Averaging of elastic constants for polycrystals

Blaschke, Daniel N.

doi:10.1063/1.4993443

Title: Averaging of elastic constants for polycrystals

Abstract

Many materials of interest are polycrystals, i.e., aggregates of single crystals. Randomly distributed orientations of single crystals lead to macroscopically isotropic properties. Here in this paper, we briefly review strategies of calculating effective isotropic second and third order elastic constants from the single crystal ones. Our main emphasis is on single crystals of cubic symmetry. Specifically, the averaging of third order elastic constants has not been particularly successful in the past, and discrepancies have often been attributed to texturing of polycrystals as well as to uncertainties in the measurement of elastic constants of both poly and single crystals. While this may well be true, we also point out here shortcomings in the theoretical averaging framework.

Authors:

Blaschke, Daniel N. ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: 2017-10-13

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1407891

Alternate Identifier(s):: OSTI ID: 1399293

Report Number(s):: LA-UR-17-24288
Journal ID: ISSN 0021-8979; TRN: US1703059

Grant/Contract Number:: AC52-06NA25396

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Applied Physics

Additional Journal Information:: Journal Volume: 122; Journal Issue: 14; Journal ID: ISSN 0021-8979

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 36 MATERIALS SCIENCE

Citation Formats


                    Blaschke, Daniel N. Averaging of elastic constants for polycrystals.  United States: N. p., 2017. 
Web.  doi:10.1063/1.4993443.

Copy to clipboard


                    Blaschke, Daniel N. Averaging of elastic constants for polycrystals.  United States.  https://doi.org/10.1063/1.4993443

Copy to clipboard


                    Blaschke, Daniel N. Fri .  
"Averaging of elastic constants for polycrystals".  United States.  https://doi.org/10.1063/1.4993443.  https://www.osti.gov/servlets/purl/1407891.

Copy to clipboard


                    
@article{osti_1407891,

  title        = {Averaging of elastic constants for polycrystals},

  author       = {Blaschke, Daniel N.},

  abstractNote = {Many materials of interest are polycrystals, i.e., aggregates of single crystals. Randomly distributed orientations of single crystals lead to macroscopically isotropic properties. Here in this paper, we briefly review strategies of calculating effective isotropic second and third order elastic constants from the single crystal ones. Our main emphasis is on single crystals of cubic symmetry. Specifically, the averaging of third order elastic constants has not been particularly successful in the past, and discrepancies have often been attributed to texturing of polycrystals as well as to uncertainties in the measurement of elastic constants of both poly and single crystals. While this may well be true, we also point out here shortcomings in the theoretical averaging framework.},

  doi          = {10.1063/1.4993443},

  journal      = {Journal of Applied Physics},

  number       = 14,

  volume       = 122,

  place        = {United States},

  year         = {2017},

  month        = {10}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (Publisher)

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1063/1.4993443

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 17 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Elastic anisotropy and yield surface estimates of polycrystals
journal, August 2009

Brenner, R.; Lebensohn, R. A.; Castelnau, O.
International Journal of Solids and Structures, Vol. 46, Issue 16
DOI: 10.1016/j.ijsolstr.2009.04.001

Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle .
journal, January 1929

Reuss, A.
ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 9, Issue 1
DOI: 10.1002/zamm.19290090104

Anharmonicity in Noble Metals; Higher Order Elastic Constants
journal, April 1966

Hiki, Yosio; Granato, A. V.
Physical Review, Vol. 144, Issue 2
DOI: 10.1103/PhysRev.144.411

Die experimentelle Ermittlung der elastischen Konstanten höherer Ordnung
journal, December 1960

Seeger, Alfred; Buck, Otto
Zeitschrift für Naturforschung A, Vol. 15, Issue 12
DOI: 10.1515/zna-1960-1205

Finite Deformations of an Elastic Solid
journal, April 1937

Murnaghan, F. D.
American Journal of Mathematics, Vol. 59, Issue 2
DOI: 10.2307/2371405

Second- and third-order elastic coefficients in polycrystalline aluminum alloy AMg6
journal, November 2015

Volkov, A. D.; Kokshaiskii, A. I.; Korobov, A. I.
Acoustical Physics, Vol. 61, Issue 6
DOI: 10.1134/S1063771015060147

Calculation of nonlinear effective elastic constants of polycrystalline materials
journal, January 1996

