Discontinuous Material Properties Research Articles

Computation of Microsegregation and Microstructure in Solidification with Fluid Convection

AbstractThe effects of fluid flow on the dendritic solidification morphology of pure materials is studied using a computational methodology based on a two-dimensional front tracking/finite difference method. A general single-field formulation is presented for the full coupling of phase change, fluid flow, and heat transport in both the solid and liquid phases. This formulation accounts for interfacial rejection/absorption of latent heat, interfacial anisotropies, discontinuities in material properties between the liquid and solid phases, shrinkage/expansion upon solidification and motion and deformation of the solid. Numerical results are presented for the dendritic solidification of pure succinonitrile in a shear flow. Comparison with solidification into a quiescent liquid indicates that fluid convection increases the overall rate of solidification while the growth rate of the leading dendrite tip is unchanged.

Short Communication: A numerical method for diffusive transport with moving boundaries and discontinuous material properties

A method for the numerical simulation of diffusive transport with moving boundaries is developed and tested. The variable domain is mapped onto a fixed region, which introduces a term of convective form to the transformed governing equation. The resulting convection/diffusion equation is solved by a finite-difference method. An 'Immersed Interface' Method (IIM) is introduced in order to retain second-order accuracy near discontinuities in material properties, where the solution is not smooth. The method performs well in benchmark calculations against an analytical solution. The IIM scheme is capable of treating a strong discontinuity in the gradient, and it is readily extended to two or three dimensions. The methods are illustrated through a calculation for the temperature profile in a growing continental ice sheet, in which the thermal properties are discontinuous at the rock/ice interface.

Short Communication: A numerical method for diffusive transport with moving boundaries and discontinuous material properties

A method for the numerical simulation of diffusive transport with moving boundaries is developed and tested. The variable domain is mapped onto a fixed region, which introduces a term of convective form to the transformed governing equation. The resulting convection/diffusion equation is solved by a finite-difference method. An ‘Immersed Interface’ Method (IIM) is introduced in order to retain second-order accuracy near discontinuities in material properties, where the solution is not smooth. The method performs well in benchmark calculations against an analytical solution. The IIM scheme is capable of treating a strong discontinuity in the gradient, and it is readily extended to two or three dimensions. The methods are illustrated through a calculation for the temperature profile in a growing continental ice sheet, in which the thermal properties are discontinuous at the rock/ice interface. © 1997 by John Wiley & Sons, Ltd.

Some results on the scattering of guided elastic SH waves from material and geometric waveguide discontinuities

The scattering of guided elastic SH waves from step discontinuities in the material properties and/or geometry of otherwise uniform flat layer waveguides is analytically/numerically analyzed. Analytical results for the case of a step discontinuity in material properties in a layer of uniform thickness are presented for comparison with later numerical results. The case of a sudden change in thickness of the waveguide, possibly in conjunction with a change in material properties, is solved numerically. The solution is based on an expansion of the wave fields on either side of the discontinuity in terms of the normal modes permissible in the analogous infinite layer and subsequent determination of the modal expansion coefficients. A ‘‘non-coupling’’ condition is proposed and proven which explains why particular modes are not generated in the scattering process. A ‘‘perfect-coupling’’ condition is also introduced, and the exact transmission amplitude of any mode satisfying the condition is given explicitly.

Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation

SUMMARY Artefacts and local instabilities are common in numerical solutions of seismic wave equations. Although many of these are already known, most are not well documented in an accessible form. One form of artefacts is a consequence of partial derivative operators that are non-local and are associated with the interaction between the propagating wavefield and the medium at discontinuities in material properties. When pseudo-spectral solutions are used, heterogeneous wave equations may be accurately solved in a wide dynamic range using even-based Fourier transforms on a staggered grid. Regardless of the implementation of a spatial differentiator, use of homogeneous wave equations or of standard (non-staggered) grids gives less accurate results. Synthetic examples include heterogeneous acoustic, elastic and poroelastic (Biot) equations.

