Application of the Jk integral to mixed mode crack problems for anisotropic composite laminates

The J_k integral method for determining mixed mode stress intensity factors separately in the cracked anisotropic plate is developed. Stress intensity factors are indirectly determined from the value of J₁ and J₂. The J₂ integral can be evaluated efficiently from a finite element solution, neglecting the contribution from the portion of the integration contour along the crack faces, by selecting the integration contour in the vicinity of the crack tip. Using functions of a complex variable, the complete relations between J₁, J₂ and K_I, K_II for anisotropic materials are derived conveniently by selecting narrow rectangular contours shrinking to the crack tip. Compared to the existing path independent integral methods, the present method does not involve calculating the auxiliary solution and hence numerical procedures become quite simple. Numerical results to various propblems are given and demonstrate the accuracy, stability and versatility of the method.     

Mixed-mode two-dimensional crack problem by fractal two level finite element method

Fractal two level finite element method (F2LFEM) has been extended to calculate the mixed mode stress intensity factor in a two-displacement cracked body. The complete eigenfunction expansion of displacement by Williams is employed for the global interpolation function, the factors K_I and K_II can be easily computed for any arbitrary loading on any boundary. Results are obtained for some slant crack problems in finite sheets and are compared with known results where available.     

Crack surface relative displacement analysis of mixed-mode fracture mechanics problems

In this paper a unique criteria, crack surface relative displacement, is used to evaluate mixed-mode (mode I and mode II) fracture mechanics problems. Using a conic-section simulation of a crack surface, relationships among the energy release rate G, the stress intensity factors (K₁ and K₂), and crack surface relative displacement are developed. Because the crack surface relative displacement criterion makes direct use of the displacements on the crack surface, instead of the stress field in the region of the crack tip, it simplifies numerical analysis of crack problems. A finite element model of a slant-center-cracked plate is employed to demonstrate the applicability of crack surface relative displacement to mixed-mode problems. The numerical results obtained agree well with analytical solutions. In addition, it is illustrated that similar to K₁, K₂, and G (J in LEFM), crack surface, relative displacement can serve as a fracture criterion for general mixed-mode I and II fracture mechanics problems.     

In-plane bending problem of a double edge cracked plate

In this paper the boundary collocation method is used for evaluating the stress intensity factors (SIF) of a double edge cracked plate under in-plane bending. For the case with a large ratio of the plate height to the width, h/b, the results obtained compare very favorably with existing solutions for an infinite strip. Moreover, this method has been used for different finite plates, and a series of conclusions is provided for application.     

Mode I stress intensity factors and weight functions for short plates under different boundary conditions

Stress intensity factors and weight functions are available in literature predominantly for cracked components under stress boundary conditions. In order to provide weight function solutions including displacement boundary conditions, rectangular plates with different length-to-width ratios are studied using the boundary collocation method. The results are reported in the form of figures and tables. The influence of Poisson’s ratio is discussed in detail. It can be concluded from the numerical data and theoretical considerations that the mode I stress intensity factors and weight functions are independent of Poisson’s ratio ν for mixed boundary conditions at the plate ends, but depend on ν in case of pure displacement conditions.     

An effective numerical method for an interfacial crack in a finite bi-material plate

In this paper an effective numerical method is presented for analyzing the stress intensity factors associated with the stress field near a partially debonded interface in a finite bi-material plate. The strees functions are assumed such that they can represent the stress singularity at the crack tips, satisfying not only the equilibrium equations in the domain, but also the stress and displacement conditions on the crack surfaces and across the interface. Therefore, only the boundary conditions of the plate need be considered, and they can be satisfied approximately by the Boundary Collocation Method. Numerical examples demonstrated that the proposed method gives satisfactory results and has many advantages compared to other methods.     

