂xδ|1/r, |α−1 ∂2α/(∂xδ ∂xγ)|1/r2, |n−1 ∂n/∂xδ|1/[r|ln(r/A)|] and |n−1 ∂2n/(∂xδ ∂xγ)|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, xδ are Cartesian coordinates, and A is a length parameter. For linear hardening materials, the crack tip field dominates at points that satisfy |Etan−1 ∂Etan/∂xδ|1/r, |Etan−1 ∂2Etan/(∂xδ ∂xγ)|1/r2, |E−1 ∂E/∂xδ|1/r, and |E−1 ∂2E/(∂xδ ∂xγ)|1/r2, where Etan is the tangent modulus and E is Young’s modulus.      13. Crack problems of piezoelectric materials      Wang Tzuchiang  Han Xueli 《Acta Mechanica Solida Sinica》1999,12(2):95-105 This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in piezoelectrics or on the interfaces of piezoelectric bimaterials. A class of boundary problems involving many cracks is also solved. For homogeneous materials it is found that the normal electric displacementD 2 induced by the crack is constant along the crack faces which depends only on the applied remote stress field. Within the crack slit, the electric fields induced by the crack are also constant and not affected by the applied electric field. For the bimaterials with realH, the normal electric displacementD 2 is constant along the crack faces and electric fieldE 2 has the singularity ahead of the crack tip and a jump across the interface. The project is supported by the National Natural Science Foundation of China(No. 19704100) and the Natural Science Foundation of Chinese Academy of Sciences(No. KJ951-1-201).      14. Rheo-optical measurements of the first and third normal stresses of homopolymer poly(vinyl methyl ether) melt      Ajay Kulkarni  Semen Kharchenko  Rangaramanujam M. Kannan 《Rheologica Acta》2006,45(6):951-958 Normal stresses play a key role in polymer processing, yet accurate measurements are still challenging. Simultaneous rheo-optical measurements are conducted on a poly(vinyl methyl ether) homopolymer melt over a wide range of temperatures and oscillatory shear frequencies, in an effort to measure the normal stresses, by using quantitative flow birefringence measurements. The stress optical rule holds well for this polymer as expected, with the value of the stress optic coefficient of (6.38±0.19)×10−11 cm2/dyn at 30°C. The first and third normal stress difference coefficients, calculated using a single memory constitutive equation applied to the stress and birefringence data, are in excellent agreement. The ratio of the measured third and first normal stress difference coefficients, (1−β)=0.71±0.05, agrees well with the result of the Doi–Edwards model with independent alignment approximation (β=0.28). The measurement of normal stress difference coefficients with such small deviations proves the robust nature of the improved rheo-optical instrument and its ability to measure complete stress tensor.      15. Elastic stresses in bodies with elliptic,parabolic, and tunnel cracks      V. S. Kirilyuk  O. I. Levchuk 《International Applied Mechanics》2007,43(8):874-885 The paper gives explicit expressions of the elastic T-stress components T I, T II, and T III for an elliptic crack in an unbounded body under uniform pressure and bending and expressions of all the T-stress components for parabolic and tunnel cracks under uniform loading. These formulas are derived by analyzing the asymptotic behavior of the stress components near the crack front using special harmonic functions. The dependence of the T-stresses on Poisson’s ratio, semiaxes and parametric angle of the elliptic crack is studied. The expressions of T I, T II, and T III for a penny-shaped crack under arbitrary uniform pressure and bending follow as a special case from the respective expressions for an elliptic crack __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 57–70, August 2007.      16. Three dimensional K-T z stress fields around the embedded center elliptical crack front in elastic plates      Junhua Zhao  Wanlin Guo  Chongmin She  Bo Meng 《Acta Mechanica Sinica》2006,22(2):148-155 Through detailed three-dimensional (3D) finite element (FE) calculations, the out-of-plane constraints Tz along embedded center-elliptical cracks in mode I elastic plates are studied. The distributions of Tz are obtained near the crack front with aspect ratios (a/c) of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. Tz decreases from an approximate value of Poisson ratio ν at the crack tip to zero with increasing normalized radial distances (r/a) in the normal plane of the crack front