Kiewel, H.; Fritsche, L.; Reinert, T.
Journal of Applied Physics, Vol. 79, Issue 8
DOI: 10.1063/1.361823

Elastic anisotropy and yield surface estimates of polycrystals
journal, August 2009

Brenner, R.; Lebensohn, R. A.; Castelnau, O.
International Journal of Solids and Structures, Vol. 46, Issue 16
DOI: 10.1016/j.ijsolstr.2009.04.001

Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect
journal, February 1961

Toupin, R. A.; Bernstein, B.
The Journal of the Acoustical Society of America, Vol. 33, Issue 2
DOI: 10.1121/1.1908623

Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls
journal, August 1958

Kr�ner, Ekkehart
Zeitschrift f�r Physik, Vol. 151, Issue 4
DOI: 10.1007/BF01337948

The Elastic Behaviour of a Crystalline Aggregate
journal, May 1952

Hill, R.
Proceedings of the Physical Society. Section A, Vol. 65, Issue 5
DOI: 10.1088/0370-1298/65/5/307

The third-order elastic constants of quasi-isotropic materials
journal, August 1968

Hamilton, R. A. H.; Parrott, J. E.
Journal of Physics C: Solid State Physics, Vol. 1, Issue 4
DOI: 10.1088/0022-3719/1/4/301

Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper
journal, January 1889

Voigt, W.
Annalen der Physik, Vol. 274, Issue 12
DOI: 10.1002/andp.18892741206

A variational approach to the theory of the elastic behaviour of polycrystals
journal, October 1962

Hashin, Z.; Shtrikman, S.
Journal of the Mechanics and Physics of Solids, Vol. 10, Issue 4
DOI: 10.1016/0022-5096(62)90005-4

A variational approach to the theory of the elastic behaviour of polycrystals
journal, October 1962

Hashin, Z.; Shtrikman, S.
Journal of the Mechanics and Physics of Solids, Vol. 10, Issue 4
DOI: 10.1016/0022-5096(62)90005-4

Estimates of Nonlinear Elastic Constants and Acoustic Nonlinearity Parameters for Textured Polycrystals
journal, June 2015

Kube, Christopher M.; Turner, Joseph A.
Journal of Elasticity, Vol. 122, Issue 2
DOI: 10.1007/s10659-015-9538-1

Texture and anisotropy
journal, July 2004

Wenk, H-R; Houtte, P. Van
Reports on Progress in Physics, Vol. 67, Issue 8
DOI: 10.1088/0034-4885/67/8/R02

Note on a variational approach to the theory of composite elastic materials
journal, April 1961

Hashin, Z.; Shtrikman, S.
Journal of the Franklin Institute, Vol. 271, Issue 4
DOI: 10.1016/0016-0032(61)90032-1

Third-Order Elastic Constants of Aluminum,
journal, May 1969

Thomas, J. F.
Physical Review, Vol. 181, Issue 3
DOI: 10.1103/PhysRev.181.1370

Finite deformations of an elastic solid
journal, May 1952

Wahl, A. M.
Journal of the Franklin Institute, Vol. 253, Issue 5
DOI: 10.1016/0016-0032(52)90724-2

Thermodynamics of Crystals
journal, November 1972

Wallace, Duane C.; Callen, Herbert
American Journal of Physics, Vol. 40, Issue 11
DOI: 10.1119/1.1987046

On the third order elastic constants of polycrystalline media and their relationships to the single crystal constants
journal, September 1968

Chang, Roger; Graham, L. J.
Materials Research Bulletin, Vol. 3, Issue 9
DOI: 10.1016/0025-5408(68)90042-1

On the determination of the fourth‐order elastic constants
journal, September 1977

Markenscoff, Xanthippi
Journal of Applied Physics, Vol. 48, Issue 9
DOI: 10.1063/1.324294

New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals
journal, April 1997

Lubarda, V. A.
Journal of the Mechanics and Physics of Solids, Vol. 45, Issue 4
DOI: 10.1016/S0022-5096(96)00113-5

Third-Order Constants of a Cubic Quasi-Isotropic Solid
journal, June 1990

Wasserbäch, W.
physica status solidi (b), Vol. 159, Issue 2
DOI: 10.1002/pssb.2221590216

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media
journal, March 1964

Thurston, R. N.; Brugger, K.
Physical Review, Vol. 133, Issue 6A
DOI: 10.1103/PhysRev.133.A1604

