Arbitrary Stress Field Research Articles

Stress Intensity Factors for Corner Cracks under Arbitrary Stress Fields in the Finite Plates Based on the Point Weight Function Method

Probabilistic damage tolerance assessment is an essential method to evaluate the safety of aeroengine rotors. The stress intensity factors (SIFs) are the core parameters. The weight function method can calculate SIF efficiently. The available weight functions for corner cracks are suitable for cracks with universal stresses. However, when cracks are under bivariant stress distributions, the lack of the weight function database makes the damage tolerance assessment impractical. Therefore, this paper derives the point weight functions for corner cracks, which are suited for cracks in real-world components with two-directional stress distributions. The response surface method is used to build the surrogate model of the point weight functions. By integrating the weight functions with the stress distributions, SIFs can be calculated with high accuracy. In sum, 81% of the differences between the point weight function method and finite element results are less than 10%.

A parallel discrete dislocation dynamics/kinetic Monte Carlo method to study non-conservative plastic processes

Non-conservative processes play a fundamental role in plasticity and are behind important macroscopic phenomena such as creep, dynamic strain aging, loop raft formation, etc. In the most general case, vacancy-induced dislocation climb is the operating unit mechanism. While dislocation/vacancy interactions have been modeled in the literature using a variety of methods, the approaches developed rely on continuum descriptions of both the vacancy population and its fluxes. However, there are numerous situations in physics where point defect populations display heterogeneous concentrations and/or non-smooth kinetics. Here, a kinetic Monte Carlo (kMC) approach for modeling vacancy transport in response to arbitrary stress fields is used. Vacancies are treated as point particles and are coupled to the dislocation substructure representing a deformed material via an advection term defined by the local stress gradients. The stress fields and the dislocation substructure are evolved using a discrete dislocation dynamics (DDD) module. To extend the coupled model to the treatment of large systems, we have implemented it in the massively-parallel DDD code ParaDiS. To avoid numerical incompatibilities associated with merging deterministic (DDD) and stochastic (kMC) integration algorithms, we cast the entire elasto-plastic-diffusive problem within a single stochastic framework, taking advantage of a parallel kMC algorithm to evolve the system as a single event-driven process. The large-scale implementation enables the study of the evolution of a variety of dislocation-defect scenarios governed by non-conservative transport kinetics. After carrying out an exhaustive numerical and computational analysis of our parallel algorithm, we show results that emphasize situations where inhomogeneous vacancy dynamics are of relevance, and compare discrete kinetics to continuum solutions for several cases.

Accurate and efficient method for analyzing mixed-mode SIFs for inclined surface cracks in semi-infinite bodies by using numerical influence function method

An effective technique to compute mixed-mode stress intensity factors (MM-SIFs) for inclined surface cracks in semi-infinite bodies under arbitrary stress fields is proposed and validated. The proposed technique in this study is accurate and handiness for calculating MM-SIFs of different crack angles by using only one non-inclined cracked model. The benefit of this technique is achieved by using the influence function method (IFM) to evaluate MM-SIFs. A fully-automated influence coefficient database (ICDB) calculation system is developed based on the IFM. The interaction integral method (IIM) including the crack face traction (CFT) integral, which can improve the accuracy of SIF solutions, is employed in the developed ICDB. Once the ICDB for unit distributed load (UDL) is established for a target finite element (FE) model, MM-SIFs under arbitrary stress distributions of different crack angles can be calculated accurately in a short time. The developed ICDB is validated by comparing mode-I and MM-SIF solutions obtained by IFM with well-established reference solutions for different crack angles and Poisson’s ratios. The effect of Poisson’s ratio on MM-SIFs is also discussed based on the SIF solutions.

