Effects of Heterogeneity on the Strength of 3D Title:...

7
Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the University of California for the U.S. Department of Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royalty- free license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos National Laboratory strongly supports academic freedom and a researcher’s right to publish; as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness. FORM 836 (10/96) LA-UR-01-0843 Approved for public release; distribution is unlimited. Title: Effects of Heterogeneity on the Strength of 3D Composites Author(s): Sivasambu Mahesh, Irene J. Beyerlein, and S. Leigh Phoenix Submitted to: http://lib-www.lanl.gov/la-pubs/00357131.pdf

Transcript of Effects of Heterogeneity on the Strength of 3D Title:...

Page 1: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the University of California for the U.S. Department of Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royalty- free license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos National Laboratory strongly supports academic freedom and a researcher's right to publish; as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness.

FORM 836 (10/96)

LA-UR-01-0843 Approved for public release; distribution is unlimited.

Title: Effects of Heterogeneity on the Strength of 3D Composites

Author(s): Sivasambu Mahesh, Irene J. Beyerlein, and S. Leigh Phoenix

Submitted to: http://lib-www.lanl.gov/la-pubs/00357131.pdf

Page 2: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

����������������� ���������������������������

�! " "�$#&%('*)+ ,-�!%.�!/0)213�!4-�657%98 ):4 %.,-� ';%./<�!4=1>%., )+ ?A@#:):B C&)2'D57%.�9'EGFIHKJMLNJKOQPSR6T!JMUSVWLXUZYW[S\^]_Va`SVcbedgf�VihjVa]_klVWFl`Sm�[gJK`SnoEpdrqsVaFltMU6uAUSvGVa`eFxwgY

Y^y�USVavj]_VWzXFl{�JKk}|�~��e�SklFIVan6T6Va{�USJM`SFl{WLa[Z��vj]_`SVakIk}�;`SFxHjVa]_LXFxz^hj[r\�zNUSJM{�J��������j���M�ed���E�~cdm�T!JKzNVW]XFlJMklLDE�{WFlVW`S{aV�|=y�Va{�US`evjklvjtKhj[gqsvjL�~�klJMO�vjL���J�zNFlvM`rJMksq}JMPpvj]NJ�zNvj]�hj[qsvjL�~�klJMO�vjLa[e��T��j�j�K�7��d���E�~�d�"���������"�c�T�vj`�zNV���JM]XkIv�L_FlOQRek�JKzXFlvj`QFI`�zNVa]_�S]XVizNVWn� �FIzXU¡zNUSVWvj]XVizNFl{aJMkrO�v�nSVWklFI`St¢FlLARSL_Van�zNv�L_zNReneh¡zNUSVDL_zNJKzNFIL_zXFl{�JKke£¤JMFlk¦¥Re]XVDO�vGnSVaL�FI`�Re`SFlnSFI]XVW{WzNFIvj`rJMk�{avjO��pvjLXFxzNVaL�{Wvj`SLXFIL_zXFl`St§vM£�J�USV¨weJMtjvj`rJKkZJK]X]NJ�h¡vK£�VaklJML_zXFl{�©rPpVa]_LVWOQPpVanSnSVWnFI`�JM`ªVak�JKL_zNFI{¢O«J�zN]XF¦wZd���vjO��ZvjL_FIzXV�L_zX]XRS{izNRS]_V�FIL�FlnSVaJMklFI¬aVan"ReLXFl`et�zXUSV§{�UrJMFl`G¥^vM£�¥�PSRS`SnSkIVaL�O�vGnSVak�FI`"zNVa]_O�LvK£APSRS`SneklVaL;vM£�klVW`StMzXUo­®JM]X]XJM`StjVWn¯JMkIvj`St®zNUSV�©rPZVW]�nSFI]XVW{WzNFIvj`�dD°±FIPZVW]XLDVaklVWO�VW`�z�L�zN]XVW`StMzXUSL�Fl`!­�¥�PSRS`SneklVaLJK]XV¡z�JM²KVa`9zNv�PZV¡³´VaFlPeRSklkAnSFIL_zX]XFlPeRezNVWn´JK`Sn�µDVanStjVW�ZVizNU.JM`Sn$¶�JM`.·�hG²KV�kIv�JMn$LXUrJK]XFl`et6FIL�JKLXLXReO�VWn9£¸vj]zX]NJM`eL_HjVW]XL_V�©rPpVa]�PS]_V�JM²"JK]X]NJ�hGLadEGFlOQRSklJKzNFIvj`SL¢vM£�­�¥�PSRS`SnSkIV�£¤JKFlklRe]XV®]XViHjV�JKkz^ vo]XVWtjFlO�VaLWdª³2UeVa`.©rPpVa]�L_zX]XVa`etMzNU.HKJM]_F�JMPeFlklFxz^hoFIL�kIv� §[�zNUSVnevjO�FI`rJM`�z�£¤JMFIklRS]_V±O�v�neV�FlL�P�h¢tM]Xv� �Fl`St�{WklRSL�zNVW]XLsvM£G©rPpVa]sPS]_V�JM²7L}RS�¢zNv�Fl`eL_z�JKPSFlkIFIz^hjdZ³2USVa`§zNUSFIL¹HKJM]XFlJMPSFIklFIz^hFIL�USFltjU¹[G{WklRSL�zNVW]£¸vj]_O«JKzXFlvj`¡FILL_RS�S�S]_VaL_LXVanªP�h�J�O�vj]_VDneFlLX�pVa]_LXVWn«©rPpVa]£¤JKFlklRe]XV�O�v�nSVKdA��vj]_]XVWLX�pvj`SnSFI`St�zXvzXUSVaL_V�z^ �v¡{aJMLXVWLa[S V¢{avj`eL_zN]_RS{Wz�LXFlO��SkIV�O�v�nSVWklL�zNUrJ�z��S]XVWnSFl{iz�zXUSV�L_zX]XVW`StMzNUªnSFIL_zX]XFlPeRezNFIvj`«vM£�Jc­�¥�PSRS`SnSkIVMdy�UeVaFl]��S]_VanSFI{WzXFlvj`SL�{WvjO��rJM]XV�HMVa]�h"£¤J�HMvj]NJKPSkIh� �FIzNU�LXFIOQRSklJKzNFIvj`SL�FI`"zNUSV¢z^ v�{aJMLXVWLadº�»�¼"½¿¾ �"À!�Re`SFlnSFI]XVW{WzNFIvj`rJMk�©rPpVa]}{avMO��pvjLXFxzNVWLa[�PS]_FIzXzXklV�ÁKnSRS{izNFlkIVÂzX]NJM`SL_FIzXFlvj`�[�V¨wGzN]XVWO�V�HKJMkIRSVA�S]_vjPSklVWO6[�{�USJMFl`§vM£GPSRS`enSklVWLO�vGnSVakÃ[eLXUeV�JM]�¥^k�JKt¡O�vGnSVakÃdÄWÅ ��� ¾ ÀªÆo�c� Ä�¾�ÅÇ�RrJKLXFlL�z�JKzXFl{�£¤JKFlklRe]XV�vK£ZRe`SFlnSFI]XVW{WzNFIvj`rJMkG{avMO��pvjLXFxzNV�O�JKzXVa]XFlJMklLW[K �USFI{�U®{avj`SL_FlL�z�vM£¹kIvj`St¢JKklFltM`SVanQ]_VaFI`e£¸vj]X{WFl`St©SPZVW]XL¢VaOcPZVWnSnSVan$FI`¯J�O«J�zN]XF¦w�FlL�J�{avjO��SklV¨w6L_zNvG{�UrJML�zNFI{¡�S]XvG{aVWLXLad¢³2USFIklV�{avjO��SklV¨w�FIz^h�L_zXVaO�L�£¸]XvjOÈzNUSVvG{a{WRS]X]_Va`S{WV�vM£�HKJM]_FlvjReLnrJKO«JMtMVDViHjVW`7zXL��S]XVW{aVWnSFl`St�£¸vj]XO�JKzNFIvj`®vM£}J�{�JKzNJML_zX]Xvj�SUeFl{D{a]NJK{�²p[�L_zNJKzNFIL_zNFI{�JMkrHKJM]_Fx¥J�zNFlvM`¡Fl`¡L_zX]XVW`StMzNU®�S]_FlO�JM]XFIkIh�JM]_FlL_VaLAnSRSV�zXv�zNUSV�HKJM]XFlJMPSFIklFIz^h�Fl`Q©rPpVa]�L_zX]XVW`StMzNU¹d���vj`SL_VaÉ7RSVa`�zXkIh®{avMO��pvjLXFxzNVzXVa`SL_FlkIV�L�zN]XVW`StMzXU.FlL�FIzXLXVWkI£�JªL_zNJKzNFIL_zXFl{�JKkAÉGRSJM`�zNFIz^h´JM`en.O�VWzNUev�nSL§zNv6nSVWzXVa]_O�FI`SV�FxzNL§nSFIL_zN]_FlPSR�zNFlvM`´JK]XV�vM£{Wvj`SL_FlnSVW]NJMPSkIV¢LXFItj`SFI©S{�JM`S{WV§Fl`�JMLXL_RS]XFI`St®{avjO��pvjLXFxzNV�]_VaklFlJMPSFIklFIz^hMd\^neV�JMkIFl¬�J�zNFlvM`:vM£§{WvjO��ZvjL_FIzXV6L�zN]XRe{WzNRe]XV$JM`Sn<O�JKzXVa]XFlJMk��S]_vj�ZVW]_zXFlVaL«JM]XV6£¸vjRS`Sn+zXv�PpVoFl`SViHGFIz�JKPSklV�PZVi£¸vj]XV£¸Re]_zNUeVa]DJM`rJMkxhGLXFlL�{aJM`6PpV�JKzXzXVaO��ezNVWn�d�\^`�zXUSFlL�L_zXRSneh" VcJKLXLXReO�V§L_zXFIÊ$kIFl`SVaJM]�VaklJML_zXFl{�©rPpVa]_L�JK]X]NJK`StjVan�Fl`