On the determination of the fourth‐order elastic constants
journal, September 1977

Markenscoff, Xanthippi
Journal of Applied Physics, Vol. 48, Issue 9
DOI: 10.1063/1.324294

Relation between Third‐Order Elastic Constants of Single Crystals and Polycrystals
journal, July 1968

Barsch, G. R.
Journal of Applied Physics, Vol. 39, Issue 8
DOI: 10.1063/1.1656855

On some variational principles in anisotropic and nonhomogeneous elasticity
journal, October 1962

Hashin, Z.; Shtrikman, S.
Journal of the Mechanics and Physics of Solids, Vol. 10, Issue 4
DOI: 10.1016/0022-5096(62)90004-2

Theoretical multiple-scattering calculation of nonlinear elastic constants of disordered solids
journal, February 1992

Ballabh, T. K.; Paul, M.; Middya, T. R.
Physical Review B, Vol. 45, Issue 6
DOI: 10.1103/PhysRevB.45.2761

Micromechanics of Defects in Solids
journal, June 1983

Mura, T.; Barnett, D. M.
Journal of Applied Mechanics, Vol. 50, Issue 2
DOI: 10.1115/1.3167076

Linear and volume compressibilities and isothermal third‐order elastic constants
journal, September 1984

Powell, B. E.; Skove, M. J.
Journal of Applied Physics, Vol. 56, Issue 5
DOI: 10.1063/1.334114

Third‐Order Elastic Constants of Polycrystalline Media
journal, March 1968

Juretschke, H. J.
Applied Physics Letters, Vol. 12, Issue 6
DOI: 10.1063/1.1651957

Symmetry of Mixed Third‐Order Elastic Constants
journal, January 1967

Skove, M. J.; Powell, B. E.
Journal of Applied Physics, Vol. 38, Issue 1
DOI: 10.1063/1.1708987

Relation between Isothermal and Mixed Third‐Order Elastic Constants
journal, January 1967

Powell, B. E.; Skove, M. J.
Journal of Applied Physics, Vol. 38, Issue 1
DOI: 10.1063/1.1708988

Third-Order Elastic Constants of Aluminum
journal, November 1968

Thomas, J. F.
Physical Review, Vol. 175, Issue 3
DOI: 10.1103/PhysRev.175.955

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media
journal, September 1964

Thurston, R. N.; Brugger, K.
Physical Review, Vol. 135, Issue 7AB
DOI: 10.1103/PhysRev.135.AB3.2

Thermodynamic Definition of Higher Order Elastic Coefficients
journal, March 1964

Brugger, K.
Physical Review, Vol. 133, Issue 6A
DOI: 10.1103/PhysRev.133.A1611

Note on a variational approach to the theory of composite elastic materials
journal, April 1961

Hashin, Z.; Shtrikman, S.
Journal of the Franklin Institute, Vol. 271, Issue 4
DOI: 10.1016/0016-0032(61)90032-1

Elastic Stiffness Coefficients of Single‐Crystal Iron from Room Temperature to 500°C
journal, July 1968

Leese, John; Lord, A. E.
Journal of Applied Physics, Vol. 39, Issue 8
DOI: 10.1063/1.1656884

Third‐Order Elastic Constants of Single‐Crystal and Polycrystalline Columbium
journal, June 1968

Graham, L. J.; Nadler, H.; Chang, Roger
Journal of Applied Physics, Vol. 39, Issue 7
DOI: 10.1063/1.1656726

The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals
journal, September 1954

Hershey, A. V.
Journal of Applied Mechanics, Vol. 21, Issue 3
DOI: 10.1115/1.4010899

Works referencing / citing this record:

Properties of dislocation drag from phonon wind at ambient conditions
text, January 2019

Blaschke, Daniel N.
arXiv
DOI: 10.48550/arxiv.1902.02451

Fourth-order elastic moduli of polycrystals
journal, October 2019

Krasilnikov, O. M.; Vekilov, Yu. Kh.
Physical Review B, Vol. 100, Issue 13
DOI: 10.1103/physrevb.100.134107

Dislocation drag from phonon wind in an isotropic crystal at large velocities
journal, December 2019

Blaschke, Daniel N.; Mottola, Emil; Preston, Dean L.
Philosophical Magazine, Vol. 100, Issue 5
DOI: 10.1080/14786435.2019.1696484