Exploring the fracture toughness KIE and KIC of a medium-Strength steel plate using the surface-flaw test

The fracture toughness of a 30 CrMnSiA steel plate of three thicknesses (10,8 and 5 mm) and three widths (110,80 and 56 mm) has been investigated by using surface-flaw method under room temperature. It is not easy to compute the value of K_IE by the maximum applied load. But the values of K_IE and K_IC could be obtained easily, if the computation of the conditional applied load P₁₀ and P₅ based on the relative effective extension Δa/a₀ = 10% and 5% were adopted, together with the conditions of P_max/P₁₀ 1.2 and P_max/P₅ 1.3. The K_R — Δ_a curve, i.e. the resistance-curve described by the parameter K, has been plotted. The values of K_IC and K_IE are then the resistances corresponding to the real extensions of flaws of Δ/a₀ = 2 and 7%, respectively. These values so obtained are in good agreement with the computed values of K_IC and K_IE by using the conditional applied loads. The values of K_IC and K_IE so obtained are also in agreement with the value of K_IC converted from the J-integral and the effective value of K_IE computed by the maximum applied load, respectively.

An approximate relation between K_IC and K_IE has been found to be: K_IC = (0.85˜0.95)K_IE.

The requirements for the dimensions of specimens are: Thickness of plate: B 1.0(K_IC/σ_0.2)² or 1.25(K_ICσ_0.2)²]; Width of plate: 8 W/B 10, 4 W/2c 5; Effective length: l 2W.     

Experimental evaluation of stress field around crack tip by caustic method

The caustic method is an optical technique which is useful to determine stress intensity factor values. In this paper, the caustics method was applied to specimens which have an oblique crack, various thicknesses and an open notch to investigate the stress field around the crack tip. The results are summarized as follows:

1. The caustic method is a useful technique to determine the stress intensity factor values of the specimens which have an oblique crack or various thickness and an open notch.

2. The conventional theory of measurement concerning this method is effective when the initial curve r₀ is larger than the minimum initial curve r₀^min which was obtained in this study. It is observed that the values of r₀^min decrease as the ratio of K_II to K_I increases under mixed-mode loading, the one increases with an increase of thickness and notch opening angle.

3. The 3D stress field exists in the vicinity of crack tip; however, the stress state is nearly plane strain deformation in the case of mode I loading. In the case of mixed-mode loading, the stress state approximates to plane stress deformation as the ratio of K_II to K_I increases.

4. A method based on the distribution of the three-dimensional (3D) stress field is proposed to expediently yield the values of K_I using the caustic method in the case of r₀<r₀^min.

Fracture mechanical characterisation of mixed-mode toughness of thermoplast/glass interfaces

According to studies conducted by, e.g. Liechti and Chai [J. Appl. Mech. 58 (1991) 680], Yuuki et al. [Eng. Fract. Mech. 47 (3) (1994) 367] and Ikeda and Miyazaki [Eng. Fract. Mech. 59 (6) (1998) 725], a significant increase of interfacial toughness is observed, whenever the magnitude of the bond tangential shear load of the asymptotic elastic mixed-mode state is increased in either direction. Between these extremes the interfacial toughness curve exhibits a pronounced minimum, which, according to Hutchinson and Suo [Mater. Sci. Eng. A 107 (1989) 135] is believed to represent the so-called intrinsic adhesion, i.e. the failure toughness under pure local mode I loading. Within linear elasticity, the biaxial, singular near-tip solution for an open interface crack may be employed for characterising the local stress state as long as non-linearities such as, e.g. crack-wall contact and plastic flow are contained within a zone small enough compared to the extension of the singular opening-dominated fields. Then, the critical stress state is given in terms of bimaterial stress intensity factors K_1,c, K_2,c and the fracture toughness under mixed-mode loading may be expressed in terms of the critical energy release rate as a function of the mode-mixity ψ=tan⁻¹K_2,c/K_1,c. The stress intensities have to be extracted from a stress analysis of the specimen under the critical load, which in the present work is performed by means of an FE-model of the loaded sample.     

