Showing papers on "Isotropy published in 1990"

Interface crack between two elastic layers

Abstract: A semi-infinite interface crack between two infinite isotropic elastic layers under general edge loading conditions is considered. The problem can be solved analytically except for a single real scalar independent of loading, which is then extracted from the numerical solution for one particular loading combination. Two applications of the basic solution are made which illustrate its utility: interface cracking driven by residual stress in a thin film on a substrate, and an analysis of a test specimen proposed recently for measuring interface toughness.

Singularities, interfaces and cracks in dissimilar anisotropic media

Abstract: For a non-pathological bimaterial in which an interface crack displays no oscillatory behaviour, it is observed that, apart possibly from the stress intensity factors, the structure of the near-tip field in each of the two blocks is independent of the elastic moduli of the other block. Collinear interface cracks are analysed under this non-oscillatory condition, and a simple rule is formulated that allows one to construct the complete solutions from mode III solutions in an isotropic, homogeneous medium. The general interfacial crack-tip field is found to consist of a two-dimensional oscillatory singularity and a one-dimensional square root singularity. A complex and a real stress intensity factors are proposed to scale the two singularities respectively. Owing to anisotropy, a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields, however defined, does not recover the classical stress intensity factors as the bimaterial degenerates to be non-pathological. Collinear crack problems are also formulated in this context, and a strikingly simple mathematical structure is identified. Interactive solutions for singularity-interface and singularity interface-crack are obtained. The general results are specialized to decoupled antiplane and in-plane deformations. For this important case, it is found that if a material pair is non-pathological for one set of relative orientations of the interface and the two solids, it is non-pathological for any set of orientations. For bonded orthotropic materials, an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized. A complex-variable representation is presented for a class of degenerate orthotropic materials. Throughout the paper, the equivalence of the Lekhnitskii and Stroh formalisms is emphasized. The article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids.

Constitutive modelling of orthotropic plasticity in sheet metals

Abstract: T he classical quadratic yield criterion for orthotropic metals is known not to be sufficiently flexible in practice. By the simple expedient of admitting non-integer exponents, however, an improved criterion was devised for sheet with in-plane isotropy (so-called normal anisotropy). On the other hand, an acceptable proposal has not been forthcoming for sheet with in-plane anisotropy (so-called planar anisotropy). It is suggested here that improvement should be sought by incorporating a compatible dependence on orientation in a homogeneous yield function of arbitrary degree. In so doing, the practicalities of forming technology are respected by keeping the number of arbitrary parameters as small as possible. A new criterion is constructed along these lines and its implications are explored in detail. Additionally, a simple means of representing anisotropic yield criteria of any kind is presented with supporting general theorems.

Estimation of surface area and length with the orientator.

Abstract: SUMMARY The orientator is a new technique for the estimation of length and surface density and other stereological parameters using isotropic sections. It is an unbiased, design-based approach to the quantitative study of anisotropic structures such as muscle, myocardium, bone and cartilage. A simple method for the practical generation of such isotropic planes in biological specimens is described. No special technical equipment is necessary. Knowledge of an axis of anisotropy can be exploited to optimize the efficiency. To randomize directions in space, points are selected with uniform probability in a square using various combinations of simple random, stratified random, and systematic random sampling. The point patterns thus produced are mapped onto the surface of a hemisphere. The mapped points define directions of sectional planes in space. The mapping algorithm ensures that these planes arc isotropic, hence unbiased estimates of surface and length density can be obtained via the classical stereological formulae. Various implementations of the orientator are outlined: the prototype version, the orientator-gencrated ortrip, two systematic versions, and the smooth version. Orientator sections can be generated without difficulty in large specimens; we investigated human skeletal muscle, myocardium, placenta, and gut tissue. Slight practical modifications extend the applicability of the method to smaller organs like rat hearts. At the ultrastructural level, a correction procedure for the loss of anisotropic mitochondrial membranes due to oblique orientation relative to the electron beam is suggested. Other potential applications of the orientator in anisotropic structures include the estimation of individual particle surface area with isotropic nucleators, the determination of the connectivity of branching networks with isotropic disectors, and generation of isotropic sections for second-order stereology (three-dimensional pattern analysis).

