CH.4. STRESS Multimedia Course on Continuum Mechanics Overview Forces Acting on a Continuum Body Lecture 1 Cauchy’s Postulates Lecture 2 Stress Tensor Lecture 3 Lecture 4 Stress Tensor Components Scientific Notation Lecture 5 Engineering Notation Sign Criterion Properties of the Cauchy Stress Tensor Cauchy’s Equation of Motion Lecture 6 Principal Stresses and Principal Stress Directions Mean Stress and Mean Pressure Lecture 7 Spherical and Deviatoric Parts of a Stress Tensor Stress Invariants 2 Overview (cont’d) Stress Tensor in Different Coordinate Systems Cylindrical Coordinate System Lecture 8 Spherical Coordinate System Mohr’s Circle Lecture 9 Mohr’s Circle for a 3D State of Stress Lecture 10 Determination of the Mohr’s Circle Mohr’s Circle for a 2D State of Stress 2D State of Stress Lecture 11 Stresses in Oblique Plane Direct Problem Lecture 12 Inverse Problem Mohr´s Circle for a 2D State of Stress Lecture 13 3 Overview (cont’d) Mohr’s Circle a 2D State of Stress (cont’d) Construction of Mohr’s Circle Mohr´s Circle Properties Lecture 14 The Pole or the Origin of Planes Lecture 15 Sign Convention in Soil Mechanics Lecture 16 Particular Cases of Mohr’s Circle Lecture 17 4 4.1. 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 f= ρ bx(),t dV body force per unit V ∫V mass Surface forces. (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 fx= t (),t dS surface force S ∫∂ (traction vector) V per unit surface 6 4.2. Cauchy’s Postulates Ch.4. Stress 7 Cauchy’s Postulates st 1. Cauchy’s 1 postulate. REMARK The traction vector t remains unchanged The traction vector (generalized to for all surfaces passing through the point P internal points) is not influenced by and having the same normal vector n at P . the curvature of the internal surfaces. tt= ()P, n 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. tn()()PP,,=−− t n REMARK Cauchy’s fundamental lemma is equivalent to Newton's 3rd law (action and reaction). 8 4.3. Stress Tensor Ch.4. Stress 9 Stress Tensor The areas of the faces of the tetrahedron are: S11= nS T S22= nS with n ≡ {}n123 ,n ,n S33= nS The “mean” stress vectors acting on these faces are ** ()1* * ()2* * ()3* * t= tx(SS , n ), −= t tx ( , − eˆˆˆ123 ), −= t tx ( S , − e ), −= t tx ( S , − e ) 12 3 xx **∈Si =1, 2, 3 ; ∈→S mean value theorem Sii S The surface normal vectors of the planes perpendicular to the axes are =−=−=−ˆˆˆ ne11;; ne 2 2 ne 33 REMARK Following Cauchy’s fundamental lemma: The asterisk indicates an not ()i mean value over the area. tx()(),,−=− eˆˆii txe =− t(){} x i ∈ 1, 2, 3 10 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 ab < . Then, there exists some x * in () a,b such that: 1 fx()* = ∫ fx()dΩ Ω Ω I.e.: f : [] a,b → R gets its “mean value” fx () * at the interior of [] a,b 11 Stress Tensor From equilibrium of forces, i.e. Newton’s 2nd law of motion: Rf ==m aρ b dV+= t dS aρρ dV = a dV ∑∑i ii ∫ ∫∫ ∫ ii V∂ VV V resultant dm body forces ∫ρρ bt dV+ ∫∫ dS +− t()12 dS +− ∫ t() dS +− ∫ t() 3 dS = ∫ a dV V SS S S V 12 3 resultant Considering the mean value theorem, surface forces **()()()12 * * 3 * * (ρρb )V+− tt S S123 − t S − t S = ( a )V 1 Introducing S = nS i ∈ {} 1, 2, 3 and V = Sh , ii 3 11 ()ρρb*h S+− tt ** S()()12 n S − t * n S − t() 3 * n S = () a * hS 33123 12 Stress Tensor If the tetrahedron shrinks to point O, * ()ii** () xS→= xO lim t xeSi , ˆˆ t(){}O, ei i∈ 1, 2, 3 iih0→ () *→=** xSS xO lim t() xn , t()O , n h0→ 11** lim (ρρ b ) h = lim ( a0 ) h = h0→→33 h0 The limit of the expression for the equilibrium of forces becomes, ()2 3 = t()1 = t = t() 11()()()12 3 i ()ρρb**h+− tt * nnn − t * − t * =() a * h t()On, nt−=() 0 33123 i = tn()O, 13 Stress Tensor Considering the traction vector’s