Optimisation Method for Determination of Crack Tip Position Based on Gauss-Newton Iterative Technique

In the digital image correlation research of fatigue crack growth rate,the accuracy of the crack tip position determines the accuracy of the calculation of the stress intensity factor,thereby affecting the life prediction.This paper proposes a Gauss-Newton iteration method for solving the crack tip position.The conventional linear fitting method provides an iterative initial solution for this method,and the preconditioned conjugate gradient method is used to solve the ill-conditioned matrix.A noise-added artificial displacement field is used to verify the feasibility of the method,which shows that all parameters can be solved with satisfactory results.The actual stress intensity factor solution case shows that the stress intensity factor value obtained by the method in this paper is very close to the finite element result,and the relative error between the two is only−0.621%;The Williams coefficient obtained by this method can also better define the contour of the plastic zone at the crack tip,and the maximum relative error with the test plastic zone area is−11.29%.The relative error between the contour of the plastic zone defined by the conventional method and the area of the experimental plastic zone reached a maximum of 26.05%.The crack tip coordinates,stress intensity factors,and plastic zone contour changes in the loading and unloading phases are explored.The results show that the crack tip change during the loading process is faster than the change during the unloading process;the stress intensity factor during the unloading process under the same load condition is larger than that during the loading process;under the same load,the theoretical plastic zone during the unloading process is higher than that during the loading process.

ANALYSIS OF STRESS FIELD NEAR CRACK TIP BASED ON ELECTRIC SATURATION MODEL

Within the framework of nonlinear eleetroelasticity, the stress field near to the crack tip in an infinite piezoelectric media subject to a far field uniform loading is studied by using an electrical strip saturation model and the complex variable method. And the emphasis is placed on the stress field near to the crack tip. The obtained solutions show that the normalized stress components at an arbitrary point near to the crack tip are determined by the angle of the point. Moreover, the stress components are independent of the distance from the point to the ori- gin of the coordinate. The distributions of in-plane stress components near to the crack tip are analyzed based on numerical results for PZT-SH. Compared with some related solutions, results show that the solutions are valid.

CRACK TIP FIELD AND J-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 conventional J_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-I K-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,the J-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases.

MAPPING THE DAMAGED ZONE AROUND THE CRACK TIP IN HIGH DENSITY POLYETHYLENE WITH SYNCHROTRON MICROFOCUS SMALL ANGLE X-RAY SCATTERING TECHNIQUE

The structural changes around a crack tip in a high density polyethylene were investigated by means of scanning synchrotron microfocus small-angle X-ray scattering technique. The scattering data confirm the process of craze structure development near a crack tip based on the evolution of voids. In addition, it was found that the main stress in the plastic zone near a crack tip exhibited a gradient distribution with respect to its strength and direction. The whole damaged area showed a strain distribution indicating a flow behavior toward the crack tip.

Anodic dissolution of a crack tip at AA2024-T351 in 3.5wt% NaCl solution

The anodic dissolution process of a crack tip at 2024-T351 aluminium alloy （AA2024-T351） was determined by means of scanning Kelvin probe （SKP）. Wedge-open loading （WOL） specimens were immersed in a 3.5wt% NaCl solution. After various durations of immersion, the Volta potential distributions around the crack were measured by SKP and the surface morphologies were observed by scanning electron microscopy （SEM）. It is found that there is a nonuniform distribution of Volta potential around the crack. Before immersion, the Volta potential at crack tip is more negative than that in other regions. However, after immersion, a converse result occurs with the most positive Volta potential measured at the crack tip. SEM observations demonstrate that the noticeable positive shift of Volta potential results fi：om the formation of corrosion products which deposit around the crack tip. Energy-dispersive spectrometry （EDS） analysis shows that the corrosion products are mainly A1 oxide and Cu-rich particles. These observations implicate that the applied stress contributes to the preferential anodic dissolution of the crack tip and the redistribution of Cu.

Stress field near interface crack tip of double dissimilar orthotropic composite materials

In this paper, double dissimilar orthotropic composite materials interfacial crack is studied by constructing new stress functions and employing the method of composite material complex. When the characteristic equations＇ discriminants △1 〉 0 and △2 〉0, the theoretical formula of the stress field and the displacement field near the mode I interface crack tip are derived, indicating that there is no oscillation and interembedding between the interfaces of the crack.

MODE Ⅰ CRACK TIP FIELD WITH STRAIN GRADIENT EFFECTS

A plane strain mode 1 crack tip field with strain gradient effects is investigated.A new strain gradient theory is used.An elastic-power law hardening strain gradient material is considered and two hardening laws,i.e.a separation law and an integration law are used respectively.As for the material with the separation law hardening,the angular distributions of stresses are consistent with the HRR field,which differs from the stress results;the angular distributions of couple stresses are the same as the couple stress results.For the material with the integration law hardening,the stress field and the couple stress field can not exist simultaneously,which is the same as the conclusion,but for the stress dominated field,the an- gular distributions of stresses are consistent with the HRR field;for the couple stress dominated field,the an- gular distributions of couple stresses are consistent with those in Ref.However,the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only,while the crack tip field of mode 1 is dominated by the tension gradient,which will be shown in another paper.