A relaxation technique for evaluating stress intensity factors by the finite element method

A simple two-step corrective technique is presented in this paper for evaluating stress intensity factors in crack problems. In the first step an approximate evaluation of the stress intensity factor was made by considering the cracked plate to be of infinite size. The stresses of the problem were relaxed by the stresses of the infinite body which corresponds to the approximate value of the stress intensity factor. The expected discrepancy in the value of SIF by the infinite plate approximation was corrected in the second step where the existing residual stresses are equilibrated at the cracked plate by using any of the conventional finite element techniques and the corrective value of the stress intensity factor is calculated by using an appropriate collocation formula. The method was applied to three typical plane problems of cracked plates with satisfactory results.     

Estimation of fracture parameters and stress field for edge cracks in finite elastically graded plates using boundary collocation

Summary The fracture parameters, stress intensity factor and T-stress are obtained for edge cracks aligned along the gradient in finite size elastically graded plates using the technique of boundary collocation. A scheme for extending the recently derived crack tip stress field for elastically graded materials is proposed. Using this extended stress field, the fracture parameters are evaluated for edge cracks subjected to far field tension and three point bending. The results for far field tension agreed well with published theoretical results over a good range of elastic gradients. The maximum shear stress calculated over the entire domain of the cracked plate using boundary collocation agrees very well with that obtained from finite element analysis. The efficacy of the extended stress field in capturing the effects of the elastic gradient on the stresses and fracture parameters is thus established in this study.     

Growth mechanism of stress corrosion cracking in high strength steel

Stress corrosion crack growth rates are measured at sveral stress intensity levels for low-tempered 4340 steel in 0.1N H₂SO₄ solution. The characteristics of the growth rates are divided into three regions of stress intensity factors: Region I near K_1SCC; Region III near unstable fracture toughness, K_1SC; and Region II, which lies between the two. K_1SCC is the value of K at which no crack growth can be detected after 240 hr.

In order to explain these experimental results, the crack initiation analysis reported in a previous paper is extended to the growth rates. A detached crack initiates and grows at the tip of an already existing crack. When the detached crack reaches the tip of the main crack, the process repeats as a new existing crack.

A relationship between crack growth rate, v, and stress intensity factor, K, is obtained as a function of b/a and a = b + d, where b is the distance from the tip of the main crack to the detached crack, and d is the ydrogen atom saturated domain.

The experimental data are in good agreement with the theoretical values in Region II when a = 0.02 mm, b/a = 0.8, c₁/c₀ = 2.8 for 200°C tempered specimens and a = 0.015 mm, b/a = 0.7, c₁/c₀ = 3.0, ρ_b = 0.055 mm for 400°C tempered specimens, where ρ_b is a fictitious notch radius. The plateau part in Region II for 400°C tempered specimens is also successfully explained by the present theory. For Region III, the value of b/a will be almost equal to 1 because v → ∞ for b/a → 1. On the other hand, for Region I, b/a will be zero, since the value of v becomes negligibly small and no crack growth is observable.     

Combination of finite element analysis and analytical crack tip solution for mixed mode fracture

A finite element program was developed which combines the analytical crack tip solution with a conventional finite element analysis and evaluates various crack tip parameters as part of the solution. This program was used to analyze cracked specimens subjected to mixed mode loading. The importance of retaining the second term of the series expansion for local stress, a contribution which is independent of the distance from the crack tip, was demonstrated. It was first shown analytically that the presence of a load applied parallel to the crack reveals itself only through this constant second term, which vanishes only for specific loading conditions. The results of the numerical analysis demonstrate that the stress intensity factor K_I is independent of the load applied parallel to the crack only when this term is included in the analytical crack tip solutions. Failure to include the constant term has the effect that K_I varies with the horizontal load. The parameter K₁₁ is independent of this load in both cases. This indicatesonce again that it is this constant term which accounts solely and entirely for the presence of a load applied parallel to the crack.     

