CH.4. STRESS. Continuum Mechanics Course (MMC)

CH.4. STRESS Continuum Mechanics Course (MMC)

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 Lecture Lecture Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7

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 D State of Stress D State of Stress Stresses in Oblique Plane Direct Problem Inverse Problem Mohr s Circle for a D State of Stress Lecture 8 Lecture 9 Lecture 0 Lecture Lecture Lecture 3 3

Overview (cont d) Mohr s Circle a D 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 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 4

4.. Forces on a Continuum Body Ch.4. Stress 5

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 fv = ρb( x, t) dv Surface forces. V body force per unit mass (specific body 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 fs = t ( x, t) ds V (traction vector) surface force per unit surface 6

4.. Cauchy s Postulates Ch.4. Stress 7

Cauchy s Postulates. Cauchy s st postulate. The traction vector tremains unchanged for all surfaces passing through the point P and having the same normal vector n at P. ( P, ) t = t n REMARK The traction vector (generalized to internal points) is not influenced by the curvature of the internal surfaces.. 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. ( P, ) = ( P, ) t n t n REMARK Cauchy s fundamental lemma is equivalent to Newton's 3 rd law (action and reaction). 8

4.3. Stress Tensor Ch.4. Stress 9

Stress Tensor The areas of the faces of the tetrahedron are: S S S = ns = ns = ns 3 3 with n { n } T,n,n3 The mean stress vectors acting on these faces are ( ) ( ) ( ) * * * * * * 3* * t = tx ( ˆ ˆ ˆ S, n), t = tx ( S, e ), t = tx ( S, e ), t = tx ( S, e 3 3) * * xs S,, 3 ; x i i i = S S mean value theorem The surface normal vectors of the planes perpendicular to the axes are n = eˆ ; n = eˆ ; n3 = eˆ3 REMARK Following Cauchy s fundamental lemma: The asterisk indicates an not i t x, eˆ = t x, eˆ = t x i,, 3 mean value over the area. ( ) ( ) i i ( ) ( ) { } 0

Mean Value Theorem [ ] Let f : a,b R be a continuous function on the closed interval [ a,b], and differentiable on the open interval ( a,b), where a < b. * Then, there exists some x in a,b such that: ( ) ( ) f x = f x dω Ω * Ω ( ) [ ] I.e.: f : a,b R gets its * mean value f ( x ) at the interior of a,b [ ]

Stress Tensor From equilibrium of forces, i.e. Newton s nd law of motion: resultant body forces R = fi = miai ρ b dv + t ds = a ρdv = ρ adv i i V V V dm V ( ) ( ) ( 3) ρ + + + + = b dv t ds t ds t ds t ds ρ a dv V S S S S V 3 Considering the mean value theorem, ( ) ( ) ( 3) ( ρb)v+ t S t S t S t S = ( ρa)v * * * * * * 3 i i 3 * * ( ) * ( ) * ( 3) * * ( ρb) h S + t S t ns t ns t n3s = ( ρa) hs 3 3 Introducing S = ns i {,, 3} and V = Sh, resultant surface forces

Stress Tensor If the tetrahedron shrinks to point O, * * * S O h 0 ( i) ( i ( ˆ ) ) ( O, ˆ ) { } * * * xs x lim i O t xs, e i i = t ei i,,3 h 0 ( ) S ( O ) x x lim t x, n = t, n lim ( ) h = lim ( ) h = 3 3 * * ρ b ρ a 0 h 0 h 0 The limit of the expression for the equilibrium of forces becomes, = t = t = t ( ) ( ) ( 3) * * * * * * ( ρb) h+ t t n t n t n3 = ( ρa) h 3 3 ( O, ) = t n ( ) ( ) ( 3) (, ) ( i) t O n t n i = 0 3