Finite deformation analysis of crack tip fields in plastically compressible hardening–softening–hardening solids

Crack tip fields are calculated under plane strain small scale yielding conditions. The material is characterized by a finite strain elastic-viscoplastic constitutive relation with various hardening-softening-hardening hardness functions. Both plastically compressible and plastically incompressible solids are considered. Displacements corresponding to the isotropic linear elastic mode I crack field are prescribed on a remote boundary. The initial crack is taken to be a semi-circular notch and symmetry about the crack plane is imposed. Plastic compressibility is found to give an increased crack opening displacement for a given value of the applied loading. The plastic zone size and shape are found to depend on the plastic compressibility, but not much on whether material softening occurs near the crack tip. On the other hand, the near crack tip stress and deformation fields depend sensitively on whether or not material softening occurs. The combination of plastic compressibility and softening (or softening-hardening) has a particularly strong effect on the near crack tip stress and deformation fields.

MODE Ⅰ AND MODE Ⅱ CRACK TIP ASYMPTOTIC FIELDS WITH STRAIN GRADIENT EFFECTS

The strain gradient effect becomes significant when the size of frac- ture process zone around a crack tip is comparable to the intrinsic material length l, typically of the order of microns. Using the new strain gradient deformation theory given by Chen and Wang, the asymptotic fields near a crack tip in an elastic-plastic material with strain gradient effects are investigated. It is established that the dom- inant strain field is irrotational. For mode Ⅰ plane stress crack tip asymptotic field, the stress asymptotic field and the couple stress asymptotic field can not exist si- multaneously. In the stress dominated asymptotic field, the angular distributions of stresses are consistent with the classical plane stress HRR field; In the couple stress dominated asymptotic field, the angular distributions of couple stresses are consistent with that obtained by Huang et al. For mode Ⅱ plane stress and plane strain crack tip asymptotic fields, only the stress-dominated asymptotic fields exist. The couple stress asymptotic field is less singular than the stress asymptotic fields. The stress asymptotic fields are the same as mode Ⅱ plane stress and plane strain HRR fields, respectively. The increase in stresses is not observed in strain gradient plasticity for mode Ⅰ and mode Ⅱ, because the present theory is based only on the rotational gradi- ent of deformation and the crack tip asymptotic fields are irrotational and dominated by the stretching gradient.

TEM OBSERVATION OF LOCAL DEFORMATION BAND AT CRACK TIP

The thin foil specimen of a ferrite-austenite duplex stainless steel was tensiled under transmis- sion electron microscope(TEM).It was found that both in ferrite and austenite the local deformation band at crack tip was formed near to the crack propagating direction.Its forma- tion was related with the crack tip Schmid factor,dislocation shielding,latent hardening and hardening coefficient.When the crack tip emitted dislocations to a slip system by the action of pure mode Ⅱ stress resolute,and the decreasing rate of hardening coefficient was suitable,lo- cal deformation band was easy to form.

CRACK TIP STRUCTURE AND DUCTILE-BRITTLE TRANSITION IN BULK SINGLE CRYSTALS

Based on the crack tip structure a new model of ductile -brittle transition was proposed. Using this new model we calculated the dependence of the transition temperature-strain rate over a wide range of strain rate. Finally the significance of this new model is discussed in detail.

Digital image correlation based high-speed crack tip locatingmethod and its application

Based on a digital image correlation（DIC）method with the measurements of a high speed crack＇s displacement and strain fields,a technique to accurately and automatically locate its crack tip has been developed.The crack tip is identified by finding the abrupt jump on the opening（or dislocation）curve of a point on the trace of the crack propagation,while the opening is measured through a virtual extensometer technique and the abrupt jump is identified by finding the peak on the curve.The method was verified using a specially designed experiment and applied to measure the critical crack tip opening angle of a rock sample.Because the involvement of analytical models has been avoided and then the good performance could be ensured for low resolution speckle images,this technique is expected to be very useful in the quantitative study of high speed cracks in experiments using high speed cameras.

CRACK TIP SUPERBLUNTING:EXPERIMENT,THEORY AND NUMERICAL SIMULATION

Experimental revelation of superblunting along plane strain predominant crack tip segment is re- ported here for modified polyproplene.As elucidated by a heuristic model of progressive circumferential cold-draw,the formation of superblunting crack tip depends critically on the ratio between the cold-draw prop- agation speed and the loading speed,and contributes significantly to the material toughening,especially for the improvement in impact toughness.Detailed numerical calculations are conducted based on a hyperelastic-viscoplastic and anisotropic damage constitutive model at finite deformation.The simulated results recapitulate the essential features of crack tip superblunting.

