高等土力学——No.9 Advanced Soil Mechanics
主讲老师：郭莹
土木工程学院岩土工程研究所
3. 6土的强度理论
3.6.2 土的经典强度理论
1. 特雷斯卡（Tresca）准则及其广义准则
2. 米泽斯（Von Mises）准则及其广义准则
3. 莫尔－库仑（Mohr-Coulomb）强度准则
4. 三个强度准则的讨论
1. 特雷斯卡（Tresca ）准则与广义特雷斯卡（Extended Tresca ）准则——单剪应力
132k
σσ?=0
2πsin 2=???????
+k J θ0
212πsin 12=???????
?
+I k J αθ()1
231I
k ασσ+=?广义形式
α、I 1反映平均
主应力影响
金属材料
或或
πsin =?????
+k J θ六棱柱的表面：
π平面上的特雷斯卡与米泽斯准则两个破坏面交点，数学处理时有困难
锥面——广义六棱柱面——特
雷斯卡
2. 米泽斯（Von Mises ）和广义米泽斯（extended Von Mises ）准则——三剪应力
()()()
2
2
132
322
216k
=?+?+?σσσσσσ2
2k
J =k
J =2k
q 3=oct
23
k
τ=或
或
12=??k I J α033
3
=??k p q α广义米泽斯——Drucker-Prager 准则
α、I 1反映平均
主应力影响
σ1
σ3
σ2
圆锥面——广义米泽斯准则
圆柱面——米泽斯准则
圆形应用起来更方便
3. 莫尔－库仑强度准则——单剪切角
()
f f τσ=莫尔（Mohr ）单值函数
13
13sin 2c tan c σσ?σσ??=++f tan c τσ?
=+在一定的应力范围，线性关系－库仑公式
莫尔－库仑强度准则
（二维应力状态）
0cos cos sin sin 31
sin 321=+??
????+??θ?θ?c J I 0
cos cos sin sin 3131sin =+??
?
???+??θ?θ?c q p 莫尔－库仑强度准则的应力不变量表达式
三轴平面
莫尔－库仑强度准则的破坏面与破坏轨迹
三维空间
π平面
非规则六面体
非规则六边形：拉压不等
4. Tresca、Mises和Mohr-Coulomb
三个强度准则的讨论
1）Tresca和Mises都没有考虑平均主应力对土的抗剪强度的影响，不能反映土的摩擦特性；2）广义形式考虑了p，但没有考虑破坏面上正应力的影响；
3）Tresca准则是最大剪应力准则；Mises是最大八面体剪应力准则，两者与土的摩擦强度不同；4）三轴压缩和伸长试验，用Tresca和Mises（拉压相等）预测的强度相同，显然与实际不符；
三个准则在常规三轴压缩试验测得抗剪强度相等。
（θ=-30o）
三维应力空间及π平面（I
或p为常数）
1
π平面
用广义米泽斯及广义特雷斯卡准则预测伸长试验，得出不合实际的结果
o
36.9?′≥（θ= -30o ）
（θ=30o ）
砂土
用广义米泽斯及广义特雷斯卡准则，将有一个主应力为拉应力（<0）——不可能
Mohr-Coulomb强度准则反映了剪切面上剪应力与正应力的关系以及土作为散粒体的摩擦特性，是比较合理的，在土力学中得到了广泛的应用；
存在问题：
1）与中主应力无关；
2）包线假定线性，即?与围压无关；
应力水平很大时引起较大误差；
3）因为破坏面非光滑连接，数值计算不够方便.
广义Mises在应力空间的曲面和π平面的轨迹光滑，数值计算方便；为避免圆半径过大与实际不符，有时采用伸长圆或折中圆。
π平面－
圆
规则六边形不规则六边形
以内切圆来替代
内切圆
广义米泽斯三轴压缩试验
三轴伸长试验
－
不同破坏准则在π平面上的轨迹
3. 6土的强度理论
3.6.3 近代的强度理论
1.莱特－邓肯(Lade-Duncan)破坏准则
2.松冈元-中井照夫（Matsuoka-Nakai）
破坏准则
3.双剪应力强度理论
4.隐式的破坏准则.