3D eight-order elastic finite-difference modeling of refraction and strong-motion data from the Coyote Lake region, California

Abstract 3D elastic modeling of (vertical component, explosive source) refraction data is performed to constrain the shallow part of a velocity model, followed by 3D modeling of three-component strong-motion data from a (spatially and temporally extended) earthquake source in that model. Modeling is by eighth-order finite differencing on a staggered grid, which automatically generates all body and surface waves, including multiples and conversions. Application is to the 1981 USGS refraction survey and the 6 August 1979 Coyote Lake event in central California. 3D modeling was able to reproduce the main observed features in both the refraction and earthquake data. For the refraction data, the synthetics explain local variations in first-arrival travel times, reverberations in the near-surface sediments, and surface waves. For the strong-motion data, the initial arrival times and amplitudes are determined by the position and orientation of the initial rupture; the shapes of the trailing edges of the pulses are modified by the actual source distribution. Changes of source position of 1 to 2 km are easily resolvable through changes in the position of nodal planes. Discontinuities in material properties (e.g., layering) produce a coda of multiples and converted waves. Lateral variations in velocities, especially in the upper few kilometers, change the energy distribution among the trace components, by 3D refraction, and so shift the apparent source directivity and position, as well as contributing to the coda. Remaining differences between the observed and synthetic data are attributed to local seismometer coupling effects, very local site structures, and phenomena such as anelasticity and anisotropy, which are not explicitly modeled.

A Front-Tracking Method for Dendritic Solidification

We present a front-tracking method to simulate time dependent two-dimensional dendritic solidification of pure substances. The method is based on a finite difference approximation of the heat equation and explicit tracking of the liquid–solid interface. Discontinuities in material properties between solid and liquid phases as well as topology changes and interfacial anisotropies are easily handled. The accuracy of the method is verified through comparison with exact solutions to a two-dimensional Stefan problem. Convergence under grid refinement is demonstrated for dendritic solidification problems. Experimentally observed complex dendritic structures such as liquid trapping, tip-splitting, side branching, and coarsening are reproduced. We also show that a small increase in the liquid to solid volumetric heat capacity ratio markedly increases the solid growth rate and interface instability.

Difference approximations for wave equations via finite elements. I. Construction and performance

AbstractMany practical applications of wave equations involve media in which there are interfaces, or discontinuities in material properties. The accurate numerical representation of these interfaces is important in mathematical models. One can develop generalizations of standard finite‐difference methods that accommodate sharp interfaces by modifying a straightforward finite‐element approach. In two space dimensions, these methods yield explicit, 5‐point or 9‐point difference schemes that accurately capture reflection, transmission, and refraction at interfaces. The approach also extends readily to the simulation of waves in elastic media. A companion article presents an error analysis for the approach. © 1995 John Wiley & Sons, Inc.

Fundamentals of Residual Stresses in Joints Between Dissimilar Materials

When a discontinuity in material properties exists across a bonded interface, stresses are generated as a result of any thermal or mechanical loading. These stresses significantly affect strength and failure characteristics and may be large enough to prevent successful fabrication of a reliable joint. The use of an interlayer material to successfully reduce mismatch stresses, thereby preventing joint failure or improving joint strength and reliability, requires knowledge of failure mechanisms and of the effects of interlayer properties on the critical stress components.The origin of residual stresses developed during cooling of a ceramic-metal joint from an elevated fabrication temperature is illustrated qualitatively in Figure 1. Away from edges, the in-plane (parallel to interface) stresses are typically compressive in the ceramic and tensile in the metal. These stresses can cause cracking perpendicular to the interface, leading to spalling or delamination failures. Such failures are frequently observed in thin-film and coating geometries. Where the interface intersects a free edge, large shear and axial (perpendicular to the interface) stresses are generated. The edge stresses are typically tensile within the ceramic and tend to promote crack propagation within the ceramic parallel and adjacent to the interface. This is the most commonly observed failure mode in bonded structural components.