Stress Tensor ( i ) ( ) ( ) Considering the traction vector s Cartesian components : () i t P = t ( P) eˆ = σ eˆ ( i σ ) ij P = tj ( P) j j ij j σ = σ eˆ eˆ ij i j In the matrix form: T t = n σ = σ n j {,,3} T [ t ] = [ ] [ n ] j i ij ji i { } i, j,,3 Cauchy s Stress Tensor ( ) t σ ( ) ( ) t ( P, ) (, ) ( i) t n = t n () i t j P n = t j ni = niσ ij σij t( P, n) = n σ ( P) ( 3) t i P t ( ) t ( 3) t 4

Stress Tensor REMARK The expression t( P, n) = n σ ( P) t( P, n) = n σ ( P, ) = t n n σ is consistent with Cauchy s postulates: ( P, ) = ( P, ) t n t n REMARK The Cauchy stress tensor is constructed from the traction vectors on three coordinate planes passing through point P. σ σ σ σ3 σ σ σ 3 σ3 σ3 σ 33 Yet, this tensor contains information on the traction vectors acting on any plane (identified by its normal n) which passes through point P. 5

4.4.Stress Tensor Components Ch.4. Stress 6

Scientific Notation Cauchy s stress tensor in scientific notation σ σ σ σ σ σ σ σ σ σ 3 3 3 3 33 Each component σ ij 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. 7

Engineering Notation Cauchy s stress tensor in engineering notation σ σx τxy τ xz τ yx σ y τ yz τzx τzy σ z Where: σ a τ ab is the normal stress acting on plane a. is the tangential (shear) stress acting on the plane perpendicular to the a-axis in the direction of the b-axis. 8

Tension and compression σ ij or σ The stress vector acting on point P of an arbitrary plane may be resolved into: a vector normal to the plane ( σ = σ n) an in-plane (shear) component which acts on the plane. ( τ ; τ = ) n n τ The sense of with respect to n defines the normal stress character: σ = σ n n σ n >0 tensile stress (tension) <0 compressive stress (compression) The sign criterion for the stress components is: a positive (+) negative ( ) τ ab positive (+) negative ( ) tensile stress compressive stress positive direction of the b-axis negative direction of the b-axis n 9

4.5.Properties of the Cauchy Stress Tensor Ch.4. Stress 0

Cauchy s Equation of Motion Consider an arbitrary material volume, Cauchy s equation of motion is: σ+ ρ b= ρa x σ ij + ρbj = ρaj j xi {,, 3} In engineering notation: σ τ x yx τzx + + + ρbx = ρax x y z τxy σ y τzy + + + ρby = ρay x y z τ τ xz yz σ z + + + ρbz = ρaz x y z V ( ) ( ) b x,t x V * t x x,t V REMARK Cauchy s equation of motion is derived from the principle of balance of linear momentum.

Equilibrium Equations For a body in equilibrium a= 0, Cauchy s equation of motion becomes σ+ ρ b= 0 x V σ ij + ρbj = 0 j,,3 xi { } The traction vector is now known at the boundary * (, ) σ (, ) (, ) * {,, 3} n x t x t = t x t x V ni σ ij = tj j internal equilibrium equation equilibrium equation at the boundary The stress tensor symmetry is derived from the principle of balance of angular momentum: T σ = σ σ = ij σ ji i, j {,,3}

Cauchy s Equation of Motion Taking into account the symmetry of the Cauchy Stress Tensor, Cauchy s equation of motion σ+ ρ b= σ + ρ b= ρa x σij σ ji + ρbj = + ρbj = ρaj j xi xi Boundary conditions * n σ = σ n= t (,) x t x V {,, 3} V * ni σ ij =σ ji ni = tj t V i j ( x, ) x, {,, 3} ( ) ( ) b x,t x V * t x x,t V 3

Principal Stresses and Principal Stress Directions 4 Regardless of the state of stress, it is always possible to choose a special set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system. The three planes perpendicular to the principal axes are the principal planes. The normal stress components in the principal planes are the principal stresses. [ σ ] σ 0 0 0 σ 0 0 0 σ 3 = x σ x σ 33 3 3 σ 3 σ 3 σ 3 σ σ σ σ x x x x 3 x x 3 σ x σ 3 σ