Page 3: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

JªUSV¨wSJKtjvj`rJMk�JK]X]NJ�h$JK`Sn´VWOQPpVanSnSVWn´FI`9J�]_Vak�J�zNFIHMVakxho{avMO��SkIF�JM`�z�klFl`eV�JM]¢VWk�JML�zNFl{c`Svj`�¥�nSVaPpvj`SnSFI`StªO«J�zN]XF¦wL_vczNUSJKz�O�JKzNVW]XFlJMkrnrJMO�JMtjVDFl`�vMRS]�FInSV�JMkIFl¬WVan�{avjO��ZvMLXFIzXV;FIL�kIFlO�FIzNVWn¡zNv�©SPZVW]�PS]XVaJM²�JMtjVWL�JMkIvj`SVMd±y�USV�k�JK]XtjV©SPZVW]�¥^O�JKzX]XFxw"L_zNFxʹ`eVaLXL;]NJKzXFlv�FlO��SklFIVaL�zXUrJKz;O�vML_z;vM£�zNUeVQJM�S�eklFlVWn6kIv�JMnª �FlkIksPZV�PZvM]X`SV�P7hªzXUSV�©rPpVa]_L�JM`enzXUSV¡]XvMklVcvK£�zXUSVQO�JKzN]_Fxw6FlL�klFlO�FIzXVanªzNv"{Wvj`SnSRS{izNFI`St"klv�JKnSL�£¸]XvMO¿PS]_vj²MVW`¯©rPZVW]XL�zXv"`SV�JM]_P�hoFl`�zNJM{Wz�©SPZVW]XLady�UeFlLckIv�JMn$zN]XJM`SL_£¸VW]cvG{a{WRS]XL�zNUS]_vjRStjU�L_USV�JK]cnSVi£¸vj]XO�JKzNFIvj`$zNUrJKz�zNVW`SnSLczXv!vG{a{WRS]cv�HjVW]¡J6{WVa]�z�JMFI`.klVW`StMzXUL_{�JMkIV�­Gd�­cFIL�z^h��eFl{�JKklkIh�vj`Skxh"JQ£¸VW 3©rPpVa]�nSF�JKO�VizNVa]_L�JM`Sn�FlL�OQRe{�U6kIVaLXL�zXUrJM`ªzNUSV§{avMO��pvjLXFxzNV�kIVa`StKzNU�Ë�d~;L�UrJKL�PZVWVa`o{WvjO�O�vj`6Fl`�zNUSV�klFIzXVa]XJKzNRS]_VM[r VQFlnSVaJMklFI¬aV¢zNUeV�£¤JMFIklRS]_V��e]XvG{aVaL_L�Fl`�zNVW]XO�L�vM£�J�klvj`StMFIzNRenSFl`rJKk�SJM]_zXFIzNFIvj`ÌFl`�zNv´Í ÎÏËÐK­¯zX]NJM`eL_HjVW]XL_V¯LXk�JKPSL�vM]�L_USvj]_z«PSRS`SneklVaL�vM£¢kIVa`StKzNU0­7[�{aJMklkIVan:­�¥�PSRS`SnSkIVaLWdÑy�UeV£¤JKFlklRe]XVc�e]XvG{aVaL_L¢ �FIzNUeFl`$J�tjFxHjVa`9­�¥^PSRe`SnSklV¡FIL�zN]_V�JKzXVan9JML¢O�Va{�UrJK`SFl{aJMklkxhoJM`en¯L_z�J�zNFlL�zNFI{�JMkIkIh�Fl`SneVa�pVa`SnSVW`�zvK£�zXUrJKz�Fl`¯`SVWFltjU7Ppvj]XFI`St�­�¥^PSRe`SnSklVWLad§y�USV¡{WvjO��pvjL_FIzNV�FlLDzNUSVW`$zX]XVaJKzNVWn$JML§J� V�JM²KVaL_z�¥^kIFl`S²!JM]X]XJM`StjVWO�VW`7zvK£±zNUeVaLXV�­�¥^PSRS`enSklVWLaÒSzNUSJKzDFlLa[SzXUSV§{avjO��pvjLXFxzNV�£¤JMFIklL� �USVW`6zNUeV§ V�JK²MVaL�z§­�¥^PSRS`enSklV�£¤JMFIklLWd�y�U7RSL�zNUeV�{�UrJKFl`�¥vK£�¥^PSRS`enSklVWL�JML_LXRSO��ezNFIvj`!{Wvj`�HjVW]_zNL�zXUSV¡�j·��S]XvjPeklVaOÈvM£�{avjO��ZvMLXFIzXV�£¤JKFlklRe]XV�Fl`�zNv�zNUSV��S]XvjPeklVaOÈvM£A£¤JKFlklRe]XVvK£�zNUSV« V�JK²MVaL�z®vM£�LXViHjVW]NJMk�ÓK·Ô­�¥�PSRS`SneklVaLWd!³´V"JMklL_voJML_LXRSO�V«zXUrJKzc©SPZVW]QL_zX]XVa`etMzNU�Õ FIL�]XJM`SnSvMOÖJM`enneFlL_zX]XFIPSRezNVWn�JM{a{Wvj]XnSFI`St¡zNv¡zNUeV§³´VaFIPSRSkIk�nSFlL�zN]_FlPSRezXFlvj`×�ظÙZÚ Î3u�]iÛ�ÕÏÜ<ݱާÎ���ß&V¨w���Û7ß Ø Ý}ÐKÝrà Ú^á Þ Ø � Ú �UeVa]XV�Ý+FIL�zXUSV«L�zN]XVWLXL�Viw��ZVW]XFIVa`S{WVan&P�h$zNUSV®©rPpVa]a[±Ýrà�FIL§zXUSV«L_{�JMkIV��SJM]NJMO�VWzXVa]§£¸vj]�J�©rPpVa]�VakIVaO�Va`�z�vM£kIVa`StKzNU�­¡JM`Snªâ�zXUSV§LXUrJK�ZV§�rJM]XJMO�VWzNVW]�vM£�zNUSV§nSFIL_zN]_FlPSR�zNFlvM`�d³´V�RSL_Vªµ;VWnStjVa�pVWzXU+JK`Sn�¶�JM`Ì·�hG²MVKãäL�åæ�aç�kIv�{aJMk�klv�JMn.LXUrJM]_Fl`St!O�vGnSVak Ø µ�¶¢q}q�E Ú zNv¯neVWzNVW]XO�Fl`SV�L_zX]XVaL_L{Wvj`S{WVa`�zN]XJKzNFIvj`SL«FI`(zXUSV!�ek�JM`SVªvM£¢J9zN]XJM`�HjVW]XLXVoJM]X]XJ�hÌvK£�©rPpVa]�PS]XVaJM²7Lad2³2USFlkIVª V!nev.`SvMz�nSVWkIHjV6Fl`�zNvzXUSV!neVWz�JKFlklL�vM£�zXUSVaFI]�JM�e�S]Xv�JK{�U�[� �V!`SvMzNV�zNUrJ�z�RS`SneVa]�µ�¶¢q}q�EZ[zXUSV!L�zN]_VaLXL«{avj`S{WVa`�zN]XJKzNFIvj`<JM]_vjRS`Sn0J�pVa`e`7h�¥�LXUrJM�pVan6{a]XJM{�²"vM£Âèc©rPpVa]�PS]XVaJM²7L;FIL�JM�S�S]_v�weFIO«J�zNVakxh