Application of critical distances to fatigue at pores

AbstractThis work applies Taylor's theory of critical distance to quantify the effect of defects on fatigue initiation in an additively manufactured metal. We focus on hollow pores that are ideal spherical, prolate, and oblate spheroids isolated in an otherwise homogeneous linear‐elastic material. These conditions support the development of exact solutions using the exterior Eshelby tensor for a pore in a remote, arbitrary stress field. For spheres, this solution process admits simple closed‐form solutions for principal stresses disturbed by the pore. For prolate and oblate spheroids, we present the solutions as graphical curves showing stress variations under uniaxial tension. This report then extends the analysis to determine the effect of defects on a parametric, power‐law stress range vs fatigue life model. By propagating the distributed stress fields through this model, this study demonstrates the effect of pore size, pore shape, stress, and parametric fatigue properties on the life reduction due to porosity. These results suggest several approaches to increasing fatigue lives in porous materials, eg, reducing the pore size, promoting spherical pores, and increasing the microstructural parameter (comparable to the El Haddad parameter). Results presented in this work may be useful to inform trends of fatigue strength and fatigue initiation lives in metallic alloys with limited porosity, eg, additively manufactured materials that have been HIP'ed.

影響関数法による任意分布応力場における表面き裂の応力拡大係数と疲労き裂進展評価手法の開発

We present an evaluation method of the crack growth rate and behavior for a semi-elliptical surface crack in a plate subjected to arbitrarily stress distribution. To calculate stress intensity factors, the influence coefficients were developed on the basis of the influence function method with the finite element method. Although the arbitrarily stress distribution such as weld residual stress on the crack surface are difficult to approximate by using a polynomial formula, the stress intensity factor could be calculated by direct mapping of arbitrary stress field obtained from doing finite element analysis of the crack surface. To determine whether the evaluation method based on the influence function method was valid, fatigue crack growth tests were conducted for flat plates with a semi-elliptical surface crack. Flat plate specimens had a distributed residual stress by the non-filler welding before the fatigue crack growth tests. The results of a comparison of estimations and the tests show that both the growth rate and behavior agreed well. From the calculated stress intensity factors based on the influence function method, the asymmetrical crack growth rates and behaviors can be calculated quickly.

A general numerical method to solve for dislocation configurations

The shape of a mechanically equilibrated dislocation line is of considerable interest in the study of plastic deformation of metals and alloys. A general numerical method for finding such configurations in arbitrary stress fields has been developed. Analogous to the finite-element method (FEM), a general dislocation line is approximated by a series of straight segments (elements) bounded by nodes. The equilibrium configuration is found by minimizing the system energy with respect to nodal positions using a Newton-Raphson procedure. This approach, termed the finite-segment method (FSM), confers several advantages relative to segment-based, explicit formulations. The utility, generality, and robustness of the FSM is demonstrated by analyzing the Orowan bypass mechanism and a model of dislocation generation and equilibration at misfitting particles. Energy differences from previous analytical methods based on simple loop shapes are significant, up to 80 pct. Explicit expressions for the coordinate transformations, energies, and forces required for numerical implementation are presented.

Shear cracks in thermoplastic and poroelastic media

T o understand the mechanism of quasi-static plane strain fracture in poro- and thermoelastic materials for arbitrary stress fields, it is necessary to consider not only tensile fracture but also shear fracture. An impulsively sheared semi-infinite crack is considered for the mathematically analogous cases of coupled quasi-static thermo- and poroelastic materials. The exact solution is obtained in Laplace and Fourier transform domains using the Wiener—Hopf technique. The crack tip behaviour is then analysed. The stress intensity factors as a function of the Laplace transform variable are identified for cracks with either permeable (conducting) or impermeable (insulating) crack faces, and are inverted numerically for all times and asymptotically for small times. The case of a steadily propagating shear crack with either permeable or impermeable crack faces is also examined; the crack tip behaviour is examined in detail and compared with the result for a permeable fault. Analytical results are found in the neighbourhood of the crack tip. The pore pressure fields are found explicitly for all x and y.The case where the entire fault is assumed impermeable is reworked and an analytical solution is given for the pore pressure. The relevance of the results for stabilising shear faults and earthquake mechanics is briefly discussed. For the most part, the solutions in this paper and an earlier one refer to situations involving a complete coupling of the poroelastic equations and boundary conditions. This occurs when the material ahead of the crack is continuous and is relevant to the fracture of “virgin” rock. In the other papers cited in this article and elsewhere this has not usually been the case; the interface along which the crack propagates has usually been assumed to have a particular property as far as the pore pressure is concerned. It is worth stressing that the most complete situation is considered here; the pore pressure condition ahead of the crack is set only by the anti-symmetry (or symmetry) of the problem and the condition on the crack faces is set by the physical situation.

