Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 1: Theory

1980 ◽  
Vol 47 (4) ◽  
pp. 775-780 ◽  
Author(s):  
W. C. Johnson ◽  
Y. Y. Earmme ◽  
J. K. Lee
Keyword(s):  
Integral Equation ◽  
Strain Field ◽  
Equivalent Stress ◽  
Taylor Series Expansion ◽  
Transformation Strain ◽  
Coherent Precipitate ◽  
Precipitate Shape ◽  
Polynomial Series ◽  
Elastic Strain Field

Two independent methods are derived for the calculation of the elastic strain field associated with a coherent precipitate of arbitrary morphology that has undergone a stress-free transformation strain. Both methods are formulated in their entirety for an isotropic system. The first method is predicated upon the derivation of an integral equation from consideration of the equations of equilibrium. A Taylor series expansion about the origin is employed in solution of the integral equation. However, an inherently more accurate means is also developed based upon a Taylor expansion about the point of which the strain is to be calculated. Employing the technique of Moschovidis and Mura, the second method extends Eshelby’s equivalency condition to the more general precipitate shape where the constrained strain is now a function of position within the precipitate. An approximate solution to the resultant system of equations is obtained through representation of the equivalent stress-free transformation strain by a polynomial series. For a given order of approximation, both methods reduce to the determination of the biharmonic potential functions and their derivatives.

Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 2: The Cuboidal Inhomogeneity

1980 ◽  
Vol 47 (4) ◽  
pp. 781-788 ◽  
Author(s):  
W. C. Johnson ◽  
Y. Y. Earmme ◽  
J. K. Lee
Keyword(s):  
Integral Equation ◽  
Strain Field ◽  
Taylor Series Expansion ◽  
Integral Equation Method ◽  
Transformation Strain ◽  
Shear Moduli ◽  
Integral Equation Methods ◽  
Central Regions ◽  
Cuboidal Precipitate

The modified equivalency and integral equation methods for determination of the constrained strain field associated with a precipitate that has undergone a dilatational stress-free transformation strain as developed in Part 1, are applied to the case of a cuboidal inhomogeneity within an isotropic matrix. Agreement between the two methods is good for small and moderate differences in the shear moduli between precipitate and matrix. For large differences in the shear moduli, some divergence is observed in that fluctuations in the constrained strain field become quite pronounced near the cube edge and corner when considering the integral equation method. Although some error is inevitable due to the cutoff of higher-order terms in the Taylor series expansion, the modified equivalency method yields fair results under such circumstances. With the latter method, the constrained strain field of a cuboid is analyzed as a function of position and orientation. Although the strain field behaves as expected in the central regions of the cube in that the harder the precipitate the larger the constrained strain, its behavior becomes complicated as the precipitate-matrix interface is approached, demonstrating a strong dependency on precipitate rigidity. As a result, the dilatation in the inhomogeneous cuboidal precipitate is found not to be a constant as contrasted with the homogeneous case.

An Integral Equation Approach to the Inclusion Problem of Elastoplasticity

1982 ◽  
Vol 49 (2) ◽  
pp. 312-318 ◽  
Author(s):  
W. C. Johnson ◽  
J. K. Lee
Keyword(s):  
Integral Equation ◽  
Stress Field ◽  
Yield Condition ◽  
Transformation Strain ◽  
Iterative Solution ◽  
Inclusion Problem ◽  
Flow Rule ◽  
Equation Approach ◽  
Coherent Precipitate ◽  
Integral Equation Approach

An integral equation approach is derived for the calculation of the elastoplastic strain field associated with a transformed inclusion of constant stress-free transformation strain and for an inhomogeneity when the far stress field remains elastic. The assumptions of a coherent precipitate and the deformation theory of plasticity are employed although any yield condition and flow rule can be chosen. The complexity of the integral equation is such that an iterative solution scheme is necessary. The technique is applied to a spherical precipitate in a uniform elastic stress field where associated stress and strain fields and plastic zone are calculated. The nature of the plastic relaxation process compares qualitatively with two-dimensional plane stress behavior. Extension of this technique to the nonaxisymmetric problem is also examined.

