章順虎,姜興睿
(蘇州大學(xué) 沙鋼鋼鐵學(xué)院,江蘇 蘇州 215021)
MY準(zhǔn)則解析受內(nèi)壓薄圓環(huán)極限壓力
章順虎,姜興睿
(蘇州大學(xué) 沙鋼鋼鐵學(xué)院,江蘇 蘇州 215021)
目的 探明受內(nèi)壓薄圓環(huán)極限承壓能力。方法 首次以MY(平均屈服)準(zhǔn)則對(duì)受內(nèi)壓薄圓環(huán)進(jìn)行彈塑性分析,克服Mises準(zhǔn)則數(shù)學(xué)求解的困難性,導(dǎo)出塑性區(qū)內(nèi)的應(yīng)力場,并獲得塑性極限壓力的解析解。此外,還給出了彈塑性臨界半徑與內(nèi)壓之間的依賴關(guān)系,并分析了二者間的變化規(guī)律。結(jié)果 塑性極限壓力的解析解表明,塑性極限壓力是材料屈服強(qiáng)度、半徑比值的函數(shù);與已有的Tresca、TSS準(zhǔn)則獲得的結(jié)果比較表明,Tresca準(zhǔn)則給出極限壓力下限,TSS屈服準(zhǔn)則給出極限壓力上限,MY準(zhǔn)則給出極限壓力居于兩者之間,可作為Mises解的替代。結(jié)論 文中結(jié)果對(duì)于充分發(fā)揮材料性能,進(jìn)而對(duì)薄圓環(huán)的設(shè)計(jì)、選材以及安全評(píng)估具有實(shí)際工程意義。
MY準(zhǔn)則;薄圓環(huán);應(yīng)力場;極限壓力;解析解
金屬薄圓環(huán)是一種典型的工程結(jié)構(gòu)件,在建筑、航空、機(jī)械等領(lǐng)域具有較為廣泛的應(yīng)用[1]。薄圓環(huán)受內(nèi)壓作用是其常見的受力形式,求解該載荷作用下的塑性極限壓力對(duì)充分發(fā)揮材料潛力具有實(shí)際工程意義[2]。目前通過聯(lián)解平衡微分方程、屈服條件及邊界條件求解薄圓環(huán)極限壓力的傳統(tǒng)方法已近成熟[3—4],Tresca準(zhǔn)則因忽略中間主應(yīng)力的影響而常給出偏低結(jié)果;TSS準(zhǔn)則給出結(jié)果偏高,浪費(fèi)材料;Mises屈服條件由于其非線性,獲得解析解困難,目前鮮見報(bào)道[5—6]。基于以上考慮,文中采用與Mises非線性屈服準(zhǔn)則非常逼近的線性 MY屈服準(zhǔn)則對(duì)受內(nèi)壓作用薄圓環(huán)進(jìn)行彈塑性極限分析,獲得了薄圓環(huán)全部進(jìn)入塑性狀態(tài)時(shí)極限壓力解析解,并定量分析了彈塑性臨界半徑與內(nèi)壓之間的變化規(guī)律。文中研究可為薄圓環(huán)的選材、設(shè)計(jì)以及安全評(píng)估提供理論依據(jù)。
MY準(zhǔn)則[7]已在材料成形[8—10]和爆破壓力[11]等領(lǐng)域獲得應(yīng)用。設(shè)主應(yīng)力其表達(dá)式見式(1)。該準(zhǔn)則在π平面上屈服軌跡見圖1,其中雙剪應(yīng)力(TSS)屈服軌跡是 Mises圓的外切正六邊形[12],Tresca屈服軌跡為Mises圓的內(nèi)接正六邊形,而MY屈服軌跡是非常逼近Mises圓的十二邊形[7],其屈服函數(shù)為TSS與 Tresca屈服函數(shù)的平均值。平面應(yīng)力下的屈服軌跡見圖2。由式(1)可得MY準(zhǔn)則在平面應(yīng)力下(σ2=0)的表達(dá)式見式(2)。
圖1 π平面上的MY屈服軌跡Fig.1 MY yield locus in π-plane
圖2 雙軸應(yīng)力的MY屈服軌跡Fig.2 MY yield locus in biaxial stress
受內(nèi)壓p作用的薄圓環(huán)見圖3。其中,ra和rb分別為薄圓環(huán)的內(nèi)徑與外徑,rc為彈塑性臨界半徑。
圖3 受內(nèi)壓薄圓環(huán)Fig.3 Thin ring under internal pressure
當(dāng)內(nèi)壓較小時(shí),此時(shí)整個(gè)薄圓環(huán)均處于彈性狀態(tài),此時(shí)圓環(huán)的應(yīng)力場為[13]:
由式(5)可見,載荷p在區(qū)間[ra,rb]為r的單調(diào)增函數(shù),因此,圓環(huán)內(nèi)壁處對(duì)應(yīng)的內(nèi)壓最小,彈性極限壓力pe在r=ra處取得,為:
彈性極限壓力隨著半徑比值的變化曲線見圖4??梢姡S著半徑比值的增加,彈性極限壓力增加。
圖4 彈性極限壓力與半徑比值的關(guān)系Fig.4 Relationship between elastic limit load and radius ratio
當(dāng)內(nèi)壓p大于彈性極限壓力后,圓環(huán)的塑性區(qū)將從內(nèi)壁向外壁擴(kuò)展,形成如圖 3所示的內(nèi)層塑性區(qū)和外層彈性區(qū)其中,在塑性區(qū)中,應(yīng)力分量滿足如下平衡微分方程和邊界條件:
聯(lián)立式(2)的第一式、式(7)和式(8),可得塑性區(qū)應(yīng)力場為:
在彈性區(qū)內(nèi),參照應(yīng)力場表達(dá)式(3),可設(shè)預(yù)先滿足微分方程(7)的通解表達(dá)式為:
式(10)在r=rc時(shí),σr連續(xù),且滿足應(yīng)力邊界條件于是,待定系數(shù)A,B如下:
將式(11)、(12)代入到式(10),可得塑性區(qū)范圍內(nèi)應(yīng)力場為:
隨著內(nèi)壓p增加,塑性區(qū)逐漸從rc范圍內(nèi)擴(kuò)展到外徑rb,因此,極限壓力在整個(gè)圓環(huán)進(jìn)入塑性狀態(tài)時(shí)求得,見式(14),該式表明,塑性極限壓力是屈服強(qiáng)度與半徑比值的函數(shù)。
對(duì)于本文求解對(duì)象,采用相同的解析方法,趙均海給出的Treaca解[5]和劉協(xié)權(quán)給出的TSS解[6]如下:
