# 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 1-D T(x) u(x) Heat flow, etc. Truss u(x) u(x) = v(x) F(x) Beam 2-D T(x,y) Heat flow, etc. u(x) = u(x,y) v(x,y) 2D - solid 3-D 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 z-directions)

9 Stresses Through the same point: t=s y make a cut in the xz-plane so that n = (0,1,0) call this traction vector t=s y Components of make a cut in the xy-plane 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 z-component of a traction vector on a surface with the normal vector in the y-direction

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 x-direction Equilibrium in y-direction Equilibrium in z-direction

16 Equilibrium equations ( analogy with heat balance ) Study the equation in x-direction 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 z-direction 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 xy-plane, 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 x-axis 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 z-axes 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 A-B dy=dz=0 and along A-C 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 yz-planes 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 z-direction

29 Plane Strain Displacements at plane strain The strains e zz = g xz = g yz =0

30 Summary stresses and strains, 3-dim Stresses Equilibrium Strains Kinematics

31 Summary stresses and strains, 2-dim 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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Math 263 Assignment #4 Solutions 1. Find and classify the critical points of each of the following functions: (a) f(x,y,z) = x 2 + yz x 2y z + 7 (c) f(x,y) = e x2 y 2 (1 e x2 ) (b) f(x,y) = (x + y) 3 (x