Stress intensity factors of square crack inclined to interface of transversely isotropic bi-material

This paper analyzes a square crack in a transversely isotropic bi-material solid by using dual boundary element method. The square crack is inclined to the interface of the bi-material. The fundamental solution for the bi-material solid occupying an infinite region is incorporated into the dual boundary integral equations. The square crack can have an arbitrary angle with respect to the plane of isotropy of the bi-material occupying either finite or infinite regions. The stress intensity factor (SIF) values of the modes I, II, and III associated with the square crack are calculated from the crack opening displacements. Numerical results show that the properties of the anisotropic bi-material have evident influences on the values of the three SIFs. The values of the three SIFs are further examined by taking into account the effect of the external boundary of the internally cracked bi-material.     

A simple procedure for the accurate determination of stress intensity factors by finite element method

A simple procedure for the accurate determination of stress intensity factors K_I, K_II by the conventional finite element method is proposed. The first step of the method is to calculate the stress σ₂ of the plate without a crack. The second step is to calculate the stress σ_tip, of the plate with the crack. The value of (σ_tip−σ_g) at the crack tip element is regarded to have the intimate relation with K_I, K_II K_I, and K_II are determined from the value of (σ_tip−σ_g) and a standard solution. It is shown that the results obtained for many problems by the proposed method are in excellent coincidence with the analytical solutions. The error is below 1–3% for the most cases.     

Several intact or broken stringers attached to an orthotropic sheet with a crack

Several intact or broken stringers which are continuously attached to a cracked orthotropic sheet through an adhesive are considered. The effect of orthotropy on the stress intensity factors is investigated. The stringers are assumed to be partially debonded due to high stress concentrations. The shear stress distribution between the stringers and the plate and the stress intensity factors are obtained from an integral equation which represents the continuity of displacements along the bond lines.     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.     

Computation of membrane and bending stress intensity factors for thin,cracked plates

The crack tip stress fields for plate bending and membrane loading problems are reviewed and the four stress intensity factors that determine these fields are defined. These four stress intensity factors arise from use of Kirchhoff plate theory to account for the bending loads and two dimensional plane stress elasticity to account for the membrane loads. The energy release rate is related to the stress intensity factors and to the stress resultants of plate theory. Virtual crack extension, nodal release and modified crack closure integral methods are discussed for computing components of the energy release rate from finite element analyses of cracked plates. Sample computations of stress intensity factors for single and mixed mode cases are presented for a crack in an infinite plate. Sample computations of stress intensity factors for a double edge notched tension-torsion test specimen are given as well.School of Civil and Environmental Engineering, Cornell University     

Evaluation of the T-stress and stress intensity factor for a cracked plate in general case using eigenfunction expansion variational method

This paper investigates the T-stress and stress intensity factor for a cracked plate in general case. In the general case, the shape of boundary and the applied loading are arbitrary. The eigenfunction expansion variational method (EEVM) is developed to evaluate the T-stress and stress intensity factor. For the traction boundary value problem, the EEVM is equivalent to the theorem of least potential energy in elasticity. Therefore, the EEVM possesses a clear physical meaning and it does not depend on any boundary collocation scheme. Several numerical examples are presented, which include: (1) a line crack in circular plate and (2) a line crack in rectangular plate. Numerical examination for convergence in an example is carried out.     

Analysis of cracked anisotropic plates using the hybrid finite element method

This paper presents a convenient and efficient method to obtain accurate stress intensity factors for cracked anisotropic plates. In this method, a complex variable formulation in conjunction with a hybrid displacement finite element scheme is used to carry out the stiffness and stress calculations of finite cracked plates subjected to general boundary and loading conditions. Unlike other numerical methods used for local analysis such as the boundary element method, the present method results in a symmetric stiffness matrix, which can be directly incorporated into the stiffness matrix representing other structural parts modeled by conventional finite elements. Therefore, the present method is ideally suited for modeling cracked plates in a large complex structure.