CH.4. STRESS Continuum Mechanics Course (MMC) - ETSECCPB - UPC - PowerPoint PPT Presentation

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