The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid are investigated under plane strain and plane stress conditions. In each case, by expanding the plastic fields and the governing equations in the coordinate y, the problem is reduced to solving a system of nonlinear ordinary differential equations which is similar to that of mode III derived by Achenbach and Z.L.Li. An approximate solution for small values of x is obtained and matched with the elastic field of a blunt crack at the elastic-plastic boundary. The crack growth criterion of critical strain is employed to determine the value of K _II of the far-field that would be required for a steadily growing crack.     

On the formulation of plane strain problems for elastic perfectly-plastic medium

Starting from the 3-dimensional basic equations for elastic perfectly-plastic medium, we have derived in this paper the strain-history-dependent yield conditions and constitutive equations for plane strain problems. For the elastic-plastic evolutionary problems with time-varying boundaries, two contiguity conditions for displacements are given in simple forms. It is proved that for the case $\text{v =}\text{1}\text{2}\text{(v—Poisson's ratio)}$, the equations are hyperbolic, and so, some discontinuities of solutions may occur inside the plastic region. On the other hand, when $\text{v <}\text{1}\text{2}$, the equations are elliptic, and so, the solutions will be sufficiently smooth inside the plastic region, and discontinuities can occur only across the boundaries between the elastic and the plastic regions. As $\text{v →}\text{1}\text{2}$, there will occur inner boundary layers within which abrupt transition takes place. The order of magnitude of the width of the inner boundary layer is estimated. It is proved that the stress discontinuities are impossible disregarding the exceptional case of the traces of elastic cores which shrinks to a line in the limiting state. The various relations for the discontinuity gaps (or jumps) are given, both for the case of weak and the case of strong discontinuities. Finally, the basic equation and the corresponding four contiguity conditions are formulated for the stress-rate function, and a theorem for unloading is proved. The results of this paper has been successfully applied to the analysis of the elastic-plastic near-tip fields of cracks. [4].     

Stress in the vicinity of a griffith crack at the interface of a layer bonded to a half plane—an approximate method

Approximations to the stress field in the vicinity of a Griffith crack located at the interface of a layer bonded to a dissimilar half plane are determined. A systematic use of Fourier transforms reduces the problem to that of solving a set of simultaneous dual integral equations with trigonometric kernels and weighting functions. This latter problem is reduced to the solution of an uncoupled pair of singular integral equations. An approximate technique using Legendre polynomial expansions is discussed. The analysis shows that when a constant pressure is applied to the faces of the crack, the stress components have the distinctive oscillatory singularities at the crack tip. Expressions up to the order of $\text{h}{}^{\mathrm{?4}}$, where $\text{h}$ is the thickness of the layer and is much greater than 1, are derived for the stress components.     

Precise elastic-plastic analysis of crack line field for mode II plane strain crack

The near crack line analysis method has been used to investigate the exact elastic-plastic solutions of a mode II crack under plane strain condition in an elastic-perfectly plastic solid. The significance of this paper is that the assumptions of the conventional small scale yielding theory have been completely abandoned. The inappropriateness of matching conditions formerly taken at the elastic-plastic boundary ths been corrected as well. By eatching the general solution of the plastic stress (but not the special solution that was adopted) with the exact elastic stresses (but not the crack tip K-dominant field) at the elastic-plastic boundary near the crack line, the plastic stresses, the length of the plastic zone and the unit normal vector of the elastic-plastic boundary, which are sufficiently precise near the crack line region, have been given. The solutions are suitable not only under the condition that the plastic region is sufficiently small but also under the condition that the plastic region is large.     

Three-dimensional elastic-plastic finite element analysis of small surface cracks

Three-dimensional, elastic and elastic-plastic finite element analysis of small surface cracks was performed. The elastic analysis is in good agreement with other solutions. For a round surface with a radius equal to six times the crack depth, the $\text{K}$ at the surface is about 4% higher than the $\text{K}$ for a flat surface. The results of the elastic-plastic analyses show a unique variation of the effective $\text{K}$ (J-integral) along the crack front with a decrease in $\text{K}$ at the surface due to a lack of plane strain constraint and an increase in the effective $\text{K}$ at the maximum depth point with increasing plasticity. Similar behavior was observed for a semielliptical crack. Increasing the strain hardening exponent from 10 to 20 produced similar results with slightly higher effective A's for high applied strains. These results are useful in understanding the fracture behavior of small surface cracks.     