A Proposed K1 Solution for Long Surface Cracks in Complex Geometries

The finite element method is used to derive a general stress intensity factor expression for straight-fronted and circumferential surface cracks subjected to an arbitrary stress field. The superposition technique is used: the stress field normal to the crack face is resolved into uniform tension and linear bending stress distributions. The K1 expression is given in terms of magnification factors that are function of these two types of stress field, the geometry considered and the opening of the crack at the free surface under loading. The results obtained using this method are in good agreement with other numerical and experimental K1 solutions of cracks under simple or complex conditions.

Propagation of fluid-driven fractures in jointed rock. Part 1—development and validation of methods of analysis

We have developed a new method of analysis to describe the propagation of induced fluid-driven fractures in rock masses which contain pre-existing discontinuities such as joints, bed interfaces, lens boundaries, etc. The analysis is based on a 2-D model, with coupled solid mechanics, fracture mechanics and fluid mechanics. The solid and fracture mechanics are solved by an implicit finite-element approach which provides for mixed-mode (I and II) fracture propagation in arbitrary stress fields. The fluid mechanics capability first was established for steady-state conditions, using a finite-element formulation for flow between parallel surfaces. This initial coupling resulted in the version 1.0 of the FEFFLAP model (Finite Element Fracture and Flow Analysis Program). This version was verified against analytical solutions, and tested for validation against results of hydrofracturing in blocks of rock simulants containing an interface, under biaxial loading, as described in Part 2, the companion paper to this publication. The developments were then extended to the time-dependent domain by replacing the original fluid flow model with a model based on the FAST fluid dynamics module. The pressure profile inside the crack and the crack velocity are provided by a 1—D analytical formulation, corresponding to a constant-height fracture. Both arbitrary flow rate and borehole pressure conditions can be simulated; these may correspond, respectively, to conventional two-wing hydrofracturing and gas-driven tailored-pulse loading for multiple fractures. The model has been verified against analytical time-dependent solutions for fracturing in permeable and impermeable media. The new coupled model also has been validated against controlled physical tests involving hydrofracturing of blocks containing sandstone lenses, with the blocks loaded independently in three orthogonal directions. These tests are also described in Part 2 (Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 27, 255–268, 1990). These new analysis tools can be used for a wide variety of applications such as obtaining a better understanding of the stimulation of unconventional gas reservoirs, i.e. lenticular sandstones, coal beds and Devonian shales, or making improvements in the design of underground waste disposal by hydrofracturing or enhancing the fracturing og geothermal reservoirs.

Stress intenstty factors for cracks at holes in arbitrary stress fields

The value of Green's function (or weight function) techniques for calculating stress intensity factors in fracture mechanics has been demonstrated [i] and is now a well-established technique. Grandt [2] has studied the problem of a radial crack of length i in an arbitrary stress field o(x) at the edge of a circular hole of radius R. He showed that if the stress could be represented as a simple polynomial,

Cyclic thermal loading in the creep range—II. High levels of thermal loading

An upper displacement bound is presented for thermal loading problems that allows energy to be dissipated as a result of the cyclic variation of strain in parts of the structure. Use of the bound requires a knowledge of the thermo-elastic stress distribution resulting from the cycle of temperature and arbitrary stress fields that are in equilibrium with the primary load and a dummy load applied in the direction of the required displacement rate. Certain limitations are imposed on the choice of equilibrium stress fields and, as a result, the bound can only be applied to a limited range of problems. Situations for which the bound can and cannot be used are examined.The bound is used to obtain a solution for the plate problem originally analysed by Bree [10]. The results are compared with the calculations of O'Donnell and Porowski [11] who adopt a different material model.

The elastic field around an elliptical crack in an anisotropic medium under an applied stress of polynomial forms

A general form is presented for the stress disturbance caused by an elliptical crack in an anisotropic medium under an arbitrary stress field in the form of polynomials. Somigliana's dislocation method is employed for analysis as Willis [4] did, but a different integral process is taken. The results are expressed in the integral forms defined on the subspace of the surface of a unit sphere. The following theorem is proved: If the displacement discontinuity of the elliptical Somigliana's dislocation has the form P N(x′ 1,x′ 2) (1−x′ 1 2/a 1 2−x′ 2 2/a 2 2) 1/2 where P N(x′ 1,x′ 2) is a homogeneous polynomial of degree N in the coordinates x′ 1 and x′ 2 and a 1 and a 2 are the semi-axes of the elliptical Somigliana's dislocation, the stresses on the plane of the Somigliana's dislocation are inhomogeneous polynomials in the coordinates, whose terms are of degree N, (N − 2), (N − 4),4.

