CH.7. PLANE LINEAR ELASTICITY Continuum Mechanics Course (MMC) - - PowerPoint PPT Presentation

CH.7. PLANE LINEAR ELASTICITY Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Overview  Plane Linear Elasticity Theory  Plane Stress  Simplifying Hypothesis  Strain Field  Constitutive Equation  Displacement Field  The Linear Elastic Problem in Plane Stress  Examples  Plane Strain  Simplifying Hypothesis  Strain Field  Constitutive Equation  Stress Field  The Linear Elastic Problem in Plane Stress  Examples 2

Overview (cont’d)  The Plane Linear Elastic Problem  Governing Equations  Representative Curves  Isostatics or stress trajectories  Isoclines  Isobars  Shear lines  Others: isochromatics and isopachs  Photoelasticity 3

7.1 Plane Linear Elasticity Theory Ch.7. Plane Linear Elasticity 4

Introduction  A lineal elastic solid is subjected to body forces and prescribed traction: Initial actions:   b x ,0 t  0   t x ,0   b x , t Actions   t x through time: , t  The Linear Elastic problem is the set of equations that allow obtaining the evolution through time of the corresponding         displacements , strains and stresses . u x , t x , t x , t 5

Governing Equations  The Linear Elastic Problem is governed by the equations: 1. Cauchy’s Equation of Motion. Linear Momentum Balance Equation.     2 u x , t           x , t b x , t  0 0 2 t 2. Constitutive Equation. Isotropic Linear Elastic Constitutive Equation. This is a PDE system of     15 eqns -15 unknowns:        1 x , t Tr 2   u x , t 3 unknowns    6 unknowns 3. Geometrical Equation. x , t    Kinematic Compatibility. 6 unknowns x , t Which must be solved in 1                S x , t u x , t u u  3 the space. R R 2  6

Plane Linear Elasticity  For some problems, one of the principal stress directions is known a priori :  Due to particular geometries, loading and boundary conditions involved.  The elastic problem can be solved independently for this direction.  Setting the known direction as z , the elastic problem analysis is reduced to the x - y plane PLANE ELASTICITY  There are two main classes of plane linear elastic problems:  Plane stress REMARK  Plane strain The isothermal case will be studied here for the sake of simplicity. Generalization of the results obtained to thermo-elasticity is straight-forward. 7

7.2 Plane Stress Ch.7. Plane Linear Elasticity 8

Hypothesis on the Stress Tensor  Simplifying hypothesis of a plane stress linear elastic problem: 1. Only stresses “contained in the x - y plane” are not null     0   x xy         0 xy y xyz   0 0 0   2. The stress are independent of the z direction.      x y t , , x x      x y t , , REMARK y y      The name “plane stress” arises x y t , , xy xy from the fact that all (not null) stress are contained in the x - y plane. 9

Geometry and Actions in Plane Stress  These hypothesis are valid when:  The thickness is much smaller than the typical dimension associated to the  plane of analysis: e L        The actions , and are contained in the plane of * * u x , t b x , t t x , t analysis (in-plane actions) and independent of the third dimension, z .   is only non-zero on the *  t x , t contour of the body’s thickness: 10

Strain Field in Plane Stress  The strain field is obtained from the inverse Hooke’s Law:     1 1 2(1 )           x x y xy xy xy   E 2 E  1         1 Tr   1 1          E E 0  y y x xz xz  E 2 z 0      xz 0 1             0 0 yz z x y yz yx E 2      x x y t , ,  As   x    x y t , ,      y x y t , , y  And the strain tensor for plane stress is:   1   0   x xy 2            with  1           x y t , , 0 z x y  1 xy y 2      0 0    z  11

Constitutive equation in Plane Stress  Operating on the result yields: E            1 2(1 )               x   x y 2 2 1 x x y xy xy xy E E   E 1                     2 0   y y x y y x xz xz 2 1 E              E 2 0    z x y yz yx   E   xy xy plane 2 1  C stress          1 0 x x       E         1 0     y y 2   1         1     xy   xy 0 0     2       Constitutive equation     plane     stress C in plane stress (Voigt’s notation) 12

Displacement Field in Plane Stress  The displacement field is obtained from the geometric equations,        S . These are split into: x , t u x , t  Those which do not affect the displacement : u z  u      x x y t , , x x     u u x y t , , u Integration      y x x x y t , ,    y in . y u u x y t , , y y   u u        y x x y t , , 2   xy xy y x  Those in which appears: u z     u           z x y t , , ( , , ) x y t u x y z t ( , , , )    z x y z 1 z     u ( , ) x y u u   Contradiction !!!        x z z x y t , , 2 0     xz xz z x x        0   u z t ( , )    z u ( , ) x y u u            y z z x y t , , 2 0    yz yz z y y        0 13

The Lineal Elastic Problem in Plane Stress  The problem can be reduced to the two dimensions of the plane of analysis.  The unknowns are:         u x x               x          u x y t , ,   x y t , , x y t , , u y y       y       xy xy  The additional unknowns (with respect to the general problem) are either null, or independently obtained, or irrelevant: REMARK           0 z xz xz xz yz This is an ideal elastic problem because it          cannot be exactly reproduced as a particular   z x y 1 case of the 3D elastic problem. There is no   does not intervene   guarantee that the solution to and u x y t , , u x y z t , , , in the problem x   z will allow obtaining the solution to u x y t , , y   for the additional geometric eqns. u x y z t , , , z 14

Examples of Plane Stress Analysis  3D problems which are typically assimilated to a plane stress state are characterized by:  One of the body’s dimensions is significantly smaller than the other two.  The actions are contained in the plane formed by the two “large” dimensions. Slab loaded on Deep beam the mean plane 15

7.3 Plane Strain Ch.7. Plane Linear Elasticity 16

Hypothesis on the Displacement Field  Simplifying hypothesis of a plane strain linear elastic problem: 1. The displacement field is   u x      u u y     0 2. The displacement variables associated to the x - y plane are independent of the z direction.    u u x y t , , x x    u u x y t , , y y 17

Geometry and Actions in Plane Strain  These hypothesis are valid when:  The body being studied is generated by moving the plane of analysis along a generational line.        The actions , and are contained in the plane * * b x , t u x , t t x , t of analysis and independent of the third dimension, z .  In the central section, considered as the “analysis section” the following holds (approximately) true: u  0 z  u  x 0  z  u  y 0  z 18