应力应变关系:
弹性模量 || 广义虎克定律 1.弹性模量
a 弹性模量 单向拉伸或压缩时正应力与线应变之比,即
E?? ?
b 切变模量 切应力与相应的切应变 之比,即
G?
?
? c 体积弹性模量 三向平均应力
(?x??y??z)
?0?3
与体积应变θ(=εx+εy+εz)之比, 即
?0
K??
d 泊松比 单向正应力引起的横向线应变ε1的绝对值与轴向线应变ε的绝对值之比,即
???1
? 2.广义虎克定律 a.弹性力学基本方程
在弹性力学一般问题中,需要确定15个未知量,即6个应力分量,6个应变分量和3个位移分量。这15个未知量可由15个线性方程确定,即 (1)3个平衡方程(或用脚标形式简)写 为:
??ij?f??2uii?(?
?xj?t2)?0
(i,j?x,y,z)
(2)6个变形几何方程,或简写为:
E1?uiij?
2(?x??uj)j?xi
(i,j?x,y,z)
(3)6个物性方程简写为: 1
E?2G?3?ij?E?0?ij
?ij?2G?ij????ij
(i,j?x,y,z)
(?ij??1(i?j)0(i?j))
2.边界条件
Fx??xlxx??xylxy??xzlxz
Fy??yzlxx??ylxy??yzlxz
Fz??zzlxx??xylxy??zlxz
式中,lnj=cos(n,j)为边界上一点的外
法线n对j轴的方向余弦 b 位移边界问题
在边界Sx上给定的几何边界条件为
ux?u*x u?u*y
uz?u*yz 式中,ui为表面上给定的位移分量
Cauchy公式: T x = σ xl + τ xym +τ zx n T y = τ xy l+σ y m +τ zy n T y =τ xz l+τ y zm +σ z n
?n?T(n)gn?Txl?Tym?TznT(n)?T22x?T2y?Tz ?2n?T(n)??2n边界条件:
(l?x?m?xy?n?xz)s?Tx(l?xy?m?y?n?yz)s?Ty (l?xz?m?yz?n?z)s?Tz平衡微分方程:
??x?????x?yx?y???zx?z?Fx?0xy??y???x??y?zy?z?Fy?0 ??xz?x???yz??z?y??z?Fz?0 主应力、不变量,偏应力不变量
?3?I1?2?I2???I??3?0I1x??y?I?zx?xy?y?yz?z?zx2?????yxy?zy?z?xz???x x?xyxzI3??yx?y?yz?zx?zy?z?1m?3(?1??2??3);si??i??m
J1?s1?s2?J?1s3?0?6????x??2y????y??2z????z??2x??6(?2222xy??yz??zx)???J3?偏应力张量行列式的秩八面体
??18(?1??2??3 ?13)28?3(?1??2)2?(?2??3)2?(?3??1)2?3J2等效应力??3J2 体积应变???x??y??z
?1?2vv?E(?1??2??3)
几何方程:
??ux??x;???u?vxy?y??x??v?v?w y??y;?yz??z??y???w?u?wz?z;?xy??z??x?1ij?2?ij
变形协调方程?2?22x??y??xy?y2??x2??xy
物理方程
??12(1?vxE???x?v??y??z??)?;?xy??xy?1y?E???y?v??x??z???;?2(1E?v)yz??yz ?12(1E?z?E???z?v??y??x???;?v)zx?E?zx
偏应力与偏应变的关系 ?m?3K?m;sij?2Geij
平面应变问题
?11x?E'??x?v'?y??1?v???1?v??x?v?y???1'1y?E'??y?v?x?????1?v??y?v?x??2(1?v')2(11??v?vE'xy?)xy??E?xy;?z?0 E'?E'v?1??v2;v??1?vzyzx??zy??zx?0;?z?v??x??y?平面应力问题
?1E??;?1x??x?v?yy???y?v?x??2(1?v)E?xy?E?xyzy??zx??zy??zx?0?1z??v??x??y?;?z?0平面问题方程: 平衡方程:
??x?????yx?Fx?0?x?y xy?x???y?y?Fy?0
几何方程
??v?u?vx??u?x;?y??y;?xy??y??x 边界条件
l?x?m?yx?Tx;l?xy?m?y?Ty
位移边界条件ux?ux;uy?uy
协调方程 2平面应变
??x??2?y?2?xy?y2?x2??xy
?2?2平面应力z??z?2?z?x2?0;?y2?0;?xy?0
平面问题应力解(直角坐标系)
??2?x??y2?Fxx???2?y?x2?Fyy
???2?xy??xy协调方程:
?2?22?2?x?y)??(?2(2?2?x2??y2)(?x??y)?0 平面问题应力解(极坐标系) 平衡微分方程:
??r1???r?r??????r?r1?????r?Fr?0 r??2??r?r???r?r?F??0几何方程:
??urr??;?u1?u???r?r??1r?u?ur?r?? r?u?r????r?r本构方程:
?1r?E??v?1r???;???E????v?r???r?2(1?v) E??r变形协调:(?21?1?22?r2?r?r?r2??2)?0
已知应力函数?,求应力
??1??1?2r?r??2??2;??2?r???r22r?1??1????1?? r???r2?r???r2?????r(r??)平面应变下:
?Er??1?u??1?2u??(1?u)?r?u??? ?E???1?u??1?2u??(1?u)???u?r?
屈服条件
Tresca屈服条件
f???2ij???1?2?k1?0单轴拉伸:k? s1?2;纯剪切:k1??sMises屈服条件
f??ij??J2?k22?0J?1?226????x??y????y??z????z??2x??6(?2222xy??yz??zx)???单轴拉伸:K12?3?s;纯剪切:K2??s
1、理想弹塑性材料的加卸载准则:
f???fij??0,df???d?ij?0;加载f??df??fij
ij??0,??d?ij?0;卸载ij2、硬化材料的加卸载准则:
f???fij,????0,??d?ij?0;加载f???fij ij,????0,??d?ij?0;中性加载f???fijij,????0,??d?ij?0;卸载ij
(完整版)弹塑性力学公式