Fracture Mechanics of Functionally Graded Materials

In today's highly demanding technological environment, one of the main challenges in new material design is combining seemingly irreconcilable thermomechanical properties in the same component (e.g., high heat and corrosion resistance, high strength in elevated-temperature applications and high resistance to wear, and high toughness in load-bearing elements). In many cases, the problem may be solved by using coatings or by layering dissimilar materials. From a structural viewpoint, a major disadvantage of these techniques, particularly in ceramic coating of metals, has been the resulting high thermal and residual stresses and relatively poor bonding strength. Thus, in thin films, coatings, and layered materials, surface cracking and debonding or delamination have been common forms of mechanical failure. One effective way of reducing residual and thermal stresses and enhancing bonding strength has been to eliminate material-property discontinuities by grading the material composition near the interfaces or through the coating. These new materials, with continuously varying compositions or volume fractions, are known as functionally graded materials (FGMs).In developing FGMs, research on the mechanics, and particularly on the fracture mechanics of these inhomogeneous materials, is needed to provide technical support to materials scientists and to manufacturing and design engineers. In the past, fracture mechanics has been useful both as a screening tool during material processing and as a design and maintenance tool for service-life assessment. Broadly speaking, fracture mechanics involves studying the effect of the applied loads, the component/flaw geometry, and the environmental conditions on the fracture of engineering materials.

A new subseismic governing system of equations and its expansions

By appealing to the subseismic approximation (SSA), we establish a new system of two first-order partial differential equations involving displacement field only, which govern the normal modes of a rotating self-gravitating stratified compressible inviscid liquid core. To treat a slightly ellipsoidal Earth model we use a coordinate transformation method which introduces modified spheroidal and toroidal fields so as to avoid trouble with the validity of Taylor expansions near a surface of discontinuity in material properties. The governing equations are expanded in these modified spheroidal and toroidal fields, so that the governing partial differential equations in three-dimensional space become systems of ordinary differential equations in one-dimensional space. As one application of the new subseismic governing system of equations (SGSE) and its expansions, we make an analytical study of the effects of the liquid core on the Chandler wobble (CW) and free core nutation (FCN) using an Earth model which consists of a much simplified mantle and inner core and a more realistic liquid outer core. The results show that the periods of the CW and FCN are independent of the structure of the Earth's liquid core when the radial variation of ellipticity of isopycnal surfaces in the liquid outer core is neglected. As a second application of the SGSE we calculate eigenperiods of some gravity/inertia modes for both non-rotating and rotating stably spherically stratified liquid core with a rigid mantle and inner core. This computation shows the advantage of the new SGSE over the subseismic wave equation (SSWE).

A fracture mechanical treatment of free edge stress singularities applied to a brazed ceramic/metal compound

The theoretical fracture mechanical treatment of craek problems in regular stress fields is extended to account for singular stresses as occur in bi-material systems whenever a discontinuity in material properties and geometry exists. The singular stress fields are derived for arbitrary material combination and geometry and the global stress state as occurs in a real compound is obtained by Finite Element (FE) calculations. Then especially the mode I stress intensity factors are calculated for semi-elliptical surface cracks in the ceramic component of a brazed ceramic/metal joint. Critical crack sizes are determined for failure analysis of the compound.

Numerical analog to plate equation‐including shear and rotary inertia for discontinuous materials

A numerical analog to the wave equation for a plate is presented and results of propagation studies of Gaussian and sinusoidal wave packets across the plate reinforced by ribs of four times the thickness of the plate are discussed. Results are presented for selected rib numbers, widths, and spacings. The wave equation is based on theory presented by Skudrzyk, modified to consider Kelvin‐type viscoelastic behavior and discontinuous material properties and plate thickness. The plate equation includes consideration of shear and rotary inertia. It is able to describe the propagation of flexural waves whose wavelengths vary from being long compared to the plate thickness to waves whose wavelength is 20% the plate thickness. Plate distortions are assumed to be antisymmetric about a plane through the center of the plate, the points of this central plane moves up and down, but not to the left or right. The wave equation in terms of the plate displacement w along the direction normal to the plate, of thickness h, is in the operator form, Λ2Λ1w + ρh [1 + 〈βμ〉 (∂/∂t)] (∂2w/∂t2) = Υq, where t is time and the operator Λ1 = ∇〈μ*〉h⋅∇ − ρh(∂2/∂t2), the operator Λ2 = ∇D*⋅∇1〈μh〉 − 〈ρh2〉12〈μ〉 ∂2∂t2(1 + 〈βμ〉∂∂t), and the source operator acting on applied pressure q is Υ = 1 + 〈βμ〉∂∂t − ∇D*⋅∇1〈μh〉 + 〈ρh2〉12〈μ〉 ∂2∂t2(1 + 〈βμ〉∂∂t), where 〈 〉 indicates that the material property has been averaged over a small volume of material which can be justified from the integral definition of the divergence theorem and represents a first‐order approximation to variations in the material property. The variable D* is defined by D* = Y*h3/[12(1 − v2)], where v is Poisson's ratio and Y* = Y[1 + βy(∂/∂t)] is Young's modulus. The shear modulus μ* = μ[1 + βμ(∂/∂t)]. The effects of viscoelastic relaxation of the material in the Kelvin form are expressed by βy(∂/∂t) and βμ(∂/∂t) where the β's are the viscoelastic relaxation times of the material. The material and plate parameters are a function of spatial position. A satisfactory numerical analog of the plate equation above has been developed by approximating the spatial and time derivatives by centered finite differences that divide the time domain into equal levels of time and the spatial domain into a grid where the functional values are computed at the grid nodes. The numerical analog to the plate equation is evaluated by the alternating direction implicit (ADI) technique. Current studies aim to identify the more significant parameters that determine the transformation of acoustic waves at beam/beam and plate/beam junctions.

