Upper bound. Circle. Lower bound. σ 3
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1 ÝÓ Ó ÐÙÖ Ò ÒØÐ ÝÐÒ ØÖÑÒØÓÒ ØÖÚÐ ÓÖ ÙÒÜÐ ÐÓÒ ÐÑÓ Øµ ÑØÖÐ ÒÒÓØ Ò Ó Ø ÓÓ Ò Á ÓØÖÓ Ý ØÑ ÓÓÖÒØ ÛÖ Á Á Ò Á ÛÒØ ØÓ ÖØ ÝÖÓ ØØ ØÖ ÖÓÑ Ï ØÖ ÚØÓÖ ÛÖ Â Ò Â ÁÄÍÊ Æ ÁÆÁÌÁÄ ÁÄ ÊÁÌÊÁ ÁËÇÌÊÇÈÁ ÅÌÊÁÄË ÁÒ ÒÖÐ µ ¼ ÓÖ Ò Ò Òµ ¼º ÐÙÖ ÒØÐ ÝÐ µ ¼ ÓÖ Á Á Á µ ¼ ÏØ ÓÙØ ÒÖÐ ¹ÐÓÒ µ ¼ ÐØÖÒØÚÐÝ Á   µ ¼ ÓÓÖÒØ Ý ØÑ ÖØÖÖÝ µ ¼ Ò ÐÐ ÓÓÖÒØ Ý ØÑ Á Â Ó µ ¼ µ Ò ÒÚÖÒØ ß ÐÖ Ò ÒÚÖÒØ ÓÖ ÜÑÐ µ ¼ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
2 ÇÇÊÁÆÌ ÀÁÀ¹ÏËÌÊÊ ß ÇÅÌÊÁÄ ËËÌÅ Ò Ì ÇÈ Ò Á ÇÆ ÝÖÓ ØØ Ü Ó ÓÓÖÒØ µ ÁÒ Ø ÑÝ Ù ÓÓÖÒØ µ Û ÆÈ ÇÈ ÇÆ Á Á Á Â Ó Â ÝÖÓ ØØ ØÖ Ó ¹ ÑÓÙÒØ ØÖ ÚØÓÖ ÕÙÚÐÒØ ÜÖ ÓÒ ÚØÓÖ ØÖ ÁÆÌÊÈÊÌÌÁÇÆ Ç ËÌÊËË ÁÆÎÊÁÆÌË ÁËÇÌÊÇÈÁ ÅÌÊÁÄË µ ¼ È Ò È Á Á Á µ ¼ Ó µ ¼ Æ Ç Á Â Ó µ ¼ Æ ÐÐÝ ÓÒÚÒÒØ ÚØÓÖ ÐÒ Á Â Ó µ ¼ Ó ÖØÓÒ ÓÙÐÒ «Ø Ó ¹ØÖÑ µ ÝÑÑØÖÝ ÖÓÖØ Â Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
3 ¼ Æ ¼ Æ ¼ Æ Ó ¼ Æ µ Ó Ó ¼ Æ µ ¼ Æ µ Ó ¼ Æ µ Ó Ó µ ¼ Æ µ Ó ¼ Æ µ Ó Ó ÝÑÑØÖÝ ÓÙØ ¼ Æ ÒÖÐ ÖÓÖØ ÓÑÐØ Üº غµ ÓÒÚÜØÝ ËÅÅÌÊ ÈÊÇÈÊÌÁË ÁÒ ÚØÓÖ ÐÒ Á ÓÒ ØÒØ Á Â Ó µ ¼ ËÅÅÌÊ ÈÊÇÈÊÌÁË ÁÒ ÚØÓÖ ÐÒ Á ÓÒ ØÒØ Á Â Ó µ ¼ ¼ Æ ÖÓ Ó Ó µ ÝÑÑÖØÝ ÓÙØ ¼ ÓÙØ ¼ Æ ÝÑÑØÖÝ ¼Æµ Ò Ó ¼ Æ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
4 ØÓ Ø ÖÒÙÐÖ Ü Á ÝÖÓ ØØ ËÌÄ ÅÌÄË ÒÒÒØ ÓÒ ÐÒ ËÇÁÄË ÊÇÃË ÇÆÊÌ Ò ØÖÓÒÐÝ ÐÙÖ Â ÐÒ ÅÖÒ ÓÒØÒÒ Ø ÐÒ Ü ÝÖÓ ØØ ÓÒ ØÒØ Â Â Á Á Á ÆÓ Ò ÙÒ Ó Á Ñ Ö ÙÐØ ØÒ ÓÒ ÓÑÖ ÓÒµ Á ¹ØÖÑ ÓÛ ¼ Æ ¹ ÝÑÑØÖÝ Ó ¼ Æ ÅÌÄË Æ ËÌÄ ÆÊÄ ÈÊÁÅÆÌÄ ÎÁÆ Â Ó µ ¼ ÝÑÑØÖÝ ÐÒ Úº ÐÒ ÐÒ ÓÒ ØÒØ ØÖ Á ÝÖÓº Ó µ ¼  Upper bound Lower bound Circle Á ÓÒ Â Ó µ ¼ Á 2 3 ÓÑÖ ÓÒ ØÒ Ð Ó ØÖÓÒ Ò ÙÒ Ï ÓÙ ÓÒ ÐÓÛÖ ÓÙÒ Ò ÖÐ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
