Strain energy density fracture criterion in elastodynamic mixed mode crack propagation

Sih's fracture criterion based on strain energy density, S, for mixed mode crack extension under static loading is extended to dynamic mixed mode, K_I and K_II, crack propagation. Influence of the second order term, σ_ox, which represents the non-singular constant stress acting parallel to the direction of crack propagation, on the S distribution surrounding the crack tip, is demonstrated. Numerical studies show that positive σ_ox enhances the fracture angle and negative σ_oxreduces the fracture angle irrespective of the sign of K_II/K_I, when S is measured at a critical distance r_c from the crack tip. This fracture criterion is verified by the crack curving results of dynamic photoelastic fracture specimens. Omission of σ_ox term leads to predicted fracture angles which are at variance with experimental data.     

Fatigue crack initiation and growth under mixed mode loading in aluminum alloys 2017-T3 and 7075-T6

Fatigue crack initiation and growth characteristics under mixed mode loading have been investigated on aluminum alloys 2017-T3 and 7075-T6, using a newly developed apparatus for mixed mode loading tests. In 2017-T3, the fatigue crack initiation and growth characteristics from a precrack under mixed mode loading are divided into three regions—shear mode growth, tensile mode growth and no growth—on the ΔK_I-ΔK_II plane. The shear mode growth is observed in the region expressed approximately by ΔK_II > 3MPa√m and ΔK_II/ΔK_I > 1.6. In 7075-T6, the condition of shear mode crack initiation is expressed by ΔK_II > 8 MPa√m and ΔK_II/ΔK_I > 1.6, and continuous crack growth in shear mode is observed only in the case of ΔK_I/ΔK_II, 0. The threshold condition of fatigue crack growth in tensile mode is described by the maximum tensile stress criterion, which is given by Δσ_θmax √2πr 1.6MPa√m, in both aluminum alloys. The direction of shear mode crack growth approaches the plane in which K_I decreases and K_II increases towards the maximum with crack growth. da/dN-ΔK_II relations of the curved cracks growing in shear mode under mixed mode loading agree well with the da/dN-ΔK_II relation of a straight crack under pure mode II loading.     

Weight function calculations for mixedmode fracture problems with the virtual crack extension technique

An efficient finite element method is presented for calculating the stress intensity factors (K_I and K_II) and the weight functions for mixed-mode cracks with one virtual crack extension. The computational efficiency is enhanced through the use of singular elements and the application of colinear virtual crack extension (VCE) technique to symmetric mesh in cracktip neighborhood. This symmetric mesh in crack-tip vicinity permits the analytical separation of strain energy release rate into G_I for Mode I and G_II for Mode II for the mixed fracture problems with the colinear virtual crack extension.

Rice's displacement derivative representation of weight function vector for symmetric crack has been extended to the mixed fracture mode at nodal location (x_i,y_i) with crack length (a) and inclination angle (β) as h_I(II)(x_i, y_i, a, β) = (H/2K_I(II)(∂U_I(II)(x_i, y_i, a, β/∂a).

This equation permits explicit determination of weight functions for the entire structure of a given asymmetric crack geometry with colinear VCE technique. The explicit weight functions for mixed fracture mode depend strongly on the constraint conditions. The method of obtaining the required stress intensity factors of a given asymmetric crack geometry, from the weight function concept under the selected constraint conditions, which are different from constraint conditions used in the available weight functions for the same crack geometry, is also presented in this paper. This is accomplished by combining the predetermined explicit weight functions with the self-equilibrium forces at their application locations. These self-equilibrium forces include both the applied surface tractions and the reaction forces induced from the constraint conditions.     

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.     

A new method for evaluation of stress intensities for interface cracks

A new method is presented for calculating the values of K_I and K_II in the elasticity solution at the tip of an interface crack. The method is based on an evaluation of the J-integral by the virtual crack extension method. Expressions for calculating K_I and K_II by using the displacements and the stiffness derivative of the finite element solution and asymptotic crack tip displacements are derived. The method is shown to produce very accurate solutions even with coarse element mesh.     

