连续介质力学_固体2
of the few people who knew how to successfully mix pleasure with business, without sacrificing either, and he had the extraordinary knock of carrying on his scientific thinking while apparently conducting other activities. He never compartmentalized his time, but he would disappear for an one hour from a party to note down an equation or work up a paper and then return to his guests to take up where he left off. 思考题:
p /3 / K V / V 1 2 3 1 2 2 1 E = G 3 1 2 3 1 2
作业:每个量的量纲
对材料弹性常数的要求或限制
E 1 K 0 1 2v 0 31 2v 2
连续介质力学中最为常用的 ij
数学力学大师庞加莱称康托尔的思想为感染着数学界的一种“烈性传染病 ”("grave disease" infecting the discipline of mathematics),克罗内克则形容康 托尔为一位“科学骗子 (scientific charlatan)”、“叛徒 (renegade)”和“年轻 人的腐蚀剂 (corrupter of youth)”. 记住马克思的名言吧:“在科学的入口处,正像在地狱的入口处一样,必须 提出这样的要求:‘这里必须根绝一切犹豫;这里任何怯懦都无济于事 .’” 对于第一个去征服科学某一领域征途上的险隘雄关的科学家来说,更需要有 超乎常人的勇气和毅力,有时甚至需要为其献身!
数学算子在连续介质力学中的应用:预热
雷诺输运定理是连续介质力学最重要的关系式之一:
d dv v dv v t dt
上式中, 为一标量场,而 v 则为速度矢量。 板书证明如下连续介质力学最为常用的关系式之一:
v div v v v divv v grad
连续介质力学中最为常用的 ij
克罗内克 (Leopold Kronecker, 1823~1891)是一个极其罕见的专 业数学家中的实业家 (businessmen),他在 30 岁以前就在经济 上实现了完全独立,换句话说,他可以不再为经济收入等担 忧,从而可以在经济上无忧无虑地进行数学研究. 克罗内克身 材不高 (约 1.52 米) 但结实,晚年还喜欢登山.
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力学大师冯· 卡门的作息时间
选自《冯· 卡门传》
力学大师冯· 卡门的作息时间
His schedule was incredible. He rose at six and breakfasted at seven or eight with a guest — sometimes with as many as twelve guests. He dictated correspondence until about noon and then glanced through samples of the vast numbers of books and papers which usually came to him from scientific workers from all over the world. Lunch, preceded by a strong whisky, was usually another business session, or a time for greeting visiting dignitaries or former students. He napped at three in the afternoon and usually arose at five to prepare for the evening engagement, which started typically with several Jack Daniels and ran through a large meal followed by afterdinner drinks until midnight. It was in the evening that von Ká rmá n was in his top form as he told stories, usually in the company of the prettiest young ladies present.
连续介质力学(固体)
Continuum Mechanics Mechanics of Continuous Media
第 4~6 讲
中国科学院大学工学院,雁栖湖校区 214,2018 秋季
赵亚溥
中国科学院力学研究所,非线性力学国家重点实验室
本讲大纲
绝大部分内容为板书,本 PPT 只起提纲挈领之作用 数学算子的一些重要应用 连续介质力学最核心的概念之一——应变的多种板书推导例子 作业:板大变形的 Fö ppl-von Ká rmá n 应变的推导 力学大师冯· 卡门在功成名就后的作息时间表
E G 0 1 v 0 1 21 v
1 因此,材料的 Poisson 比需要满足: 1 ν 2
极端情况,当 = 0.5 时, K
E 31 2 1/ 2
对应于不可压缩材料 ( = 0.5 )。
当然,材料的杨氏模量 E > 0.
根据年轻的天才可能会用脑太多而过于劳累的理论,数学是禁忌的 (Mathematics was taboo, on the theory that the young genius might overstrain himself by using his head)!老帕斯卡是一位极好的教练,但又是一位蹩脚的 心理学家. 老帕斯卡对小帕斯卡学习数学的禁令很自然地激发了小帕斯卡对 数 学 的 好 奇 心 (Pascal senior was an excellent drillmaster but a poor psychologist. His ban on mathematics naturally excited the boy’s curiosity).
少年的帕斯卡到底数学有多强?帕斯卡的姐姐吉尔伯特声称:她的弟弟 独自重新发现了欧几里得的前 32 个命题,而且是以欧几里得的同样次序发 现它们的 (Gilberte declared that her brother had rediscovered for himself the first thirty two propositions of Euclid, and that he had found them in the same order as that in which Euclid sets them forth). 帕斯卡还发明了世界上第一个机械计算器 (mechanical calculator),如下 图所示.
In 1848 he married his cousin Fanny Prausnitzer, and the couple had six children.
克罗内克曾对康托尔 (Georg Cantor, 1845~1918) 的集合论进行 了“猛烈地、恶毒地攻击”,从而使康托尔进入了疯人院. 康 托尔在 40 岁时第一次经历了精神崩溃,在其以后的漫长岁月 中,以不同的强度和周期发作. 克罗内克于 1891 年去世,享 年 69 岁. 康托尔则于 1918 年在精神病院去世,时年 73 岁, 他最后获得了自己应该享有的荣誉. 下图给出了科学辩论社针 对两个数学家之间斗争的场景.
帕斯卡正在进行桶裂实验
帕斯卡是一位杰出的英年早逝的数学家
可怜的帕斯卡在与他聪明的大脑一起继承了一个极差的体格 (But poor Blaise inherited a wretched physique along with his brilliant mind). 老帕斯卡又 是如何对神童小帕斯卡如何培养的呢?
在受拉杆中截出一个各边长为单位值的立方体, 其体积的相对变化,定义为体应变:
V 1 1 1 1 2 V 1
2
上式适用于小应变条件,即忽略掉高阶小量。 体 模 量 (bulk modulus) 定 义 为 静 水 压 强 (hydrostatic pressure, p = σ/3) 和体应变之比,则有:
帕斯卡是一位杰出的英年早逝的数学家
帕斯卡是一位杰出的英年早逝的数学家
应变能密度的表达式
u 1 1 2 2 2 2 2 2 2 x y z x y y z z x xy yz zx 2E 2G
If the principal stresses are used:
u
1 2 2 2 1 2 2 3 3 1 1 2 3 2 2E
where 1, 2, 3 = the principal stresses
Volume strain & bulk modulus
力学中应力的单位为什么是帕斯卡?
法国数学家布莱士· 帕斯卡 (Blaise Pascal, 1623~1662, 终年 39 岁) 于 1647~1648 年 间 , 研 究 了 受 限 的 不 可 压 缩 的 静 止 流 体 (confined, incompressible fluids at rest) 所产生的正应强 (normal pressure),创建了帕斯卡定律. 帕斯卡定律 (Pascal’s law):在受限静止液体的内部,在一个给 定的点,压强在所有方向上均相等 (In a confined fluid at rest, pressure acts equally in all directions at a given point).