A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom

A higher-order elastic parabolic equation (HEPE) is derived for wave propagation in depth-dependent and weakly range-dependent fluid/solid media. Galerkin’s method is used to discretize the depth operators in the HEPE within layers in which depth variations in the Lamé constants and density are continuous. Discontinuities in material properties are handled with centered differences for the interface conditions between layers. The numerical solution of the HEPE also involves the method of alternating directions and Crank–Nicolson integration. The HEPE is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range-dependent problem. The accuracy of the HEPE is demonstrated with benchmark calculations.

Fatigue strength characteristics of S35C/S35C friction welded tubular butt joints: Ogawa, K.-i., Nakayama, H., Oh-Ue, T. and Hasui, A. J. Soc. Mater. Sci., Jpn. Oct. 1988 37, (421), 1209–1215 (in Japanese)

Offshore wind turbines (OWT) are subjected to harsh environmental conditions in addition to the variable service loads. The present study is aimed at performing a realistic fatigue life estimation of the monopile structure using operational service loads recorded by online monitoring systems. Fatigue damage analysis has been conducted at the circumferential weld joints using finite element (FE) method by considering geometrical and material property discontinuities. Global-local modelling of the OWT was performed in as-welded condition to capture the local stress range at the weld toe, which acts as the critical site where cracks are most likely to initiate and propagate. The S-N fatigue design approach and maximum stress range at the weld toe have been used to determine the fatigue crack initiation life in monopiles. The results from the proposed approach show that a realistic life assessment can be made on monopile structures by accounting for the geometrical effects at the circumferential welds.

Elastic Gaussian wave packets in isotropic media

The theory of Gaussian wave packet (GWP) propagation in elastic materials is developed. The GWP solutions are in the form of localized disturbances with Gaussian spatial envelopes at any instant in time. The method is explicitly time dependent, but is conceptually no more difficult than time harmonic ray theory. The equations of propagation and evolution are very similar to those of standard, elastodynamic ray theory, but include an extra degree of freedom not considered previously: the temporal width of the pulse. The theory is valid if the carrier wavelength is short in comparison with typical length scales in the medium. Interfaces of discontinuity in material properties give rise to reflected and transmitted GWPs. Explicit expressions are presented that relate the incident GWP to the reflected and transmitted GWPs. These results are illustrated by numerical simulations of a pulse incident upon a spherical interface.

Inverse problems for nonabsorbing media with discontinuous material properties

One-dimensional electromagnetic and elastic inverse problems are formulated for media with discontinuous material properties. In addition, an impedence mismatch between source and medium is allowed. The measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem. The solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media.

Reflection of elastic waves from a linear transition layer

abstract A separation of P- and S-wave potentials is achieved for an inhomogeneous medium in which density is constant and Lame's parameters, λ and μ, are assumed to vary as λ/λ1 = μ/μ1 = (1 + bz)2 where λ1, μ1 and b are constants. The resulting equations are solved for an arbitrary angle of incidence. Plane wave reflection coefficients are obtained for the situation when the material mentioned above forms a transition layer between two homogeneous, elastic half-spaces. First and/or second-order discontinuities in material properties are permitted at the boundaries of the transition layer. Some numerical results are given.