Tension-field theory

Abstract: A general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations. The principal contribution is a partial differential equation describing a geometrical property of tension trajectories. This is one of a system of two equations which describes the state of stress independently of the deformation. This system is strongly elliptic at any stable solution, whereas the deformation is described by a system of parabolic type. Controllable solutions, i.e. those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone, are obtained. The general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples. Existing small strain theories are shown to correspond to a singular limit of the general theory, at which the underlying system changes from elliptic to parabolic type.

A stochastic model for the motion of particle pairs in isotropic high-reynolds-number turbulence, and its application to the problem of concentration variance

Abstract: A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.

Perturbations of Friedmann-Robertson-Walker cosmological models

Abstract: The mathematical principles underlying the theory of gauge-invariant perturbations of homogeneous isotropic cosmological models are discussed and a collection of useful formulae is given.

Isotropic and anisotropic damage variables in continuum damage mechanics

Abstract: The present work analyzes the implication and limitation of some scalar damage models. In particular, the elastic‐damage thermodynamic potential and the effective stress concept are reexamined. It is demonstrated that isotropic damage does not necessarily imply scalar damage representation in general. The notion of isotropic and anisotropic damage variables in continuum damage mechanics is then discussed. In addition, some results from micromechanical analyses are applied to show the direct relationship between the fourth‐order damage tensor and the damage‐induced compliance tensor characteristic of microcrack‐weakened brittle materials. It is shown that even for isotropic damage one should employ an isotropic fourth‐order damage tensor (not a scalar damage variable) to characterize the state of damage in materials, in accordance with the effective stress concept. In general, however, a damage tensor is anisotropic and should be derived from micromechanical analysis when possible.

Delamination Specimens for Orthotropic Materials

Abstract: A semi-infinite crack in an infinite strip of orthotropic material is analyzed. Analytic expressions for mixed-mode stress intensity factors are derived with only parameter undetermined, which is then extracted from numerical solutions to integral equations. The results are relatively simple and complete, and provide the flexibility to simulate a wide range of practical problems, such as fracture specimens and edge delamination phenomena of woods and fiber-reinforced composites. As an illustration, specimens with transverse splitting from notches are analyzed based on the general solution. The validity of using solutions for an isotropic material to calibrate some testing geometries of orthotropic materials is discussed.

Lode Dependences for Isotropic Pressure-Sensitive Elastoplastic Materials

Abstract: Experimental investigations indicate that the third stress invariant; Lode angle α affects significantly the behavior of pressure sensitive materials. The present communication presents a formulation to account for α in isotropic pressure-sensitive elastoplastic materials. Seven Lode dependences are reviewed

Finite element procedures for strain estimations of sheet metal forming parts

Abstract: A finite element algorithm is developed to predict the strain distribution of sheet forming parts and to evaluate their formability at the design stage. The algorithm is simply called the inverse approach since we know the positions of the material points on the final product and we determine the positions of the corresponding material points in the initial blank. This is also a direct algorithm which avoids the path dependent incremental procedure of plasticity and contact. This paper presents some theoretical aspects related to the large logarithmic strains with membrane triangular elements, the deformation theory of plasticity, the isotropic and anisotropic material properties of metal sheets and the non-linear solution techniques. Some particular problems are also studied, such as the influence of friction forces under the punch and blankholders, the ‘optimization’ of the initial blank's contour and the deep drawing in several steps. The results are compared with those obtained by experiments and by the more classical incremental approach.

Residual stresses and stored elastic energy of composites and polycrystals

Abstract: R esidual Stresses in heterogeneous materials may arise because of differential or anisotropic thermal expansion of constituents. The paper is concerned with thermoelastic solids whose material properties fluctuate on the microscopic scale. Rigorous general relations between stored elastic energy and statistical averages (mean values and fluctuations) of residual stresses are derived. These results are applied to two-phase composites and to materials where the fluctuations of elastic constants can be neglected. One obtains exactly the stored energy, certain conditional mean values and the covariance matrix of the residual stresses. Under the assumptions of statistical homogeneity and isotropy, the results hold for any type of heterogeneous microstructure.