Strain analysis in composite ceramics

1986 ◽  
Vol 44 ◽  
pp. 494-497
Author(s):  
W. M. Kriven
Keyword(s):  
Strain Field ◽  
Strain Analysis ◽  
Processing Temperature ◽  
Transformation Toughening ◽  
Zirconia Ceramics ◽  
Volume Increase ◽  
Analysis Technique ◽  
Fundamental Understanding ◽  
Contrast Analysis ◽  
Elastic Strain Field

Significant progress towards a fundamental understanding of transformation toughening in composite zirconia ceramics was made possible by the application of a TEM contrast analysis technique for imaging elastic strains. Spherical zirconia particles dispersed in a large-grained alumina matrix were examined by 1 MeV HVEM to simulate bulk conditions. A thermal contraction mismatch arose on cooling from the processing temperature of 1500°C to RT. Tetragonal ZrO2 contracted amisotropically with α(ct) = 16 X 10-6/°C and α(at) = 11 X 10-6/°C and faster than Al2O3 which contracted relatively isotropically at α = 8 X 10-6/°C. A volume increase of +4.9% accompanied the transformation to monoclinic symmetry at room temperature. The elastic strain field surrounding a particle before transformation was 3-dimensionally correlated with the internal crystallographic orientation of the particle and with the strain field after transformation. The aim of this paper is to theoretically and experimentally describe this technique using the ZrO2 as an example and thereby to illustrate the experimental requirements Tor such an analysis in other systems.

Laser interferometric method for determination of elastoplastic strain field

2003 ◽  
Author(s):  
Shibin Wang ◽  
Jingwei Tong ◽  
Mario Cottron ◽  
Linan Li ◽  
Zhiyong Wang
Keyword(s):  
Strain Field ◽  
Elastoplastic Strain ◽  
Interferometric Method ◽  
Laser Interferometric

Strain field determination in III–V heteroepitaxy coupling finite elements with experimental and theoretical techniques at the nanoscale

2017 ◽  
Vol 26 (1-2) ◽  
pp. 1-8
Author(s):  
Nikoletta Florini ◽  
George P. Dimitrakopulos ◽  
Joseph Kioseoglou ◽  
Nikos T. Pelekanos ◽  
Thomas Kehagias
Keyword(s):  
Finite Elements ◽  
Strain Field ◽  
Strain Distribution ◽  
Elastic Strain ◽  
Growth Direction ◽  
Current Status ◽  
Interface Plane ◽  
Fe Method ◽  
Semiconductor Nanostructure

AbstractWe are briefly reviewing the current status of elastic strain field determination in III–V heteroepitaxial nanostructures, linking finite elements (FE) calculations with quantitative nanoscale imaging and atomistic calculation techniques. III–V semiconductor nanostructure systems of various dimensions are evaluated in terms of their importance in photonic and microelectronic devices. As elastic strain distribution inside nano-heterostructures has a significant impact on the alloy composition, and thus their electronic properties, it is important to accurately map its components both at the interface plane and along the growth direction. Therefore, we focus on the determination of the stress-strain fields in III–V heteroepitaxial nanostructures by experimental and theoretical methods with emphasis on the numerical FE method by means of anisotropic continuum elasticity (CE) approximation. Subsequently, we present our contribution to the field by coupling FE simulations on InAs quantum dots (QDs) grown on (211)B GaAs substrate, either uncapped or buried, and GaAs/AlGaAs core-shell nanowires (NWs) grown on (111) Si, with quantitative high-resolution transmission electron microscopy (HRTEM) methods and atomistic molecular dynamics (MD) calculations. Full determination of the elastic strain distribution can be exploited for band gap tailoring of the heterostructures by controlling the content of the active elements, and thus influence the emitted radiation.

Experimental Determination of Side Boundary Effects on Stress Intensity Factors in Surface Flaws

1975 ◽  
Vol 97 (1) ◽  
pp. 45-51 ◽  
Author(s):  
M. Jolles ◽  
J. J. McGowan ◽  
C. W. Smith
Keyword(s):  
Stress Intensity ◽  
Crack Length ◽  
Taylor Series ◽  
Stress Intensity Factors ◽  
Taylor Series Expansion ◽  
Correction Method ◽  
Intensity Factors ◽  
Surface Flaws ◽  
Plate Width