é�ê�ë�ì Ó�í èî�ï�ð mQñ ��Î ì ò î ñ � Ø Ó Ú �UeVa]XV ò FILDzXUSV�VWÊZVa{WzXFIHMVcnSFlJMO�VizNVW]�vM£±zXUSVc�pVa`S`�h�¥^L_UrJM�pVano{W]NJM{�²�JM`en!è�Î î ò m¨ÐK�ed�~�kILXvS[pJML�zNUeV�{W]NJM{�²L_Fl¬WVcPpVa{avMO�VWL�k�JK]XtjVK[gzXUSVcL�zN]_VaLXL¢{Wvj`S{WVa`�zN]XJKzNFIvj`!nSVW{�J�hGL§JMLc��Ð í ó FI`6zXUSVc`SVaJM]�¥ô©rVakIn! �USVW]XV ó FIL�nSFIL_zNJM`S{aV£¸]_vjOõzXUSV§{a]XJM{�²�zNFI�6JK`SnªLXUrJM]_VaL�zNUSFIL�{�UrJM]XJM{WzXVa]XFIL_zXFl{§vM£�zNUSV§`SVaJM]�¥ôzNFl�"L�zN]_VaLXLD©rVakIn" �FIzNU�q}ö�°AT´dy�UeV�FI`�¥^�ek�JM`SV;£¤JMFlkIRS]XV�vK£±vjRe];­�¥^PeRS`SnSkIV¢FlL�FlneV�JMkIFl¬aVWnªJML�£¸vjklkIv� �Lad���vj`eLXFlneVa]DJ¡]XUSvjOcPSRSL�¥�LXUrJK�ZVWn6�rJ�zN{�U�vM£÷ m�USViweJMtMvj`rJMkIkIh�¥�JM]_]NJM`etjVanQ³´VaFIPSRSklk�©rPZVW]XL�LXRePGø_Va{izNVan�zNv�J§kIv�JMn�ø_RSL�zL_Reù�{aFIVa`�zzXv¢£¤JMFlkGzNUSV� V�JM²KVaL_z�©rPZVW]ad�DRSO�Va]XFI{�JMk�L_zX]XVWLXL§]XVWnSFlL�zN]XFIPSRezXFlvj`oFIL�{avMO��SR�zNVan¯ReLXFl`etªµD¶¢q}q�EZd�\�£�zXUSVQ]XVWLXRSkxzNFI`St�©rPpVa]�L�zN]XVWLXL_VaL�Viw�{aVWVanzXUSV�L�zN]_Va`StMzXUSL�vK£}JM`�h�vMzNUSVW]�©SPZVW]XLzXUSVa`�zNUSVWLXV�zXv�v�JM]XVD{avM`7HMVa]�zNVan"zXv�£¤JMFIklRS]_VaL�JM`Sn«L_zN]_VaL_L�]XVWnSFlL�zN]XFIPSRezXFlvj`£¸vM]§zXUSV®`SVW ú{avj`e©StjRS]NJ�zNFlvM`9FIL§{avjO��SRezXVan�d�y�USFlL§PrJM{�²j¥^JM`Sn�¥ô£¸vj]_zXU´�S]_v�{WVaL_L�vM£©rPpVa]¢£¤JMFlkIRS]XVWL§JM`Sn9L_zX]XVaL_L]_VanSFIL_zX]XFlPeRezNFIvj`$FlL§{avj`�zXFl`7RSVan´RS`�zNFIkAVaFIzXUSVa]§L�z�JMPSFIklFxz^hoFlL§]_V�JM{�USVWn�vj]¢zNUeV«­�¥^PSRe`SnSklV¡£¤JMFIklLWd�\^`9zNUSV�{�JKLXV�vM£L�z�JMPeFlklFxz^hj[7J§klv�JKn�FI`S{a]_VaO�Va`�z�FIL�JM�S�SkIFlVan®zNv§zNUeV�­�¥^PSRS`enSklV�zNUSJKz�FIL¹ø_RSL_zVa`SvMRStjU«zXv�£¤JMFIkgJM`SvMzXUSVa]�©rPpVa]W[eJM`enzXUSV�JMPpv�HMV§�S]XvG{aVWLXLDvM£±L�zN]_VaLXL;]XVWnSFlL�zN]XFIPSRezXFlvj`ªJM`enª£¸Re]_zNUeVa]�©rPpVa]�£¤JMFlkIRS]XVWL�FlL�]_Va�pV�JKzXVan�d�ö�HjVW`7zXRrJMkIkIhj[gJ�£ûzNVa]L_vjO�V�JK�S�SklFIVanªklvjJMnªFl`S{W]XVaO�Va`�za[ZJ®{�JKLX{�JKnSV�vK£±©rPpVa]�£¤JMFlkIRS]XVWL�v�{W{aRS]_L�JKL�zXUSV�­�¥�PSRS`SnSkIV¢£¤JMFlkIL�{�JKzNJML_zX]Xvj�eUSFx¥{aJMklkxhjd�y�USV�]XVWLXRSkxzNFl`et�JK�S�SklFIVanQ©rPpVa]AL�zN]_VaLXL�PpVa{avMO�VWL�zXUSV�L_zX]XVW`StMzNU�vM£pzNUSVD­�¥^PSRe`SnSklVKd³´V�`Sv� 0�S]_v�{WVaVWn�zXvneVaLX{W]XFIPZV�nevjO�FI`rJM`�z�£¤JKFlklRe]XVDO�VW{�UrJM`SFILXO�L�vM£�{avMO��pvjLXFxzNV�£¤JMFlkIRS]XVDvjPSL_Va]�HjVan�Fl`«LXFlOcRSk�JKzXFlvj`eLJK`Sn�zNv¡O�v�neVakzXUSVaOüFl`ªvj]XneVa]�zNv�JM`rJKkIh7zNFI{�JMkIkIh��S]XVWnSFl{iz�zNUSV§L�z�JKzXFlL_zXFl{aJMk�L_zX]XVW`StMzNU�nSFIL_zN]_FlPSR�zNFlvM`"vM£±{WvjO��ZvjL_FIzXV�L_zX]XVa`etMzNU�dý � Äaþ Æ"� » ÿ-» � � � Å�Ä � ÿ �(� Å À ÿú¾ À »�þ � ÄWÅ ­�� �¡Æ Å À þ�» ���� �������� ���������������������� ý ��� �!�6�"�����#�%$��'&)(*����'$+� â�,EG`rJM�SL_USvMzNL§vM£�zXUSVQnrJMO�JMtjVcVWHjvMklRezXFlvj`oFI`ozNUSV¡â"Îü�.-�O�VWnSF�JK` Ø JMO�vj`Stª�/-0-«LXFIOQRSklJKzNFIvj`SL Ú ­�¥^PSRS`enSklVQVW`]_vjRezXV«zNv!£¤JKFlklRe]XV�JM]XV�L_USv� �`ÌFl`�°ÂFltjRS]_V!�jd!y�USV"k�JML�zQL�z�JMtjV«LXUSv� �`Ì{avj]_]XVWLX�pvj`SnSLQzXv!zNUSV�JM]X]XJM`StjVWO�VW`7z

