CH.4. STRESS Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Overview Forces Acting on a Continuum Body Cauchy’s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion Properties of the Cauchy Stress Tensor Cauchy’s Equation of Motion Principal Stresses and Principal Stress Directions Mean Stress and Mean Pressure Spherical and Deviatoric Parts of a Stress Tensor Stress Invariants 2 MMC - ETSECCPB - UPC 03/11/2014
Overview (cont’d) Stress Tensor in Different Coordinate Systems Cylindrical Coordinate System Spherical Coordinate System Mohr’s Circle Mohr’s Circle for a 3D State of Stress Determination of the Mohr’s Circle Mohr’s Circle for a 2D State of Stress 2D State of Stress Stresses in Oblique Plane Direct Problem Inverse Problem Mohr´s Circle for a 2D State of Stress 3 MMC - ETSECCPB - UPC 03/11/2014
Overview (cont’d) Mohr’s Circle a 2D State of Stress (cont’d) Construction of Mohr’s Circle Mohr´s Circle Properties The Pole or the Origin of Planes Sign Convention in Soil Mechanics Particular Cases of Mohr’s Circle 4 MMC - ETSECCPB - UPC 03/11/2014
4.1. Forces on a Continuum Body Ch.4. Stress 5 MMC - ETSECCPB - UPC 03/11/2014
Forces Acting on a Continuum Body Forces acting on a continuum body: Body forces. Act on the elements of volume or mass inside the body. “Action-at-a-distance” force. E.g.: gravity, electrostatic forces, magnetic forces f b x , t dV body force per unit V V mass (specific body forces) Surface forces. Contact forces acting on the body at its boundary surface. E.g.: contact forces between bodies, applied point or distributed loads on the surface of a body surface force t f x , t dS (traction vector) S V per unit surface 6 MMC - ETSECCPB - UPC 03/11/2014
4.2. Cauchy’s Postulates Ch.4. Stress 7 MMC - ETSECCPB - UPC 03/11/2014
Cauchy’s Postulates Cauchy’s 1 st postulate. 1. REMARK The traction vector t remains unchanged The traction vector (generalized to for all surfaces passing through the point internal points) is not influenced by P n and having the same normal vector at . the curvature of the internal surfaces. P t t , n P 2. Cauchy’s fundamental lemma (Cauchy reciprocal theorem) The traction vectors acting at point P on opposite sides of the same surface are equal in magnitude and opposite in direction. REMARK t , n t , n P P Cauchy’s fundamental lemma is equivalent to Newton's 3 rd law (action and reaction). 8 MMC - ETSECCPB - UPC 03/11/2014
4.3. Stress Tensor Ch.4. Stress 9 MMC - ETSECCPB - UPC 03/11/2014
Stress Tensor The areas of the faces of the tetrahedron are: S n S 1 1 T n n ,n ,n with S n S 1 2 3 2 2 S n S 3 3 The “mean” stress vectors acting on these faces are 1 * 2 * 3 * * * * * * ˆ ˆ ˆ t t x ( ), t t x ( , e ), t t x ( , e ), t t x ( , e ) 1 2 3 S S S S 1 2 3 * * x 1,2,3 ; x mean value theorem S i S S i S i The surface normal vectors of the planes perpendicular to the axes are ˆ ˆ ˆ n e ; n e ; n e 1 1 2 2 3 3 REMARK Following Cauchy’s fundamental lemma: not The asterisk indicates an i ˆ ˆ t x , e t x e , t x i 1,2,3 i i mean value over the area. 10 MMC - ETSECCPB - UPC 03/11/2014
Mean Value Theorem R Let be a continuous function on the closed interval : a,b f a,b , and differentiable on the open interval , where . a,b a b Then, there exists some * in such that: a,b x 1 * d f x f x R I.e.: gets its : a,b f * “mean value” at the interior f x a,b of 11 MMC - ETSECCPB - UPC 03/11/2014
Stress Tensor From equilibrium of forces, i.e. Newton’s 2 nd law of motion: R f a b t a a m dV dS dV dV i i i i i V V V V dm resultant body forces 1 2 3 b t t t t a dV dS dS dS dS dV V S S S S V 1 2 3 resultant surface forces Considering the mean value theorem, 1 2 3 * * * * * * ( b ) V t S t S t S t S ( a ) V 1 2 3 1 Introducing and , 1,2,3 V Sh S n S i 3 i i 1 1 1 2 3 * * * * * * ( b ) t S t S t S t S ( a ) h S n n n hS 1 2 3 3 3 12 MMC - ETSECCPB - UPC 03/11/2014
Stress Tensor (h 0) If the tetrahedron shrinks to point O , i i * * * ˆ x x lim t x t e i 1,2,3 O, S S i O i h 0 i * * * x x lim t x n , t , n O S S O h 0 1 1 * * lim ( b ) lim ( a ) 0 h h 3 3 h 0 h 0 The limit of the expression for the equilibrium of forces becomes, t 2 3 t 1 t 1 1 1 2 3 * * * * * * ( b ) t t t t ( a ) t , n t i 0 h n n n h O n 1 2 3 3 3 i t , n O 13 MMC - ETSECCPB - UPC 03/11/2014
Stress Tensor Considering the traction vector’s Cartesian components : ( ) ˆ ˆ t i ( ) e e t , n t i i P t P P n j j ij j , 1,2,3 i i j i ( ) P t P t , n t i P n n ij j j j i i ij ij n Cauchy’s Stress Tensor t , n P P ˆ ˆ e e ij i j P In the matrix form: 1 1 1 t t t 2 3 1 11 21 31 t 1 n 1 T t n n j i ij ji i 12 22 32 {1,2,3} t 2 n 2 j T 13 23 33 t n t 3 n 3 1 2 3 t t t 14 MMC - ETSECCPB - UPC 03/11/2014
Stress Tensor REMARK 1 The expression is consistent with Cauchy’s postulates: t , n n P P t , n n P t , n t , n P P P t , n n REMARK 2 The Cauchy stress tensor is constructed from the traction vectors on three coordinate planes passing through point P . 11 12 13 21 22 23 31 32 33 Yet, this tensor contains information on the traction vectors acting on any plane (identified by its normal n ) which passes through point P . 15 MMC - ETSECCPB - UPC 03/11/2014
4.4.Stress Tensor Components Ch.4. Stress 16 MMC - ETSECCPB - UPC 03/11/2014
Scientific Notation Cauchy’s stress tensor in scientific notation 11 12 13 21 22 23 31 32 33 ij Each component is characterized by its sub-indices: Index i designates the coordinate plane on which the component acts. Index j identifies the coordinate direction in which the component acts. 17 MMC - ETSECCPB - UPC 03/11/2014
Engineering Notation Cauchy’s stress tensor in engineering notation x xy xz yx y yz zx zy z Where: a is the normal stress acting on plane a . ab is the tangential (shear) stress acting on the plane perpendicular to the a -axis in the direction of the b -axis. 18 MMC - ETSECCPB - UPC 03/11/2014
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