Chapter_2高等材料力学课件.ppt

* 2.5 Solution (Continued) Equilibrium yields: Combine these two deductions, we can infer that: where A is a constant and F(y) is an unknown function of y. As a simple trial, let F(y) = By (B is a constant). Then: where A, B are constants. * 2.5 Solution (Continued) * 2.5 Solution (Continued) The additional term Ay3 in ?x creates an additional Ay3/E term in ?x, consequently causes an Axy3/E term in u(x, y) and an 3Axy2/E term in ?u/?y. Thus: It can be shown that this is still NOT the exact solution, but is a quite accurate one. * 2.5 Compatibility (III) Considering the xy plane only, Compatibility can be defined rigorously as following: Recall the strain-displacement equations: (2.5-2a) * 2.5 Compatibility Equations It is called the compatibility equation. There are totally 6 compatibility equations, given by: * 2.5 Compatibility Equations The first three equations are of in-plane dependence. The last three equations are of out-of-plane dependence. These compatibility equations provide the check on whether a given displacement field is compatible or not. * 2.5 Example 2.5-4 Example 2.5-4 Show that the strains given by Eqs.(d) to (f) of Example 2.5-2 do not satisfy the compatibility equations. Solution: * 2.5 Solution to Example 2.5-4 That is to say, the compatibility is only valid at y=0, i.e., along the centroidal axis. * Homework Problem: 2.3 2.8 2.13 2.23 2.40 2.41 2.42 2.48 * * * * * * * * * 2.3 Stress-Strain Relations (IV) * The 6?6 Compliance Matrix can be shown to be symmetric where c12=c21, c13=c31, c23=c32, …, etc. This is called Maxwell’s Reciprocity Theorem. Thus, in general (except when cross-shear stresses are unequal), there are at most 21 possible independent elastic constants. If a material has one plane of symmetry, there can be no interaction between the out-of-plane shear stresses and the remaining strains. If the xy plane is assumed to be a plane of symmetry, then Eq.(2.3-2) becomes: 2.3 Stress-Strain Relations (V) * Now the