Principal Stresses and Principal Stress Directions 5 The Cauchy stress tensor is a symmetric nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers. For the eigenvalue λ and its corresponding eigenvector v: σ v =λv [ σ λ] v = 0 [ σ λ ] not det = σ λ = 0 λ σ λ σ λ σ 3 3 REMARK The invariants associated with a tensor are values which do not change with the coordinate system being used. x INVARIANTS λ I ( σ) λ I ( σ) λ I ( σ) = 0 3 3 σ x σ 33 3 3 σ 3 σ 3 σ 3 σ σ σ σ x x x x 3 x x 3 σ characteristic equation x σ 3 σ

Mean Stress and Mean Pressure Given the Cauchy stress tensor and its principal stresses, the following is defined: Mean stress σ σm = Tr ( σ ) = σii = ( σ+ σ + σ3) 3 3 3 Mean pressure p = σm = σ+ σ + σ3 3 ( ) A spherical or hydrostatic state of stress: σ σ = σ = σ 3 σ σ 0 0 σ 0 0 = σ 0 0 σ σ σ σ σ σ σ σ σ σ 3 3 3 3 33 REMARK In a hydrostatic state of stress, the stress tensor is isotropic and, thus, its components are the same in any Cartesian coordinate system. As a consequence, any direction is a principal direction and the stress state (traction vector) is the same in any plane. 6

Spherical and Deviatoric Parts of a Stress Tensor The Cauchy stress tensor σ=σ σ sph + σ The spherical stress tensor: can be split into: Also named mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body σsph : = σm= Tr ( σ) = σii REMARK 3 3 The principal directions of a stress tensor The stress deviator tensor: and its deviator stress component coincide. Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body σ = devσ = σ σ m 7

Stress Invariants Principal stresses are invariants of the stress state: invariant w.r.t. rotation of the coordinate axes to which the stresses are referred. The principal stresses are combined to form the stress invariants I : I = Tr ( σ) = σ = σ + σ + σ ii 3 I = I = σσ + σσ 3+ σσ 3 I 3 = det ( σ) ( σ: σ ) ( ) These invariants are combined, in turn, to obtain the invariants J : J = I = σ ii J = ( I + I) = σσ ij ji = ( σ: σ) 3 J3 = ( I + 3II + 3I3) = Tr ( σ σ σ) = σσ ij jkσki 3 3 3 REMARK The I invariants are obtained from the characteristic equation of the eigenvalue problem. REMARK The J invariants can be expressed in the unified form: i Ji = Tr ( σ ) i {,, 3} i 8

Stress Invariants of the Stress Deviator Tensor The stress invariants of the stress deviator tensor: I = Tr ( σ ) = 0 I = σ : σ I = σ σ σ σ σ σ + + I 3 = det σ = σ σ σ 33 + σ σ 3σ 3 σ σ 33 σ 3σ σ 3σ = σ σ σ 3 These correspond exactly with the invariants J of the same stress deviator tensor: J ( ) 3 3 3 3 = I = 0 ( ) ( ij jk ki ) J = ( I + I ) = I = ( σ : σ ) 3 J 3 = ( I + 3I I+ 3I 3) = I 3 = Tr ( σ σ σ ) = ( σσ ij jkσ ki ) 3 3 3 9

4.6. Stress Tensor in Different Coordinate Systems Ch.4. Stress 30

Stress Tensor in a Cylindrical Coordinate System The cylindrical coordinate system is defined by: dv = r dθ dr dz x= r cos θ x( r, θ, z) y = r sin θ z = z The components of the stress tensor are then: σx τx y τ x z σr τr θ τrz σ = τxy σ y τ yz τr θ σθ τ = θz τx z τ y z σ z τrz τθ z σ z 3

