Analytical Elastic Solution Research Articles

Elastic solutions of a cylindrical rod containing periodically distributed inclusions with axisymmetric eigenstrains

In this paper, we give the elastic solution for a special type of microstructure – a circular cylindrical rod containing periodically distributed inclusions along its axial direction. Each inclusion has the same uniform axisymmetric transformation strain (eigenstrain). Analytical elastic solutions are obtained for the displacements, stresses and elastic strain energy of the rod. The effects of microstructure and its evolution (growth of inclusions) on the elastic stress and strain fields as well as the strain energy of the rod are quantitatively demonstrated. As a result of such microstructure evolution nominal stress-strain relation with strain softening is derived for a rod under uniaxial tension.

Analysis of a transversely isotropic rod containing a single cylindrical inclusion with axisymmetric eigenstrains

This paper studies a transversely isotropic rod containing a single cylindrical inclusion with axisymmetric eigenstrains. The analytical elastic solution is obtained for the displacements, stresses and elastic strain energy of the rod. The effects of microstructural parameters and its evolution on the elastic stress and strain fields as well as the strain energy of the rod are quantitatively demonstrated through examples.

Phase-field model of domain structures in ferroelectric thin films

A phase-field model for predicting the coherent microstructure evolution in constrained thin films is developed. It employs an analytical elastic solution derived for a constrained film with arbitrary eigenstrain distributions. The domain structure evolution during a cubic→tetragonal proper ferroelectric phase transition is studied. It is shown that the model is able to simultaneously predict the effects of substrate constraint and temperature on the volume fractions of domain variants, domain-wall orientations, domain shapes, and their temporal evolution.

Inherent damage zone model for strength evaluation of small fatigue cracks

An inherent damage zone model is presented to explain the fatigue properties of small sized cracks near the fatigue limit and the crack growth threshold. A special feature of the paper is using the exact stress distributions of notched and cracked specimens at the strength evaluations. Analytical elastic solutions by Neuber and Westergaard are employed for this purpose. A crack propagation law is newly developed based on the inherent damage zone model, and the propagation properties of small sized cracks and the mechanism of non-propagating crack are discussed based on the model. The estimated propagation rates of small sized fatigue cracks, the crack size effect on threshold strength and the occurrence condition of non-propagating crack in elliptic notches and circular holes show good agreements with the previous experimental results.

Elasto-plastic finite element analysis of fretting stresses in pre-stressed strip in contact with cylindrical pad

An elasto-plastic analysis of fretting stresses in a pre-stressed strip in contact with a cylindrical pad is studied using a finite element solver, ABAQUS. The pad and the strip are made from a titanium alloy. A bilinear elasto-plastic isotropic hardening model with a von Mises yield surface is employed to characterize the material behavior of the titanium alloy. This model is first verified through comparison with an analytical elastic solution. Various parameters, such as friction coefficient, the normal force applied on the top of the pad, tangential force applied to the left side of the pad, and bulk tension applied on the right edge of the strip, are adopted to study the influences of these parameters on fretting stresses to understand the implication and importance of elasto-plastic analysis in fretting fatigue experiments.

Phase-Field Simulation of Domain Structure Evolution in Ferroelectric Thin Films

ABSTRACTA phase-field model for predicting the domain structure evolution in constrained ferroelectric thin films is developed. It employs an analytical elastic solution derived for a constrained film with arbitrary eigenstrain distributions. In particular, the model is applied to the domain structure evolution during a cubic→tetragonal proper ferro- electric phase transition. The effect of substrate constraint on the volume fractions of domain variants, domain-wall orientations, and domain shapes is studied. It is shown that the predicted results agree very well with existing experimental observations in ferroelectric thin films.