The plastic stress ahead of a mixed mode I and II plane stress crack

The small scale yielding for mixed mode I and II plane stress crack problems in elastic perfectly-plastic solids is analysed by considering the stress field near the crack line. By expanding the stresses near the crack line and matching the stress field in the plastic zone with the elastic dominant field for a blunt crack near the crack line at the elastic-plastic boundary, the problem is reduced to solving a system of nonlinear algebraic equations. The relationship between the near-field mixity parameter M^p and the far-field mixity parameter M^e is detennined by solving the system of equations numerically. Analogous to Shih's calculation by the finite element method for the small scale yielding of mixed mode plane strain crack problems, the numerical results indicate that the shift from a mixed mode to a pure mode may not be a smooth one.     

Vertical crack inside a semi-infinite plane

In this paper we consider the problem of determining the stress-intensity factors and the crack energy in a semi-infinite plane containing an inside crack perpendicular to the straight boundary of the plane. By the use of Mellin transform, we reduce the problem to solving a single singular integral equation. Approximate solution of the integral equation is obtained as a series of Chebyshev polynomials of the first kind. The coefficients $\text{B}{}_{\mathrm{n}}$ of the series are determined from a system of linear algebraic equations. Expressions for the stress-intensity factors at the edges of the crack, the shape of the crack and the crack energy are derived in terms of the coefficients $\text{B}{}_{\mathrm{n}}$. The numerical values of these quantities have been displayed graphically for three particular cases.     

The crack tip opening angle (CTOA) of the plane stress moving crack

Recently, Crack Tip Opening Angle (CTOA) was proposed by C.F. Shih et al. to describe the instability criterion of ductile crack propagation during plane strain (flat crack) conditions, and was derived by J. R. Rice analytically by means of the slip line field theory and the incremental theory of plasticity. CTOA appears to be applicable in (some or most) cases, but does not accurately describe the plane stress growing crack (slant crack).Unstable ductile crack propagation of the plane stress crack is widely studied for the safe design of highly pressurized gas pipelines. The impact absorption energy of the Charpy test is well correlated to the fracture arresting properties of the structures, but the mechanics of the fracture are not yet well established.In this paper, CTOA of the plane stress growing crack is derived from the plane stress plasticity of perfectly plastic materials by Sokolovsky's approach. Our proposed modification of CTOA expressed as follows: $\text{CTOA}\text{= (α/δ}{}_0\text{)(}\text{d}\text{J/}\text{d}\text{l) + β(δ}{}_0\text{/E)}\text{ln}\text{(eR/r)}$ where $\text{β = 1.40}$ under the plane stress conditions.CTOA in the Dugdale model is also defined and compared with the results of laboratory test. The results show that $\text{α = 0.5}$, and $\text{β = 1.27}$ for plane stress crack growth. These analyses give similar results to those obtained by Rice et al. for CTOA under plane strain conditions, that is, $\text{α = 0.65}$ from the experimental results and $\text{β = 5.08}$ from the slip line theory.The CTOA obtained for plane stress ductile crack growth is applied to the wide plate tensile crack growth test. The results of the present analysis coincide well with those of the plane stress finite element method (FEM) computed by T. Kanazawa et al. The phenomena of plane stress ductile crack propagation are also explained by the CTOA criterion under plane stress conditions.     

Near-tip behavior of a crack in a plane anisotropic elastic body

The asymptotic form of the stress and displacement components near the tip of a straight crack in a generally rectilinear anisotropic plane elastic body are resolved. As in the isotropic analysis, the solutions for the stresses display a $\text{r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ dependence, where r is the distance from the tip, while the angular dependence depends upon the anisotropy in a complicated way. The effect of some special anisotropies upon these solutions is fully explored. Finally, these solutions are used to solve the problem of a finite length straight crack in an anisotropic elastic plane when uniform stresses are applied far from the crack. This solution includes obtaining the stress intensity factors, and the nature and magnitude of the crack face displacements.     