Stress Tensor in a Spherical Coordinate System The cylindrical coordinate system is defined by: dv = r sen θ dr dθ dϕ The components of the stress tensor are then: σx τxy τ xz σr τr θ τr φ σ τxy σ y τ yz = τrθ σθ τθφ τxz τ yz σ z τr φ τφθ σ φ x x= r sen θ cosφ r, θϕ, y = r sen θ senφ z = r cosθ ( ) 3

4.7. Mohr s Circle Ch.4. Stress 33

Mohr s Circle Introduced by Otto Mohr in 88. Mohr s Circle is a two-dimensional graphical representation of the state of stress at a point that: will differ in form for a state of stress in D or 3D. illustrates principal stresses and maximum shear stresses as well as stress transformations. is a useful tool to rapidly grasp the relation between stresses for a given state of stress. 34

4.8. Mohr s Circle for a 3D State of Stress Ch.4. Stress 35

Determination of Mohr s Circle Consider the system of Cartesian axes linked to the principal directions of the stress tensor at an arbitrary point P of a continuous medium: x 3 The components of the stress tensor are σ 0 0 σ 0 σ 0 with 0 0 σ 3 The components of the traction vector are σ 0 0 n σ n t = σ n= 0 σ 0 n = σ n 0 0 σ 3 n 3 σ3 n 3 ê σ 3 x σ ê x ê 3 where n is the unit normal to the base associated to the principal directions 36

Determination of Mohr s Circle The normal component of stress σ is n σ = tn = [ σ n, σ n, σ n ] n = σ n + σ n + σ n T n 3 t 3 3 3 3 The squared modulus of the traction vector is n σ n = σ n t t = σ + τ : τ : = τn = t t = σn + σn + σ3n3 σn + σn + σ3n3 = σ + τ The unit vector n = n must satisfy n n n + + 3 = Mohr's 3D problem half - space Locus of all possible ( στ, ) points? 37

Determination of Mohr s Circle The previous system of equations can be written as a matrix equation which can be solved for any couple σ σ σ 3 n σ + τ σ σ σ3 n = σ n 3 A x b n 0 T A feasible solution for x requires that n 0 n for the, n, n 3 expression n + n + n = to hold true. 3 0 n3 Every couple of numbers ( στ, ) which leads to a solution x, will be considered a feasible point of the half-space. The feasible point is representative of the traction vector ( στ, ) on a T plane of normal n n, n, n 3 which passes through point P. The locus of all feasible points is called the feasible region. 38

Determination of Mohr s Circle The system σ σ σ 3 n σ + τ σ σ σ3 n = σ n 3 A x b can be re-written as A ( I) σ + τ σ + σ σ+ σσ n = 0 ( ) 3 3 ( ) ( ) ( σ σ3) A ( II) σ + τ σ + σ σ+ σσ n = 0 3 3 ( σ σ3) A ( III) σ + τ σ + σ σ+ σσ n = 0 3 ( σ σ) with ( ) ( 3) ( 3) A = σ σ σ σ σ σ 39

Determination of Mohr s Circle Consider now equation ( III) : A σ + τ σ + σ σ+ σσ n = 0 It can be written as: ( ) σ a + τ = R which is the equation of a semicircle of center and radius R : C R ( ) 3 = ( σ + σ ), 0 = + 4 3 ( σ σ) with ( σ σ ) ( σ σ ) ( σ σ ) 3 3 3 3 n with a = ( σ + σ) R= + 4 ( ) ( 3) ( 3) A = σ σ σ σ σ σ ( σ σ ) ( σ σ ) ( σ σ ) 3 3 3 n C3 3 REMARK A set of concentric semi-circles is obtained with the different values of n 3 with center C and radius R 3 3( n3) : min n3 = 0 R3 = ( σ σ) n3 = max R3 = ( σ + σ) σ3 40