C0 Zigzag Kinematic Displacement Models for the Analysis of Laminated Composites

Three new theories for analysis of laminated composite and sandwich structures are presented and assessed for accuracy and for suitability for finite-element analysis. These theories are based on assumed displacement fields that consist of a smooth polynomial expansion (quadratic, fully cubic, and partly cubic) in the thickness coordinate with a piecewise (layerwise) linear field superposed upon them. Interlaminar transverse shear continuity is enforced to make the number of degrees of freedom in the theories independent of the number of layers (plies) in the structure. In the case of the quadratic and partly cubic theories, a special formulation is developed that yields a theory in which all variables are C0continuous. In addition, a new transverse normal strain field is derived by assuming the transverse normal stress to be constant through the thickness of the laminate. The final form of the model uses only engineering-type degrees of freedom-displacements and rotations. To verify the accuracy of the proposed theories, Navier-type solutions are developed for a simply supported beam subjected to a sinusoidally varying transverse load and the results are compared with an analytical elasticity solution. The results show that all the proposed theories predict local stresses very accurately compared to the first-order shear deformation theory.

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 µm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone.

Mechanical modeling and analysis of the impact testing of wire

A new experimental test method and its associated mechanics description is reported for the instrumented impact of small diameter rod and wire. The use of this test lies in its ability to quickly and effectively measure impact fracture energy at various dynamic strain rates while indirectly providing a measure of the material's dynamic yield stress. The basic outline of the test is similar to that of an instrumented drop-weight test, albeit with two major differences: (i) the test material (rod or wire) is axially loaded to 60 percent of its yield stress prior to impact and (ii) the rod remains unnotched and is in no other way modified from its original condition. Between the grips, the rod is supported laterally by two hardened steel anvils having a radius of 12.5 mm and is impacted laterally at midspan by a hardened steel tup having a radius of 8 mm. Fracture occurs in a cup and cone manner in the region directly the rod during impact, analytical and numerical solutions were developed. Elastic analytical solutions were first investigated and then used to partially verify subsequent nonlinear finite element analyses. Nonlinearities arose as a consequence of both large deformation and elastic-plastic behavior in the rod during impact. The experimental testing program consisted of both quasi-static\((\dot \in = 10^{ - 4} )\) and dynamic\((\dot \in = 9)\) tests on preloaded rods. Excellent agreement was found between the numerical and experimental results for impact fracture energy and for peak load at failure. Numerical and experimental results indicate that significant strain hardening occurs in the rod as the strain rate is increased from 10−4 to 9. Based on these models, the mechanical behavior of the rod under impact loading is discussed.

Comparison between an elastic-perfectly plastic finite element model and a purely elastic analytical model for a spherical indenter on a layered substrate

The contact problem of an elastic sphere indenting an elastic-perfectly plastic substrate having an elastic layer is analysed using the finite element method and compared with a purely elastic analytical solution. The case of a layer stiffer and harder than the substrate is investigated and solutions for the contact pressure, subsurface stresses and strains are presented for various indentation depths. The results show good agreement until the threshold of plasticity is reached. Thereafter, as the system is subjected to higher penetrations, the two solutions diverge as plastic deformation initiates in the finite element model. The analysis also demonstrates that the influence of a deformable indenter upon the plastic solutions cannot be neglected.

Determination of stress field in an elastic solid weakened by parallel penny-shaped cracks

This paper investigates the stress intensity factors of two penny-shaped cracks with different sizes in a three-dimensional elastic solid under uniaxial tension. The two cracks are symmetrically parallel located in the isotropic solid. Based on Eshelby's equivalent inclusion method and the superposition pronciple of the elasticity theory, a closed-form analytical elastic solution for the stress intensity factors (SIFs) on the boundaries of the cracks is obtained when the center distance between the two cracks is much larger than the crack sizes. A numerical method is employed to extract the solution for small center distance case. It is found that, due to the interaction between the two cracks, the first and second kinds of SIFs exist at the same time even if the applied stress is pure tension. Numerical examples are given for different configurations and it is clearly shown that the SIFs are strongly determined by the distance between the centers of the two cracks.

Deformation of two-phase materials: Application of analytical elastic solutions to plasticity

Deformation of two-phase materials: Application of analytical elastic solutions to plasticity

Spheroidal Sliding Inclusion in an Elastic Half-Space

An analytical elasticity solution for a half-space having a spheroidal sliding inclusion is obtained. The inclusion is subjected to either a uniform plane hydrostatic loading applied at infinity or a uniform transformation strain (eigenstrain). The interface between the inclusion and the surrounding material allows sliding and does not sustain shear tractions. Boussinesq’s displacement potentials in infinite integral form and in infinite series form are used in the analysis. Numerical examples are included.

