Solutions for a viscoelastic axisymmetric plane problem involving time-dependent boundary regions under mixed boundary condition

The stress and displacement fields for a viscoelastic axisymmetric plane problem involving time-dependent boundary regions under mixed boundary condition are presented in this paper. The viscoelastic fields are determined by solving two unknown functions; one is governed by a resulting second kind Volterra integral equation from the stress condition at the outer radius of an annular region, which is dependent on the other by the displacement condition at the inner radius. The integral equation has an analytical solution for the case of an infinite region with large enough outer radius while it can be solved using a numerical integral method for the case of a finite region. Numerical examples for the Boltzmann viscoelastic model are given, and the responses of the displacement and stresses are presented in detail to illustrate the effects of velocity of change in the inner radius and the magnitude of the outer radius. Meanwhile, the effect of the aspect ratio (void concentration parameter) on the viscoelastic fields is presented. When the ratio of the outer radius to the inner radius is large enough, the responses can be evaluated by directly adopting the resulting analytical solution to avoid a complex numerical procedure. The resulting solutions and computational results are helpful to a better understanding of mechanical behaviors for (large) excavation or finite void growth in viscoelastic media.     

On Boussinesq’s problem for a cracked halfspace

This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.     

Stress analysis of multilayered plates around circular holes

In this paper, a technique to study the 3-dimensional stress state around a circular hole in laminated plates is developed. First, the 3-dimensional elasticity problem for a thick plate with a circular hole is formulated in a systematic fashion by using the z-component of the Galerkin vector and that of Muki's harmonic vector function. This problem was originally solved by Alblas[1]. The reasons for reconsidering it are to introduce a technique which may be used in solving the elasticity problem for a multilayered plate and to verify and extend the results given by Alblas. Among the additional results of particular interest, one may mention the significant effect of the Poisson's ratio on the behavior and the magnitude of the stresses. Secondly, the elasticity problem for a laminated thick plate, which consists of two bonded dissimilar layers and which contains a circular hole, is considered. The problem is formulated for arbitrary axisymmetric tractions on the hole surface. Through the expansion of the boundary conditions into Fourier series, the problem is reduced to an infinite system of algebraic equations which is solved by the method of reduction. Of particular interest in the problem are the stresses along the interface as they relate to the question of delamination failure of the composite plate. These stresses are calculated and are observed to become unbounded at the hole boundary. An approximate treatment of the singular behavior of the stress state is presented, and the stress intensity factors are calculated. It is also observed that, the results compare rather well with those obtained from the finite element method.     

Stress field near interface edge of elastic-creep bi-material

Stress fields on elastic-creep bi-material interfaces with different geometry of the interface edge are analyzed by finite element method. The results reveal that the stress highly concentrates near the interface edge at the loading instant and it gradually decreases as the creep-dominated zone expands from the small-scale creep to the large-scale creep. The stress singularity due to creep which resembles the HRR stress singularity appears near the interface edge in all cases. The stress intensity near the interface edge time-dependently decreases and becomes constant when the transition reaches the steady state. The magnitude is scarcely influenced by the edge shape of elastic material, though it depends on the edge shape of creep material. The stress intensity during the transition can be approximately predicted by the J-integral at the loading instant.     

Residual stress release characteristics of hole-drilling determined by in-plane three-directional optical interference moiré

By adopting a new stress calibration method, a grating rosette moiré interferometry and incremental hole-drilling combined system is developed to determine the magnitud of the residual stress in aluminium plate subjected to a uniform uniaxial tensile load. Performing in-plane three-directional fringe analysis, the optical data contained in the moiré interferograms are converted into values of the strains in three directions corresponding to the grating rosette. The evaluation is carried out through the measurement of the in-plane displacement field in three directions generated by the introduction of the small incremental hole. The in-plane three-directional displacement fields are determined from the calculation of the optical in-plane three-directional phase distribution by means of a phase-shifting method. A new finite element calibration analysis that is general axisymmetric elements instead of 3-D block element and harmonic axisymmetric element is adopted to relate the relieved displacement field to the magnitude of the residual stress. The magnitude of the principal stresses is finally evaluated through a least-squares calculation and is also compared with the stress value applied to the specimen measured with strain gauges.     