本构关系－包括应力应变与强度关系
土的强度，或者破坏是其应力应变过程的最后阶段，即在微小的应力增量下，会产生很大（或者不可控制）的应变增量。因而破坏是应力应变关系的最后阶段。——现代强度理论基本思想。
1.莱特－邓肯(Lade-Duncan)破坏准则
适用于砂土的弹塑性模型
不相关联的流动法则
屈服面、塑性势面、破坏面性状一致
北京城市学院 2014-2015学年第一学期 《大学英语3》期中口试阅读部分试题 （学生版）
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1 How do you feel about trying new things? I like trying new things because it enables me to experience more and learn more about others, by which I can become more liberal and less biased. they often let me down. Trying new things I don’t like trying new things because sometimes means venture or even danger, so I don’t like trying new things and would like to stick to the familiar 2 What stops you from trying new things? Trying new things may have too much demand on me. Furthermore, trying new things may bring me disappointment or even failure. 3 What have you achieved in your life that makes you feel proud? I take great pride in winning first place in the speech contest held in my university. 4 Who do you admire for their achievements and why? I admire Winston Churchill, who led his country from the verge of defeat to victory. 1Do you believe that a good memory is critical to success? Why or why not? Yes, I firmly believe that a good memory determines a person’s success. Firstly, a good memory is a great help in the process of learning. Secondly, a good memory can help you build relationships, which is also a decisive factor for success. Thirdly and most importantly, a good memory can provide you with quick, reliable access to essential thinking tools. No, I don’t think a good memory is critical to success.In my opinion, it is diligence, creativity and persistence that eventually determine one’s success. 2 Have you ever tried any method to improve your memory? Do you think that useful? Yes, I have tried some methods to improve my memory. For example, I learned the method of location from a book and I liked it very much. Yes, I once believed that there might be some special shortcuts to improving my memory. But when I tried a couple of methods, I felt it was inconvenient to apply them and it is a waste of time. I think I would never try any method again
大连理工大学2013量子力学真题（完整版） 一．简答题 1已知[A,B]=iC,问AB可否同时有确定值？为什么 2已知[A,B]=C,C为常数，问AB可否同时有确定值，为什么 3 0xxx 位置，动量的不确定度 4写出泡利矩阵及其对易关系 5在量子力学中问什么说静止的粒子不存在？ 6写出坐标，动量的不确定度并解释其物理意义 7试估算一维谐振子的零点能 8举例说明，AB同时与C対易，但AB可能不対易（ABC都是算符） 9为什么幺正变换不改变力学量的本征值？ 10AB均为厄米算符，证C=-i[A,B]也是厄米算符 11用全同性原理解释泡利不相容原理 12一个混态能否幺正演化为一个纯态？为什么？二.计算题 1一电子处于自旋态|Ψ〉= 2 1 ﹛|↑x〉＋|↓x〉﹜求在自旋态下，XS的可能测值及相应几率在自旋态下，ZS的可能测值及几率 2设一个质量为m的粒子束缚在势场V(x)中做一维运动，其能量本征值和本证函数分别为nE ，n ，n=1,2……，求证n)0(mx dnm 3证明，两个厄米算符满足关系 iBA , 算符A、B将满足Heisenberg不确定关系 4磁矩为 =的电子置于方向为)cos,sinsin,cos(sin n的均匀磁场 nBB0中求该系统的能量本征值和本征态 设t=0时刻，电子的自旋处于 01态，求任意时刻粒子处于xS=2 的几率 5在阱宽为a的一维无限深方势阱内放入两个质量均为m的无自旋粒子，两个粒子之间相互作用势)(),(202xxvxxv （1）写出不考虑两个粒子的相互作用时，体系的H，能量本征值和本证函数。 （2）试求在微扰 210XXVH 的作用下基态能量的一级修正值
8.1 复习笔记 一、氢原子光谱与Bohr 理论 1．氢原子光谱 氢原子光谱是人们认识原子结构的实验基础，原子光谱是线状光谱。 每种元素的原子辐射都具有由一定频率成分构成的特征光谱，是一条条离散的谱线，称为线状光谱。 每一种元素都有各自不同的原子光谱。氢原子光谱的频率的经验公式：，n=3，4，5，615122113.28910()s 2v n -=?-2．Bohr 理论 Bohr 理论（三点假设）： （1）核外电子只能在有确定半径和能量的轨道上运动，且不辐射能量； （2）通常，电子处在离核最近的轨道上，能量最低——基态；原子获得能量后，电子被激发到高能量轨道上，原子处于激发态； （3）从激发态回到基态释放光能，光的频率取决于轨道间的能量差。 氢原子光谱中各能级间的能量关系式为： 21 h E E ν=-氢原子能级图如图8-1所示。
图8-1 能级间能量差为 H 2212 11 (E R n n ?=-式中，R H 为Rydberg 常数，其值为2.179×10-18 J 。 当时，，即氢原子的电离能。 121n n ==∞或182.17910J E -?=?二、微观粒子运动的基本特征 1．波粒二象性 微观粒子具有粒子和光的特性，即具有波粒二象性。 微观粒子的波长为： h h mv p λ==式中，m 为实物粒子的质量；v 为粒子的运动速度；p 为动量。
