Interior Stress Field Research Articles

Relationships between multiaxial stress states and internal fracture patterns in sphere-impacted silicon carbide

Internal fracture patterns developed in silicon carbide cylindrical targets as a result of dynamic indentation (63–500 m/s) by tungsten carbide spheres are defined. Microscopy of recovered and sectioned targets delineate into three regions, each associated with distinct cracking modes, i.e., shallow cone macrocracking at and near the impact surface, steep interior cone macrocracks that radiate into the target from the impact region and local grain-scale microcracking directly underneath the impact region. The observed fracture patterns are found to maintain a noticeable degree of self-similarity upto the impact velocity of 500 m/s. Linear elastic analysis of the full (surface and interior) stress field developed under static (Hertz) contact loading delineate the target into four regions, based on the number of principal stresses that are tensile (none, 1, 2 or all 3). A strong correlation is found between the principal stress conditions within each region and the forms of cracking, their locations and orientations present therein. This correlation has a number of implications, including non-interaction of crack systems, which are discussed. Illustrative linear elastic fracture mechanics analyses are performed for three regions, and calculated and observed macrocrack lengths are found to be in reasonable agreement.

Examination of the Optimal Local Model Size in the Three-Dimensional Local Hybrid Method

The interior stress field of the surface crack specimen subjected to a uniform tensile load has been successfully analyzed by the 3-D local hybrid method. In this study, it was examined whether the 3-D local hybrid method was applicable to the specimen subjected to the bending load. It has been understood that the decision of the optimal size of the local model is indispensable in the improvement of accuracy from the previous research. Then, the width and thickness of the local model were changed widely, and analyses were carried out to find the optimal hybrid local model size. Moreover, it was examined how to decide the optimal size of the hybrid local model with various crack length and aspect ratio of the specimen. The optimal size of the hybrid local model was examined from the comparison with the J integral of the whole model.

Some analytical results for plane contact problems

Traditionally, two-dimensional elasticity problems have been solved using an Airy stress function or a Muskhelishvili complex potential approach. The principal advantage of the Muskhelishvili approach is that the interior stress field may be found directly from the boundary tractions, unlike the Airy approach where a degree of guesswork is necessary to obtain a suitable potential function. This paper demonstrates the equivalence of these two approaches to contact problems. The paper then extends the Muskhelishvili method to out-of-plane contact problems. To demonstrate the usefulness of this method, it is applied to two important problems motivated by fretting fatigue studies. The stick/slip behaviour of a Hertzian indenter, when the substrate is subjected to a bulk tensile stress field, is determined and the complete stress fields for both a flat-ended punch and a parabolic punch, when the punch is sliding in an out-of-plane direction, are found.

On the uniformity of stresses inside an inhomogeneity of arbitrary shape

We consider an inhomogeneity of 'arbitrary' shape embedded within an infinite isotropic elastic medium (matrix) subjected to antiplane shear deformations under the assumption of uniform remote loading. The inhomogeneity-matrix interface is assumed to be imperfect, characterized by a single interface function. Under these assumptions, we present a novel method leading to the solution of the problem concerned with identifying the shape of the inhomogeneity and the form of the corresponding interface function which leads to a uniform interior stress field. The analysis is based on complex variable methods. Specific solutions are derived in closed form and verified by comparison with existing solutions. As a consequence of our analysis, we present an interesting result on the uniqueness (within a certain class of smooth curves) of the circular inhomogeneity as the only inhomogeneity which, under the given conditions, leads to a uniform interior stress field when the interface function is constant (homogeneously imperfect interface).

Simultaneous Calculation of Roll Deformation and Polymer Flow in the Calendering Process