Direct limit analysis via rigid-plastic finite elements

This work concerns the limit analysis of rigid-perfectly plastic plane structures. The continuum is discretized into finite elements, thereby permitting the theoretical nonlinear problem to be transformed to an equivalent linear elastic problem for which the corresponding collapse load may be found in a reasonable number of analysis iterations; as such, the analysis may be termed a quasi-direct approach. The duality concepts for linear elastic-analysis by the finite element method are extended to the plasticity problem using classical variational principles; in this way, both primal and dual quasi-direct approaches to the limit analysis problem are identified. Generalization of the variational principles permits the analysis to be formulated for not only compatible elements or equilibrium finite elements, but also for hybrid elements having two independent fields (i.e. stress and strain). Some numerical examples concerning classical academic problems for plane structures illustrate the efficiency of the method. Finally, a new hybrid element with an arbitrary stress field in the interior and a quadratic displacement field on the boundary is described in detail in an appendix.

Analytic solution for linear dislocation arrays in an arbitrary stress field

Applying the continuum approximation, the behaviour of a dislocation pile-up in an arbitrary periodic stress field σ( x) is examined. The distribution function, aswell as the length of pile-up the individual dislocation positions and stresses outside the pile-up that results from it, can be derived in an analytical way. The main result of this work shows that the dislocation distribution depends not only on the kind of σ( x) function but also additionally on the position of σ( x) relative to the pile-up. Therefore, in the case where σ( x) is given by a single arbitrary periodic stress field, the phase shift ϕ x of the stress amplitude of σ( x) with respect to the center of pile-up appears as a parameter only. Further, the conditions for splitting of pile-ups depends also on the wavelength and stress amplitude of the stress field additionally from ϕ x . The consequences of these results are discussed and compared with earlier results of an asymmetric periodic as well as uniform stress field.

Statistical Considerations on the Strength and Fracture of Rocks

A considerable difference between the fracture stresses theoretically calculated and those actually observed is usually explained by the presence of minute flaws, the so-called Griffith cracks, in the material.Assuming that the fracture is caused by the stress concentration around Griffith cracks, a statistical theory of brittle fracture is given which could be applied to arbitrary stress field. In this theory, the distribution of strength of test piece is related to the distribution of shapes of Griffith cracks.In this paper, it is supposed that the probability density function of the shape of Griffith cracks isf(ξ0)=δkδ/ξ0(δ+1)·exp{-(k/ξ0)δ}where ξ0 is the index of the shape or flatness of Griffith cracks.Based on this probability density function of the shape of Griffith cracks and some assumption concerning the stress concentration around cracks, the size effects of the uniaxial tensile and compressive strengths are calculated and given in the equations which coincide with the formula derived from Weibull's distribution function, that is, St=Ct·k·s·n-1/mfor the uniaxial tensile strength of the test piece which contains n cracks, where Ct is a constant, and s is the theoretically calculated fracture stress.In the calculation of the uniaxial compressive strength, besides Griffith's theory, a modified Griffith theory by McClintock and Walsh is considered.As the result of these calculations, the relations between the parameter of the distribution function of the shape of Griffith cracks δ and Weibull's coefficient of uniformity m are given in graphs. And the relations between the parameter δ and the“brittleness index”(the ratio between the uniaxial compressive and tensile strength) is given in another graph.Using these graphs, the dispersions of the actually observed uniaxial compressive and tensile strengths of rocks are discussed.To conclude, the following suggestions are given:(1) The brittleness index is not always a constant, but is a function of the distribution of shapes of Griffith cracks.(2) Besides the“inherent”dispersion discussed in this paper, there may be some experimental errors. Based on the theoretical aspect given in this paper, it is possible to evaluate the difference of experimental errors between the uniaxial compressive and tensile strength.