Anisotropic Nonlinear Elastic Model for Particulate Materials

This paper presents the development of an elastic model for particulate materials based on micromechanics considerations. A particulate material is considered as an assembly of particles. The stress–strain relationship for an assembly can be determined by integrating the behavior of the interparticle contacts in all orientations and using a static hypothesis which relates the average stress of the granular assembly to a mean field of particle contact forces. Hypothesizing a Hertz–Mindlin law for the particle contacts leads to an elastic nonlinear behavior of the particulate material, we were able to determine the elastic constants of the granular assembly based on the properties of the particle contacts. The numerical predictions, compared to the results obtained during experimental studies on different granular materials, show that the model is capable of taking into account both the influence of the inherent anisotropy and the influence of the stress-induced anisotropy for different stress conditions.     

Anisotropic Damage Model for Concrete

In this paper, a damage constitutive model accounting for induced anisotropy and bimodular elastic response is applied to two-dimensional analysis of reinforced concrete structures. Initially, a constitutive model for the concrete is presented, where the material is assumed as an initial elastic isotropic medium presenting anisotropy and bimodular response (distinct elastic responses, whether tension or compression stress states, prevail) induced by damage. Two damage tensors govern the stiffness under prevailing tension or compression stress states. Criteria are then proposed to characterize the dominant states. Finally, the proposed model is used in plane analysis of reinforced concrete beams to show its potential for use and to discuss its limitations.     

Energy approach to the problem of calculating the stresses at the initial stages of plastic deformation of crystalline substances and the appearance of structural defects

The critical shear stress and its temperature dependence are calculated for 12 simple substances with different structures and types of bonding. The shear stress for stage II–III of deformation of single crystals (τ^II–III) and σ_{0, y}, i.e., the Hall–Petch relation extrapolated to an infinitely large grain size, are estimated. The energy of formation of lattice defects (vacancies) is calculated using a proposed expression. The results of calculation of the elastic shear energy of a matrix and regions with a high elastic anisotropy are used to estimate the role of elastic anisotropy in lattice stability and fracture. The calculated and experimental results agree satisfactorily with each other.     

Glass Fiber Reinforced Polymer Pultruded Members: Constitutive Model and Stability Analysis

The increasingly widespread use of fiber-reinforced polymers as an alternative to conventional materials makes it necessary to formulate theoretical models which adequately evaluate the influence of the anisotropy of such composites on the structural behavior. While the cross section shapes adopted for compressed members are generally the same as in steel structures, the anisotropy which characterizes these polymers may reduce the critical loading threshold due to local buckling phenomena. A procedure to study the buckling of glass fiber reinforced polymer pultruded members by means of an homogenization approach is proposed here. A two-stage buckling model permits the determination of both global and local critical loads as explicit functions of the member geometry and its material behavior. These functions may be used for optimization of the shape of the above-mentioned members. Besides the model shows its reliability as it fits the results of experimental testson members with different slenderness ratios.     

Effect of elastic strain energy on the core-shell structures of the precipitates in Al-Sc-Er alloys

The effect of the elastic strain energy on the core-shell structures was studied in an Al-0.06Sc-0.02Er (at.%) alloy. A theoretical model for the calculation of the elastic strain energy caused by core-shell precipitates, which is applicable to materials with weak elastic ani-sotropy, was adopted. It was demonstrated that the partitioning of Er to the precipitate core did not reduce the elastic strain energy as expected in the previous study. The resistance due to the elastic strain energy to form an Al3(Sc0.36Er0.64)-Al3(Sc0.8Er0.2) core-shell precipitate was quite small, and could be easily overcome by the decrease of the total interfacial energy, which was consistent with the previous experimental re-sults. On the other hand, the resistance due to the elastic strain energy to form an Al3Er-Al3Sc core-shell precipitate was much larger than that to form an Al3(Sc0.36Er0.64)-Al3(Sc0.8Er0.2) core-shell precipitate, thus the partitioning of all the Er atoms to the core was strongly hindered by the elastic strain energy and was not observed in the experiment of the previous study.)     