IN-SITU OBSERVATION OF CRACK TIP DEFORMATION IN A DUPLEX STAINLESS STEEL

The development of dislocation structures in the plastic zone ahead of a crack tip has been in- vestigated in a duplex stainless steel during in-situ deformation experiments in a scanning transmission electron microscope.It was found that the dislocation distribution was significantly different in the ferrite and in the austenite.In the ferrite grains,the dislocations emitted by the crack tip may cross-slip out of the original slip planes and form a broad plastic zone.However,in the austenite,the dislocation free zone is small and the dislocations emitted by the crack pile up in its slip plane.The selection of slip systems at the crack tip depends on the crack tip Schmid factors in both phases.But after large deformation,the selection of the second slip systems at the craek tip in austenite does not depend on the Schmid factors.

Dislocation Behaviour near the Crack Tip in Bulk Fe-3% Si Single Crystal

An investigation has been made of the disloca- tion distribution and dislocation free zone near the crack tip in bulk Fe-3% Si single crystal during deformation in SEM.It has been found that a number of dislocations were emitted from the crack tip during deformation.After that,the dislocations moved rapidly away from the crack tip,which indi- cated that they were strongly repelled by the stress field at the crack tip.Between the crack tip and the plastic zone there is a region of dislocation-free, which is referred to as dislocation-free zone (DFZ). The length of DFZs is roughly estimated 100 μm which is much longer than that found in thin foil specimen.The variation of dislocation density as a function of the distance from the crack tip was measured,which showed that the dislocations are inversely piled up in the plastic zone.The length of DFZs increased with both the length of pre-crack and the amplitude of applied stress.

A WEIGHTED RESIDUAL METHOD FOR ELASTIC-PLASTIC ANALYSIS NEAR A CRACK TIP AND THE CALCULATION OF THE PLASTIC STRESS INTENSITY FACTORS

In this paper, a weighted residual method for the elastic-plastic analysis near a crack tip is systematically given by taking the model of power-law hardening under plane strain condition as a sample. The elastic-plastic solutions of the crack lip field and an approach based on the superposition of the nonlinear finite element method on the complete solution in the whole crack body field, to calculate the plastic stress intensity factors, are also developed. Therefore, a complete analvsis based on the calculation both for the crack tip field and for the whole crack body field is provided.

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material （FGM） plate by Reissner＇s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson＇s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner＇s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner＇s effect when the in-homogeneity parameter approaches zero.

AN APPLICATION OF SPI TECHNIQUE FOR STUDING STRA DISTRIBUTION AT CRACK TIP

A computerized digital speckle-interferometry(SPI)system has been set up to determine the in-plane distribution of displacement and strain at crack tip of CF specimens.The principle and the experimental method of this technique are described in detail.The effect of corrosion factors on plastic deformation at crack tip in CF process has been examined by comparing the near-tip strain fields of steel A537 before and after hydrogen charging,and of pure copper before and after applying an anodic current in 3.5% NaCl. The results shows that the clasic finite element mathematical models are not suitable to describe the near-tip deformation under the experimental conditions.Hydrogen charging made the crack tip plasticity decreased for steel A537 and the anodic dissolution enhance the near tip deformation of pure copper specimen.

Crack tip higher order stress fields for functionally graded materials with generalized form of gradation

A generalized form of material gradation applicable to a more broad range of functionally graded materials(FGMs) was presented.With the material model,analytical expressions of crack tip higher order stress fields in a series form for opening mode and shear mode cracks under quasi-static loading were developed through the approach of asymptotic analysis.Then,a numerical experiment was conducted to verify the accuracy of the developed expressions for representing crack tip stress fields and their validity in full field data analysis by using them to extract the stress intensity factors from the results of a finite element analysis by local collocation and then comparing the estimations with the existing solution.The expressions show that nonhomogeneity parameters are embedded in the angular functions associated with higher terms in a recursive manner and at least the first three terms in the expansions must be considered to explicitly account for material nonhomogeneity effects on crack tip stress fields in the case of FGMs.The numerical experiment further confirms that the addition of the nonhomogeneity specific terms in the expressions not only improves estimates of stress intensity factor,but also gives consistent estimates as the distance away from the crack tip increases.Hence,the analytical expressions are suitable for the representation of crack tip stress fields and the analysis of full field data.

THE CRACK TIP FIELDS OF ORTHOTROPIC COMPOSITE PLATE UNDER BENDING

In this paper, mechanics analysis for crack tip fields of linear elastic orthotropic composite plate under symmetric bending load was done. By using a complex variable method, the equations for bending moment, twisting moment, stress, strain and displacement fields near crack tip are derived.