The mixed mode I/II fracture behaviour of lightly tempered HY130 steel at room temperature

The mixed mode I/II fracture behaviour at room temperature of HY130 steel tempered at 350°C has been investigated using edge-cracked bend bar specimens loaded in anti-symmetric and symmetric four point bend configurations. In all cases fracture occurred by a localized shear decohesion mechanism that could not be characterized by the stress intensity factors, K_I and K_II, but for which the crack tip displacements, δ_I, and δ_II, appear to provide a first level of characterization. The results suggest that fracture is described by a maximum shear criterion, and this is consistent with the present understanding of fibrous fracture micro-mechanisms in the material.     

The micropolar thermoelastic problem of uniform heat-flow interrupted by an insulated penny-shaped crack

The classical problem of uniform heat-flow disturbed by an insulated penny-shaped crack is solved in the context of micropolar elasticity. The mode II stress intensity factor, K_II is found to depend on two new non-dimensional parameters N and τ − N is a measure of the coupling of the displacement field with the microstructure or the medium (0 N √2) and τ is the ratio of a material characteristic length to the crack radius. K_II remains higher than its classical value when N > 0, τ > 0 and attains the classical value as N and τ vanish. A closed-form expression to K_II is obtained in the physically important limiting case of τ → 0 with N fixed. In this limit the relative increment in K_II, over its classical value, is found to be (1 − v')N² where v' is the micropolar Poisson's ratio.     

First and second order stress extrapolation for mode I and mode II edge cracks

The use of linear and second order stress extrapolation to obtain K_I and K_II in two-dimensional finite element models of a thick plate containing an edge crack was examined. Three loading cases were studied, including classical Mode I and Mode II problems and a problem of tribological contact. Linear extrapolation was observed to yield accurate predictions of K_I in cases of dominant Mode I loading. In Mode II situations, notably where the crack faces experienced compressive normal stresses, second order extrapolation was observed to improve estimates of K_II.     

The solution of a displacement discontinuity for an anisotropic half-plane and its applications to fracture mechanics

By imaginal method the solution of displacement discontinuity for anisotropic half-plane is derived in this paper. According to the Betti's reciprocal theorem, the relationship between the stress intensity factor K_I and the increment Δv/Δa is established and numerical results of K_I are obtained by the twice calculation method. Furthermore, the problem of edge crack chopping is studied in great detail, and the area of contaction and stress intensity factor K_I are found. These results are very worthwhile for research of chopping fracture in theory.     

Theoretical and experimental study of superimposed fracture modes I, II and III

The authors wish to present an all-fracture mode specimen with which it is possible to conduct fracture mechanics tests for pure mode I, pure mode II, pure mode III, as well as for all possible combinations of the above-mentioned. By means of a finite element analysis of this specimen, the stress intensity factors K_I, K_II, and K_III were computed. It was discovered that K_II and K_III are coupled for in-plane shear and anti-plane shear loading, i.e. a mixed state occurs locally. The integral mean along the crack front yields however only to a K_II factor for in-plane shear and to a K_III factor for anti-plane shear loading. Fracture experiments under mixed-mode loading, using this new specimen, demonstrate the influence of the loading type on the orientation and on the structure of the fracture surface.     

Magneto-elastic interchange in ferromagnetic alloys

The internal friction δ, exchange integral A, magnetocrystalline anisotropic constant K_I and saturation magnetization M_s of Fe–Cr–Al and Fe–Cr–Al–Si alloys annealed at 1373 and 1473 K are measured. The energy density and volume fraction of domain walls (DWs) of these alloys are calculated based on the theories of ferromagnetism and the magnetic parameters measured. The physical process of irreversible movement of 90° DWs is suggested. The results indicate the dissipated elastic energy per unit volume due to the irreversible movements of 90° DWs is equal in value to the energy density of DWs, that is γ_w/δ_w=λ_sE/2. It is an effect of magneto-elastic interchange in ferromagnetic alloys.     

Verification of various methods for fatigue notch effect estimations in case of aircraft materials

There is as yet no general agreement regarding the proper method of analyzing situations involving stress raisers, in conjunction with combined static and alternating loadings. In this investigation fatigue tests were performed in pulsating tension and tension-compression on internally-notched sheet specimens (K_T = 2–4.6) for comparison of the fatigue notch factor K_F and notch sensitivity index K_F/K_T of aluminum-alloy sheet materials with and without cladding, and tor verification of various methods of notch-effect estimation, especially in the case of pulsating tension. The ratios K_F/K_T showed, for all investigated specimens, some dependence on the notch radius r, which was more regular than the dependence of the notch sensitivity index q = f(r). An analytical formulation of the functions K_F/K_T = f(r) permitted separate consideration of the investigated cases of tension-compression and pulsating tension. The method used here is more exact than the commonly used classical method of Neuber, especially in the case of pulsating tension.     