Abstract Calender units with rotating rolls are used to produce foils or sheets from polymer material with viscous or elastomeric flow properties. Up to now, the design process of calender units, the calculation of flow and pressure fields in the calender gap and the calculation of resulting deformations of the rotating rolls are implemented with different computer programs. In this work a design concept that integrates design and calculation programs is being developed in order to achieve integrated design and calculation. With this concept, it is possible to design calenders and calculate flow fields and deformations. After designing the rolls with the Computer Aided Design program Pro/Engineer, the flow field in the calender gap is calculated with the Finite Volume Method program CFX4, and the deformation of the rolls is calculated with the Finite Element Method program ANSYS. The iterative process of design, calculation and modification of shape is continued until sheets or foils of constant thickness leave the calender gap. The design concept has an object-oriented operational structure. The data transfer between the programs takes place via a central data base in STEP format. As examples, calenders with combinations of solid rolls are designed. The flow field in the calender gap is calculated for Newtonian and for viscoelastic fluids with use of an integral rheological equation of state. The flow field is compared to a measured flow field. The resulting deformations of the rolls are calculated and represented graphically. Furthermore, a design process for elastomer-coated rolls is implemented and their deformations as well as the interior stress fields are calculated.

New analytical and numerical results for two-dimensional contact profiles

Recently, a generalized Coulomb law for elastic bodies in contact has been developed by the author, which assumes that the tangential traction is the difference of the slip stress of the contact and the stick area, whereby each stick area corresponds to a smaller contact area. It holds for multiple contact regions also. Several applications for elastic half planes, half spaces, thin and thick layers and impact problems have been published. For plane contact of equal bodies with friction, it provides exact solutions, and the interior stress field can be expressed with analytical results in closed form. In this article, a singular superposition of flat punch solutions is outlined, in which the punches are aligned with an edge of the contact area. It is shown that this superposition satisfies Coulomb's inequalities directly, and new results for the Muskhelishvili potentials of several profiles are presented. It is illustrated how problems of singularity and multi-valuedness of complex functions can be solved in closed form, and the Chebyshev approximation used by earlier authors can be avoided. For comparison, some previous solutions for symmetric profiles are appended. Some results for the interior stress field, the pressure, the frictional traction and the surface displacements are compared with FEM solutions of an equivalent problem. The small differences between both methods show characteristic features of the FEM model and the theoretical assumptions, and are shortly explained. Further, this example can be used as benchmark test for FEM and BEM programs.

New Analytical Solutions For A Flat Rounded Punch Compared With FEM

Recently, a generalized Coulomb law for elastic bodies in contact was developed by the author and several applications for elastic half-planes, half-spaces, thin and thick layers and impact problems have been published. This theory assumes that the tangential traction is the difference of the slip stress of the contact and the stick area, and each stick area is a smaller contact area. It holds for multiple contact regions also. For plane contact of equal bodies with friction, it provides exact solutions and the interior stress field can be expressed with analytical results in closed form. In this article, a symmetric superposition of flat punch solutions is outlined and some useful formulas are listed. The necessary assumptions and simplifications are discussed. It is shown that this superposition satisfies Coulomb’s inequalities directly and can also be used for torsion or shift of axisymmetric profiles. A new formula for the Muskhelishvili potential of a twodimensional flat rounded punch is presented, which avoids the Chebyshev expansion used by other authors. Similar equations can be derived for different profiles. Some results for the interior stress field, the pressure, the frictional traction and the surface displacements are compared with FEM solutions of an equivalent problem. The small differences between both methods show some characteristic features of the FEM model and the theoretical assumptions, and are shortly explained. Further, this example can be used as benchmark test for FEM and BEM programs.

State space approach for stress decay in laminates

Stress decay in laminates due to edge boundary effects are studied through a state space formulation. A self-equilibrium eigenstress field accounting for the multilayer construction of the laminate is derived using the state variables and the transfer matrix method. The eigenvalue determination requires only the solution of 6×6 determinants irrespective of the number of laminae. Through combinations of the eigenstress field and the interior stress field a complete solution valid in the boundary layer as well as in the interior region of the laminate can be obtained. For verification, the formulation is first applied to determining the eigenstress in a homogeneous anisotropic layer, and then the free edge stress decay in laminates under uniform extension is examined.

On non-symmetrical plane contacts

Plane elastic contact problems are considered, with particular emphasis on asymmetrical punch profiles, in the case of ‘complete’, ‘partially complete’ and ‘incomplete’ contact. An explicit, analytical solution is presented for the case of a single area of contact where the overlap is described by a generic spline function, and examples presented. The interior stress field and strength of the contact, under full or partial slip conditions, are also discussed, and some example shown for representative cases. It is found also that the direction of sliding has a significant effect for the strength of non-symmetrical contacts.