A model of the thermoelastic growth of martensite

Eshelby's continuum elastic model of the strain field of a coherent inclusion in an infinite matrix is adapted to describe the strain field associated with a circular plate of ellipsoidal cross-section transformed martensitically. Initially, the plate and the austenitic matrix are assumed to be elastically Isotropic, but with different moduli. The model is also used to calculate the strain field of a pair of parallel, ellipsoidal, martensite plates and of a single, isolated, square, flat plate. Finally, the treatment is extended to reveal the effects of elastic anisotropy.A criterion for thermoelastic growth of an isolated plate of martensite in terms of the permissible plastic yielding is proposed. Using the model of the strain fields, contours of the yield envelopes associated with an isolated plate of martensite formed in Fe₃Pt are calculated as a function of the atomic order in the parent, austenitic phase. It is concluded that thermoelastic growth is to be expected when the long-range order, S, is greater or equal to 0.2, in agreement with experimental observations. The equilibrium thickness of an isolated thermoelastic plate at M_s and the associated chemical driving force are found to be much smaller than for plates formed irreversibly. Quantitatively, the predictions of the model applied to the ‘ordered’ alloy are in good agreement with experimental measurements. Clearly, Invar elastic softening is a major factor determining thermoelastic behavior in Fe₃Pt.In addition, the model is used to explore the reduction in elastic accommodation strain which can be achieved by the interactions of two or more plates arranged edge-to-edge or in stacks of self-accommodating plates. The influence of the shape of the individual plates is also discussed.Calculations using the anisotropic model for an isolated plate in the form of an oblate spheroid show that the effects of the large elastic anisotropy, which develops on cooling partially ordered Fe₃Pt, on the equilibrium thickness of the thermoelastic plate and the chemical driving force at _Ms are surprisingly small.     

Numerical determination of the elastic driving force for directional coarsening in Ni-superalloys

We have developed a general methodology, in the framework of the finite element method, for locally evaluating the generalized force acting on a material interface which is work-conjugate with the normal displacement of the interface itself. This methodology has been applied to the study of directional coarsening of γ′ precipitates in Ni-superalloys. The flexibility of the proposed method has allowed us to closely model the actual microstructural morphology of the alloys and to account for the effects of applied boundary conditions, lattice misfit, elastic anisotropy and inelastic behavior of the crystals. We have positively compared the indications of our model with available experimental data for a few alloys, and a circumscribed parametric study has lead us to formulate a more general interpretation of the rafting phenomenon, which appears to give a satisfactory explanation for all the available experimental observations.     

The histopathology of nonsteroidal anti-inflammatory drug-associated colitis

We combined three techniques--mechanical testing, three-dimensional imaging, and finite-element modeling--to distinguish between the contributions of architecture and tissue modulus to mechanical function in human trabecular bone. The objectives of this study were 2-fold. The first was to assess the accuracy of micromechanical modeling of trabecular bone using high-contrast x-ray images of the trabecular architecture. The second was to combine finite-element calculations with mechanical testing to infer an average tissue modulus for the specimen. Specimens from five human L1 vertebrae were mechanically tested along the three anatomic axes. The specimens were then imaged by synchrotron x-ray tomography, and the elastic moduli of each specimen were calculated from the tomographic image by finite-element modeling. We found that 23-microm tomographic images resolved sufficient structural detail such that the calculated anisotropy in the elastic modulus was within the uncertainties of the experimental measurements in all cases. The tissue modulus of each specimen was then estimated by comparing the calculated mean stiffness of the specimen, averaged over the three anatomical directions, with the experimental measurement. The absolute values of the experimental elastic constants could be fitted, again within the uncertainties of the experimental measurements, by a single tissue modulus of 6.6 GPa, which was the average tissue modulus of the five specimens. These observations suggest that a combination of mechanical testing, three-dimensional imaging, and finite-element modeling might enable the physiological variations in tissue moduli to be determined as a function of age and gender.     