Page 4: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

vK£Pe]XV�JK²GL§FIO�O�VanSFlJKzNVWkIhªJ�£ûzNVa];zNUSVQ£¸vj]_O«J�zNFlvM`!vM£JM`¯RS`eL_z�JKPSklV¡L�h�L�zNVWO vM£�©rPpVa]¢PS]_V�JM²7L§JM`Sn$PpVW£¸vj]_VQzNUSV{aJKz�JKL_zN]_vj�SUSFI{Q£¤JMFlkIRS]XV�vM£�zNUSV¡]_VaO�JMFl`eFl`St«©SPZVW]XLadQ�;vMzXVQzNUrJKz�zNUSV®PpvjRS`SnrJK]_ho{Wvj`SnSFxzNFIvj`SL¢JM]_V¡�ZVW]XFIv�nSFI{cL_vzXUrJKz«J´PS]_V�JM²Ì{WklRSL�zNVa]«JM�S�pV�JM]_Fl`St´JKz�vj`SV6VanStjV6O«J�hÌPpV6{Wvj`�zNFI`GReVan0vj`:zNUSV�vj�S�pvjLXFxzNV�VanStMVMd>��klRSL�zNVa]£¸vM]XO�JKzNFIvj`"JM`Sn�tj]_v� �zXU6FIL�{akIV�JM]_kIh�zNUSV§nSvjO�Fl`SJM`�z�£¤JMFlkIRS]XV;O�vGnSV¢Fl`"zXUSV§LX�pVa{aFIO�VW`6L_USv� �`�d\

