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  • Finite Deformations and Internal Forces in Elastic-Plastic

    Crystals: Interpretations From Nonlinear Elasticity and Anharmonic Lattice Statics

    by J. D. Clayton and D. J. Bammann

    ARL-RP-274 September 2009

    A reprint from Journal of Engineering Materials and Technology, Vol. 131, pp. 041201-1041201-15, October 2009.

    Approved for public release; distribution is unlimited.

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    Disclaimers The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. Citation of manufacturers or trade names does not constitute an official endorsement or approval of the use thereof. Destroy this report when it is no longer needed. Do not return it to the originator.

  • Army Research Laboratory Aberdeen Proving Ground, MD 21005-5066

    ARL-RP-274 September 2009

    Finite Deformations and Internal Forces in Elastic-Plastic Crystals: Interpretations From Nonlinear Elasticity

    and Anharmonic Lattice Statics

    J. D. Clayton Weapons and Materials Research Directorate, ARL

    D. J Bammann Mississippi State University

    A reprint from Journal of Engineering Materials and Technology, Vol. 131, pp. 041201-1041201-15, October 2009.

    Approved for public release; distribution is unlimited.

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    Finite Deformations and Internal Forces in Elastic-Plastic Crystals: Interpretations From Nonlinear Elasticity and Anharmonic Lattice Statics

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    U.S. Army Research Laboratory ATTN: RDRL-WMT-D Aberdeen Proving Ground, MD 21005-5066

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    A reprint from Journal of Engineering Materials and Technology, Vol. 131, pp. 041201-1041201-15, October 2009. *Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39762

    14. ABSTRACT

    Large deformation kinematics and internal forces arising from defects in crystalline solids are addressed by a nonlinear kinematic description and multiscale averaging concepts. An element of crystalline material with spatially uniform properties and containing defects such as dislocation lines and loops is considered. The average deformation gradient for this element is decomposed multiplicatively into terms accounting for effects of dislocation flux, recoverable elastic stretch and rotation, and residual elastic deformation associated with self-equilibrating internal forces induced by defects. Two methods are considered for quantifying average residual elastic deformation: continuum elasticity and discrete lattice statics. Average residual elastic strains and corresponding average residual elastic volume changes are negligible in the context of linear elasticity or harmonic force potentials but are not necessarily inconsequential in the more general case of nonlinear elasticity or anharmonic interactions.

    15. SUBJECT TERMS

    elasticity, plasticity, dislocations, lattice statics, molecular dynamics, multiscale modeling, nonlinear elasticity

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    J. D. ClaytonMem. ASME

    Impact Physics,US Army Research Laboratory,

    Aberdeen Proving Ground, MD 21005e-mail: jclayton@arl.army.mil

    D. J. BammannFellow ASME

    Center for Advanced Vehicular Systems,Mississippi State University,

    Starkville, MS 39762e-mail: bammann@cavs.msstate.edu

    Finite Deformations and InternalForces in Elastic-Plastic Crystals:Interpretations From NonlinearElasticity and Anharmonic LatticeStaticsLarge deformation kinematics and internal forces arising from defects in crystallinesolids are addressed by a nonlinear kinematic description and multiscale averagingconcepts. An element of crystalline material with spatially uniform properties and con-taining defects such as dislocation lines and loops is considered. The average deforma-tion gradient for this element is decomposed multiplicatively into terms accounting foreffects of dislocation flux, recoverable elastic stretch and rotation, and residual elasticdeformation associated with self-equilibrating internal forces induced by defects. Twomethods are considered for quantifying average residual elastic deformation: continuumelasticity and discrete lattice statics. Average residual elastic strains and correspondingaverage residual elastic volume changes are negligible in the context of linear elasticityor harmonic force potentials but are not necessarily inconsequential in the more generalcase of nonlinear elasticity or anharmonic interactions. DOI: 10.1115/1.3183773

    Keywords: nonlinear elasticity, plasticity, lattice statics, dislocations, multiscalemodeling

    IntroductionBy typical definition, plastic deformation in crystalline solids

    akes place by the motion or flux of distributions of dislocations1,2. An element of material through which a net flux of disloca-ions has passed exhibits a permanent plastic shape change 3.he dislocation lines within the element, i.e., displacement dis-ontinuities across the slip plane in the context of Volterra defectsn elastic continua, may also contribute to this plastic deformation47. However, plastic deformation in this context does not ex-licitly account for the additional change in dimensions of theody due to residual elastic deformation associated with localtress fields i.e., eigenstresses 7 induced by defects. For ex-mple, dislocation glide preserves the volume of the crystal, andangential displacement i.e., slip discontinuities do not alter theolume occupied by the material. Yet ample evidence suggestshat dislocation lines affect the volume of crystals 811. Storednergies associated with elastic fields of defects are importantecause they affect recrystallization 9,12 and the fraction oftress power converted to temperature rise at high deformationates 13 that can lead to strain softening and shear localization inetals during dynamic failure events. Large numbers of disloca-

    ions, twins, and stacking faults can be generated during shockoading 14, and presumably the corresponding volume changes,hape changes, and stored energies associated with local stresselds of these defects affect the observed response e.g., pressure-olume or stress-strain profiles under such conditions 15.

    Defects of interest in the present work are those that can beescribed in a continuum sense by closed displacement disconti-uities tangential to an internal surface i.e., crystallographiclane in a volume element of material. These include gliding

    Contributed by the Materials Division of ASME for publication in the JOURNAL OFNGINEERING MATERIALS AND TECHNOLOGY. Manuscript received January 14, 2009;nal manuscript received March 30, 2009; published online August 27, 2009. Assoc.

    ditor: Hussein Zbib.

    ournal of Engineering Materials and TechnologyCopyright 20

    aded 27 Aug 2009 to 128.63.66.91. Redistribution subject to ASME

    straight or curved dislocations lines, dislocation loops, as well aspartial dislocations. Motion of dislocation defects through a re-gion of the crystal results in plastic shape change as mentionedabove but preserves the lattice spacing 3; this requires coopera-tive motion of leading and trailing partials for the case of partialdislocations. Residual elastic stress fields from disclination linesand loops are also considered 6,16,17, though their contributionto the plastic shape change 18 is not addressed explicitly. Alsonot considered are voids, fractures, and point defects 19,20 thatmay have finite volume and affect the lattice differently. Disloca-tion climb is also not considered here, because it generally in-volves vacancy migration 21.

    Modeling eff