line, and increases gradually when the elliptical parameter angle ϕ changes from 0° to 90°at the same r/a. With a/c rising to 1.0, Tz is getting nearly independent of ϕ and is only related to r/a. Based on the present FE calculations for Tz, empirical formulas for Tz are obtained to describe the 3D distribution of Tz for embedded center-elliptical cracks using the least squares method in the range of 0.2≤a/c≤1.0. These Tz results together with the corresponding stress intensity factor K are well suitable for the analysis of the 3D embedded center-elliptical crack front field, and a two-parameter K-Tz principle is proposed. The project supported by the National Natural Science Foundation of China (50275073) The English text was polished by Keren Wang.      17. The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity      Craig Cowan  Pierpaolo Esposito  Nassif Ghoussoub  Amir Moradifam 《Archive for Rational Mechanics and Analysis》2010,198(3):763-787 We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D2 u=\fracl(1-u)2{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbRN{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u λ with 0 < u λ < 1 exists for l ? (0,l*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution ul*{u_{\lambda^*}} is regular (supB ul* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while ul*{u_{\lambda^*}} is singular (supB ul* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C0|x|4/3 \leqq ul* (x) \leqq 1-|x|4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C0:=(\fracl*[`(l)])\frac13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.      18. On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases      David Ruiz 《Archive for Rational Mechanics and Analysis》2010,198(1):349-368 This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem: - Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      19. Improved model of the griffith crack      E. E. Deryugin  G. V. Lasko 《Journal of Applied Mechanics and Technical Physics》1998,39(6):934-942 The plane-stress state of a cracked continuous medium in tension is determined using relaxation elements. The stress state is analyzed at the tip of a crack surrounded by a plastically deformed material as a band of localized plastic deformation (LPD) shaped like an elongated ellipse. The plastic deformation considerably decreases the stress concentration at the crack tip. As the localization of the plastic deformation increases, the stresses at the crack sides decrease to zero. The decrease in stresses at the tip is accompanied by an increase in the concentration and gradients of the stresses at the end of theLPD band. Here the region of perturbation of the stress field is comparable with the width of the band. Institute of Physics of Strength and Materials Science, Tomsk 634055. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 132–141, November–December, 1998.      20. Plane-strain crack tip field and a dual-parameter fracture-criterion for non-linear fracture mechanics      H.W. Liu  T. Zhuang 《Theoretical and Applied Fracture Mechanics》1987,7(3) Hancock and Cowling measured the critical crack tip opening displacements, δf, at fracture initiation in HY-80 steel specimens of six different configurations. δf varied from 90 μm in a deeply double-edge-cracked tensile panel to 900 μm in a single-edge-cracked tensile panel.McMeeking and Parks, and Shih and German have shown by their finite element calculations that the characteristics of the plane strain crack tip fields in both large scale yielding and general yielding are strongly dependent on specimen geometry and load level.In this study, the plane strain crack tip fields in the specimens tested by Hancock and Cowling were calculated using the finite element method. The crack tip triaxial tensile stress field is strongly affected by specimen geometric constraint, and the state of the triaxial tensile stress in a crack tip region is monitored by the ratio between the local tensile stress and the effective stress, i.e., ( ), at a distance x=2δ from the crack tip. The values of ( ) vary from 3.1 for the double-edge-cracked tensile panel to 1.7 for the single-edge-cracked tensile panel. The δf measured by Hancock and Cowling correlates very well with the ratio ( ). δf is a measure of the fracture ductility of the material ahead of the crack tip, and the ductility decreases with an increase in the triaxial tensile stress, i.e., the ratio ( ).