Dugdale plastic zone of a penny-shaped crack in a piezoelectric material under axisymmetric loading

The Dugdale plastic zone ahead of a penny-shaped crack in a piezoelectric material, subjected to electric and axisymmetric mechanical loadings, is evaluated analytically. Hankel transform is employed to reduce the mixed boundary-value problem of the penny-shaped crack to dual integral equations, which are solved exactly under the assumption of electrically permeable crack face conditions. A closed-form solution to the mixed boundary-value problem is obtained to predict the relationship between the length of the plastic zone and the applied loading. The stress distribution in and outside of the yield zone has been derived analytically, and the crack opening displacement has been investigated. The electric displacement has a constant value in the strip yield zone. The current Dugdale crack model leads to non-singular stress and electric fields near the crack front, and it is observed that the material properties affect the crack opening displacement.     

Axisymmetric contact problem for an elastic layer and an elastic foundation

This paper is concerned with the axisymmetric problem of an elastic layer lying on a semi-infinite foundation. The layer is pressed against the foundation by a uniform clamping pressure applied over its entire surface and a uniform vertical body force due to the effect of gravity. In addition, an axisymmetric vertical line load is applied to the layer. It is assumed that the contact between the layer and the foundation is frictionless and that only compresive normal tractions can be transmitted through the interface. The contact along the interface will be continuous if the value of the line load is less than a critical value. However, interface separation takes place if it exceeds this critical value. The problem is formulated and solved for the cases of tensile and compressive line loads. Numerical results for contact stress distributions are given for different material combinations.     

Complex variable method of computing Jk for bi-material interface cracks

A method using functions of a complex variable is developed for evaluation of J₁ and a modified J₂ integrals for bi-material interface cracks. This method, used in conjunction with the finite element method, would be useful in the prediction of stress intensity factors for cracks lying between the interface of two dissimilar materials. Since the direct evaluation of J₂ poses difficulties in modeling the singular behavior in the near vicinity around the crack tip for bi-material crack problems, it is modified by evaluating it around a contour path of small radius from the crack tip within the singularity dominated zone. It is shown that the stress intensity factors for a bi-material interface crack can be accurately evaluated using these j_k integrals.     

Stress field equations for U and blunt V-shaped notches in axisymmetric shafts under torsion

Analytical solutions are developed for the stress fields induced by circumferential U- and blunt V-shaped notches in axisymmetric shafts under torsion, with a finite value of the notch root radius. The boundary value problem is formulated by using complex potential functions and the real boundary notch shape. Shear stress components are then written as a function of the maximum shear stress evaluated at the notch tip. Considering different global and local geometries the obtained equations are compared with a large bulk of finite element results, showing a very good agreement. Due to their reduced complexity, such equations turn out to be particularly useful in practice.     

On the local variation of the crack driving force in a double mismatched weld

A material inhomogeneity in the direction of crack extension causes a difference between the near-tip crack driving force, J_tip, and the nominally applied far-field crack driving force, J_far. This difference can be quantified by a material inhomogeneity term, C_inh, which is evaluated by a post-processing procedure to a conventional finite element stress analysis. The magnitude of the material inhomogeneity term is evaluated for cracks in an inhomogeneous welded joint made of a high-strength low-alloy steel. Both a crack proceeding from the under-matched (UM) to the over-matched (OM) and from the OM to the UM weld metal are treated. The effects of the inhomogeneity of the different material parameters (modulus of elasticity, yield strength, and strain hardening exponent) on C_inh and J_tip are systematically studied. The results demonstrate that the material inhomogeneity term is primarily influenced by the inhomogeneity of the yield strength. A crack growing towards an OM/UM interface experiences an accelerated crack growth rate or a pop-in, an UM/OM interface leads to a reduced crack growth rate or a crack arrest. The application of global assessment methods of the mismatch effect which are included in the Engineering Treatment Model (ETM) or in the Structural Integrity Assessment Procedures for European Industry (SINTAP) is discussed.     