5 ÅÌÄË Æ ËÌÄ Èʺ ÅÌÄË Æ ËÌÄ 0.7 ÌÝÐÓÖ ² ÉÙÒÒÝ µ 2 ÚÓÒ Å ÌÖ Hydrostatic axis; = 2= 3 von Mises cylinder 3 ( yo, 0) ( 0, yo) 2 45 o (0, yo ) 2 Ellipse ( yo,0) τ/ yo Tresca Copper Aluminium Mild steel yo von Mises / 2 Hydrostatic axis; = 2=3 Tresca surface 3 ( yo,0) ( yo, yo) (0, yo ) (0, yo ) ( yo, (yo,0) yo ) ÄÒ ² ÓÖ µ von Mises Tresca θ=30 o Experimental data for aluminium 3 2 Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
6 ÜÖ ÓÒ ÖÙÖ ² ÈÖÖ µµ ÄÒÖ «Á ¼   Á Ã Ó ÖÓ Ã Ó µ Ó ¼  Á ¹ÈÊÅÌÊ ÊÁÌÊÁÇÆ Â Â µ Á ÅÆÌÁÌÁÇÍË ÅÌÊÁÄË ¼ Á Â Ó µ ¼ ÛÖ ËÑÐ Ø Ó Ð ËÐÖ µµ Ã Ó ÖÓ Ã Ó µ Ó ¼ Á  µ ¼ µ ÈÖÑØÖ Ã Ò Ã º Á à ¼ µ ÖÙÖ ² ÈÖÖ Á à ¼ µ ÚÓÒ Å ÓÒÚÒÒØ ÑØÑØÐ ÜÖ ÓÒ Í Ø ÛØ Ö ÎÖÝ ÓÓÖ ¹ ÜØ ÓÖ ÑÐÐ ÖØÓÒ ÒÐ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
7 ÊÁÌÊÁÇÆ ¹ÈÊÅÌÊ ßÇÆÊÌß ÒØÐ ÝÐ ÒÒÒØ ÓÒ ÝÖÓ¹ ÙÑ ØÖ ØØ È Ð È Æ Æ Ð Æ Ð Æ µ ÝÓ ÝÓ µ ¼ ÝÓ ÆÁËÇÌÊÇÈÁ ÊÁÌÊÁ ÁÒ ÒÖÐ µ ¼ Ò ÒÚÖÒØ ρ/ c Compressive meridian (-5.4) * Tensile meridian ÙÑ È Ð Ð ¼ ÓÖ Ì È ¼ ÛÖ Ì ξ/ c ÓÓ / c / 2 c ¼ ¼ ¼ Ò ÑØÖÜ ÓÖÑØ ÛÖ ¼ Ù µ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ Krupfer et al ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ -.0 ºº ÚÓÒ Å ÖØÖ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
8 È È È À È È È À È È È È È È Ä È È È È È È Å È ¼ ¼ ¼ À ¼ ¼ ¼ À ¼ ¼ ¼ ÐÒ ÓÙÐ ÓÒÐÝ Ò ÓÒ ÚØÓÖ ÙÑ ÐØ ØÖ ØÖ ¼ À ¼ À ¼ ¼ ¼ ¼ À À ¼ ¼ ¼ À À ¼ ¼ ¼ ÆÁËÇÌÊÇÈÁ ÊÁÌÊÁ ÆÁËÇÌÊÇÈÁ ÊÁÌÊÁ ÓÒ Ö ÚÓÒ Å µµ Ì È Ì µè ¼ È Ð Ð ¼ ÓÖ Ì È ¼ ÁØ ØÙÖÒ ÓÙØ ØØ È È Ì ºº ÖÑØÖ Ò Ò Ö ÒÒÒØ È ¼ È Ì È ¼ ÓÒ ØÖÒØ È ¼ È È È È È Æ ÙÑ ÇÖØÓØÖÓÝ ÝÑÑÖØÝ ÐÒ µ ÇÖØÓØÖÓÝ ß ÓÒÐÝ Ò ÙÒ Ó ÚØÓÖ ØÖ È ¼ ¼ ¼ Ä ¼ ¼ È ¼ ¼ ¼ ¼ Å ¼ ¼ ¼ ¼ Ä ¼ ¼ ¼ ¼ ¼ ¼ ¼ Æ ¼ ¼ ¼ ¼ Å ¼ ¼ ¼ ¼ ¼ ¼ Æ ÖÑØÖ ØÓ ØÖÑÒº ÖÑØÖ ÀÐÐ ¼µ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