2．不确定原理 Heisenberg 不确定原理： 2h x p π ???≥ 式中，Δx 为微观粒子位置的测量偏差；Δp 为微观粒子的动量偏差。 微观粒子的运动不遵循经典力学的规律。 微观粒子的波动性是大量微粒运动表现出来的性质，即具有统计意义的概率波。 三、氢原子结构的量子力学描述 1．薛定谔方程与波函数 式中，ψ为量子力学中描述核外电子在空间运动的数学函数式，即原子轨道；E 为轨道能量（动能与势能总和）；V 为势能；m 为微粒质量；h 为普朗克常数；x ，y ，z 为微粒的空间坐标。 2．量子数 主量子数n ：n =1，2，3…正整数，它决定电子离核的远近和能级。 角量子数l ：l =0，1，2，3…，（n -1），以s ，p ，d ，f 对应的能级表示亚层，它决定原子轨道或电子云的形状。n 确定后，l 可取n 个数值。 磁量子数m ：原子轨道在空间的不同取向。在给定角量子数l 的条件下， m =0，±1，±2，±3…，±l ，一种取向相当于一个轨道，共可取2l +1个数值。m 值反映
大工18秋《大学英语3》在线测试1答案 (单选题) 1: The wild flowers looked like a soft orange blanket _____ the desert. A: covered B: covering C: cover D: to cover 正确答案: B (单选题) 2: The poor police had never ____ of winning. A: stand a chance B: get a chance C: have a chance D: stood a chance 正确答案: D (单选题) 3: We have only one vacancy, but so far have received more than 30 () for the job. A: applications B: applicants C: applicators D: appliances 正确答案: A (单选题) 4: I was surprised by the ______ of so many people at the meeting. A: example B: record C: goal D: presence 正确答案: D (单选题) 5: It's important for you to make a list of your tasks in order of _______ when you work on a project. A: suggestion B: goal C: priority D: tip
大工15秋《大学英语4》在线测试1 一、单选题（共 20 道试题，共 80 分。） 1. You don't have to be in such a hurry. I would rather you _________ on business first. A. would go B. will go C. went D. have gone 正确答案 ：C 2. Who would you rather ______ with you, George or me? A. going B. to go C. have gone D. went 正确答案 ：D 3. You would be _____ a risk to let your child go to school by himself. A. omitting B. attaching C. affording D. running 正确答案 ：D 4. When the railway is completed, we _________ get to town much easily. A. must B. would
C. are able to D. will be able to 正确答案 ：D 5. It’s cold outside, you'd better _________ your coat. A. wear B. dress C. put on D. in 正确答案 ：C 6. Jane’s dress is similar in design __________ her sister’s. A. like B. with C. to D. as 正确答案 ：C 7. She __________ the man as her attacker and reported him immediately to the police nearby. A. knew B. realized C. identified D. regarded 正确答案 ：C
2011年 数理 第一大题：10个小题 б函数的定义和用法，格林函数法求解步骤，极坐标系下柯西黎曼条件证明，简单的洛朗展开的计算，勒让德函数自然边界条件以及本征值本证函数，用拉普拉斯变换把一个数学物理方程变换式变换出来（含边界和初始条件） 二：写出贝塞尔函数的母函数，并由此推导出贝塞尔函数的递推公式（这个公式就是书上的一个公式，我暂时想不出来了） 三：两道计算题： 第一是用留数定理计算积分（好像是第二种情况）；第二是用柯西积分公式计算积分。 四：稳定场方程在指定条件下的求解（边界是其次的）。 五：球函数的应用题，很常规的，跟ΘΦ有关。 六：利用傅里叶变换求解半无界区域的数理方程。 量子 一.空间自由粒子t=0时候波函数为ψ（0）=coskx 1、求任意时间的波函数表达式 2、求任意时刻的动量可能值和相应的概率 二.设一维无限深势阱中运动粒子的波函数为ψ（x）=4/√a sin（πx/a）cos2（πx/a），求在此任意态下，粒子能量的可能值和相应的概率练习册p40 三.
四.求证：P×L+L×P=2ihP p50 五.求氢原子1s电子的动能，势能的平均值。（1s的波函数给出）练习册p87 六.求在Sz的本征态I↑z>=﹙10）下，求σ?n的可能值及相应几率p110 七.有一量子态体系，其hamilton量为Ho，并已知Ho的本征值和本证函数分别为En和ψn，（n=1,2,3…..）.在初始时刻t=0，体系处于ψo态，当t>0时体系开始受到一微扰H′=F（x）exp（-βt）的作用。在一级近似下求 1、经过充分长的时间后，体系跃迁到ψn的几率 2、如果该体系为一维谐振子，且F（x）=x，结果将如何？P164
【奥鹏】[大连理工大学]大工18春《大学英语4》在线测试1 试卷总分:100 得分:100 第1题,You are late. If you ()a few minutes earlier, you ()him. A、come; would meet B、had come; would have met C、come; will meet D、had come; would meet 正确答案:B 第2题,Common interests have formed a _______ between us, which may bring about cooperation with each other in more fields. A、trade B、bond C、unit D、fantasy 正确答案:B 第3题,Seeing their teacher ()into the classroom, they stopped __()at once. A、walk; telling B、entering; to speak C、enter; to tell D、walking; talking 正确答案:D
第4题,You _______ such a serious mistake if you had followed his advice. A、may not make B、might not make C、shouldn't have made D、might not have made 正确答案:D 第5题,I would never have _____ a court of law if I hadn't been so desperate. A、sought for B、accounted for C、turned up D、resorted to 正确答案:D 第6题,Mark's latest work ________ the learning theories of the 1980s as well as his own experience in teaching English to foreigners. A、looked out B、left out C、took place D、drew on 正确答案:D