\X\

\_\X\

\�¶

°ÂFltjRS]_V"�01�203!4�57698;:.<96=:.>?<98;@A>B4.CEDEF;G9@H5!GI:KJL@�696MCN3O<98;@�PA@�Q!CN4.3SRB4.PA:'3;TVU ��� 6ICEPWF#DE4�<9CX:'3!6ZY\[�]�^#F!3!Q#DX@H_�CX<98a` ���b @�CX^#F!DEDdc!^�@eGI6\:.>gfih ��� F!3!Q;@eG=5�@eGICX:0Q!CEJM^�:'F!3#Q!4�G9jiJL:'3!Q#CX<9CX:'3!6ek � 5�@�3OJeCXGZJeDX@�6mlnQ;@�3;:.<o@ACE3/<94.JL<Mc!^�@eGI6eprqQ;@�3;:.<o@m^!GI:.s.@�3tc!^�@eGI6ek°rvjkIklv� �Fl`et¡µ�JM]_klv� >JM`Sn�uAUSv�VW`SFxw9åäÓ�çÃ[S V�USJ�HMV��eklvMz_zNVan"zXUSV§VaO��SFl]_Fl{�JKkZ V�JM²KVaL_z�¥^kIFl`S²�neFlL_zX]XFIPSRezNFIvj`SLuvxwrØ Ý Ú Î���ß2åæ�Dß uy wSØ Ý Ú_Ú ç Y ð w Ø � ÚvM`�³.VWFlPSRSkIk¹�rJK�ZVW]a[SFI`ª°±FltjRe]XV¢ÓQvjP�z�JMFI`SVan�£¸]_vjOõvMRS]DT6vj`�zNV¨¥���JM]_klv¡L_FlOQRSklJKzNFIvj`SL�vM£}£¤JMFlkIRS]XV;vM£Â­�¥�PSRS`SneklVaLRe`SnSVa]�µD¶�q�q�EZdpµDVa]XV uy w neVa`SvMzXVaLDzNUSV�VaO��SFl]_Fl{aJMk�L_zX]XVW`StMzNU!nSFlL�zN]XFIPSRezXFlvj`ªvM£ÂzNUSV�zp¥ô©rPZVW]�­�¥^PeRS`SnSkIVMd�°rvj]â|{úÓ�zXUSV uv}wSØ Ý Ú {aRS]�HjVWL�£¸vj]Wz&Î ÓjÓM��[�~�ÓM��JM`SnS�0-0-�{WvjklklJM�SL_Vcvj`�zNvªvj`eVQ{�UrJM]XJM{WzXVa]XFIL_zXFl{Q{WRS]_HMV uv Ø Ý Ú dy�UeFlL�zg¥^FI`SnSVa�pVa`enSVa`�zQ{WvjklklJM�SL_V�Uev� VWHMVa]Q£¤JMFIklL¢zNv6USvjkIn´£¸vj]§â9Î*�jdªy�USV�{avjkIk�JM�SL_V�vK£ uv}w FI`7zXvoJ�LXFl`etjklV{WRS]_HMV�£¸vj]�â�{ÑÓ;L_RStjtjVWL_zNL�zXUrJKz�zNUSV�{aklReL_zNVW]±tj]_v� �zNUQ£¤JMFIklRS]_V�O�vGnSVFIL�JM{izNFxHjV£¸vM]±â¢]NJK`StjV�FI`czNUeV�{avMO��pvjLXFxzNVL_Fl¬WVaL�zNUrJ�zD Va]_V�L_FlOQRek�JKzXVan�d³´VQO�v�neVak}zNUSVQ{aJMLX{aJMnSV¡ViHjVa`�z�nSVi©r`SFl`et v Ø Ý Ú JKL;zNUeVQ£¸vj]XO�JKzNFIvj`�vM£�J«PS]_V�JM²6{aklReL_zNVW]¢JKz�L_zN]_VaL_L�Ý´zNUrJKztMv�VWL«RS`SL�z�JMPeklVMd>y�USV¯nSF�JKO�VizNVa] ò vM£�J9zNFltMU7z«{aFI]X{aRek�JM]�{akIRSL_zXVa]«vK£§è´PS]_V�JM²7L� �JML�nSVi©r`SVWnÑVaJM]XkIFlVW]«JMLî ò m¨ÐK�®Î>èGd�y�USV�{aFl]_{aRSO®£¸Va]_Va`S{WV�vK£�zNUSV�{aFI]X{WklVM[ î ò Î í � î è¡FIL�JK�S�S]Xv�w�FlO�JKzXVakIh�zNUSV�`7RSOQPpVa]DvM£�FI`7zNJM{Wz©SPZVW]XL«L_RS]X]_vjRS`SneFl`St9zNUSFIL�è�¥^{WklRSL�zNVW]ad2qsVWz�� ê PZVªzNUeV6`7RSOQPpVa]�vM£�zXUSVaL_V!`SVWFltjU7Ppvj]XL®zNUSJKz�JK]XV�LXVWHMVa]_VakIhv�HMVa]XkIv�JMnSVWn�d�y�USV¢©r]XL�z�L�zNVa��FlL�zXUSV¢£¤JMFlkIRS]XV�vK£�J®tjFIHMVa`ª©rPpVa]�Fl`"zNUeV�­�¥^PeRS`SnSkIV¢RS`SnSVW];Ý�[S£¸vMklklv� VanªP�h"zNUSV£¤JKFlklRe]XVQvM£�vj`SV®vM£FIzXL�� Y Î�~"VaÉ7RrJMkIkIh¯v�HMVa]XkIv�JMnSVWn´`eVaFltMUGPpvj]_L�Re`SnSVa]�L_zX]XVWLXL é Y Ý�d®y�USV�]XVWLXRSkxzNFl`et��SJMFl]vK£�©rPZVW]cPS]_V�JM²7LQUSJMLcVWFltjU�zQFI`�z�JM{izQ`SVWFltjU7Ppvj]XL�vM£� �USFI{�U&vj`SkIh�� m ÎÈÓ!JM]XV�LXViHjVa]_Vakxh�v�HjVW]XklvjJMnSVan.RS`SnSVW]L�zN]_VaLXL é m ݱd:y�USV�`SV¨w�z�klFI²MVWkIh�ViHjVW`7z«FlLQzXUSV�£¤JKFlklRe]XV�vM£�vj`SVªvM£;zNUSVWLXVM[zNv¯£¸vj]XO J9PS]_V�JM²�zX]XFI�SklVizQ �FIzXU� ï Î��"LXViHjVW]XVakxh$v�HjVa]_klv�JKnSVan9`SVaFItjU7PZvM]XLa[}vM£� �USFI{�U´vj`eV®£¤JKFlklLW[sJM`Sn9LXv�vj`�d¡y�UeV�{W]XFxzNFl{aJMk±ViHjVW`7zcFlL�zXUGReLzXUSV�VWHjvMklRezXFlvj`9vM£�Jªtj]_v� �FI`St���zNFItjU�zL�ªè�¥�{akIRSL_zXVa]a[} �FIzXU´VaJM{�U�JKnSnSVan.PS]XVaJM²¯PZVWFl`StªzNUeV�£¤JMFIklRS]_V¡vM£�vM`SV®vM£zXUSV�� ê LXViHjVW]XVakxhªv�HjVa]_klv�JKnSVanª©rPpVa]XL�L_RS]X]_vjRS`SnSFI`St�FIz�d³.V¢ �]XFxzNV¢zNUeFlL�JML

vxwrØ Ý ÚAë-×«Ø Ý Ú Û7��ß<å¦��ß ×�ØÃé Y Ý Ú ç��"��Þ� Û7�Dß<å¦��ß ×�ØÃé m Ý Ú ç��+�WÞ��.�.��Û7��ßÑå¦��ß ×�ØÃé�w/� Y Ý Ú ç��+���K��Þ!� Ø � Ú

Page 5: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

−1.2 −1 −0.8 −0.6 −0.4 −0.2−14

−12

−10

−8

−6

−4

−2

á�� �Y�� �m�� �ï � ���� �*� Y��Y��sm*� �d�*� ï �

â¡Î � Ó � � �.- ?¡I¢L£�¤ Y ?¡I¢L£�¤ mw�¥ Y �d¦ Y �w�¥ m �d¦ m �w�¥ ï � ¦ ï �

klvjt Ø Ý}ÐKÝrà Ú§©¨ª«/¬§ ¨ª« ­¬®«�¯°°°

°ÂFltjRS]_V�Ó71 � :'PA5#4�GZCE6o:'3t:.>±<98;@²_³@�4�s.@�6o<MDECN3;s�Q#CE6o<oGICX^7F;<9CX:'3�5!G9@�Q!CNJL<o@�Qt^/jt<98!@W´�µ¶JeDEF#6o<o@eG�T.G9:·_¸<98�PA:KQ!@�D�_�CX<98<98;@H@�PA5#CXGICNJe4.Dr_�@�4�s�DNCE3;s�Q!CE69<oGICX^#F;<9CE:'3!6º¹» R�¼�Y�:.^!<94.CE3!@�QO>½G9:'P¿¾i:'3�<o@L] � 4�GIDX:i69CNPWF!DN4�<9CX:'3!6ekm¾i:KQ;@�D � 4.696IF!PA@�6À hÂÁ Ã'ÄS4.3!QÆÅSh �0Ç U�>�:.GW4.DND±f#k � 3Ⱦ�:0Q;@�D³´Kp"_�@�4.QÊÉ9F!6o<m<98;@�5#4�GI4.PA@e<o@eGI6 À 4.3!QÆÅ}6o:i4.6W<o:iT.@e<W<98;@Ë^�@�6o<c!<²_�CE<98a<98;@�69CEPÌF!DE4�<o@�QÍQ#4�<940kiθ8;@�6o@�ÏÊ4.DNF;@�6²:.> À 4.3#QaÅS4�G9@ÐDECE6o<o@�QÍCE3a<98;@�^�:.<o<o:'P�GZCXT'8�<WJL:.GI3;@eG�k b @�4�s.@�69<DECE3;s�Q!CN6o<oGICX^#F!<9CX:'3!6�JL:.GIG9@�6o5�:'3!Q!CE3!T�<o:�fih �0Ç UÐ4�G9@Ë3;:.<m698;:·_�3�^�@�Je4.F!6o@A^�:.<98a¾�:0Q;@�D � 4.3!Qa¾i:KQ!@�D%´�5!G9@�Q!CEJL<Q!CE6o<oGICE^#F;<9CX:'3!6�<98!4�<²4�G9@�:'F;<M:.>³<98!@ËGI4.3;T.@A:.>�<98!CE6=5#DE:.<ek � DE69:t:'PËCX<o<o@�QaCE6=<98!@�¾�:0Q;@�D � DECN3;@H>½:.GmfVh � _�8#CEJI84.DE6o:ËDECX@�6?:'F!<969CEQ;@M<98;@²GI4.3;T.@M:.>�<98!CE6?5#DE:.<ek �UeVa]XV é®ê FlLQzNUeV"L_zN]_VaL_L�{Wvj`S{aVW`�zN]NJ�zNFlvM`(vM`ÌzXUSVÍ� ê O�vjL_zQLXVWHMVa]_VakIhÌv�HMVa]XkIv�JMnSVWnÌ`SVaFItjU7PZvj]_L¡vM£�J¯zXFltjU�zè�¥^{WklRSL�zNVW]�JM`Sn®FlLAtjFxHjVW`�P�h®öAÉ7`�dGÓ�d±³´V�Fl`�zX]XvGnSRS{aV�z^ v��SJM]NJMO�VWzXVa]_La[;Ñ¡JM`eniÒ�[jzNv�JM{W{avjRe`7z�£¸vj]�zXUSVDJM{WzXRrJMk`7RSOcPZVW]DvK£Â©rPpVa]_LDJKzDUSFltjU"]_FlL_²�vM£�£¤JMFlkIRS]XVKdAqsVWz