Properties of Dislocation Drag from Phonon Wind at Ambient Conditions
journal, March 2019

Blaschke, Daniel
Materials, Vol. 12, Issue 6
DOI: 10.3390/ma12060948

Elasticity and yielding of mesophases of block copolymers in water–oil mixtures
journal, January 2019

Qavi, Sahar; Firestone, Millicent A.; Foudazi, Reza
Soft Matter, Vol. 15, Issue 28
DOI: 10.1039/c8sm02336k

Properties of dislocation drag from phonon wind at ambient conditions
text, January 2019

Blaschke, Daniel N.
arXiv
DOI: 10.48550/arxiv.1902.02451

Dynamical spin phenomena generated by longitudinal elastic waves traversing ${CoFe}_{2} O_{4}$ films and heterostructures
journal, December 2019

Azovtsev, Andrei V.; Pertsev, Nikolay A.
Physical Review B, Vol. 100, Issue 22
DOI: 10.1103/physrevb.100.224405

Mesomechanical Modeling and Numerical Simulation of the Diffraction Elastic Constants for Ti6Al4V Polycrystalline Alloy
journal, October 2018

Chen, Qiang; Liu, Li; Zhu, Changjun
Metals, Vol. 8, Issue 10
DOI: 10.3390/met8100822

Similar Records in DOE PAGES and OSTI.GOV collections:

Inversion of Ultrasonic Attenuation for Textural Information of Polycrystals.

Conference Ahmed, Salahuddin ; Taylor, Tom T

The mechanical properties of polycrystalline materials depend on the individual physical properties of the constituent grains. When grains are randomly oriented with respect to a fixed coordinate system, the average elastic properties are isotropic. A polycrystalline aggregate possesses macroscopic texture, and thus exhibit anisotropic elastic properties, when the grains are preferentially oriented. Accurate knowledge of texture is important for a number of engineering applications. For example, it plays an important role in determining their subsequent formability into finished parts of complex shape by deep drawing. Textural information can also be exploited to assess plastic damage in a component due tomore »« less
Evaluating Bounds and Estimators for Constants of Random Polycrystals Composed of Orthotropic Elastic Materials

Journal Article Berryman, J. G. - International Journal of Engineering Science

While the well-known Voigt and Reuss (VR) bounds, and the Voigt-Reuss-Hill (VRH) elastic constant estimators for random polycrystals are all straightforwardly calculated once the elastic constants of anisotropic crystals are known, the Hashin-Shtrikman (HS) bounds and related self-consistent (SC) estimators for the same constants are, by comparison, more difficult to compute. Recent work has shown how to simplify (to some extent) these harder to compute HS bounds and SC estimators. An overview and analysis of a subsampling of these results is presented here with the main point being to show whether or not this extra work (i.e., in calculating bothmore »« less
https://doi.org/10.1016/j.ijengsci.2012.03.021

Full Text Available
Determining elastic anisotropy of textured polycrystals using resonant ultrasound spectroscopy

Journal Article Evans, Jordan A. ; Sturtevant, Blake T. ; Clausen, Bjørn ; ... - Journal of Materials Science

Abstract Polycrystalline materials can have complex anisotropic properties depending on their crystallographic texture and crystal structure. In this study, we use resonant ultrasound spectroscopy (RUS) to nondestructively quantify the elastic anisotropy in extruded aluminum alloy 1100-O, an inherently low-anisotropy material. Further, we show that RUS can be used to indirectly provide a description of the material’s texture, which in the present case is found to be transversely isotropic. By determining the entire elastic tensor, we can identify the level and orientation of the anisotropy originated during extrusion. The relative anisotropy of the compressive (c ₁₁ /c ₃₃ ) and shearmore »« less
https://doi.org/10.1007/s10853-021-05827-z
Lattice strains and diffraction elastic constants of cubic polycrystals

Journal Article Zhang, Yin ; Chen, Wen ; McDowell, David L. ; ... - Journal of the Mechanics and Physics of Solids

In situ synchrotron X-ray and neutron diffraction experiments provide a powerful approach to measure lattice strains in bulk polycrystalline materials. They are being increasingly used for quantitative characterization of microscale deformation within and between grains and phases. Here we use a self-consistent micromechanics model to obtain a general analytic solution of the grain-level lattice strains and diffraction elastic constants for a broad class of elastically isotropic polycrystals with cubic crystal symmetry. This analytic solution reveals a direct linear relationship between the reciprocal of the elastic diffraction constant and the orientation index parameter along the direction of any diffraction vector, includingmore »« less
https://doi.org/10.1016/j.jmps.2020.103899