MY解與Tresca解、TSS解的對(duì)比情況見圖5。可見,隨著半徑比值的增大,極限壓力均增大,其中TSS提供極限壓力上限,Tresca提供下限,MY準(zhǔn)則居于二者之間??紤]MY準(zhǔn)則對(duì)Mises準(zhǔn)則具有較高的線性逼近程度,因此本文MY解可作為Mises解的替代。時(shí)的值從1開始,每隔0.5(增量)遞增至比值為5條件下的關(guān)系曲線見圖6。圖6表明,內(nèi)壓隨著彈塑性臨界半徑的增大而增大,當(dāng)rc=rb,內(nèi)壓達(dá)到最大,為塑性極限壓力條件下,塑性區(qū)內(nèi)應(yīng)力場分布見圖7,可知,徑向應(yīng)力σr為負(fù),為壓應(yīng)力,且隨著半徑r的增大而減?。恢芟驊?yīng)力σθ為正,為拉應(yīng)力,且隨著半徑r的增大而增大。此外,因最大主應(yīng)力為σθ,兩應(yīng)力始終存在著的大小關(guān)系。
圖5 依賴于屈服準(zhǔn)則的塑性極限壓力Fig.5 Plastic limit pressure depending on yield criteria
圖6 內(nèi)壓與彈塑性臨界半徑間的變化關(guān)系Fig 6. Relationship between internal pressure and elastic-plastic critical radius
圖7 塑性區(qū)內(nèi)應(yīng)力場分布Fig.7 Stress field distribution in plastic zone
1) 圓環(huán)內(nèi)壁處最先達(dá)到彈性極限狀態(tài),求得的彈性極限壓力依賴于半徑比值與屈服強(qiáng)度,隨著半徑比值的增大而增大。
2) 理論導(dǎo)出了受內(nèi)壓作用薄圓環(huán)塑性區(qū)內(nèi)的應(yīng)力場。分析表明,塑性區(qū)內(nèi)的徑向應(yīng)力為負(fù),周向應(yīng)力為正。
3) 首次以 MY準(zhǔn)則獲得圓環(huán)內(nèi)壓條件下塑性極限壓力解析解。結(jié)果表明,極限壓力是半徑比值與屈服強(qiáng)度的函數(shù),隨著半徑比值的增大而增大。MY準(zhǔn)則預(yù)測的塑性極限壓力介于Tresca與TSS極限壓力之間,可作為Mises解的替代。
4) 內(nèi)壓與彈塑性臨界半徑的關(guān)系表明,內(nèi)壓隨著臨界半徑由內(nèi)徑向外徑推移,不斷增大,最終在外徑處獲得塑性極限壓力。
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Analysis of Limit Pressure for Thin Ring Subjected to Internal Pressure with MY Criterion
ZHANG Shun-hu,JIANG Xing-rui
(Shagang School of Iron and Steel, Soochow University, Suzhou 215021, China)
To clarify the ultimate loading capacity of thin ring subjected to internal pressure, the elastic-plastic analysis of thin ring under internal pressure was first carried out with MY criterion to overcome the difficulty of mathematical solving of Mises criterion. The stress field in the plastic zone was derived, and an analytical solution of plastic limit load was then deduced.The dependency relationship between elastic-plastic critical radius and internal pressure was given. And the change rules of them were analyzed. It was shown in the solution that the plastic limit pressure was a function of yield stress and radius ratio. By comparing the plastic limit pressure with those obtained based on Tresca and TSS, Tresca criterion provided a lower bound, TSS provided an upper bound, while the solution of the MY criterion lay between them, and could be taken as the approximation of Mises solution. The present result has realistic engineering significance in full using of material properties, and further in guiding of the design, material selecting, and safety assessment of thin ring.
MY criterion; thin ring; stress field; limit pressure; analytical solution
2017-11-15
國家自然科學(xué)基金(51504156);江蘇省基礎(chǔ)研究計(jì)劃(自然科學(xué)基金)(BK20140334);江蘇省高校自然科學(xué)研究(14KJB460024);中國博士后科學(xué)基金(2014M561707)
章順虎(1986—),男,副教授,主要研究方向?yàn)樗苄猿尚卫碚撆c工藝。
10.3969/j.issn.1674-6457.2018.01.015
TG301
A
1674-6457(2018)01-0122-05