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Characterization of crack-tip strain singularity in brittle-microcracking materials by means of the photoelastic-coating method

The strain fields ahead of crack tips in rock, mortar, and graphite were measured using a photoelastic coating method. A transparent ferroelectric ceramic was used as a coating material to observe the photoelastic fringe pattern. A coating plate of 110–150 μm thick was placed on single-edge-notch specimens, and photoelasticity experiments were conducted in three-point bending under a polarizing microscope. The results show that the principal-strain difference ahead of the crack tip is given by $$\Delta \in = \Delta \in _o [(J/\sigma _{ult} )/r]^m $$ whereσ _ult is the ultimate tensile strength,r is the distance from the crack tip, and9? _o andm are material constants. Based on this observation, the use of theJ _Ic concept in determining the fracture toughness of brittle-microcracking materials is discussed.     

<Emphasis Type="Italic">T</Emphasis>-stress analysis for a Griffith crack in a magnetoelectroelastic solid

T-stress as an important parameter characterizing the stress field around a cracked tip has attracted much attention. This paper concerns the T-stress near a cracked tip in a magnetoelectroelastic solid. By applying the Fourier transform, we solve the associated mixed boundary-value problem. Adopting crack-faces electromagnetic boundary conditions nonlinearly dependent on the crack opening displacement, coupled dual integral equations are derived. Then, the closed-form solution for the T-stress is obtained. A comparison of the T stresses for a cracked magnetoelectroelastic solid and for a cracked purely elastic material is made. Obtained results reveal that in addition to applied mechanical loading, the T-stress is dependent on electric and magnetic loadings for a vacuum crack.     

Numerical simulation of stress intensity factors via the thermoelastic technique

The purpose of this study is to investigate the accuracy of the least squares method for finding the in-plane stress intensity factorsK _I andK _II using thermoelastic data from isotropic materials. To fully understand the idealized condition ofK _I andK _II calculated from thermoelastic experiments, the total stress field calculated from finite element analysis is used to take the place of data obtained from real thermoelastic experiments. In the finite element analysis, theJ-integral is also calculated to compare with (K _I ² +K _II ² )/E evaluated by the least squares method. The stress fields near the crack tip are dominated by the two stress intensity factors; however, the edge effect will cause inaccuracy of the thermoelastic data near the crack tip. Furthermore, the scan area of thermoelastic experiments cannot be too small. Therefore, we suggest that three or four terms of stress function be included in the least squares method for evaluating stress intensity factors via the thermoelastic technique. In the idealized condition, the error can be smaller than 3 percent from our numerical simulations. If only ther ^–1/2 term (K _I andK _II ) is included in the least squares method, even in the idealized case the error can be up to 20 percent.     

An Experimental Study of Mixed Mode Crack Initiation and Growth in Functionally Graded Materials

Quasi-static mixed mode crack initiation and growth in functionally graded materials (FGMs) was studied through fracture experiments on polymer-based FGMs manufactured by selective ultraviolet irradiation poly(ethylene carbon monoxide)—a photo-sensitive copolymer that becomes more brittle and stiffer under ultraviolet irradiation. The objective of the study was to determine whether crack kinking criteria for homogeneous materials, e.g., maximum hoop stress criterion, also hold for FGMs. Single edge notched tension specimens with different spatial variations of Young's modulus, failure stress and failure strain, were tested. Near tip mode mixity was introduced either by inclining the crack to the remote loading direction, as in the case of homogeneous materials, or to the direction of material gradient, or both. A full-field digital image correlation technique was used to measure in real-time the displacement field around the crack tip while it propagated through the graded material, and to extract the fracture parameters of stress intensity factor K _I and K _II , and the T-stress. It was found that the nonsingular T-stress term in the asymptotic expansion for stresses plays a very important role in accurately measuring fracture parameters. It was also found that the maximum tangential stress criterion can be applied to the case of FGMs to predict crack kinking provided that the effect of the T-stress is accounted for and the process zone size is small compared to the intrinsic material gradient length scale. However, for accurate crack path prediction at a length scale comparable to the material gradient, detailed material property information is required. In general, the crack will propagate towards a region that exhibits less fracture toughness, but, unlike the case of homogeneous materials, along a path where K _II is not necessarily equal to zero.     

Crack tip field andJ-integral with strain gradient effect

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventionalJ ₂ deformation theory. No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory. Two typical crack problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-IK-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory. The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, theJ-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases. The project supported by the National Natural Science Foundation of China (19704100 and 10202023) and the Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20)     

Behaviour of the Reynolds stress on rough walls

Quadrant analysis is used to study the contributions, associated to the four quadrants of the (u,v) plane, to the production of the turbulent shear stress on rough walls. The measurements are described for a fully developed turbulent flow between two rough plates with varying the parameterλ _z(span/height ratio of roughness elements). The application of this technique indicates that the (Q ₂) events (ejections) and (Q ₄) events (sweeps) cause an intense production of the Reynolds stress —ρ as compared with the (Q ₁) and (Q ₃) quadrants. The (Q ₂) contribution to the Reynolds stress depends on the geometry factorλ _z. Variation of the parameterλ _zaffects the distributions of and forH⩽3, whereH is a particular threshold. Comparison with boundary layer flow shows that the region 0.2⩽y/h⩽0.7 is characterized by a Reynolds stress production, independent of the flow nature. The third moment of the longitudinal velocity fluctuations is found to be sensitive to the surface roughness.     

