以應變和應力的形式貯存在物體中的勢能,又稱變形能。
基本介紹
- 中文名:應變能
- 外文名:strain energy
正文,
正文
以一維問題為例,一個截面積為A、長度為L的等截面直桿在軸向外力P1的作用下伸長
(圖1)。如果不考慮變形過程中的動力效應和溫度效應,則外力作的功W全部貯存到桿中,變成了桿的應變能U,其值為:
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式中P為變形過程中與伸長量
對應的載荷。在圖2所示的P-
曲線中,曲線下方的面積相當於桿中的應變能。而和曲線上方的面積相應的為余應變能(簡稱余能),記為U*,其值為:
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式中V=LA為桿的體積;
=P/A為桿中的應力;
=
/L為桿中的應變;
1、
1分別為P1、
1對應的應力和應變。如果桿的材料為線彈性的(即應力和應變成正比),則應變能和余能相等,即
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式中E為彈性模量。
在三維問題中,有六個獨立的應力分量和六個獨立的應變分量。在小變形的情況下,每個應力分量在相應的應變分量上作功,因此應變能和余能的表達式都包括六項:
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式中
xx、
yy、
zz、
xy、
xz、
yz為物體在載入過程中的應力分量;
xx、
yy、
zz、
xy、
xz、
yz分別為與上述應力分量相應的應變分量;積分上限的下標1表示載入終點。對於線彈性體則有:
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