Determination of Mohr s Circle Following a similar procedure with ( I) and ( II), a total of three semi-annuli with the following centers and radii are obtained: C = [ ( 3),0] σ + σ C = [ ( 3),0] σ + σ a a C = 3 [ ( ),0] σ + σ a 3 R R R R R R min max max min min 3 max 3 = = σ a ( σ σ ) 3 = = σ a ( σ σ ) 3 = = σ a ( σ σ ) 3 3 4

Determination of Mohr s Circle Superposing the three annuli, 44 The final feasible region must be the intersection of these semi-annuli Every point of the feasible region in the Mohr s space, corresponds to the stress (traction vector) state on a certain plane at the considered point

4.9. Mohr s Circle for a D State of Stress Ch.4. Stress 45

D State of Stress 3D general state of stress D state of stress σ σx τxy τ xz τ yx σ y τ yz τzx τzy σ z σ σx τxy τ yx σ y 0 0 0 0 σ z 3D problem REMARK In D state of stress problems, the principal stress in the disregarded direction is known (or assumed) a priori. σ σx τ yx τxy σ y D (plane) problem 46

Stresses in a oblique plane Given a plane whose unit normal n forms an angle θ with the x axis, Traction vector σx τxy cosθ σx cosθ + τxy sinθ t = σn = τxy σ y sinθ = τxy cosθ σ y sinθ + σ n Normal stress Shear stress σx σ y τ = tm = sin θ τ cos θ σx + σ y σx σ y σθ = tn = + cos( θ) + τxy sin ( θ) θ ( ) ( ) xy cosθ sinθ n = m = cosθ sinθ 47 Tangential stress θ is now endowed with sign τ ( τ 0 or τ < 0) Pay attention to the positive senses given in the figure θ θ

Direct and Inverse Problems Direct Problem: Find the principal stresses and principal stress directions given σ in a certain set of axes. Inverse Problem: Find the stress state on any plane, given the principal σ stresses and principal stress directions. equivalent stresses 48

Direct Problem 49 In the x and y axes, τ = 0 then, σx σ y τα = sin ( α) τxy cos( α) = 0 τ xy tan ( α ) = σx σ y Using known trigonometric relations, τ xy ( α ) =± =± + σx σ y tg ( α ) + τ xy sin cos ( α ) σx σ y =± =± + tg ( α ) σx σ y + τ α xy This equation has two solutions:.. α ( sign " + ") π α = α + ( sign " ") These define the principal stress directions. (The third direction is perpendicular to the plane of analysis.)

Direct Problem The angles and θ = α are then introduced into the equation θ α = σx + σ y σx σ y σθ = + cos( θ) + τxy sin ( θ) to obtain the principal stresses (orthogonal to the plane of analysis): σ α σx + σ y σx σ y σ = + + τ σx + σ y σx σ y σ = + τ xy xy θ α 50

Inverse Problem Given the directions and principal stresses σ and, to find the σ stresses in a plane characterized by the angle β : Take the equations Replace,, and to obtain: θ β σ+ σ σ σ σβ = + σ σ τβ = sin ( β) cos ( β) 5

Mohr s Circle for a D State of Stress Considering a reference system and characterizing the inclination of a plane by, From the inverse problem equations: σ+ σ σ σ σ = σ σ τ = sin ( β) cos Squaring both equations and adding them: + σ σ σ τ σ σ + = Eq. of a circle with center C and radius R. Mohr s Circle ( β) x y REMARK This expression is valid for any value of. C R = σ σ σ+ σ =,0 5

Mohr s Circle for a D State of Stress The locus of the points representative of the state of stress on any of the planes passing through a given point P is a circle. (Mohr s Circle) The inverse is also true: ( ) Given a point στ, in Mohr s Circle, there is a plane passing through P whose normal and tangential stresses are and τ, respectively. σ R = σ σ + σ σ Mohr's D problem space σ a ( β ) = = cos sin ( β ) σ σ = τ τ σ σ = R R σ+ σ C =,0 53