On Surface Waves and Deformations in a Pre-stressed Incompressible Elastic Solid

Abstract: The propagation of infinitesimal surface waves on a half-space of incompressible isotropic elastic material subject to a general pure homogeneous pre-strain is considered. The secular equation for propagation along a principal axis of the pre-strain is obtained for a general strain-energy function, and conditions which ensure stability of the underlying pre-strain are derived

Computer simulation of a biaxial liquid crystal

Abstract: We report the first results of Monte Carlo simulations using hard ellipsoids with three distinct semi-axes a, b, c chosen such that abc = 1, c/a = 10 and b/a varies between 1 and 10 A survey of the phase diagram provides evidence for the existence of isotropic, nematic, discotic, and biaxial liquid crystal phases; this is believed to be the first simulation of a biaxial phase of a bulk liquid with full rotational and translational freedom We find that the phase diagram is approximately symmetric under the transformation {a, b, c} → {a−1,b−1,c−1}, and that the biaxial phase is most stable at about the expected (self-conjugate) value b= √ac For this value, the isotropic phase transforms directly into the biaxial phase on compression, at a density at least 50 per cent higher than that at which-the nematic-isotropic transition occurs in the corresponding uniaxial systems These results are in semi-quantitative agreement with recent theories, but there are also some significant differences

Solar radiation model for Europe

Abstract: This study is concerned with the evaluation of the isotropic and four anisotropic solar radiation models for inclined surfaces. The evaluation procedure was split in two stages— abbreviated and det...

Static and dynamic behavior of a porous solid saturated by a two‐phase fluid

Abstract: A method is presented to determine the elastic constants for an isotropic, porous, elastic solid saturated by a two‐phase fluid. Assuming that the shear modulus of the empty matrix is known, it is shown that the six additional coefficients in the stress–strain relations can be uniquely determined by performing two ideal experiments referred to as ‘‘generalized jacketed and partially jacketed compressibility tests,’’ in analogy with the single‐phase theory of Biot. Under reasonable assumptions on the behavior of the material, the experiments yield expressions for the coefficients in terms of the material properties of the individual phases and the capillary pressure function relating the pressures in the two fluid phases. Finally, numerical results showing properties of the phase velocities and attenuations for the four different types of body waves are presented and analyzed.

Stiffness and thermal expansion predictions for hybrid short fiber composites

Abstract: A model is developed to predict thermal expansion coefficients and elastic moduli of multi-component (hybrid) composites. The model includes the influences of fiber aspect ratio; isotropic and anisotropic fiber materials; planar, three-dimensional or arbitrary fiber orientation; hollow and solid spherical reinforcements; and voids. The first step in the procedure is to predict the properties of an aligned-fiber single-reinforcement composite for each reinforcement type. Various micro-mechanics approaches are used, depending on the type of reinforcement. A simplified version of Lee and Westmann's theory is found to work well for hollow spherical reinforcements. Performing an orientation average accounts for the spatial orientation of each reinforcement, then an aggregate averaging procedure combines the single-reinforcement properties to model the hybrid. Predictions of the model compare favorably to experimental elastic and thermal properties of short fiber/hollow sphere composites designed for very high speed integrated circuit (VHSIC) board applications.

A constitutive model for the dynamic response of brittle materials

Abstract: A microphysically based material model for the dynamic inelastic response of a brittle material is developed. The progressive loss of strength as well as the post‐failure response of a granular material with friction are included. Crack instability conditions (an inelastic surface in stress space) and inelastic strains are obtained by considering the response of individual microcracks to an applied stress field. The assumptions of material isotropy and an exponential distribution for the crack radius are invoked to provide a tractable formulation. The constitutive model requires a minimal number of physical parameters, is compatible with a previously developed ductile fracture model [J. Appl. Phys. 64, 6699 (1988)] that utilizes inelastic surfaces, and can be formulated as an efficient, robust numerical algorithm for use in three‐dimensional computer codes. The material model is implemented into a Lagrangian computer formulation for the demonstration of material response to dynamic loading conditions. Comparisons with one‐dimensional, uniaxial impact experiments are provided.

Ferromagnetic resonance of particulate magnetic recording tapes

Abstract: The room‐temperature ferromagnetic resonance (FMR) spectra of γ‐Fe2O3, CrO2, and barium ferrite particulate magnetic recording tapes have been measured at microwave frequencies of 9.35 and 35 GHz for various orientations of the static and high‐frequency magnetic fields with respect to the tape. For CrO2 tapes, the influence of the width of the angular distribution of the particle orientations on the FMR spectra has been studied from the nearly isotropic case up to the highly oriented case. Hysteretic behavior for a CrO2 tape as well as the effect of tape calendering for a γ‐Fe2O3 tape has been observed by FMR. Experimental results are found to be in reasonable agreement with results of theoretical calculations based on a model of an ellipsoidal single‐domain particle with both shape and magnetocrystalline anisotropy. Magnetostatic interaction inside the magnetic film has been introduced by expressing the total magnetostatic energy as a combination of a part dependent on particle shape and a part dependent...