Prediction of crack propagation in anisotropic solids

The minimum strain energy density theory, originally developed for crack growth in isotropic solids, is extended to the anisotropic case. The criterion for predicting the direction and onset of crack growth is reformulated through minimization of the ratio of the strain energy density (written as a function of the three stress intensity factors K_I, K_II and K_III) over the material critical strain energy density, assumed to have a polar variation in terms of four fracture toughnesses. Mixed models I–II and I–III crack growth are evaluated for solids with material symmetries.     

Computer simulation of martensitic textures

We consider a Ginzburg-Landau model free energy F(ε, e₁, e₂) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (ε) order parameter and quadratic e₁², e₂² in the compressional and shear strains, respectively. Random compositional fluctuations η(r) (e.g. in an alloy) are gradient-coupled to ε, ˜ − ∑_rε(r)[(Δ_x² − Δ_y²)η(r)] in a “local-stress” model. We find that the compatibility condition (linking tensor components ε(r) and e₁(r), e₂(r)), together with local variations such as interfaces or η(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective ε-ε interactions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins, and chequer-board tweed. The observed microstructure in NiAl and Fe_xPd_{1 − x} alloys can be explained on the basis of our analysis and simulations.     

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.     

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.     

A new method to calculate k1 for semi-elliptical surface cracks in finite plates

Irwin's solution of the stress intensity factor K_I for an embedded elliptical cracks was extended to solve for K_I for semi-elliptical surface cracks in finite plates. A double series was set up to express the displacement of the crack surface, and the unknown coefficients of the series were determined by the crack surface displacements of two dimensional edge cracks and center cracks. The maximum displacement was determined with an energy method. The results reflected the influence of both the relative crack depth a/t and the relative crack length c/W. The cases in which elliptical axis ratio a/c > 1 were also included.     

Statistical evaluation of fatigue crack propagation properties including threshold stress intensity factor

The relationship between fatigue crack propagation rate, da/dn, and range of stress intensity factor, ΔK, including threshold stress intensity factor, ΔK_th, is analyzed statistically. A non-linear equation, da/dn = C{(ΔK)^m-(ΔK_th)^m}, is fitted to the data by regression method to evaluate the 99% confidence intervals. Several experimental results on fatigue crack propagation properties of welded joints are compared by using these confidence intervals.     

Enhancement of delayed failure resistance by partial unloading method

The effects of two types of pre-stressing, i.e. partial unloading (0 →K_max → K) and perfect unloading (0 → K_max → 0 → K) on the delayed failure strength were investigated using pre-cracked specimens of JIS SNCM8 steel quenched and tempered, where K_max is the maximum stress intensity factor at pre-stressing, and K is the stress intensity factor under which delayed failure test is carried out.

Both pre-stress methods can markedly increase the delayed failure strength or the lower limit stress intensity factor K_ISCC. The partial unloading method is superior to the perfect unloading method in each tempering condition (200 or 400°C) and in each environment (distilled water or 3% NaCl water). The reason why K_ISCC is increased by each pre-stressing can be explained by the decrease of surface stress at crack tip, which will suppress the corrosion reaction and prevent the invasion of hydrogen atoms into the material.     

A theory of caustics for mixed-mode fast running cracks

The method of caustics (shadow spot method) has proven to be a powerful optical method to measure stress intensity factors in static and dynamic fracture mechanics problems. In this paper, a theory of caustics was developed for elastodynamically propagating cracks under inplane mixed-mode conditions. Complex potentials for the general solutions of a near-tip field which have been previously derived by the authors were used in this theoretical development. Completely analytical expressions were derived for the caustic curves as well as for the initial curves for fast running cracks under inplane mixed-mode conditions. The effects of crack velocity and mixed-mode condition on the caustic pattern and the initial curve were investigated. New procedures were also proposed for the evaluation of the dynamic stress intensity factors K_I and K_II using the overall dimensions of the caustic pattern. The method of caustics developed here enables one to study quantitatively various mixed-mode dynamic fracture phenomena such as crack branching, crack curving, and crack kinking.