Construction of Mohr s Circle Interactive applets and animations: by M. Bergdorf: http://www.zfm.ethz.ch/meca/applets/mohr/mohrcircle.htm from MIT OpenCourseware: http://ocw.mit.edu/ans7870/3/3./tools/mohrscircleapplet.html from Virginia Tech: http://web.njit.edu/~ala/keith/java/mohr.html From Pennsilvania State University: http://www.esm.psu.edu/courses/emch3d/design/animation/animation.htm 54

Mohr s Circle s Properties A. To obtain the point in Mohr s Circle representative of the state of stress on a plane which forms an angle β with the principal stress direction : σ. Begin at the point on the circle (representative of the plane where acts).. Rotate twice the angle in the sense σ σ β. 3. This point represents the shear and normal stresses at the desired plane (representative of the stress state at the plane where acts). σ 3... 55

Mohr s Circle s Properties B. The representative points of the state of stress on two orthogonal planes are aligned with the centre of Mohr s Circle: π This is a consequence of property A as β = β+. 56

Mohr s Circle s Properties C. If the state of stress on two orthogonal planes is known, Mohr s Circle can be easily drawn:. Following property B, the two points representative of these planes will be aligned with the centre of Mohr s Circle.. Joining the points, the intersection with the σ axis will give the centre of Mohr s Circle. 3. Mohr s Circle can be drawn. 3... 57

Mohr s Circle s Properties D. Given the components of the stress tensor in a particular orthonormal base, Mohr s Circle can be easily drawn: This is a particular case of property C in which the points representative of the state of stress on the Cartesian planes is known.. Following property B, the two points representative of these planes will be aligned with the centre of Mohr s Circle.. Joining the points, the intersection with the σ axis will give the centre of Mohr s Circle. 3. Mohr s Circle can be drawn. 3... σ σx τxy = τxy σ y 58

Mohr s Circle s Properties The radius and the diametric points of the circle can be obtained: σ σx τxy = τxy σ y σ σ R = + τ x y xy 59

Mohr s Circle s Properties Note that the application of property A for the point representative of the vertical plane implies rotating in the sense contrary to angle. σx τxy σ = τxy σ y 60

The Pole or the Origin of Planes The point called pole or origin of planes in Mohr s circle has the following characteristics: Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane parallel in space to that line... 6

The Pole or the Origin of Planes The point called pole or origin of planes in Mohr s circle has the following characteristics: If a straight line, parallel to a given plane, is drawn from the pole, the intersection point represents the state of stress on this particular plane... 6

Sign Convention in Soil Mechanics The sign criterion used in soil mechanics, is the inverse of the one used in continuum mechanics: In soil mechanics, σ β negative ( ) positive (+) tensile stress compressive stress τ β positive (+) negative (-) counterclockwise rotation clockwise rotation continuum mechanics But the sign criterion for angles is the same: positive angles are measured counterclockwise τ σ * β * β = τ β = σ β soil mechanics 63

Sign Convention in Soil Mechanics For the same stress state, the principal stresses will be inverted. continuum mechanics soil mechanics τ σ * β * β = τ β = σ β σ σ β * = σ * = σ * π = β + 64 The expressions for the normal and shear stresses are σ σ σ + σ * * * * * * σβ cos( β π) * * * * σ σ σ σ = + + + * * cos( ) σ + σ σ σ β = + cos β* β cos = + * * * * σ σ * σ + σ * * σ sin σ β = ( ) β = sin + * β = sin σ β σ β ( ) τ β τ β π τ β sin β* The Mohr s circle construction and properties are the same in both cases ( ) ( ) like in continuum mechanics

4.0. Particular Cases of Mohr s Circle Ch.4. Stress 65

Particular Cases of Mohr s Circles Hydrostatic state of stress Mohr s circles of a stress tensor and its deviator ( σ sph = σ m) σ = σ + σ sph σ = σm + σ σ = σm + σ σ = σ + σ 3 m 3 Pure shear state of stress 66