A simple nonlinear model for the return to isotropy in turbulence

Abstract: A quadratic nonlinear generalization of the linear Rotta model for the slow pressure‐strain correlation of turbulence is developed for high Reynolds number flows. The model is shown to satisfy realizability and to give rise to no stable nonzero equilibrium solutions for the anisotropy tensor in the case of vanishing mean velocity gradients. In order for any model to predict a return to isotropy for all relaxational flows, it is necessary to ensure that there is no nonzero stable fixed point that attracts realizable initial conditions. Both the phase space dynamics and the temporal behavior of the model are examined and compared against experimental data for the return to isotropy problem. It is demonstrated that the quadratic model successfully captures the experimental trends which clearly exhibit nonlinear behavior. Comparisons are also made with the predictions of the linear Rotta model, the quasilinear Lumley model, and the nonlinear model of Shih, Mansour, and Moin. The simple quadratic model proposed in this study does better than the Rotta model as anticipated, and also compares quite favorably with the other more complicated nonlinear models.

On a constitutive theory for materials undergoing microstructural changes

Abstract: Of particular interest is a two-network theory of polymer response. The mechanical response depends on the deformation of both the remaining portion of the original material and newly formed one. A particular constitutive equation is introduced. The original and newly formed material are both treated as incompressible isotropic nonlinear neo-Hookean elastic materials, but with different reference configurations

Overall elastic properties of isotropic materials with arbitrary distribution of circular cracks

Abstract: SUMMARY The transmission properties of the mean field in elastic material with a random distribution of circular cracks of small aspect ratio are presented here for the general case where the crack normals are distributed in any pre-determined way in space. Special distributions, where the crack normals lie in one direction only, or lie at a fixed angle to a fixed direction, reproduce established results. Expressions are given for the case of crack normals lying close to a given direction in a Gaussian distribution. All results are valid to second order in the crack number density.

Flux diffusion in high-T c superconductors

Abstract: In type-II superconductors in the flux flow (J⊥≫J c ), flux creep (J c ⊥≈J c ), and thermally activated flux flow (TAFF) (J⊥≪J c ) regimes the inductionB(r,t), averaged over several penetration depths λ, in general follows from a nonlinear equation of motion into which enter the nonlinear resistivities ρ⊥(B, J⊥,T) caused by flux motion and ρ‖(B, J‖,T) caused by other dissipative processes.J⊥ andJ‖ are the current densities perpendicular and parallel toB,B=|B|, andT is the temperature. For flux flow and TAFF in isotropic superconductors with weak relative spatial variation ofB, this equation reduces to the diffusion equation\(\dot B = (\rho _ \bot /\mu _0 ) \bar V^2 B\) plus a correction term which vanishes whenJ‖=0 (this meansBׇ×B=0) or when ρ⊥ − ρ‖ = 0 (isotropic normal conductor). When this diffusion equation holds the material anisotropy may be accounted for by a tensorial ρ⊥. The response of a superconductor to an applied current or to a change of the applied magnetic field is considered for various geometries. Such perturbations affect only a surface layer of thickness λ where a shielding current flows which pulls at the flux lines; the resulting deformation of the vortex lattice diffuses into the interior until a new equilibrium or a new stationary state is reached. The a.c. response, in particular the frequency with maximum damping, depends thus on the geometry and size of the superconductor.

An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels

Abstract: This paper is an analytical study on natural two-dimensional convection in horizontal rectangular channels filled by isotropic and anisotropic porous media. The channel walls, assumed to be impermeable and perfectly heat conducting, are nonuniformly heated to establish a linear temperature distribution in the vertical direction. The authors derive the critical Rayleigh numbers for the onset of convection and examine the steady flow patterns at moderately supercritical Rayleigh numbers. The stability properties of these flow patterns are examined against two-dimensional perturbations using a weakly nonlinear theory.