Full Text Available
Using Scattering Theory to Determine Orientation Distribution Coefficients of Polycrystals with Preferred Grain Orientations

Conference Ahmed, Salahuddin ; Taylor, Tom T

The mechanical behavior of single-phase polycrystalline materials depend on the elastic constants of individual grains. Most engineering metals are polycrystalline aggregates possess macroscopic texture resulting from preferential grain orientations. Accurate knowledge of texture is important for a number of engineering applications. For example, it plays an important role in determining their subsequent formability into finished parts of complex shape by deep drawing. Textural information can also be exploited to assess plastic damage in a component due to fatigue or external impact. Current ultrasonic methods are based on the relationship between ultrasonic phase velocities and the low-order orientation distribution coefficients (ODCs);more »« less

Similar Records

Title: Averaging of elastic constants for polycrystals

Abstract

Citation Formats

Elastic anisotropy and yield surface estimates of polycrystals journal, August 2009

Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle . journal, January 1929

Anharmonicity in Noble Metals; Higher Order Elastic Constants journal, April 1966

Die experimentelle Ermittlung der elastischen Konstanten höherer Ordnung journal, December 1960

Finite Deformations of an Elastic Solid journal, April 1937

Second- and third-order elastic coefficients in polycrystalline aluminum alloy AMg6 journal, November 2015

Calculation of nonlinear effective elastic constants of polycrystalline materials journal, January 1996

Elastic anisotropy and yield surface estimates of polycrystals journal, August 2009

Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect journal, February 1961

Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls journal, August 1958

The Elastic Behaviour of a Crystalline Aggregate journal, May 1952

The third-order elastic constants of quasi-isotropic materials journal, August 1968

Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper journal, January 1889

A variational approach to the theory of the elastic behaviour of polycrystals journal, October 1962

A variational approach to the theory of the elastic behaviour of polycrystals journal, October 1962

Estimates of Nonlinear Elastic Constants and Acoustic Nonlinearity Parameters for Textured Polycrystals journal, June 2015

Texture and anisotropy journal, July 2004

Note on a variational approach to the theory of composite elastic materials journal, April 1961

Third-Order Elastic Constants of Aluminum, journal, May 1969

Finite deformations of an elastic solid journal, May 1952

Thermodynamics of Crystals journal, November 1972

On the third order elastic constants of polycrystalline media and their relationships to the single crystal constants journal, September 1968

On the determination of the fourth‐order elastic constants journal, September 1977

New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals journal, April 1997

Third-Order Constants of a Cubic Quasi-Isotropic Solid journal, June 1990

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media journal, March 1964

On the determination of the fourth‐order elastic constants journal, September 1977

Relation between Third‐Order Elastic Constants of Single Crystals and Polycrystals journal, July 1968

On some variational principles in anisotropic and nonhomogeneous elasticity journal, October 1962

Theoretical multiple-scattering calculation of nonlinear elastic constants of disordered solids journal, February 1992

Micromechanics of Defects in Solids journal, June 1983

Linear and volume compressibilities and isothermal third‐order elastic constants journal, September 1984

Third‐Order Elastic Constants of Polycrystalline Media journal, March 1968

Symmetry of Mixed Third‐Order Elastic Constants journal, January 1967

Relation between Isothermal and Mixed Third‐Order Elastic Constants journal, January 1967

Third-Order Elastic Constants of Aluminum journal, November 1968

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media journal, September 1964

Thermodynamic Definition of Higher Order Elastic Coefficients journal, March 1964

Note on a variational approach to the theory of composite elastic materials journal, April 1961

Elastic Stiffness Coefficients of Single‐Crystal Iron from Room Temperature to 500°C journal, July 1968

Third‐Order Elastic Constants of Single‐Crystal and Polycrystalline Columbium journal, June 1968

The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals journal, September 1954

Properties of dislocation drag from phonon wind at ambient conditions text, January 2019

Fourth-order elastic moduli of polycrystals journal, October 2019

Dislocation drag from phonon wind in an isotropic crystal at large velocities journal, December 2019

Properties of Dislocation Drag from Phonon Wind at Ambient Conditions journal, March 2019