� ê ζÑGè � Ø � ÚPpVAzNUSV�`7RSOQPpVa]}vM£eLXViHjVa]_Vakxh§v�HMVa]_klv�JMneVanc`eVaFltMUGPpvj]_L}JM]XvjRe`SncJK`�è�¥^{WklRSL�zNVa]W[� �USVa]_VÓÑÆÔ�-;JM`en�-tÕ�Ò!Ü ��ÐjÓ�d³´V�©S`Sn!zNUeFlL�L_zX]XRS{izNRS]_Vc£¸vj]W� ê zNv«PpVcVWLXL_Va`�zNFlJMkÂFI`6vM]XnSVW];zNUSJKz vxwSØ Ý Ú Fl`!ö�É7`�dp��JKtj]XVWV� �FIzXU�zXUSV�£¸vM]XOvK£±zNUeV�L_FlOQRSklJKzNVWn uv Ø Ý Ú nSFlL�zN]_FlPSRezXFlvj`"VWLX�pVa{WF�JMkIkIh«£¸vM]DL_O«JKklk�ârdy±JM²7FI`StVÑ�Î í � î ë ��Ö �j�®JM`SnÈÒ6Î=��ÐjÓ�[FIO��eklFlVWL�{avjRS`�zXFl`St"JKklkszNUSV�©rPpVa]XL�vj`ozNUSV�{akIRSL_zXVa]��pVa]_Fl�SUSVW]_hªzNv«PZVcJKz�]_FlLX²ªvM£A£¤JKFlklRe]XVMd�T6vGnSVak��QkIFl`SVWL�Fl`°ÂFltjRS]_V�Ó®{Wvj]X]_VaL_�Zvj`en6zNv¡zXUSFlL�{�JKLXV�JM`Sn�nSv�`evMz�©Sz�zNUSV�LXFIOQRSk�J�zNVan uv Ø Ý Ú HjVa]�h" �VWklkÃdµDv� �ViHjVa]�O�vGnSVaksÓkIFl`SVWL�FI`! �USFl{�Uo VQHKJM]_hÈÑoJK`SnxÒ�JKL�£¸RS`S{izNFlvM`SL�vM£�â"©Sz�zXUSVQL_FlOQRSklJKzNVWn uv Ø Ý Ú OQRS{�U¯PpVWz_zNVa]Wd§~ú{akIvjLXVWn£¸vM]XOÔJK�S�S]Xv�w�FlO�JKzXFlvj`�£¸vj] v Ø Ý Ú FlL�nSVW]XFxHjVan�Fl`´å ��çÃd×�� þ ��'$r�È ���/����³�������Ø�Ù��� ý ��� �!���d�.�'�#�g$��'&�(*Ú������ â�,³2UeVa`«â�FlL�L_O«JMkIke{Wvj]X]_VaLX�pvj`SneFl`St§zNv§k�JK]XtjV�HKJM]_F�JMPeFlklFxz^h�Fl`¡©rPpVa]�L_zX]XVa`etMzNU�[�zXUSV�{aklReL_zNVW]�¥�nS]XFxHjVa`�PS]_V�JM²7nSv� �`O�Va{�USJM`SFlL_OüFIL�nSvjO�Fl`rJKzXVan"P�hªJQnSFILX�pVa]_LXVan¹[rL_zN]_Va`StKzNU�¥�nS]XFxHjVa`�PS]_V�JM²7nSv� �`!O�Va{�USJM`SFlL_OüvM£�zNUeV§­�¥�PSRS`SnSkIVMdy�UeFlL�FlL�{WklV�JK]XkIh�LXVWVa`6Fl`"zNUSV¢£¤JMFIklRS]_V�LX`SJM�SLXUevMzNLDvM£±J�­�¥^PeRS`SnSkIV§£¸vM]�â¡Î �M[gJML�LXUSv� �`6Fl`�°±FltMRS]XV§�ed\^`®zNUeVD{aJMLXV�vM£ZnSFILX�pVa]XL_Van®©rPpVa]A£¤JMFlkIRS]XV�Fl`®J§­�¥�PSRS`SneklVM[�zXUSV�nSVWzNJMFlkILAvM£pzNUSV�©rPpVa]�kIv�JMn�¥�LXUrJM]_Fl`St�O�vGnSVakeO«J�h`evMzDPZV¢FIO��pvj]_zNJM`�z��S]_v�HGFInSVanªzNUrJ�zDzXUSV§O�vGnSVWk�{avj`eLXVa]�HjVWL�klvjJMn�dAy�U7RSL� �V�{avj`eLXFlneVa]�PZVWUrJ�HGFlvj]�RS`enSVa]�zNUSVVWÉ7RrJMksklvjJMn�¥^L_UrJM]_Fl`StQ]_RSklV�vj]�ö�qÂEpdrö�qÂEªJML_LXRSO�VaL�zNUSJKz�zNUeV�L�zN]_VaLXL�{Wvj`S{aVW`�zN]NJ�zNFlvM`�£¤JM{izNvj]£¸vj]�V�JM{�UªFI`7zNJM{Wz©SPZVW]�FI`9JM`Ûzg¥�©rPpVa]�­�¥^PSRe`SnSklVQ �FxzNUÈÜ!PS]_vj²MVW`´©rPpVa]_L�FIL�Ý w�Þ ß Î¿zsÐ Ø zoßÙÜ Ú d�~;�S�Skxh�FI`St�J�]_VaL_RSkIz�nSRSV¡zXvEGO�FxzNU9å �Kç� �USFI{�U�L_UrJM]_�ZVW`SL�vj`SV§nSRSV�zXv�·¢JK`SFlVWklL¢å �Mç�zXv�JM`�ö�q�E�PSRS`enSklV¢vM£�³´VaFlPeRSklkp©rPZVW]XL� V§©S`Sn"zNUrJKzzXUSV�PSRe`SnSklV¢L�zN]_Va`StMzXU"nSFlL�zN]_FlPSRezXFlvj`�y wSØ Ý Ú {avj`�HMVa]XtMVaLDJML=zÈà á>[GzNvczXUSV¢`Svj]_O«JKkp£¸vj]_O�â ØXØ Ý�ßÛã�äw Ú Ð ÷ äw Ú