An investigation of the edge-sliding mode in fracture mechanics

A boundary collocation procedure has been applied to the Williams stress function to determine the elastic stress distribution for the crack tip region of a finite, edge-cracked plate subjected to mode II loading at the crack tips. The asymmetric specimen selected was particularly suitable for the determination of plane strain fracture toughness for mode II loading. Numerical solutions for stress intensity factors for the edge-sliding mode obtained by the boundary collocation method were in close agreement with values obtained from photoelastic experiments.Fracture tests of several compact shear specimens of 2024-T4 aluminum were conducted in order to experimentally investigate the behavior of the edge-sliding mode. In each case a brittle shear failure was observed and mode II fracture toughness values were obtained. The average value for K_IIc obtained from two tests was 39.5 ksi(in)^{$\text{1}\text{2}$}. No K_Ic. data for 2024-T4 were available for comparison purposes; however, K_Ic values for a similar alloy, 2024-T351, have been reported as 34ksi(in)^{$\text{1}\text{2}$} which is only about 15 per cent below the corresponding K_IIc value.     

Influence of biaxial loading on analysis of cracked stiffened panels

The elastic and elastic-plastic analysis of a biaxially loaded, stiffened panel has been carried out using the finite element method. The elastic-plastic analysis was accomplished assuming Prandtl-Reuss material behavior. The influence of biaxial load ratios on crack openings, stresses in the stringer, plastic zone size and $\text{(J)}$ was studied. The analytical results of crack openings and stresses in the stringer are compared to the experimental results. It is found that contrary to the behavior of unstiffened structures, the $\text{(J)}$ values and crack openings increase for positive biaxial load ratios in cracked stiffened structures.     

Stress intensity factor at a bilaterally-bent crack in the bending problem of thin plate

An infinite plate with an asymmetric bilaterally-bent crack is analyzed as a bending problem of a thin plate. Stress distributions and stress intensity factors are obtained for some angles and length of bent crack. These are obtained for the some Poisson's ratio. Influence of the initial crack width on the stress intensity factor are also investigated. Three loading conditions are taken into consideration: uniform out of plane bending at infinity in the $\text{x}$ and $\text{y}$ directions and uniform out of plane twist. The rational mapping function in the form of a sum of fractional expressions and the complex variable method are used for the analysis.     

Annular punch on an elastic layer overlying an elastic foundation

The problem solved here is the axisymmetric mixed boundary value problem of the isotropic homogeneous theory of elasticity, in which the normal displacement is specified inside an annular area $\text{a ≤ r ≤ b}$, the normal stress is zero in $\text{r < a, r \# b}$ and the shearing stress is zero on the whole face $\text{z = ?h}$, the upper face of the elastic layer; the continuity of the normal and radial displacements and the normal and shearing stresses is assumed at the interface $\text{z = 0}$ between the elastic layer and the elastic foundation having different elastic constants. The problem is reduced to the solution of a Fredholm integral equation of the first kind. The Fredholm integral equation is further put in terms of four simultaneous Fredholm integral equations of the second kind in four unknown functions. The iterative solution of these integral equations has been obtained for $\text{epsi = b/h ? 1}$, and $\text{λ = a/b ? 1}$ for the case of an annular cylindrical punch. The expressions for the normal stress $\text{σ}{}_{\mathrm{zz}}\text{(r, ?h)}$ for $\text{a ≤ r ≤ b}$ and the total load $\text{P}$ on the punch have been obtained.     

The analysis of stress-strain field in a finite plate with a blunt crack

The expressions of stress-strain field in a finite plate with crack-tips of different radii of curvature have been derived by the method of conformal mapping-weighted residues. The stress-strain field has been evaluated for the LY12-M plate of dimensions 300mm × 150mm × 2mm under uniaxial tension. Values of σ_v computed by the present method and by Creager's formula are in agreement within 5% in the distance range of $\text{(x ? a)}\text{a}\text{?}\text{1}\text{50}$ from the crack-tip. The radius of curvature of the crack-tip has a significant effect on the stress σ_y at the tip. While it has little effect on the stress σ_x and almost no effect on the stress σ_y in the elastic region, it has a remarkable effect in the plastic region for small-scale yielding and very little effect for large-scale yielding. The effect of the radius of curvature (for $\text{ρ ?}\text{2L}\text{250}$) on the stress intensity factor at the crack-tip is negligible.Uniaxial tensile tests on 300mm × 150mm × 2mm specimens of LY12-M with crack-tips of different radii of curvature have been performed, and strain measurements made by Moire's method and electrical-resistance gauges. Experimental results show that the radius of curvature has no effect on the strain ?_y in the elastic region; measured values of ?_x and ?_y in front of the crack from the boundary of the plastic region to the edge of the specimen agree with the computed values to an accuracy of 5%; the measured values of K_I from specimens with cracks of different radii of curvature (ρ ? 2L/250) are consistent with the computed values.     