Indentation by nominally flat or conical indenters with rounded corners

Axisymmetric indentation of a flat surface is considered: specifically, the case of flat-ended indenter with rounded edges, and the case of a shallow cone with a rounded tip. Analytical solutions are obtained for the normal and sequential tangential loading, in both full or partial slip conditions (with the Cattaneo fn9 fn9 degree polynomial in x and y. The most relevant are Hertzian elliptical contacts (as proved explicitly by Cattaneo, 1938) and plane problems ( Ciavarella, 1998a, Ciavarella, 1998b) . Indeed, for plane problems, with no dependence, say, on y, Δ V ( x) = (d 2V ( x) /d x 2) = h 1+ h 2 has a general solution V ( x) = [ ( h 1+ h 2) /2] x 2+ c. Other particular cases of such exact solutions may exist, with a more general shape of stick area, but the question of limited practical interest, as long as the surface profiles to produce that contact area and symmetry are of rather particular form, so that the property will hold only for very special values of geometry and load. Mindlin approximation) , and for the complete interior stress field in all these conditions. Implications for strength of the contact are discussed with reference to metallic or brittle materials, with the intention to shed more light in particular to the understanding of common fretting fatigue or indentation testings with nominally flat or conical indenters. It is found that the strength of the contact, which is nominally zero for perfectly sharp flat or conical indenters, is well defined even for a small radius of curvature. This is particularly true for the flat indenter, for which the strength is even significantly higher than for the classical Hertzian indenter for a wide range of geometrical and loading conditions, rendering it very attractive for design purposes.

Contact problems for a wedge with rounded apex

Elastic contact between a shallow elastic wedge, whose apex is blunted by a finite radius, and an elastically similar half-plane is studied. A closed-form contact law is found, and the interior stress field is then deduced using a Muskhelishvili’s solution in series form, for frictionless and sliding conditions. This geometry removes one of the principal objections to classical solutions to the wedge indentation problem—the unrealistic infinite stress concentration implied by an atomically sharp apex—and in the latter part of the paper the strength of the contact is evaluated explicitly. Further, cases of partial slip associated with the application of tangential load less than needed to cause sliding are considered.

The influence of rounded edges on indentation by a flat punch

The contact problem and stress state for indentation by a flat punch with rounded edges is studied. For the contact problem itself analytical solutions are obtained for both surface pressure and interior stress fields. Cases of normal indentation and frictional contact, the latter in both sliding or partial slip conditions, are all treated. The transition from the Hertzian configuration to the contact between a nominally flat pad and contacting flat surface is discussed, and it is found that the strength of the contact decays surprisingly slowly. Regarding the von Mises yield parameter, there is a range of configurations for which the strength is actually higher than the Hertzian one, and the strength decays only when the corner radii are very small. The present solution is therefore a realistic alternative to the classical rigid-flat punch idealization, and has particular application to fretting fatigue tests.

Half-planes without coupling under contact loading

Nonrotating half-planes in contact under oblique loading are investigated in this paper. The solution is based on the influence integrals of the Flamant solution. The problem is determined by two integral equations for the normal and tangential stresses, which are uncoupled for special cases, as bodies of similar material in contact. In order to simplify the singular integrals, the method of superposition of flat punches is used. The result for the symmetric case is almost identical with the axisymmetric solution. For polynomial profiles of the form x s , the Muskhelishvili potentials can be written in terms of a complex hypergeometric function. The interior stress field is illustrated for an example.

A simulation of the vertical split head failure in rails

A two-dimensional analysis is presented to assess a possible mechanism to account for the axial cracking behavior in rails which is usually called a vertical split head failure. This failure is potentially damaging and may eventually lead to derailment. The vertical split head is simulated in the present analysis as a vertical crack in an eccentrically loaded infinite strip, representing the head, loaded on its upper surface and constrained on its lower surface, approximating the web constraint. The vertical crack is modeled as distributed dislocations. By using Fourier tarnsforms, the interior stress field and stress intensity factors at the crack tip are determined from the derived coupled integral equations. The calculated stress field indicates that the initiation of a vertical defect is more likely to be caused by the influence of residual stresses upon an existing rail defect, such as an inclusion. Moreover, the growth of the crack is constrained to within the rail head by the high magnitudes of compressive stresses that occur beneath the contact loading and the head-web juncture.