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.     

Crack problems of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in piezoelectrics or on the interfaces of piezoelectric bimaterials. A class of boundary problems involving many cracks is also solved. For homogeneous materials it is found that the normal electric displacementD ₂ induced by the crack is constant along the crack faces which depends only on the applied remote stress field. Within the crack slit, the electric fields induced by the crack are also constant and not affected by the applied electric field. For the bimaterials with realH, the normal electric displacementD ₂ is constant along the crack faces and electric fieldE ₂ has the singularity ahead of the crack tip and a jump across the interface. The project is supported by the National Natural Science Foundation of China(No. 19704100) and the Natural Science Foundation of Chinese Academy of Sciences(No. KJ951-1-201).     

Rheo-optical measurements of the first and third normal stresses of homopolymer poly(vinyl methyl ether) melt

Normal stresses play a key role in polymer processing, yet accurate measurements are still challenging. Simultaneous rheo-optical measurements are conducted on a poly(vinyl methyl ether) homopolymer melt over a wide range of temperatures and oscillatory shear frequencies, in an effort to measure the normal stresses, by using quantitative flow birefringence measurements. The stress optical rule holds well for this polymer as expected, with the value of the stress optic coefficient of (6.38±0.19)×10⁻¹¹ cm²/dyn at 30°C. The first and third normal stress difference coefficients, calculated using a single memory constitutive equation applied to the stress and birefringence data, are in excellent agreement. The ratio of the measured third and first normal stress difference coefficients, (1−β)=0.71±0.05, agrees well with the result of the Doi–Edwards model with independent alignment approximation (β=0.28). The measurement of normal stress difference coefficients with such small deviations proves the robust nature of the improved rheo-optical instrument and its ability to measure complete stress tensor.     

Elastic stresses in bodies with elliptic,parabolic, and tunnel cracks

The paper gives explicit expressions of the elastic T-stress components T _I, T _II, and T _III for an elliptic crack in an unbounded body under uniform pressure and bending and expressions of all the T-stress components for parabolic and tunnel cracks under uniform loading. These formulas are derived by analyzing the asymptotic behavior of the stress components near the crack front using special harmonic functions. The dependence of the T-stresses on Poisson’s ratio, semiaxes and parametric angle of the elliptic crack is studied. The expressions of T _I, T _II, and T _III for a penny-shaped crack under arbitrary uniform pressure and bending follow as a special case from the respective expressions for an elliptic crack __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 57–70, August 2007.     

Three dimensional K-T z stress fields around the embedded center elliptical crack front in elastic plates

Through detailed three-dimensional (3D) finite element (FE) calculations, the out-of-plane constraints T_z along embedded center-elliptical cracks in mode I elastic plates are studied. The distributions of T_z are obtained near the crack front with aspect ratios (a/c) of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z decreases from an approximate value of Poisson ratio ν at the crack tip to zero with increasing normalized radial distances (r/a) in the normal plane of the crack front line, and increases gradually when the elliptical parameter angle ϕ changes from 0° to 90°at the same r/a. With a/c rising to 1.0, T_z is getting nearly independent of ϕ and is only related to r/a. Based on the present FE calculations for T_z, empirical formulas for T_z are obtained to describe the 3D distribution of T_z for embedded center-elliptical cracks using the least squares method in the range of 0.2≤a/c≤1.0. These T_z results together with the corresponding stress intensity factor K are well suitable for the analysis of the 3D embedded center-elliptical crack front field, and a two-parameter K-T_z principle is proposed. The project supported by the National Natural Science Foundation of China (50275073) The English text was polished by Keren Wang.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem:

