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Approved REPOT DCUMETATON PGE0MBNo 0704-0188Publc reporting ourden for this collection of information is estimated to average i hour per resopor.5e. inctuong thre time for reviewing instructions. searching exinsting data sources. Approved for public release; distribution is unlimited. ABSTRACT (Maximum 200 words)We study the second-order tensor equation AX + IA = G for symmetric, positive-definite A and arbitrary G.Motivated by applications in the continuum mechanics literature, we also exUamine some special cams where G depends on A and another tensor H. For arbitrary dimensions we establish relations between the solutions X for various forms of G. These results, together with Rivlin's identities for tensor polynomials in two variables, we applied in two and three dimensions to obtain explicit formulas for X in direct (component-free) notation. The results include new formulas as well as new derivations of previously known formulas. An application to the kinatics of rigid motions; is considered.

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM-YYYY) March 20052. REPORT TYPE PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) U. S. Army Research Laboratory ATTN: AMSRD-ARL-WM-TD Aberdeen Proving Ground, MD 21005-5066 PERFORMING ORGANIZATION REPORT NUMBER ARL-TR-3437 SPONSOR/MONITOR'S ACRONYM(S) 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution is unlimited. SUPPLEMENTARY NOTES ABSTRACTWe analyzed data from quasistatic, uniaxial tension tests on single-ply specimens of plain-woven, 600-denier KM2. Failure strains were found to range from 0.120 to 0.144 for warp-loaded and from 0.143 to 0.160 for fill-loaded tests. Failure stresses (strengths) ranged from 0.424 to 0.543 GPa for warp-loaded and from 0.593 to 0.642 GPa for fill-loaded tests. A least-squares fit, in the form of a rational function, was obtained to the second Piola-Kirchhoff stress vs. Green-St. Venant strain data for a representative warp-direction and a representative filldirection test. In addition, averaged least-square fits to all the warp data and to data from four out of five fill tests were obtained, also in the form of a rational function. These fits are intended to inform constitutive modeling in numerical simulations.

Some of the recent efforts to improve the ballistic performance of lightweight armors utilize functionally graded materials to provide a continuous transition in properties between dissimilar materials. It has been conjectured that the elimination of abrupt acoustic impedance changes may result in beneficial stress wave attenuation. We examine this issue for some idealized one-dimensional problems in which all materials are linear elastic. Exact solutions are compared with DYNA3D simulations of the same problems.

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