弹塑性力学复习思考题(一)

1、研究生弹塑性力学复习思考题1 .简答题：(1)什么是主平面、主应力、应力主方向？简述求一点主应力的步骤？(2) 什么是八面体及八面体上的剪应力和正应力有何其特点(3) 弹性本构关系和塑性本构关系的各自主要特点是什么？(4) 偏应力第二不变量 j2的物理意义是什么？(5) 什么是屈服面、屈服函数？ tresca屈服条件和mises屈服条件的几何 与物理意义是什么？(6) 什么是drucker公设？该公设有何作用？(能得出什么推论？)(7) 什么是增量理论？什么是全量理论？(8) 什么是单一曲线假定？(9) 什么是平面应力问题？什么是平面应变问题？在弹性范围内这两类问题之间有 和联系和区别？(10

2、)论述薄板小挠度弯曲理论的基本假定？二、计算题1、for the following state of stress, determine the principal stresses and directions and find the traction vector on a plane with unit normal n = (0,1,1)/ 72。311仃 j =102：1202、 in suitable units, the stress at a particular point in a solid is found to be一2determine the traction

3、 vector on a surface with unit normal (cos,sin 日,0) , where 日is a general angle in the range 0 w 日 w 冗。plot the variation of the magnitude of the traction vector tn as a function of 9 .3、利用应变协调条件检查其应变状态是否存在存在?d2 i i .m a 2 川n=- = -r q ddxkdxtn dx idxjjxj djtfcdx idxk(1) a=axy2, &=bx2y,%=0, a、b 为常数22

4、2，=k(x + y ),匹=ky , y = 2kxy k 为常数x2y2xz (2)输=y2zz2i2cxz z 54、the displacements in an elastic material are given by2m (1 一.)eixy,vm (1 .2ei22m(1-. ) / 2y (x2ei),w = 0where m, e, i, andl are constant parametersdetermine the corresponding strain and stress fields and show that this problem represents

5、the pure bending of a rectangular beam in thex,y plane.5、写出如下问题的边界条件(a)用直角坐标，(b)用极坐标6、express all boundary conditions for each of the problems 川ustrated in the following figure.7、show that the following stress components satisfy the equations of equilibrium ilh zero body forces, bul are noi the solu

6、tion to 口 problem iti elasticily:% =出2 +可/-/)%=+出工一目s = r(.x2 +j% = - 2iyxyt2 = ta = 0, c = constaiii 丰 0show that the airy function 0 = xy j + j-v2 solves the following cantilever beam problem, as qi。中 n in the following figute. as ilsuji for such problems, boundary condilioiis al (tie ends (j = 0

7、and l should be hnnulaied only in tenm of the resultant force na/f/rr. while ai v = r (he exact points te specification should he used. for (he cde *itha q corn pai e ihe elaslicity stress field wilh the cone spending insults ftotn strength of materials theory.8、pthe cantilever he an】 show n in the

8、figure is subjected io a diribuied sheai stress7 on the upper face. the following airy stress fijnetion is proposed for this ptoblern0 = 口9+ cy , + 门 y + qy5 + iyr + 4&白1 4-门的？ + 匚,./determine ihe constants 圻 and find the s【甑 distribution in the beant use resultant force boundary conditions a( the e

9、nch. ans: / qc/ini. rj小，/24fl )9、4、正方形薄板三边固定，另一边承受法向压力p = -posing，如图所示，设位移函数为x . ?u -0v = a2 sin一 sin一b 2b利用ritz法求位移近似解（泊松比卡0）。（第4题图）（第6题图）5、悬臂梁在自由端受集中力 p作用，如图所示。试用极小势能原理求最大挠度vy第5题图提示设梁的挠曲线为23w = a2xa3x6、对给定的应力函数：（1）51 =ax3y2,52 =bx2y2,53 =cxy3,试确定它们哪个能作为平面问题的应力函数,分析它们能解什么问题?3_x0,-cy yy c 区=ax3 bx23

10、fxy3po（2）证明中=xy -y2+y2可以作为应力函数，并求在区域4c3c24c域内的应力分量，并分析该应力函数可以解决那类平面问题。7.如图所示矩形截面柱承受偏心载荷作用，且不计其重量，若应力函数 试求:（1）应力分量；（2）应变分量；（3）假设d点不移动，且该点处截面内线单元不能转动；:yx=s, y =0=0 ）,求位移分量，j 广8、图示三角形截面梁只受重力作用，梁的质量密度为p,宽度为1,试用纯三次应力函数求解各应力分梁。9 .如图所示的楔形体两侧面上受有均布切向载荷q,试求其应力分量。p和扭矩t作用，写出10 .已知一圆形薄管，平均半径为a,厚度为t,在薄管的两端受有拉力 管内一点处的tresca屈服条件和mises屈服条件表达式。11 .如图所示的矩形薄板 oabc , oa边与bc边为简支边，oc边与ab边为自由边。板不受横向荷载，但在两个简支边上受大小相等而方向相反的均布弯矩mo试证，为了将薄板弯成柱面，即w=f(x),必须在自由边上施加以均布弯矩m。并求挠度和反力。12 .如