The plastic anisotropy of two-phase aluminium alloys—I. Anisotropy in unidirectional deformation

The changes in plastic anisotropy which accompany precipitation from a supersaturated solid solution have been investigated in textured polycrystals of three different aluminium alloys. Plastic behaviour was measured in unidirectional straining by uniaxial tension and by plane strain compression. Plastic anisotropies in the solution treated conditions were well predicted by the Taylor/Bishop and Hill model of polycrystalline plasticity used in conjunction with a series expansion method of treating the crystal orientation distribution. Significant deviations from these predicted anisotropies resulted from the introduction of semi-coherent precipitates, although the textures of the primary phases were unchanged by the ageing treatments. It is shown that changes in plastic anisotropy caused by precipitation can be explained satisfactorily by two different continuum models. The first of these, due to Hosford and Zeisloft, assumes plastic deformation of the second phase. The second model, introduced here, is based essentially on the adaptation of the transformation problem solutions of Eshelby made by Brown and Stobbs: it assumes deformation of the precipitate is elastic. It is concluded that selection of the more appropriate model requires that additional evidence be taken into account For the case of semi-coherent precipitates in the aluminium alloys investigated, the elastic inclusion model is more realistic. However, direct evidence of the magnitude of the internal stresses generated during plastic deformation is required to test some of the assumptions implicit in the simple form of the elastic inclusion model.     

A two-dimensional analysis of the evolution of coherent precipitates in elastic media

The morphological evolution of coherent inclusions in elastic media is studied in two-dimensions. The inclusions are simple dilations with isotropic surface energy in a system with homogeneous elastic constants of negative anisotropy. The equilibrium sizes at which a circular inclusion transforms to a rectangle or square, and at which a square splits into a doublet or quartet of separated inclusions are computed analytically. A finite-element model is then constructed to simulate the evolution of an arbitrary distribution of inclusions along the minimum-energy path. In the model, the circle evolves into a square, which splits into a doublet by hollowing from its center, or, if this is forbidden, by drawing in a perturbation on its surface. The sizes at which shapes spontaneously transform are compared to the equilibrium values. Finally, the simulation is used to study the evolution of a random distribution of inclusions. The first metastable state assumed by the distribution depends on the elastic interaction, surface energy and areal fraction of the inclusion phase through a single dimensionless parameter that groups these three effects. The results are compared to prior theoretical and experimental work on coarsening patterns in three dimensions.     

Inhomogeneous Strain/Stress Profiles in the Nacre Layer of Mollusk Shells

Internal strain/stresses in the nacre layer of mollusk shells are measured as a function of depth under controlled etching in selected areas. We found strain release to be enhanced when approaching the inner surface of the shell adjacent to the mollusk mantle. Strain distribution across the thickness of the shell consists of two components: a slowly changing component, which reflects elastic bending of the shell; and a short-range periodic modulation due to forces acting at interfaces between ceramic lamella. Simulations based on the multilayer structure model reproduce well the main features of the obtained experimental data.     

Lattice plane response during tensile loading of an aluminum 2 percent magnesium alloy

In-situ neutron diffraction measurements of plane specific elastic lattice strains were made during a tensile test of an aluminum 2 pct magnesium alloy. The macroscopic response exhibited a serrated flow curve, evidence of dynamic strain aging. The neutron results are compared to calculations using a self-consistent polycrystal deformation model. The relatively poor agreement with the measured data may suggest that the model has limitations with respect to face-centered cubic (fcc) alloys with low elastic anisotropy.     

Lattice plane response during tensile loading of an aluminum 2 percent magnesium alloy

In-situ neutron diffraction measurements of plane specific elastic lattice strains were made during a tensile test of an aluminum 2 pct magnesium alloy. The macroscopic response exhibited a serrated flow curve, evidence of dynamic strain aging. The neutron results are compared to calculations using a self-consistent polycrystal deformation model. The relatively poor agreement with the measured data may suggest that the model has limitations with respect to face-centered cubic (fcc) alloys with low elastic anisotropy.     

