MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
Preliminary Math Base Vectors Describe the coordinates system in 3D space. Base Vectors are Orthogonal to each other. (i.e. the normal of the plane formed by two base vectors is always the third base vector.) Let s assume the unit direction vector e 1, e 2, and e 3 are directed along the positive x, y, and z axes. 1 0 0 e1 0 2 1 3 0 e e 0 0 1 Note: base vectors are not always aligned to x, y, z. y e 2 e 3 z e 1 x
Preliminary Math Scalar/Dot Product the produce of the magnitude of one vector and the component of the second vector in the direction of the first. u u e u e u e u u u v v v ue v e 1 1 2 2 3 3 v v e v e v e uv 1 1 2 2 3 3 u v e e e e e e 1 1 2 2 3 3 1 1 2 2 3 3 i i j j e e i j i j uv i i u v u v u v 1 1 2 2 3 3 u v u v u v u v 1 1 2 2 3 3 u v u v u v cos yields a scalar quantity. where θ is the angle between the vectors
Preliminary Math Often, Thus, e iˆ e ˆj e kˆ e 1 2 3 e i j ij where ij 1 0 0 0 1 0 0 0 1 I ij Identity Matrix / Kronecker Delta
Preliminary Math Tensor Multiplication Tensor, A Vector, b Row x Column a a a A a a a a a a 11 12 13 21 22 23 31 32 33 3x3 Product of a Tensor and a Vector is a Vector! a a a b a b a b a b a a a b a b a b a b a a a b a b a b a b 11 12 13 1 21 1 22 2 23 3 21 22 23 2 21 1 22 2 23 3 31 32 33 3 21 1 22 2 23 3 3x3 3x1 3x1 b b b b 3x1 1 2 3
Preliminary Math A B Tensor Multiplication (continued) a a a a a a a a a 11 12 13 21 22 23 31 32 33 b b b b b b b b b 11 12 13 21 22 23 31 32 33 a11 a12 a13 b11 b12 b13 AB a a a b b b 21 22 23 21 22 23 a31 a32 a 33 b31 b32 b 33 a11b11 a12b21 a13b31 a11b12 a12b22 a13b32 a11b13 a12b23 a13b33 a21b11 a22b21 a23b31 a21b12 a22b22 a23b32 a21b13 a22b23 a23b33 a b a b a b a b a b a b a b a b a b 31 11 32 21 33 31 31 12 32 22 33 32 31 13 32 23 33 33
Example
Example 1 Given the following orthogonal base vectors. 0 0 1 2 2 e1 0 e2 3 2 e 2 0 2 2 2 2 Prove e e i j ij
Example 2 Show that the dot product of u i, u j, u k are equal to the projected component of each vector in the index directions.
Example 3 Given Find, A 1 2 3 4 5 6 7 8 9 b 0 1 1 Ab
Concept of Stress
Concept of Stress Traction Vector, t t lim A F A Traction can be decomposed into Normal and Shear Components t n s n n lim A F n A lim A F s A
Concept of Stress Traction relates to Cauchy Stresses when the normal of a surface aligns with a base vector. For example, the normal aligned with x axis. F F x y Fz xx lim, xy lim, xz lim A A A A A A
Example
Example 4 Using static equilibrium to prove that the sum of internal forces across the two halves are equal and opposite. (i.e. internal forces disappear when we remove the section!) Cauchy s Fundamental Lemma
Stress Components Cauchy Stress Tensor, S S xx xy xz yx yy yz zx zy zz or S 11 12 13 21 22 23 31 32 33 Notation -> Solecki Boresi S σ T Stewart
Stress Components If we know the tractions, t (x), t (y), and t (z) on the x, y, and z surfaces of a 3D element. ( y) t ( x) t We can find the Cauchy stress components as follows t i, t j, t k ( x) ( x) ( x) xx xy xz ( y) ( y) ( y) yx t, yy t, yz t ( z) ( z) ( z) zx t, zy t, zz t i j k i j k ( z) t Surface Direction
Stress Components We can also determine the relationship between traction, t (n) on an arbitrary plane, n and Cauchy stress by invoking Equilibrium!
Derive!
Stress Components Thus the relationship between the traction on an arbitrary plane, n and Cauchy stress equates to t ( n) S n
Concept of Stress Traction can be decomposed into Normal and Shear Components ( n ) t nn s n tn ( n ) ( n) 2 t t 12 n
Example 5 Select a σ yy such that there will be a traction free plane σ 1 0 1 0 yy 2 1 2 0
Example 6 2-4 from Solecki
Transformation of Stress Components When the reference coordinate system is rotated. The state of stress will transform. For example, rotating about z x, y, z to x, y, z n cos, n sin, n 0 xx xy xz n sin, n cos, n 0 yx yy yz n 0, n 0, n 1 zx zy zz N y' y x' x z z' nxx ny x 0 nx ny n z nx y ny y 0 0 0 nzz 1
Transformation of Stress Components Transformation Tensor, N cos sin 0 N sin cos 0 0 0 1 Transformed stress, S S T N SN
Example 7
Body Forces Body Forces are forces that act on every element of a material and hence on the entire volume of the material. Example: Gravitational Forces b f b dv m b dv m g f b V Force Vector x, t Density b x, t Body Force Vector x Current Configuration V
Surface Forces Surface forces act on the surface of a material. This surface may be either a part or the whole of the boundary surface f f s b Surface Force Vector S t ds t x, t Traction Vector ds Surface Increment
Equilibrium When we speak of Equilibrium, we refer to the Newton s laws of motion and equilibrium given as, s b F ma or F F ma where F represents the sum of all Forces (both surface and body), m is mass, and a is acceleration. Equilibrium ONLY EXISTS if the left hand side (LHS) and right hand side (RHS) of the equation ARE EQUAL. In the case where a=0 and v 0 this equation becomes s b F 0 or F F 0 This condition is called Static Equilibrium.
Derive!
Balance of Linear Momentum d d d 11 21 31 b f1 1 2 3 dx dx dx d d d 12 22 32 b f2 1 2 3 dx dx dx d13 d 23 d 33 b f3 dx dx dx 1 2 3 0 0 0 d dx ij j f b i a i Motion d ij b fi j dx 0 Static Equilibrium
Balance of Angular Momentum M x v dv I ω V Motion M 0 Static Equilibrium Proves, ij ji
Contact Information Calvin M. Stewart Assistant Professor Department of Mechanical Engineering The University of Texas at El Paso 500 W. University Blvd. Suite A126 El Paso, Texas 79968-0717 Email: cmstewart@utep.edu URL: http://me.utep.edu/cmstewart/ Phone: 915-747-6179 Fax: 915-747-5019