Analysis of the fixation of total hip femoral components using adina

There are well over 100,000 total hip replacements done per year in the U.S., and over 250,000 per year worldwide. Extending the service life of these implants is a high priority due to the large number of patients and the increasing use of joint replacement in younger patients. In an older population, total hip replacements usually outlast the patient, due to advanced age and the relatively low physical demands placed on the prosthesis. Younger, more active, and heavier patients often require revision of their total hip due to progressive implant loosening and the pain which results. Revised total hips do not last as long as the original, and a younger patient can sometimes require more than three revisions. In a standard total hip replacement, a steel prosthesis is fixed to the thigh bone using polymethyl methacrylate bone cement. The fatigue characteristics of the bone cement, and of the prosthesis-cement interface have been identified as the critical areas which initiate failure in this structure. In this study, the stresses within the cement and at the cement-metal interface have been studied using a large-scale linear finite element analysis. Also, to assess the effects of cement-prosthesis debonding, a 3D contact study was conducted which assessed the effect of several different areas of debonding between the prosthesis and the cement. The results of finite element analysis showed that the most likely sites for failure initiation are in the proximal antero-medial region and at the distal prosthesis tip. The loading situation which appeared to put the interface in the most danger for failure is that encountered in stair climbing. The stresses during normal walking did not seem as critical. Simulation of cement-metal debonding showed that a drastic increase in cement stresses occurred under both gait and stair climbing loads, and that stair climbing loads also produced much higher stresses with debonding. The effect of pores which often occur in bone cement was assessed using an analytical elasticity solution for a spherical void in an infinite medium, which allowed a calculation of the maximum tensile stress at the surface of a pore. These stresses were sufficient to initiate fractures near the distal tip of the implant in many cases, and near the proximal medial region of the implant in stair climbing. This study concurs to a remarkable degree with a study of well-functioning total hip replacements recovered at autopsy. The initiating failure events seen in the retrieved femoral components matched closely with the predicted areas of failure initiation. The conclusions of the study were that the cement-prosthesis interface should be strengthened, porosity in bone cement should be minimized, and that total hip patients should not use their prosthetic hip when climbing stairs.

Boundary element implicit differentiation equations for design sensitivities of axisymmetric structures

Design sensitivity analysis of axisymmetric elastic media is formulated using boundary elements. The kernels for sensitivity matrices are obtained through implicit differentiation of the corresponding boundary element elasticity kernels. The singular terms are obtained by applying the boundary displacements and tractions, and their respective sensitivities for the rigid body motion mode and the inflation mode. The equations for the recovery of sensitivities of axisymmetric boundary stresses are presented. As a check on accuracy, the approach is applied to a series of examples for which analytical elasticity solutions are available. The predictions for both displacement- and stress-sensitivities are accurate. Additional examples are provided to demonstrate the versatility of the present approach.

Effect of borehole deviation on breakout orientations

Well bore breakouts are zones of spalling and fracture that form on opposite sides of a well bore and tend to change the cross‐sectional shape of the borehole from circular to roughly elliptical. In vertical holes drilled in areas where one principal stress (Sv) is vertical, breakouts tend to form at opposite ends of the borehole diameter parallel to the least compressive horizontal principal stress direction (Sh). This paper uses an analytical elastic solution for stress at the wall of a borehole to analyze the rotation of breakout orientations away from the direction of Sh as the borehole deviates from the direction of the vertical principal stress. The calculated orientations of breakouts in deviated boreholes depend on the type of faulting regime in which the well was drilled (i.e., normal, strike‐slip, or thrust), on the deviation angle ø of the borehole axis from vertical, on the angle θ between the horizontal projection of the borehole axis and the direction of Sh, and on the relative magnitudes of the three principal stresses (Sh; SH, the greatest compressive horizontal stress; and Sv). In the strike‐slip faulting regime, regardless of the values of θ, Sv, SH, and Sh, a borehole must deviate at least 35° from vertical before the horizontal projection of the breakout orientation differs by more than 10° from Sh. In the normal and thrust faulting regimes, however, the borehole deviation angle øcrit required to rotate projected breakout orientations by 10° from Sh approaches zero as the value of Sh approaches that of SH. In the thrust faulting regime, only about 3% of all combinations of θ, Sv, SH, and Sh will produce øcrit values less than 10°, while about 12% of all such combinations will yield øcrit < 10° in the normal faulting regime. About 12% and 33% of such combinations will yield øcrit < 20° in the thrust and normal faulting regimes, respectively. Careful study of changes in breakout orientation as a function of borehole deviation may improve the resolution of inferred stress directions when studying breakouts in deviated boreholes in these two faulting regimes.