The preferred habit of a tetragonal inclusion in a cubix matrix

The preferred crystallographic habit of a plate-like precipitate in a cubic matrix is determined by minimizing the elastic strain energy of the precipitate under the conditions that 1. the precipitate and matrix behave as linear elastic media, 2. the difference between their elastic constants may be neglected, 3. the transformation strain is tetragonal, and 4. the elastic strain energy dominates the surface energy in determining the preferred habit. It is shown that the preferred habit plane is necessarily of type {hok} if the matrix has negative elastic anisotropy (Δ = c₁₁ − c₁₂ − 2c₄₄ < 0) and is of type {hhk} if Δ > 0. In both cases the normal to the habit plane varies systematically with the tetragonality of the transformation strain according to relations which are found analytically and confirmed by numerical computation. The predictions of the model are compared with experimental evidence for the cases of G P zones in Al-Cu and Cu-Au, Fe₄C precipitates in α-Fe and β-VH_0.45 precipitates in vanadium, with encouraging results.     

Acoustic Measurements of the Anisotropy of Dynamic Elastic and Poromechanics Moduli under Three Stress/Strain Pathways

A series of triaxial compression experiments have been conducted to investigate the effects of induced stress on the anisotropy developed in dynamic elastic and poroelastic parameters in rocks. The measurements were accomplished by utilizing an array of piezoelectric compressional and shear wave sensors mounted around a cylindrical sample of porous Berea sandstone. Three different types of applied states of stress were investigated using hydrostatic, triaxial, and uniaxial strain experiments. During the hydrostatic experiment, where an isotropic state of stress was applied to an isotropic porous rock, the vertical and horizontal acoustic velocities and dynamic elastic moduli increased as pressure was applied and no evidence of stress induced anisotropy was visible. The poroelastic moduli (Biot’s effective stress parameter, α) decreased during the test but also with no evidence of anisotropy. The triaxial compression test involved an axisymmetric application of stress with an axial stress greater than the two constant equal lateral stresses. During this test a marked anisotropy developed in the acoustic velocities, and in the dynamic elastic and poroelastic moduli. As axial stress increased the magnitude of the anisotropy increased as well. The uniaxial strain test involved axisymmetric application of stresses with increasing axial and lateral stresses but while maintaining a zero lateral strain condition. The uniaxial strain test exhibited a quite different behavior from either the triaxial or hydrostatic tests. As both the axial and lateral stresses were increased, an anisotropy developed early in the loading phase but then was effectively “locked in” with little or no change in the magnitude of the values of the acoustic velocities, or the dynamic elastic and poroelastic parameters as stresses were increased. These experimental results show that the application of triaxial states of stress induced significant anisotropy in the elastic and poroelastic parameters in porous rock, while under the uniaxial strain condition the poromechanics, Biot’s effective stress parameter, exhibited the largest variation among the three test conditions.     

Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap

Morphological evolutions controlled by a transformation-induced elastic strain during a solid state precipitation are systematically investigated using a prototype binary alloy as a model system. A computer simulation technique based on a microscopic kinetic model including the elastic strain effect is developed. Without any a priori assumptions concerning shapes, concentration profiles and mutual positions of new phase particles, various types of coherent two-phase morphologies such as basket-weave structures, sandwich-like multi-domain structures, precipitate macrolattices and GP zones are predicted. A wide variety of interesting strain-induced kinetic phenomena are observed during development of the above microstructures, including selective and anisotropic growth, reverse coarsening, particle translational motion, particle shape transition and splitting. In spite of all simplifications of the model, most of the simulation results are confirmed by experimental observations in various alloy systems, indicating that this kinetic model can be efficiently used for understanding, interpreting and predicting structural evolutions in real alloys.     