Page 6: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

\

\X\

\_\X\

\�¶

°ÂFltjRS]_V¡��1Ë203!4�5#698!:.<96Ó:.>³<98;@A>B4.CEDEF;G9@W5!G9:0JL@�696MCE3�<98;@ËPA@�Q#CE4.3}RB4.PA:'3;TiU ��� 69CNPWF!DN4�<9CX:'3!6ZY�[e]�^#F#3!Q!DX@H6o5�@�JeCEPA@�3_�CX<98x` ��� b @�CE^#F!DEDgc!^�@eGI6²:.>�fÈh � F!3!Q!@eGW5�@eGICX:0Q!CEJH^�:'F!3#Q!4�G9jÆJL:'3!Q!CX<9CE:'3!6ek � 5�@�3ÍJeCXGIJeDE@�6Al Q;@�3!:.<o@�CE3/<94.JL<c!^�@eGI6ep q Q;@�3;:.<o@²^!G9:.s.@�3tc!^�@eGI6�k �FxzNU�JML�h�O��ezXvMzNFI{¢O�VaJM`

ã äw ÎÑÝrà Ø â!å Ú � Y ð á\æ � ñ -�Öç�0�0~/z � m ð�ïÓè å m ð á Ð�â!é Y ð�ï0ê Ø ~ ÚJK`Sn!JKL_hGO���zNvMzXFl{¢L_zNJM`SnrJK]Xn�nSVWHGF�J�zNFlvM`

÷ äw ÎÑÝràLz � Y ð m â � Y ð á!ë å � Y ð á Ø ��ßìå � Y ð á Ú Ö Ø � Ú× �Ðí ��î+�g��oï þ î±ð0�� ÿ î±ñ%�#��D`SklFI²MVDFl`«ö�q�EZ[G �USVa]_VaFI`�O«J�zNVa]_F�JMkrnSJMO«JKtjV�JK{a{a]_RSVaL�tjklvjPSJMklkxhj[j V�LX�pVa{WRSk�JKzXV;zXUrJKzFl`�µ�¶¢q}q�E�zNUeVa]XV�FlLJâM¥�nSVa�pVa`enSVa`�z�L_Fl¬WV�L_{�JMkIV§ �FxzNUSFI`" �USFl{�U6nrJMO�JMtjV�JM{a{WRSOQRSklJKzNVWL�Fl`6J�nSFILX�pVa]_LXVan6O«JK`S`SVa]DJM`en!�S]_vj�rJMt�J�zNVaL{aJKz�JKL_zN]_vj�SUSFI{�JMkIkIh¯£¸]_vjOÏzXUSVa]_VMd$y�UrJKz®FILa[±£¤JKFlklRe]XV«FI`SFIzXF�JKzXVaL§v�HjVW]�òÍ Îóz�Ðdòz0PSRe`SnSklVWLQvM£iòz+©rPpVa]_L¡Fl`&JK`ö�q�E7¥�klFl²KVcO�JM`S`SVW]� �FIzXUSFl`oJ«klvG{�JMkIFl¬WVan!]_VatjFIvj`¯JM`en¯�S]Xvj�SJMt�JKzXVaL�{aJKz�JML�zN]_vj�SUSFI{�JMkIkIhª£¸]XvjO zXUSVa]_V¡]XVaL_RSkIzXFl`StFI`ª{avjO��ZvMLXFIzXV�L_zX]XVW`StMzNU6nSFlL�zN]_FlPSRezXFlvj`

y weØ Ý Ú Î ��ß0Û7�Dß�â¢å Ø Ý6ß�ã ä ôw Ú Ð ÷ ä ôw Ú çÃÞ ôõ � Ø � Ú\^`0°±FltjRe]XV��S[�£¸vj]�USFltjUekIh�HKJM]_F�JMPSkIV"©rPZVW]XL« �FxzNU:â2Î �j[DÓ�[���[�JM`Sn2�$ �VoUSJ�HMVo�SkIvMzXzXVan:zXUSV!L�zN]_Va`StMzXUneFlL_zX]XFIPSRezNFIvj`�vM£�zNUeV�L_O«JKklklVWL_z�LXFl¬WVan6­�¥�PSRS`SneklV Ø z Y � z Y Ú zXv¡ �USFl{�Uª �VaJM²j¥^kIFl`S²KVan�nSFlL�zN]XFIPSRezXFlvj`SLvM£Âk�JM]_tjVa]PeRS`SnSkIVaLc{WvjklklJM�SL_VMd�y�USFIL�L_O«JKklklVWL_z�­�¥^PSRS`enSklV®LXFI¬aV«JM�S�S]_v�w�FIO«JKzXVakxho{avM]X]XVWLX�pvj`SnSL�zNvªzNUSV�{a]XFxzNFI{�JMkA{WklRSL�zNVa]L_Fl¬WV�neVW©r`SVWn0�S]_VWHGFlvjReLXkIhMd(³´V6JMklL_v9L_USv� ÔzNUeV6nSFIL_zX]XFlPeRezNFIvj`SLQvM£�k�JM]_tjVa]¡PeRS`SnSkIVaL�vM£�L_Fl¬WV Ø z m � z m Ú vj]Ø z ï � z ï Ú  V�JM²j¥�klFl`e²MVan�zXvczNUeV�LXFI¬aV Ø z Y � z Y Ú d��DvMzNV�zXUrJKz�JML�â®nSVa{W]XVaJMLXVWLa[SzNUeVaLXV� V�JK²�¥�klFI`S²MVWn"nSFlL�zN]XFIPSR�¥zXFlvj`eL�PpVa{WvjO�V;FI`S{a]_V�JML_Fl`Stjkxhiö§JMRSLXL_F�JM` Ø FI`SnSFl{aJKzNVWn�P�h®zNUeV;L�zN]NJKFltjU�zN`eVaLXLvM£�zXUSV;L�zN]_Va`StMzXU�nSFIL_zN]_FlPSR�zNFlvM`�vM``evj]XO�JMk�{avGvj]XnSFI`rJKzXVaL Ú JM`Sn$JM]_VQPpVWzXzXVa]§JM�e�S]Xv�w�FlO�JKzNVWnoP�h!zXUSV¡ö�q�EoJML_hGO��ezNvMzXFl{�nSFlL�zN]XFIPSRezXFlvj`�d�·�VaLX�eFIzNVzXUSVªViw�{aVWklkIVa`�z�JMtj]XVWVaO�Va`�z�vM£DzNUSVÍ�0-0-�¥�©rPpVa]W[� V�JM²j¥�klFl`e²MVan(L_zX]XVW`StMzNU:nSFlL�zN]_FlPSRezXFlvj`´ �FIzNUÌzNUeVÈ~jÓj��¥�©SPZVW]a[

Page 7: Effects of Heterogeneity on the Strength of 3D Title: Composites/67531/metadc718363/m2/1/high_re… · Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher

0.3 0.4 0.5 0.6 0.7−4

−3

−2

−1

0

1

2

3

4

á w � w � w�÷�*� � m � ï �Y m � ï �m¯mo��m � ï �ï Y�øAm � ï �� Y¤m�m � ï �

YmâQÎ � Ó � �

ù�úüû*ýþ£�¤Ê ?¡Z¢*£�¤w?¥Ìw � ¦ w �w?¥Ìw � ¦ w �w?¥ÌwÊ÷ ¦ wÊ÷ÝsÐKÝrà

ÿ �� «���«�¯°°

°ÂFltjRS]_VQ��1 � :'PA5#4�GICE69:'3V:.>��MRoR���� äw Y� �� äw YÓT'CXÏ.@�3O^KjVµ=4.3!CX@�D�� 6=4.6oj0PA5!<o:.<9CNJH>½:.GIPÌF!DE4�>�:.G ��� 2V^7F!3!Q!DX@�6�_�CX<9869CEPÌF!DE4�<o@�Q�69<oG9@�3;T.<98�� w R��rYË:.>m4.3��¶h�� Y�� � Y���� �Z� 2Û[e]�^[email protected] � DE6o:|698;:·_�3Û4�GI@V@�PA57CXGICEJe4.D¸69<oG9@�3;T.<98Q!CE6o<oGICE^#F;<9CX:'3!6g:.>"DN4�G9T.@eG�[e]�^7F!3!Q!DX@�6?_�@�4�sÐDECE3;s.@�Q�<o:Ë<98;@m6IC��e@ � Y!� � Y CE3Ð4�T.G9@e@�PA@�3/<�_ÓCX<98�<98;:'6o@M:.>"<98;@"� Y!� � Y[e]�^#F#[email protected] V�JK²�¥�klFI`S²MVWn&nSFlL�zN]XFIPSRezXFlvj`9 �USVa`Ìâ.Îó-�Ö ��[�FxzczXRS]X`SL¡vMRezczXUrJKzQzXUSVWh´nSv¯`SvMzQJMtj]_VaV« �FxzNU&J$Ój�/-0-�¥�©SPZVW]a[ V�JK²�¥�klFI`S²MVWn$­�¥�PSRS`SnSkIVcL�zN]XVW`StMzXU¯nSFlL�zN]XFIPSRezXFlvj`�dDy�USFIL�LXRetjtjVaL�zNL�zXUrJKz�zXUSVQL_O«JMkIklVWL_z;{�JKzNJML_zX]Xvj�SUeFl{�£¤JKFlklRe]XVViHjVW`7z;vM£ÂzNUSV§PSRS`enSklV¢vG{a{WRS]XLDv�HjVa]�O�vj]XV�zXUrJM`È~�Ój�Qvj]��pVa]_UrJM�SL�VWHMVa`|�0-0-Q©rPpVa]XLWd� ¾�Å � þ Æ!� Ä�¾�Å �\^`Qö�É7`�dM�S[K V�tjFxHjVzXUSV� V�JM²KVaL�z_¥^kIFl`S²�{�UrJM]XJM{WzXVa]_FlL_zXFl{nSFlL�zN]XFIPSRezXFlvj`§£¸RS`S{izNFlvM` v Ø Ý Ú £¸vj]�­�¥^PeRS`SnSkIVaLadÂy�USVaL_VPeRS`SnSkIVaL§JM]_V¡klFl`e²GL�Fl`ozNUeV¡{�UrJMFl`G¥^vM£�¥�PSRS`SnSkIVaL�O�vGnSVak�£¸vj]�zNUeVQ£¤JMFlkIRS]XV�vM£��M· Re`SFlnSFI]XVW{WzNFIvj`rJMk}{WvjO��pvjL_FIzNVWLad°rvj];LXReù�{WFlVa`�zXkIhoklJM]XtMV�³´VaFlPeRSklk�O�vGnSRSkIRSL�âr[¹LNJ�hoâxÔ>�"Fl`¯�j·ú{WvjO��ZvjL_FIzXVaLa[ZzNUSVQL�zN]_Va`StMzXU$nSFlL�zN]XFIPSRezXFlvj`vK£�J�{avMO��pvjLXFxzNV�vM£�klVW`StMzXU¯Ë0Πͪ­�JM`en! �FIzXUxz´©rPpVa]_L�FlL�# õ Þ weظÙZÚ�ë �;ß Ø ��ß v ظÙZÚ_Ú õ w d�°rvj];â6Ü �S[Uev� VWHMVa]W[� V¡vjPSL_Va]_HMV¡zNUrJKz¢zXUSVQnSViz�JMFIklL�vK£�zXUSVQkIv�JMn�¥�LXUrJM]_Fl`St�PZVW{avjO�V¡Fl`e{a]XVaJMLXFI`Stjkxh6Re`SFlO��ZvM]_z�JK`7za[�JM`enzXUSV§­�¥^PSRS`enSklV�L�zN]_Va`StMzXU6neFlL_zX]XFIPSRezNFIvj`«£¸vj]©ew�VanÍzoFlL�`SvKz�vj`Skxh«FI`S{a]_V�JML_Fl`StjkxhÆö§JMRSL_LXFlJM`"RS�"zNv¡É7RSFxzNV¢k�JK]XtjVz$PSR�z�JMkILXv®{avM`7HMVa]_tjVaL�zXv�zXUrJKz�£¸vj]�ö�q�Eª �USvMLXV�JM`rJMkxhGzXFl{aJMk¹£¸vj]_OÔFlL�²7`Sv� �`�d�°rvj]�©ew�VWnoâr[rUSv� �ViHjVW]a[ZzXUSFlLö§JMRSL_LXFlJM`Q`rJKzXRS]XV�FlL±V¨w��ZVW{WzNVWn¡zNv¢�pVa]XL_FlL�z�vj`Skxh�RS�QzNv¢J¢­�¥^PeRS`SnSkIV�LXFI¬aV�vK£rzNUSV�vM]XnSVW]±vM£gzNUeV�{a]_FIzNFI{�JMk7{WklRSL�zNVa]L_Fl¬WVMd�°gvM]�{WvjO��ZvjL_FIzXVaL¢PpVWhjvM`Sn.zNUeFlL¢{a]_FIzXFl{�JKk�LXFI¬aV®zNUSV®nSFIL_zN]_FlPSR�zNFlvM`o£¸RS`S{WzXFlvj`$£¸vj]�­�¥^PSRS`enSklV®L_zX]XVW`StMzNU9FlLzXUrJKz�£¸vj]�J6{�UrJMFI`.vM£mö�JMReLXLXFlJM`%$ä�rJKzX{�USVaLWã±vM£�òz:©rPpVa]_LcFI`9zXUSV�­�¥^PSRe`SnSklVKd"y�U7RSLczNUeV«{avMO��pvjLXFxzNV®{�JK`�PZVHGFIVW VanoJKL;JQ V�JM²KVaL�z_¥^kIFl`S²�JM]X]XJM`StjVWO�VW`7z�vK£�Í�òz$LXRS{�Uxö�JKRSLXL_F�JM`ª�rJKzX{�USVaLWd� »¢ý�» � »¢Å � » �å¦�aç�µDVanStjVW�ZVizNU�[pbed T JM`Snª¶�JM`!·�hG²KVM[guÂd Ø �.�0~�� Ú ['&)( *�+-,�.'(0/214365879(l[ � [pÓK�K�SdåäÓ�ç�µ;JM]XkIv� §[S·Qd©ö¡dSJM`en!uAUSvGVa`SF¦wZ[pEZd qAd Ø �.���M� Ú [:&)( *�+-,�.'(0/214365879(l[ �#× [g�e���å �KçcE�O�FIzXU�[<;§däq�d Ø �.�j�jÓ Ú [>=�?�?@(BAC7D+4EF1GEH(l[ �JI [s�����Gdå �Mç�·¢JM`eFlVakILa[eµ�däö�d Ø �.�M�7� Ú [@A�7D+HKL(NMO+-PQ14RTST+HKL(BUN+-?@VW+-?X= [ �WY>Z [r�;-���dåä��çcT!JKUSVaL_U�[ZEZdI[Su�Uev�VW`SFxwZ[pEZdgqAdx[gJM`Sn�f�VWhMVa]_klVaFI`6\�d bed Ø Ó/-0-e� Ú \^`��S]XVW�rJM]XJKzNFIvj`�d