Surface effects on the near-tip stress fields of a mode-II crack

On the physical nature, most crack tips are not ideally sharp but have a small curvature radius. Both surface energy and crack-root curvature affect the stress and displacement fields in the vicinity of the crack tip. In the present paper, a numerical method, which incorporates the effect of surface elasticity into the finite element method, is employed to study the surface effects on the mode-II crack tip fields. It is found that when the curvature radius of the crack root decreases to micro-/nanometers, surface elasticity has a significant influence on the stresses near the crack tip. For a mode-II crack, surface effects alter both the magnitude and position of the maximum stresses, as is different from a mode-I crack, in which case only the stress magnitude is influence by surface stresses.     

Interface fracture analysis of joints with a ductile interlayer

Plane strain problem of an interface crack with two interface shear yield zones and one crack-face contact zone is studied. The plastic yielding of the interlayer is stimulated by the interface shear yield zones and contact zone is included near one tip of the interface crack. An interesting and important phenomenon found in this analysis is that for such an interface crack the applied compressive normal stress can increase the stress intensity factor K ₁ at one of the two crack tips, the size of the crack-face contact zone, and the maximum value of the crack opening, in a combined normal and shear stress field. Examples are given for two pairs of materials used in ceramic-metal brazed joints, Si₃N₄/Ni and Si₃N₄/Incoloy 909.     

Interface separation for an elastic layer loaded by a rigid stamp

The frictionless contact problem for an infinite elastic layer lying on a horizontal rigid plane is considered. The external load is applied to the layer through a rigid stamp. It is assumed that the layer is subjected to uniform vertical body forces due to the effect of gravity. For values of the resultant compressive force P acting on the stamp vertically which are less than a critical value P_cr the continuous contact along the layer-substrate interface is maintained. However, if tensile tractions are not allowed on the interface, for P >P_cr the layer separates from the interface along a certain finite region. The problem is formulated and solved for both cases and numerical results for P_cr, separation initiation distance, contact stresses, distances determining the separation area, and the vertical displacement in the separation zone are given.     

Effect of interface layer consisting of nanosprings on stress field near interface edge

The purpose of this study is to investigate the effect of an interface layer consisting of discretely arrayed nano-sized elements on stress intensified fields. A material where an interface layer consisting of Ta₂O₅ helical nanoelements (nanosprings) is inserted between dissimilar components is prepared and two types of crack initiation experiments, which possess radically different stress conditions, are carried out. The finite element analyses indicate that the stress fields in the components with and without the interface layer are completely different, and it is experimentally clarified that the fracture mechanics concept cannot be applied to the crack initiation at the dissimilar interface edge with the interface layer. The stress distributions at the crack initiation reveal that the crack initiation is governed by the apparent stress of the nanospring, σ′, at the edge. This signifies that the interface layer eliminates the stress singular field at the interface edge. The criterion of the crack initiation is evaluated as .     

Fictitious notch rounding concept applied to V-notches with end holes under mode 3 loading

The present technical note is aimed to provide a closed form expression for the microstructural support factor and for the fictitious notch radius in plates weakened by V-notches with root end-holes. Taking advantage of some recent closed form expressions for the stress distributions due to V-notches with end holes the fictitious notch rounding approach is applied here to mode 3 loading. The factor s for the V-notch with end holes is found to be strongly influenced by the opening angle and the new values are compared with the previous solution available in the literature and dealing with blunt V-notches. To validate the new expressions a comparison is carried out between the theoretical stress concentration factor (SCF) obtained from a rounded V-notch with a fictitiously enlarged end hole (of radius ρ_f) and the effective stress concentration factor obtained by integrating the relevant stress over the microstructural characteristic length (MCL), ρ^*, in a pointed V-notch. A sound agreement is found from the comparison. The range of validity of the present equations are limited to linear elasticity or in those cases where the plastic zone is very small with respect to the MCL of the material.     