Contact Stress Analysis of a Layered Transversely Isotropic Half-Space

The three-dimensional problem of contact between a spherical indenter and a multi-layered structure bonded to an elastic half-space is investigated. The layers and half-space are assumed to be composed of transversely isotropic materials. By the use of Hankel transforms, the mixed boundary value problem is reduced to an integral equation, which is solved numerically to determine the contact stresses and contact region. The interior displacement and stress fields in both the layer and half-space can be calculated from the inverse Hankel transform used with the solved contact stresses prescribed over the contact region. The stress components, which may be related to the contact failure of coatings, are discussed for various coating thicknesses.

Finite element analysis of Transantarctic Mountain uplift and coeval subsidence in the Ross Embayment

The Transantarctic Mountains, which mark the boundary between East and West Antarctica, arguably form the largest continental rift-shoulder structure on earth. Viscoelastic finite element analysis has been used to model the present state of uplift, and of subsidence of the adjacent Victoria Land Basin, as the response to the lithospheric tension and basal upthrust produced by the low density uppermost mantle underlying adjacent West Antarctica and extending for 50–70 km beneath the Transantarctic Mountains. The observed uplift and complementary subsidence across the Transantarctic Fault can be matched by Model TAM4 with a low density upper mantle (−50 kg/m 3) extending 70 km westwards from the fault and from 45 to 200 km depth. The East Antarctic lithosphère as modelled has an elastic thickness of 100 km which decreases to about 25 km at the eastern margin of the craton where the uplift is occurring and is 25 km thick beneath adjacent West Antarctica. Alternatively, in Model TAM5 low density upper mantle extends for only 50 km beneath the Mountains, and an upper mantle of smaller density contrast (−36 kg/m 3) extending proportionately deeper to 254 km depth is required to match the observations. A horizontal deviatoric tension of 120 MPa (TAM4) or 150 MPa (TAM5) occurs in the thin elastic lithosphere of West Antarctica as a result of the sub-lithospheric loading caused by the low density upper mantle, but this dies off into East Antarctica on passing through the Mountains. Additionally, there are substantial superimposed bending stresses due to flexure on both sides of the fault. Such high stresses suggest elastic failure to be occurring, which underscores the paradox of the apparent aseismicity of Antarctica. The result of superimposing a supplementary compression averaging 20 MPa across the edge of the 100 km thick lithosphere of East Antarctica is to reduce the amplitude of the flexural uplift by about 19% and the subsidence by about 32% as a result of the reduced deviatoric tensions. A superimposed lithospheric tension has the exact opposite effect. The models with superimposed stresses suggest that the main plate interior stress field is less important than the locally derived loading stresses in the present support of the Transantarctic Mountains.

A global collocation technique for the solution of opening mode crack problems

A review of various experimental and numerical techniques for determination of fracture mechanics calibration functions (i.e., the variation of K1 and CMOD with crack length) revealed that neither technique, employed independently, can determine K1, CMOD, and full field stresses in closed form over a wide range of crack lengths. To fill this void, a combined experimental/numerical collocation technique based on a series expansion of the modified Westergaard functions was developed. This technique uses both boundary conditions, known a priori, and interior stress field conditions, determined using a suitable experimental technique, for analysis of two dimensional, finite body, opening mode crack problems. This paper reports on an investigation of the accuracy of this technique and its sensitivity to errors in experimental data for a sample problem of practical interest.

Analysis of a Transversely Isotropic Half Space Under Normal and Tangential Loadings

This paper analyzes the response of a transversely isotropic half space subjected to various distributions of normal and tangential contact stresses on its surface. Both the interior displacement and stress fields are given in closed form. Among them, rectangular patch solutions are constructed for application to solutions to non-Hertzian contact problems.

The Interior Stress Field Caused by Tangential Loading of a Rectangular Patch on an Elastic Half Space

A solution is obtained for the tangential loading on a rectangular patch. The solution gives the internal stress field in terms of elementary functions and is a form analogous to the solution for vertical load developed by Love.