A Hybrid Finite Element Approach to Composite Laminate Elasticity Problems With Singularities

A hybrid finite element method is presented for studying composite laminate elasticity problems with singularities. The basic equations of laminate anisotropic elasticity and stress singularities in composite laminates are discussed briefly. Formulation of a singular, hybrid composite-wedge element for this class of problems is given in detail. Two numerical examples are shown to illustrate the accuracy and efficiency of the present method of approach. The first example is the well-known laminate free-edge problem, which has a rather weak stress singularity; the second one is the important composite problem, which is shown to have a strong stress singularity. Results are compared with existing analytical elasticity solutions for these problems. Excellent solution accuracy, convergence, and computational efficiency are demonstrated.

Viscosity-based numerical model for fault-zone development in drape folding

The propagation of a fault zone through homogeneous and layered sedimentary materials above a basement deflection was simulated using a finite-element solution for slow, incompressible viscous flow. The layered models represented a generalized stratigraphic section of the Paleozoic and Mesozoic strata of the Wyoming Province, U.S.A. Fractured elements within the models were determined for each increment of deformation using the Mohr-Coulomb fracture criterion and the viscosity of those elements was decreased prior to subsequent increments. In this manner the effects of local fracturing on subsequent deformation were included. In homogeneous models, failure begins near the basement deflection and propagates discontinuously upward. The length of the fault zone depends on the vertical velocity imposed on the bottom boundary and the width of the deflection. The resulting fault zone is a high-angle, rather straight, reverse fault which is in contrast to the curved, concave downward faults predicted by previous elastic analytical solutions. However, the analytical solutions were based on the initial stress field and do not take into consideration any changes that occur within the body during deformation. In contrast to this, the finite-element models can be used to study the changes in the state of a body that occur during deformation. Deformation rates determine the style of deformation and the location and intensity of fracturing. High gradients of mean pressure occur near the basement deflection and the fault zone is initiated in a region of anomalously low pressure produced by deformation. Faults develop as a consequence of the coalescense of fractured elements that occur during deformation. The width of the fault zone varied from layer to layer depending on its stiffness. Narrow fault zones develop in weak layers and wide zones in stiffer layers. In both homogeneous and layered models, extensional fractures occur near the surface on the upper block.

Elastk‐plastic dynamic analysis of structures using known elastic solutions

AbstractA numerical method is shown to analyse the dynamic elastic‐plastic responses of those structures with known elastic solutions. The displacement at one point at time t caused by a unit load applied at another point at zero time, called dynamic influence coefficient, is calculated from the known elastic solutions. Incremental plastic strain is accounted for by a set of additional incremental loads, so the stiffness matrix and the eigenvectors do not vary with time. From the incremental load including that caused by the incremental plastic strain, the displacement vs. time of the structure is obtained.This method is applied to simply supported beams with bilinear stress‐strain relations with different strain‐hardening rates and to a simply supported elastic‐ideally plastic rectangular plate. This procedure can be extended to structures with no available known analytical elastic solutions. For these structures, the elastic solutions can be obtained by the finite element method.

Crack-morphological aspects in fracture mechanics

Fracture mechanics can be regarded as a methodology to characterize the actual crack by using the parameters given by the analysis of a simple mathematical crack model. Therefore, the selection of these crack models has very significant meanings. In recent years, several precise crack models have been proposed. In this paper, we paid attention to the influences of the two-dimensional crack geometry and presented the numerical value of the elastic analytical solutions for the non-linear shaped cracks, those are, the bent, the curved and the branched crack. On the basis of these results, the characteristics of these cracks were discussed and the methods of approximate replacement to a simple straight crack were proposed. We examined a zig-zag growth of crack and the crack branching observed in actual fracture processes.