A technique consisting of stress-freezing photoelasticity coupled with a Taylor Series Expansion of the maximum local in-plane shearing stress known as the Taylor Series Correction Method (TSCM) is applied to the determination of stress intensity factors (SIF’s) in flat bottomed surface flaws of flaw depth/length ratios of approximately 0.033. Flaw depth/thickness ratios of approximately 0.20 and 0.40 were studied as were plate width/crack length ratios of approximately 2.33 and 1.25, the former of which corresponded to a nearly infinite width. Agreement to well within 10 percent was found with the Rice-Levy and Newman theories using a depth-modified secant correction and equivalent flaw depth/length ratios. The Shah-Kobayashi Theory, when compared on the same basis, was lower than the experimental results. Using a modified net section stress correction suggested by Shah, agreement with the Shah-Kobayashi Theory was greatly improved but agreement with the other theories was poorer. On the basis of the experiments alone, it was found that the SIF was intensified by about 10 percent by decreasing the plate width/crack length from 2.33 to 1.25.

scholarly journals An approach to analysis of dynamic crack growth at bimaterial interface

2009 ◽  
Vol 36 (4) ◽  
pp. 299-327 ◽  
Author(s):  
R. Nikolic ◽  
Jelena Djokovic
Keyword(s):  
Asymptotic Analysis ◽  
Strain Field ◽  
Crack Tip ◽  
Optical Data ◽  
Bimaterial Interface ◽  
Dynamic Crack ◽  
New Approach ◽  
Shear Wave Speed ◽  
Propagating Crack

In this paper is presented the new approach to asymptotic analysis of the stress and strain fields around a crack tip that is propagating dynamically along a bimaterial interface. Through asymptotic analysis the problem is being reduced to solving the Riemann-Hilbert's problem, what yields the strain potential that is used for determination of the strain field around a crack tip. The considered field is that of a dynamically propagating crack with a speed that is between zero and shear wave speed of the less stiffer of the two materials, bound along the interface. Using the new approach in asymptotic analysis of the strain field around a tip of a dynamically propagating crack and possibilities offered by the Mathematica programming package, the results are obtained that are compared to both experimental and numerical results on the dynamic interfacial fracture known from the literature. This comparison showed that it is necessary to apply the complete expression obtained by asymptotic analysis of optical data and not only its first term as it was done in previous analyses.

Roles for A Precursor Oxide Phase in The Siting, Shaping, and Shrinking of Oxygen Precipitates

1985 ◽  
Vol 59 ◽  
Author(s):  
P. Fraundorf
Keyword(s):  
Phase Transition ◽  
Strain Field ◽  
Oxide Phase ◽  
Induction Effect ◽  
First Order ◽  
First Order Phase Transition ◽  
Precursor Phase ◽  
Precipitate Shape ◽  
Oxygen Precipitates ◽  
First Order Phase

ABSTRACTThree separate “anomalous” effects in the precipitation of oxygen in silicon may be explained if typical poorly-crystallized platelet oxygen precipitates begin as tiny crystalline clusters. The first anomaly, sometimes referred to as the induction effect, may be explained if one postulates the existence of kinetically stable precipitate embryos (seeds) containing no more than one or two oxygen atoms. We show here that such a postulate, coupled with observations, places rather specific constraints on binding energy as a function of size for such tiny clusters. The second and third anomalies, arising in precipitate shape and retrogrowth behavior dependences, respectively, may be explained if one postulates the existence of a relatively dense precursor phase which undergoes first order phase transition, following otherwise classical rules, to the final-stage amorphous oxide normally found. In this case, both precipitate shape and strain field can be interpreted as a barometer of the interstitial ambient during key periods in a precipitate's history.

Elastic Strain Energy of Dislocations in Ice

1971 ◽  
Vol 49 (16) ◽  
pp. 2181-2186 ◽  
Author(s):  
W. R. Tyson
Keyword(s):  
Elastic Constants ◽  
Strain Field ◽  
Elastic Strain ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
High Mobility ◽  
Anisotropic Elasticity ◽  
Hexagonal Ice ◽  
Complete Set ◽  
Elastic Strain Field

The energy stored in the elastic strain field of dislocations in hexagonal ice is calculated using anisotropic elasticity and the most complete set of elastic constants available. Ice is elastically fairly isotropic, and it is proposed that the high mobility of dislocations on the basal plane is due to dissociation of perfect dislocations on this plane.

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