12. Stresses and Strains


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1 12. Stresses and Strains
2 Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM  Formulation
3 Classification of Problems Scalar Vector 1D T(x) u(x) Heat flow, etc. Truss u(x) u(x) = v(x) F(x) Beam 2D T(x,y) Heat flow, etc. u(x) = u(x,y) v(x,y) 2D  solid 3D T(x,y,z) Heat flow, etc. u(x) = u(x,y,z) v(x,y,z) w(x,y,z) 3D  solid
4 Fundamental Equations Heat flow and Elasticity Flux vector q n Balance Heat source Q Heat flow problem Constitutive law Gradient T Gradient Differential eq. Temperature T Elasticity problem Stresses s Constitutive law Equilibrium Body force b Differential eq. Strains e Kinematics (displacement gradients) Displacement u
5 Forces acting on solid bodies  two types of forces Body force: b(x,y,z) [N/m 3 ], (Internal forces: e.g. Gravity, accelerations) Surface force: t(x,y,z) [N/m 2 ], (Tractions, on surfaces and on cuts of the body ) ),, ( ),, ( ),, ( z y x b z y x b z y x b z y x b ),, ( ),, ( ),, ( z y x t z y x t z y x t z y x t
6 Stresses  Internal surface forces in bodies
7 Traction vector Traction vector: Surface force per unit area External forces acting on solid body Traction vector z y x t t t t 0 A, A F t [ N/m 2 ]
8 Stresses Make a cut through a point of a body in the yzplane so that n = (1,0,0) In the point, there will be a traction force vector Call this traction vector t=s x n n = (1,0,0) t=s x with components (Components of t in x, y and zdirections)
9 Stresses Through the same point: t=s y make a cut in the xzplane so that n = (0,1,0) call this traction vector t=s y Components of make a cut in the xyplane so that n = (0,0,1) t=s z Call this traction vector t=s z Components of
10 Stress tensor Collect s x, s y and s z in the stress tensor S: S contains all the stress components acting in a point s xx, s yy, s zz are normal stresses s xy, s xz, s yx, s yz, s zx, s zy are shear stresses
11 Stress tensor Notation of stress components: Example: s yz is the zcomponent of a traction vector on a surface with the normal vector in the ydirection
12 Properties of the stress tensor Moment equilibrium about an axis through E: i.e. letting dx, dy and dz => 0 also and => 0 Study the other planes and we can conclude and we have
13 Coordinate Transformation Reminder: Temperature flux on boundary Analogy with tractions (but now three) or as That may be written s s nn nm t t T T n m n T n Sn T Sm
14 Equilibrium equations Elasticity problem Stresses s Constitutive law Strains e Equilibrium Kinematics (displacement gradients) Body force b Differential eq. Displacement u
15 Equilibrium equations Equilibrium requires that (cf. heat flow) Which comprises three equations Equilibrium in xdirection Equilibrium in ydirection Equilibrium in zdirection
16 Equilibrium equations ( analogy with heat balance ) Study the equation in xdirection first: Insert the definition: Rewrite first term using Gauss divergence theorem Which should hold for arbitrary regions Writing out the equation
17 Equilibrium equations Treating the equations in y and zdirection in the same manner results in Equilibrium, 3 equations Defining the following matrices The equilibrium equations may be written as
18 Plane Stress Thin bodies. All forces and stresses are located in the plane
19 Plane Stress Plane stress state: Stress only in the xyplane, s zz =s xz =s yz =0 Stress tensor Traction vector Equilibrium equations
20 Strains  dimensionless measure of deformation
21 Displacement from A to A Strains (x+dx,y+dy,z+dz) B A (x,y,z) u+du B A (x+u x,y+u y,z+u z ) with the chain rule du is written ( since u=u(x,y,z) )
22 Normal strains Consider now a line AB, parallel to the xaxis The length from A to B is The length from A to B is or
23 Normal strains Using that Since dy=dz=0 we get: and Assuming small strains where we get and
24 Normal strains We now have that Inserting in the definition of strains Treating lines along y and zaxes in the same manner we find the normal strains as
25 Shear strains Evaluate changes of angle Ignore changes in length The angles may be written Along AB dy=dz=0 and along AC dx=dz=0, then For small angles we may assume sin(q)q
26 Shear strains The total angle change is the sum Which is the measure of shear strain Treating angle changes in the xz and yzplanes in the same manner we find the shear strains as
27 Normal strains Strains  summary Shear strains Defining the matrices: The strains may be written:
28 Plane strain No forces or displacement in the zdirection
29 Plane Strain Displacements at plane strain The strains e zz = g xz = g yz =0
30 Summary stresses and strains, 3dim Stresses Equilibrium Strains Kinematics
31 Summary stresses and strains, 2dim Plane stress Plane strain Note! e zz 0 Note! s zz 0
32 Fundamental Equations Elasticity Stresses s Equilibrium Body force b? Constitutive law Differential eq.? Strains e Kinematics Displacement u
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