Elastic constants of transversely isotropically porous (TIP) materials

We derive formulas describing the dependence of the elastic characteristics of multicapillary materials on the capillary porosity. The investigated materials are classified as transversely isotropic, and the anisotropy in their properties is the result of the directionality of the capillary pores. Analysis of the dependences obtained has shown that the elasticity moduli of these materials may be calculated using formulas suggested for reinforced materials, in which the elastic constants of the fibers are assumed to be equal to zero. We derive a relation between the Poisson's ratios and the capillary porosity.Institute of Problems of Materials Science, National Academy of Sciences of the Ukraine, Kiev. Translated from Poroshkovaya Metallurgiya, No. 5–6, pp. 104–109, May–June, 1994.     

Characterization of anisotropie elastic constants of silicon-carbide participate reinforced aluminum metal matrix composites: Part I. Experiment

The anisotropic elastic properties of silicon-carbide particulate (SiC _p ) reinforced Al metal matrix composites were characterized using ultrasonic techniques and microstructural analysis. The composite materials, fabricated by a powder metallurgy extrusion process, included 2124, 6061, and 7091 Al alloys reinforced by 10 to 30 pct ofα-SiC _p by volume. Results were presented for the assumed orthotropic elastic constants obtained from ultrasonic velocities and for the microstructural data on particulate shape, aspect ratio, and orientation distribution. All of the composite samples exhibited a systematic anisotropy: the stiffness in the extrusion direction was the highest, and the stiffness in the out-of-plane direction was the lowest. Microstructural analysis suggested that the observed anisotropy could be attributed to the preferred orientation of SiC _p . The ultrasonic velocity was found to be sensitive to internal defects such as porosity and intermetallic compounds. It has been observed that ultrasonics may be a useful, nondestructive technique for detecting small directional differences in the overall elastic constants of the composites since a good correlation has been noted between the velocity and microstructure and the mechanical test. By incorporating the observed microstructural characteristics, a theoretical model for predicting the anisotropic stiffnesses of the composites has been developed and is presented in a companion article (Part II). Formerly with the Department of Aerospace Engineering and Engineering Mechanics, Iowa State University Formerly with Westinghouse Science & Technology Center     

Dynamic properties of lymphatic pathways for the absorption of cerebrospinal fluid

Recent advances in small, linear-array transducers have opened new avenues for three-dimensional image acquisition from an intracardiac approach. The purpose of this study was to introduce a novel method of image acquisition using toroidal geometry, explore its fidelity of reproduction of three-dimensional cardiac anatomy, and determine whether a whole-heart scan is achievable. Acquisition was accomplished through 360-degree incremental rotation of a rigid endoscope with a side-mounted ultrasound transducer. The procedure was first tested with the use of a gelatin model to define far-field slice resolution with 1.8-degree rotational increments. Comparison of three-dimensional scans of cardiac specimens with corresponding photographs confirmed that toroidal geometry can provide a high-quality display of structures from all sides. We conclude that whole-heart three-dimensional scanning from within the cardiac chambers is possible with toroidal geometry. The quality of depicted anatomy depends on transducer location within the heart, distance from the transducer, density of slices, and image resolution. The potential of intracardiac three-dimensional ultrasound imaging includes detailed spatial evaluation of cardiac morphology, determination of appropriate placement of investigative or therapeutic devices (catheters, closure devices, etc.), and assessment of cardiac function.     

Robustness of the sheet metal springback cup test

The robustness of a proposed test for elastic springback characterization of sheet metal has been examined using a matrix of defined experimental errors. A series of flat bottom deep drawn cups made from AISI 1010 steel sheet were examined. It was found that misalignment of the blank over the forming tool and error in the vertical location where the springback ring was cut from the cup sidewall had the largest effect on the resulting springback opening. Other experimental errors involving cup height and ring width were found to be less important. The effect of in-plane anisotropy of mechanical properties on springback was negligible. The results are examined in terms of measured through thickness residual stresses and elastic bending of beams with circumferential thickness gradients.