Cartesian components : ()i ()i ()i t()P= tPj() eeˆˆ j =σ ij j t()Pn, nt= ⇒ ij,∈ 1, 2, 3 i ()i {} σ = ()i ij ()PtPj () t()P, nt= nn = σ j j i i ij σij Cauchy’s Stress Tensor t()()PP, nn= ⋅σ σ =σ eeˆˆ ⊗ ij i j P In the matrix form: ()1 ()2 ()3 t1 t1 t1 T tnj= iσσ ij = ji n i j ∈{1, 2, 3} T [][][]tn= σ ()1 t t()2 t()3 14 Stress Tensor REMARK 1 The expression t ()() PP , nn = ⋅ σ is consistent with Cauchy’s postulates: t()P, nn= ⋅σ tn()()PP,,=−− t n t()P,− nn =−⋅σ REMARK 2 The Cauchy stress tensor is constructed from the traction vectors on three coordinate planes passing through point P. σ11 σσ 12 13 ≡ σ σσ σ 21 22 23 σ31 σσ32 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 4.4.Stress Tensor Components Ch.4. Stress 16 Scientific Notation Cauchy’s stress tensor in scientific notation σσσ11 12 13 σ ≡ σσσ 21 22 23 σσσ31 32 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. 17 Engineering Notation Cauchy’s stress tensor in engineering notation στ τ x xy xz σ ≡ τ στ yx y yz ττσ zx zy z Where: σ a is the normal stress acting on plane a. τ ab is the tangential (shear) stress acting on the plane perpendicular to the a-axis in the direction of the b-axis. 18 Tension and compression The stress vector acting on point P of an arbitrary plane may be resolved into: a vector normal to the plane ()σ n = σ n an in-plane (shear) component which acts on the plane. (;ττnn=τ ) The sense of σ n with respect to n defines the normal stress character: >0 tensile stress (tension) σ =σ ⋅n n <0 compressive stress (compression) The sign criterion for the stress components is: positive (+) tensile stress σσor ij a negative (−) compressive stress positive (+) positive direction of the b-axis τ ab negative (−) negative direction of the b-axis 19 4.5.Properties of the Cauchy Stress Tensor Ch.4. Stress 20 Cauchy’s Equation of Motion Consider an arbitrary material volume, Cauchy’s equation of motion is: ∇⋅ σ+ρρ ba = ∀∈ x V ∂σ ij +=ρρ ∈ bjj aj{}1, 2, 3 ∂xi In engineering notation: bx(),t x∈ V ∂∂στ∂τ tx* (),t x∈∂ V x +yx +zx +=ρρba ∂∂∂xyz xx ∂∂∂τxy στy zy REMARK + + +=ρρbayy ∂∂∂xyz Cauchy’s equation of motion is derived ∂τ ∂τ ∂σ from the principle of balance of linear xz +yz +z +=ρρba ∂∂∂xyz zz momentum. 21 Equilibrium Equations For a body in equilibrium a0 = , Cauchy’s equation of motion becomes ∇ ⋅σ+ρ b0 = ∀∈ x V ∂σ ij internal equilibrium +=∈ρbj0{} 1, 2, 3 ∂x j equation i The traction vector is now known at the boundary nx()()(),,tt⋅σ x = t* x , t ∀ x ∈∂ Vequilibrium equation σ=* ∈ at the boundary nti ij j j{}1, 2, 3 The stress tensor symmetry is derived from the principle of balance of angular momentum: σσ= T σσ= ∈ ij ji ij,{} 1, 2, 3 22 Cauchy’s Equation of Motion Taking into account the symmetry of the Cauchy Stress Tensor, Cauchy’s equation of motion ∇⋅ σ+=ρ b σ⋅∇ += ρρ ba ∀∈ x V ∂∂σσ ij +=ρji += ρρ ∈ bj ba jj j{}1, 2, 3 ∂∂ xxii bx(),t x∈ V Boundary conditions tx* (),t x∈∂ V * n⋅σσ = ⋅ n = tx(,)tV∀x ∈∂ σ =σ = * ∀ ∈∂ ∈ ni ij ji ni t j ()xx, t V ij,{} 1, 2, 3 23 Principal Stresses and Principal Stress Directions 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 σ principal stresses. 33 σ 31 x3 σ 32 x3 x3′ σ13 σ 23 σ1 00 σ11 σ12 σ x1′ σ σ = σ 21 σ 22 3 [ ] 002 σ1 00σ 3 σ 2 x1 x1 x2 x2 x2′ 24 Principal Stresses and Principal Stress Directions The Cauchy stress tensor is a symmetric 2nd 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 ⋅=vvλ []σ −λ1 ⋅=v0 INVARIANTS not 32 characteristic det[]σσ−λλ11 =−=0 λλ−I1()σ − II 23 () σσ λ −= () 0 equation σ 33 λσ11≡ σ 31 x3 σ 32 x3 x3′ σ13 λσ22≡ σ 23 σ11 σ12 σ x1′ σ λσ33≡ 21 σ 22 3 σ1 σ REMARK 2 x1 x1 The invariants associated with a x x 2 ′ 2 tensor are values which do not change x2 25 with the coordinate system being used.