Surface effects on mode-I crack tip fields: A numerical study

Surface energy often significantly influences the deformation and failure behavior of materials and devices at the nanoscale. However, how it alters the local deformation around a crack tip remains unclear. In the present paper, we investigate the surface effects on the near-tip fields of a mode-I blunt crack (or notch). The theory of surface elasticity is incorporated into the finite element method. It is found that when the curvature radius of the crack root shrinks to nanometers, surface effects considerably affect the local stress distributions near the crack tip. We also calculate the J-integral, which is almost independent of surface effects except when the integral path approaches the crack tip. This demonstrates that surface effects are localized in a small zone around the crack tip, where the classical fracture mechanics solutions neglecting surface effects should be modified.     

A double receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, the axisymmetric problem of a frictionless double receding contact between a rigid stamp of axisymmetric profile, an elastic functionally graded layer and a homogeneous half space is considered. The graded layer is modelled as a nonhomogeneous medium with an isotropic stress-strain law. Assuming the double contact between the bodies to be frictionless, only compressive normal tractions can be transmitted in each contact area while the rest of the surface is free of tractions. Using an appropriate integral transform, the axisymmetric elasticity equations are converted analytically into a system of singular integral equations where the unknowns are the pressures and the radii of the receding contact area in the two contact zones. The global equilibrium conditions are supplemented to solve the problem. The singular integral equations are solved numerically using orthogonal Chebyshev polynomials. An iterative scheme based on the Newton-Raphson method is employed to obtain the receding contact radii and pressures that satisfy the equilibrium conditions. The main objectives of the paper are to study the effect of the nonhomogeneity parameter, the thickness of the graded layer and the magnitude of the applied load on the contact pressures, the radii of the receding contact zones and the indentation for the case of a spherical rigid punch.     

The statical Reissner-Sagoci problem for an internally loaded transversely isotropic halfspace

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular punch which is bonded to the surface of a transversely isotropic elastic halfspace region. The rotation of the punch is induced by a concentrated couple which acts at a finite distance from the punch along the axis of symmetry. The resulting rigid rotation of the punch is evaluated in exact closed form.     

Boundary element analysis of thermoelastic contact problems

A boundary element method (BEM) is applied to thermoelastic contact problems where thermal resistance at the contact interface is not negligible. The displacement, traction, temperature and temperature gradient in the contact zone are unknown quantities to be determined numerically. Due to the existence of thermal resistance, temperature and stress fields are mutually coupled. To solve the problem, two kinds of methods are presented. In the first method, the solution is obtained by minimizing a suitably defined objective function. In the second method, discretized equations of each of the bodies in contact are computed alternately until all prescribed boundary conditions are satisfied. The applicability of these methods to practical problems is examined through several numerical examples.     

Axisymmetric yielding of functionally graded spherical vessel under thermo-mechanical loading

Primarily the separate distributions of elastic and perfectly plastic thermal stress in spherical pressure vessels made up of Functionally Graded Material (FGM) are represented. Next, the combined elastic and perfectly plastic thermal stress analysis of a spherical pressure vessel is considered. It is assumed that no unloading is occurred and the modulus of elasticity, yielding stress and some specific material characteristic parameters are power functions of radius. In a spherical FGM vessel and for different material compositions, the effect of pressure and temperature upon the growth of plastic zone is studied. Especially the change in the position of the borderline between the elastic and plastic regions is sought. In an extensive range of material thermo-mechanical properties, the position of the elastic–plastic interface line and the yield pattern, which shows the number of plasticized layers in the wall of the vessel, are indicated.