Free vibrations of a piezoelectric layer of hexagonal (6mm) class

An asymptotic method due to “Achenbach” is used to analyze the free vibrations of a piezoelectric layer of hexagonal (6 mm) class. In this method, the displacement components, the electric potential and the frequency are expressed as power series of the dimensionless wavenumber ? = 27π × Layer Thickness/Wavelength. Substituting the expansions of field variables and the frequency in the field equations of piezoelectricity and in the boundary conditions, a system of coupled, second order, inhomogeneous, ordinary differential equations with thickness variable as the independent variable is obtained by collecting the terms of same order ^?n. Integration of such systems of differential equations yields the various terms in the series expansions for the field variables and the frequency, for all modes and in the whole range of frequencies, in a range of the dimensionless wavenumber $\text{0 < ? < ?}{}^1\text{< 1}$ where $\text{?}{}^1$ increases as more terms are retained in the expansions. The frequency coefficients reduce to the corresponding ones in the elastic case as a limit. The exact frequency equation in the case of plane strain is obtained and analyzed numerically. The results thus obtained are compared with those obtained in the asymptotic method. The results fairly agree upto three decimal places.     

Penny-shaped crack in a sphere embedded in an infinite medium

We consider the problem of determining the stress intensity factor and the crack energy in an Isotropie, homogeneous elastic sphere embedded in an infinite Isotropie, homogeneous elastic medium when there is a diametrical crack in the sphere. We assume that the crack is opened by an internal pressure and the sphere is bonded to the surrounding material. The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function φ($\text{t}$). Expressions for the stress intensity factor and the crack energy are obtained in terms of φ($\text{t}$). The integral equation is solved numerically and the numerical values of the stress intensity factor and the crack energy are graphed.     

Quasistatic thermal crack growth in unidirectionally fiber reinforced composite materials

Micromechanical investigations concerning the quasistatic thermal crack propagation in self-stressed unidirectionally reinforced composite structures with a low-fiber volume fraction have been performed. Thus, in order to gain certain microstructural informations about the thermal shock resistance of those reinforced composites different cracked unit cells of several two-phase composites as well as ensembles of such composite microcomponents are considered which are subjected to thermal loading. Therein the principal facets of composite material failure theories, emphazising matrix cracking, fiber breaking and interfacial debonding, respectively, have been studied. The resulting mixed boundary-value problems of the stationary plane thermoelasticity are solved numerically by using standard finite element programs. Further, the influences of the fiber diameters, different shapes of the fiber-matrix interfaces as well as of the external boundaries of the microcomponents on the crack opening displacement $\text{u}{}_{\mathrm{y}}{}^{\mathrm{c}}$ and the opening mode stress intensity factor $\text{K}{}_{\mathrm{I}}$, respectively, have been investigated. Numerical results are given for an Al₂O₃ matrix/Molybdenum fiber composite by consideration of different crack configurations.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Stress distribution in the neighbourhood of Griffith and external cracks under general surface loadings

The problem of determining the stress distribution in an infinite elastic medium, containing Griffith cracks and the external cracks, is considered under the assumptions of plane strain. The crack surfaces are subjected to completely arbitrary surface tractions and the cracks are located on a straight line (say, x-axis).Detailed consideration is given to the cases when the elastic medium contains, (1) a Griffith crack ($\text{0 < ¦x¦ < a, y = 0}$); (2) a pair of symmetrically located Griffith cracks ($\text{0 < a < ¦x¦ < b, y = 0}$); and (3) a pair of external cracks ($\text{0 <a < ¦x¦, y = 0}$). The method of Fourier transform is used to analyse the problem and the results are compared, in special cases, with the known results in the literature.