Elasticity and yielding of mesophases of block copolymers in water–oil mixtures journal, January 2019

Properties of dislocation drag from phonon wind at ambient conditions text, January 2019

Dynamical spin phenomena generated by longitudinal elastic waves traversing CoFe 2 O 4 films and heterostructures journal, December 2019

Mesomechanical Modeling and Numerical Simulation of the Diffraction Elastic Constants for Ti6Al4V Polycrystalline Alloy journal, October 2018

Elastic anisotropy and yield surface estimates of polycrystals
journal, August 2009

Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle .
journal, January 1929

Anharmonicity in Noble Metals; Higher Order Elastic Constants
journal, April 1966

Die experimentelle Ermittlung der elastischen Konstanten höherer Ordnung
journal, December 1960

Finite Deformations of an Elastic Solid
journal, April 1937

Second- and third-order elastic coefficients in polycrystalline aluminum alloy AMg6
journal, November 2015

Calculation of nonlinear effective elastic constants of polycrystalline materials
journal, January 1996

Elastic anisotropy and yield surface estimates of polycrystals
journal, August 2009

Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect
journal, February 1961

Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls
journal, August 1958

The Elastic Behaviour of a Crystalline Aggregate
journal, May 1952

The third-order elastic constants of quasi-isotropic materials
journal, August 1968

Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper
journal, January 1889

A variational approach to the theory of the elastic behaviour of polycrystals
journal, October 1962

A variational approach to the theory of the elastic behaviour of polycrystals
journal, October 1962

Estimates of Nonlinear Elastic Constants and Acoustic Nonlinearity Parameters for Textured Polycrystals
journal, June 2015

Texture and anisotropy
journal, July 2004

Note on a variational approach to the theory of composite elastic materials
journal, April 1961

Third-Order Elastic Constants of Aluminum,
journal, May 1969

Finite deformations of an elastic solid
journal, May 1952

Thermodynamics of Crystals
journal, November 1972

On the third order elastic constants of polycrystalline media and their relationships to the single crystal constants
journal, September 1968

On the determination of the fourth‐order elastic constants
journal, September 1977

New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals
journal, April 1997

Third-Order Constants of a Cubic Quasi-Isotropic Solid
journal, June 1990

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media
journal, March 1964

On the determination of the fourth‐order elastic constants
journal, September 1977

Relation between Third‐Order Elastic Constants of Single Crystals and Polycrystals
journal, July 1968

On some variational principles in anisotropic and nonhomogeneous elasticity
journal, October 1962

Theoretical multiple-scattering calculation of nonlinear elastic constants of disordered solids
journal, February 1992

Micromechanics of Defects in Solids
journal, June 1983

Linear and volume compressibilities and isothermal third‐order elastic constants
journal, September 1984

Third‐Order Elastic Constants of Polycrystalline Media
journal, March 1968

Symmetry of Mixed Third‐Order Elastic Constants
journal, January 1967

Relation between Isothermal and Mixed Third‐Order Elastic Constants
journal, January 1967

Third-Order Elastic Constants of Aluminum
journal, November 1968

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media
journal, September 1964

Thermodynamic Definition of Higher Order Elastic Coefficients
journal, March 1964

Note on a variational approach to the theory of composite elastic materials
journal, April 1961

Elastic Stiffness Coefficients of Single‐Crystal Iron from Room Temperature to 500°C
journal, July 1968

Third‐Order Elastic Constants of Single‐Crystal and Polycrystalline Columbium
journal, June 1968

The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals
journal, September 1954

Properties of dislocation drag from phonon wind at ambient conditions
text, January 2019

Fourth-order elastic moduli of polycrystals
journal, October 2019

Dislocation drag from phonon wind in an isotropic crystal at large velocities
journal, December 2019

Properties of Dislocation Drag from Phonon Wind at Ambient Conditions
journal, March 2019

Elasticity and yielding of mesophases of block copolymers in water–oil mixtures
journal, January 2019

Properties of dislocation drag from phonon wind at ambient conditions
text, January 2019

Dynamical spin phenomena generated by longitudinal elastic waves traversing ${CoFe}_{2} O_{4}$ films and heterostructures
journal, December 2019

Mesomechanical Modeling and Numerical Simulation of the Diffraction Elastic Constants for Ti6Al4V Polycrystalline Alloy
journal, October 2018