−
qh3 12
σy
=
− qh3
6
τxy
=
− qx h2
4
左表面：即当 x=0 时，载荷分布为：
σx
=
− 2qy3
3
σy
=
2qy3 3
Hale Waihona Puke −qy h2 4−
qh3 12
τxy = 0
右表面：即当 x=l 时，载荷分布为：
σx
=
ql2y
−
2qy3 3
σy
=
2qy3 3
−
qy h2 4
−
qh3 12
τxy
=
−qly2
Equation
physics equation: ε =(σx-μσy)/E , εy=(σy-μσx)/E , γxy=τ xy/G
x
x

u x
Geometric equation:
y

v y
（2 3）
xy

v x

u y
Solutions: displacement method(位秱法)，force method (力法)，hybrid method(混 合法)。
解：1km 处静水压力为：p = ρhA = 10 × 103 × 1 × 103 = 1 × 107Pa
叏向里为负，主应力为：σxx =σyy =σzz= − p = −1 × 107Pa 第一应力丌发量：I1=σxx +σyy + σzz = − 3p = −3 × 107Pa
根据各项同性条件下的广义胡克定律得体积应发：
1 − 2μ 1 − 2 × 0.3 e = E I1 = 210 × 106 ×
−3 × 107
= −0.05714
体积减小到：
V′ = 1 + e V = 1 − 0.05714 × 1 = 0.94286m3
根据广义胡克定律得每边应发：
ε=
εxx
= σxx
− μ（σyy E
+
σzz
）
=
e 3
=
[3] 杨桂通.弹塑性力学引论.2010.清华大学出版社.
第二次作业
1.Plane stress problem :concept,equation,solutions.
Concept ：Plane stress is a stress distribution where in all compoents in some one direction are zero.This direction is usually taken as the z direction.Hence,plane stress requires that σz=τ zx=τ zy=0.
Mechanical strength is the capability of mechanical compenents or system carry external loads without failure. It usually is expressed as bending strength, tensile strength, compressive strength, impact strength, etc.
the integral under the condition of simple loading. 全量理论：增量理论在简单加载条件下的积分得应力-应发之间的关系。
parison of total quantity theory and incremental theory 全量理
论与增量理论的比较
+ ql h2
4
参考文献 [1] 陈明祥.弹塑性力学.2010.科学出版社. [2] 杨桂通.弹塑性力学引论.2010.清华大学出版社.
第三次作业
题目：一长宽高均为 1m 的物体落入水中，沉入深度为 1km 的水底。水的相 对密度为 10kPa/������������，该物体的弹性模量为 210Mpa，泊松比为 0.3。试求该 物体落水后体积减少到多少，每边的长度减少多少？
Relations :Mechanical strength is the basic of mechanical design. We should think about all kinds of factors in mechanical design, such as strength,rigidity, heat media, stress states, vibration and work condition, and mechanical strength is a necessary factor that we must think about.
But under the condition of small deformation and simple loading, the two theories are the same, then the total quantity relationship can be derived from the incremental relationship. In the case of general loading, the method of incremental theory is reasonable. And under simple loading or similar with this situation, total theory is also available, especially due to the total theory
2. 解：由于下端是自由端，以得到等式： 当 y=-h/2 时，σy=τ xy=0.解得：C1= qh2/4；C2=- qh3/12
则，
σx
=
qx2y −
2qy3 3
σy
=
2qy3 3
−
qy h2 4
−
qh3 12
τxy
=
−qxy2
+ qx h2
4
上表面：即当 y= h 时，载荷分布为：
2
σx
=
qx2h 2
The incremental theory is a theory that describes the relationship between stress and strain increment or strain rate when the material is in a plastic state.
增量理论是描述材料处于塑性状态时，应力不应发增量戒应发速率之间关 系的理论。
2.Stress-strain total quantity theory 应力应发全量理论 Incremental theory： The relationship between the stress and strain of
Q:A body with dimension of 1m*1m*1m in length . height and width respectively is sunken 1km depth in water.The relative density of water is10kPa/m3.The elastic modulus is E=210Mpa. And possion’s ratio is 0.3. Decide the volume and each side length of the body after it sank in the water.
参考文献
[1] /link?url=eOiNpsY5rblAXzs4BPCv05VVjnZOBD1UG4lCO0MQ3EV5YexFS2WPph9Ze30joGjeDH4it_rfTfjTcfNFwDHXZ_Bs780dr8KYq7beOkIn_
[2] 陈明祥.弹塑性力学.2010.科学出版社.
In the process of loading, the ultimate strain state is not only dependent on the ultimate stress, but also on the path of the strain. According to the theory, the total strain is determined by the ultimate stress, and the solution of the two theories is not consistent with the strain path. Especially in the neutral condition, the difference between the two is the most obvious. According to the experimental observation, the neutral variable load does not produce plastic strain change, incremental theory reflects the this characteristic, and according to the theory, as long as the stress components change, plastic strain was about to change.
机械设计是根据机械的工作原理、结构、运动方式、力和能量的传递方式、 各个零件的材料和形状尺寸、润滑方法等进行构思、分析和计算机械零件和系 统，以作为制造依据的工作过程。
机械强度是机械零件或系统不失效的情况下所能承受的最大。一般用抗弯 （抗折）强度、抗拉（抗张）强度、抗压强度、抗冲击强度等来表示。