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      19. Improved model of the griffith crack      E. E. Deryugin  G. V. Lasko 《Journal of Applied Mechanics and Technical Physics》1998,39(6):934-942 The plane-stress state of a cracked continuous medium in tension is determined using relaxation elements. The stress state is analyzed at the tip of a crack surrounded by a plastically deformed material as a band of localized plastic deformation (LPD) shaped like an elongated ellipse. The plastic deformation considerably decreases the stress concentration at the crack tip. As the localization of the plastic deformation increases, the stresses at the crack sides decrease to zero. The decrease in stresses at the tip is accompanied by an increase in the concentration and gradients of the stresses at the end of theLPD band. Here the region of perturbation of the stress field is comparable with the width of the band. Institute of Physics of Strength and Materials Science, Tomsk 634055. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 132–141, November–December, 1998.      20. Plane-strain crack tip field and a dual-parameter fracture-criterion for non-linear fracture mechanics      H.W. Liu  T. Zhuang 《Theoretical and Applied Fracture Mechanics》1987,7(3) Hancock and Cowling measured the critical crack tip opening displacements, δf, at fracture initiation in HY-80 steel specimens of six different configurations. δf varied from 90 μm in a deeply double-edge-cracked tensile panel to 900 μm in a single-edge-cracked tensile panel.McMeeking and Parks, and Shih and German have shown by their finite element calculations that the characteristics of the plane strain crack tip fields in both large scale yielding and general yielding are strongly dependent on specimen geometry and load level.In this study, the plane strain crack tip fields in the specimens tested by Hancock and Cowling were calculated using the finite element method. The crack tip triaxial tensile stress field is strongly affected by specimen geometric constraint, and the state of the triaxial tensile stress in a crack tip region is monitored by the ratio between the local tensile stress and the effective stress, i.e., ( ), at a distance x=2δ from the crack tip. The values of ( ) vary from 3.1 for the double-edge-cracked tensile panel to 1.7 for the single-edge-cracked tensile panel. The δf measured by Hancock and Cowling correlates very well with the ratio ( ). δf is a measure of the fracture ductility of the material ahead of the crack tip, and the ductility decreases with an increase in the triaxial tensile stress, i.e., the ratio ( ).

Improved model of the griffith crack

The plane-stress state of a cracked continuous medium in tension is determined using relaxation elements. The stress state is analyzed at the tip of a crack surrounded by a plastically deformed material as a band of localized plastic deformation (LPD) shaped like an elongated ellipse. The plastic deformation considerably decreases the stress concentration at the crack tip. As the localization of the plastic deformation increases, the stresses at the crack sides decrease to zero. The decrease in stresses at the tip is accompanied by an increase in the concentration and gradients of the stresses at the end of theLPD band. Here the region of perturbation of the stress field is comparable with the width of the band. Institute of Physics of Strength and Materials Science, Tomsk 634055. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 132–141, November–December, 1998.     

Plane-strain crack tip field and a dual-parameter fracture-criterion for non-linear fracture mechanics

Hancock and Cowling measured the critical crack tip opening displacements, δ_f, at fracture initiation in HY-80 steel specimens of six different configurations. δ_f varied from 90 μm in a deeply double-edge-cracked tensile panel to 900 μm in a single-edge-cracked tensile panel.McMeeking and Parks, and Shih and German have shown by their finite element calculations that the characteristics of the plane strain crack tip fields in both large scale yielding and general yielding are strongly dependent on specimen geometry and load level.In this study, the plane strain crack tip fields in the specimens tested by Hancock and Cowling were calculated using the finite element method. The crack tip triaxial tensile stress field is strongly affected by specimen geometric constraint, and the state of the triaxial tensile stress in a crack tip region is monitored by the ratio between the local tensile stress and the effective stress, i.e., ( ), at a distance x=2δ from the crack tip. The values of ( ) vary from 3.1 for the double-edge-cracked tensile panel to 1.7 for the single-edge-cracked tensile panel. The δ_f measured by Hancock and Cowling correlates very well with the ratio ( ). δ_f is a measure of the fracture ductility of the material ahead of the crack tip, and the ductility decreases with an increase in the triaxial tensile stress, i.e., the ratio ( ).