9 ÊÁÌÊÁ ÆÁËÇÌÊÇÈÁ ÄÁÊÌÁÇÆ ß ß µ µ À µ ÝÐÒ ØÖØ Ø ÝÓ ÙÑ ÝÓµ Ø Ñ ÑÒÒÖ ÙÒÜÐ ÐÓÒ Ò Ò ÁÒ ÖØÓÒ ÝÓµ À ÊÁÌÊÁ ÆÁËÇÌÊÇÈÁ ÄÁÊÌÁÇÆ ß ß ÝÓµ À ÝÓµ À À ÝÓ µ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓ µ ÝÐÒ ØÖØ Ø ÝÓ ºº ÙÑ ÝÓ µ ÝÓ µ Æ Ï ÓÙÒ Ì È ¼ ÓÖ Ï ÓÙÒ ÝÓµ Ä Å Æ ¼ ÌÖ ÖÑØÖ ØÖ ÕÙØÓÒ ÝÓ µ Æ ØÓ ØÖÑÒ ÖÑØÖ ÝÓ µ ÍÒÜÐ Ø Ø ¹ÖØÓÒ ÝÓ µ ¼ ¼ ÓØÖÛ ÖÓÑ Æ ÆÜØ ÐÝÒ Ö ØÖ ÓÒÐÝ ¼ ÓØÖÛ µ ¼ ÀÐг ÝÐ ÖØÖÓÒ Ä ¼ Ä ÁÒ Ø Ñ ÑÒÒÖ ÓÖ Ø ÓØÖ ÖÑØÖ ÝÓµ À ÝÓ µ Å Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
10 À ÝÓµ ÝÓ Å Æ Ä ÝÓ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓ µ µ Å ÝÓ µ Æ ÝÓ ÝÓµ ÝÓµ ÝÓµ ÝÓµ ÝÓ ÝÓ ÝÓµ ÝÓµ ʵ ÛÒ Ê ÈÖÓÐÑ ÆÁËÇÌÊÇÈÁ ÊÁÌÊÁ ÀÁÄÄ³Ë ÊÁÌÊÁÇÆ ÀÐг ÓÖØÓØÖÓ ÝÐ ÖØÖÓÒ ¼µ µ µ À µ Ä Å Æ ¼ ÖÒÐ Ü R= ÄØ ÑØÖÐ Ü Ó ÓÖØÓØÖÓÝ ÓÒ ÛØ Ø Á À R= 5 3 Å ÚÓÒ ÁÒ ÒÖÐ ÝÓ µ 2 3 ÝÓ µ ÝÓ µ ÛÖ ÝÓ µ Ä Ê ØÖØÓÒ ÓÒ Ö Ó ÓÖØÓØÖÓÝ µ ÚÓÒ Å ÓÖ Ê Ì Ö ÓÒ Ï Ú ÙÑ ÕÙÖØ ÓÖÑ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
11 ÊÁÌÊÁ ÆÁËÇÌÊÇÈÁ ÄÁÊÌÁÇÆ ÈÈÊÇÊ ß ß ÝÓ ÅÈ À ÝÓµ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓµ ÝÓ ÅÈ ÝÓ ¼ÅÈ ÝÓ µ ÝÓ µ ÝÓ µ ÝÓµ ÝÓµ ÝÓµ ¼ ¼ ¼ À ¼ ¼ ¼ À ¼ ¼ ¼ ÓÖØÓØÖÓÝ Ò Ö ÙÖ ÒÒÒØ ÙÑÒ ¼µ ÀÐÐ È Ð Ð Õ ÓÖ Ì È Ì Õ ¼ ÆÁËÇÌÊÇÈÁ ÁÄ ÊÁÌÊÁ Ï Ú Ò ÒÖÐ ÖÓÑ ÈÖ ÓÒ µ Û Ú µ ¼ ÜÑÐ ÙÑÒ ÓÖØÓØÖÓÝ È Ð Ð ¼ ÓÖ Ì È ¼ ÈÖÑØÖ Ò ÀÐг ÖØÖÓÒ ÝÓ µ ÛÖ ÝÓ µ ÝÓ µ È ¼¼ ¼ ¼ ¼ Ä ¼ ¼ ¼ ¼ ¼ ¼ Å ¼ ¼ ¼ ¼ ¼ ¼ Æ È Ð Ð ¼ ÓÖ Ì È ¼ ÒÖÐÞ ¼ Ê ØÖØÓÒ ÓÒ Ö Ó ÓÖØÓØÖÓÝ Ì ¹ÏÙ Ò ÓØÖÓ ÝÐ ÖØÖÓÒ µ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ Úº Ó ËÓÐ ÅÒ ÄÙÒ ÍÒÚÖ ØÝ
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