Basic concepts to start Mechanics of Materials
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1 Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS
2 Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen in a given frame as a column of components V i. This is a tensor of order 1 (one index). A second order tensor can be seen in a given frame as a matrix of components M ij A fourth order tensor L can be seen in a given frame as a four index table L ijkl A scalar x is a tensor of order zero (no index) Einstein convention means repeated index summation one order less (vector. vector) gives a scalar (order_1. order_1) gives order_0 3 v i v i = (v i ) 2 = x i=1 (2nd order tensor. vector) gives a vector (order_2. order_1) gives order_1 3 M ij v j = M ij v j = w i j=1 (2nd order tensor. 2nd order tensor) gives a 2nd order tensor (order_2. order_2) gives order_2 3 M ij N jk = M ij N jk = P ik j=1 Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
3 Notations Notations (maths) (2/2) Intrinsic notation means no component marked, one dot summation one order less (vector. vector) gives a scalar (order_1. order_1) gives... order_0 v.v = x (2nd order tensor. vector) gives a vector (order_2. order_1) gives... order_1.v = w (2nd order tensor. 2nd order tensor) gives a 2nd order tensor (order_2. order_2) gives... order_2.n = P (2nd order tensor : 2nd order tensor) gives a scalar (order_2 : order_2) gives... order_0 : N = x : L : N gives? Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
4 Notations Notations (maths) (2/2) Intrinsic notation means no component marked, one dot summation one order less (vector. vector) gives a scalar (order_1. order_1) gives... order_0 v.v = x (2nd order tensor. vector) gives a vector (order_2. order_1) gives... order_1.v = w (2nd order tensor. 2nd order tensor) gives a 2nd order tensor (order_2. order_2) gives... order_2.n = P (2nd order tensor : 2nd order tensor) gives a scalar (order_2 : order_2) gives... order_0 : N = x : L : N gives? Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
5 Notations Notations (maths) (2/2) Intrinsic notation means no component marked, one dot summation one order less (vector. vector) gives a scalar (order_1. order_1) gives... order_0 v.v = x (2nd order tensor. vector) gives a vector (order_2. order_1) gives... order_1.v = w (2nd order tensor. 2nd order tensor) gives a 2nd order tensor (order_2. order_2) gives... order_2.n = P (2nd order tensor : 2nd order tensor) gives a scalar (order_2 : order_2) gives... order_0 : N = x : L : N gives? Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
6 Notations Notations (maths) (2/2) Intrinsic notation means no component marked, one dot summation one order less (vector. vector) gives a scalar (order_1. order_1) gives... order_0 v.v = x (2nd order tensor. vector) gives a vector (order_2. order_1) gives... order_1.v = w (2nd order tensor. 2nd order tensor) gives a 2nd order tensor (order_2. order_2) gives... order_2.n = P (2nd order tensor : 2nd order tensor) gives a scalar (order_2 : order_2) gives... order_0 : N = x : L : N gives? Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
7 Notations Index expansion Tensorial product Produce a 2nd order tensor from two vectors index form : m ij = n i l j intrinsic form : m = n l Produce a 4th order tensor from two 2nd order tensors index form : L ijkl = M ij N kl intrinsic form : L = N Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
8 Notations Notations (mechanics) Displacement vector : u components u i Strain tensor (second order, symmetric) : ε components ε ij Stress tensor (second order, symmetric) : σ components σ ij Stress vector for a facet of normal n : T = σ.n components T i = σ ij n j (sum on j) Stress tensor from strain tensor (elastic constitutive equations) σ = Λ : ε components σ ij = Λ ijkl ε kl (sum on k and l) Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
9 Kinematics and statics in small strains Displacement-strain The strain tensor is the symmetric part of the displacement gradient ε ij = 1 2 (u i,j + u j,i ) ε = 1 ( ) u + u T 2 Kinematically admissible field : u = u d sur Ω u Compatibility equations (ex : 6 strain components derive from 3 displacement components). For 3D cartesian coordinates ε 11,22 + ε 22,11 2ε 12,12 = 0 ε 11,23 + ε 23,11 ε 12,13 ε 13,12 = 0...and circular permutations, that is : ε inm ε ljk ε ij,km = 0 with ε ijk = 0 ε ijk = 1 if 2 indices are equal for the case of an even permutation, =-1 for an odd permutation Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
10 Kinematics and statics in small strains Geometrical meaning of the terms in the strain tensor V V = Tr ε = ε ii γ = 2ε 12 (1+ ε 11 ) dx1 dx1 dx2 (1+ ε 22 ) dx2 γ dx 2 The diagonal terms characterize the elongation of a unit segment in the direction of the axes dx 1 Non diagonal terms characterize the rotations with respect to the axes Elongation in the direction defined by n δ(n) = n.ε.n = ε ij n i n j Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
11 Kinematics and statics in small strains Stress Volumetric forces : f d in the volume Ω Surface forces : F d on the surface Ω F Statically admissible stress : in Ω divσ + f d = 0 σ ij,j + f d i = 0 on Ω F σ.n = F d σ ij n j = F d i Spherical part of the stress tensor : S = 1 3 trace(σ )I S ij = σ ll 3 δ ij Deviator associated to the stress tensor : s = σ S s ij = σ ij σ ll 3 δ ij trace(s ) = s ll = 0 Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
12 Kinematics and statics in small strains Physical meaning of the terms of the stress tensor x σ 2 22 σ12 σ 21 σ x 1 11 The diagonal terms characterize the normal forces The non diagonal terms characterize the shear forces Stress vector for a facet of direction n T (n) = σ.n T i = σ ij n j Normal stress for a facet of direction n Shear on a facet of direction n T n (n) = n.t = n.σ.n = σ ij n i n j T t (n) = T T n n T t = T 2 T 2 n Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
13 Internal/external forces Work of internal/external forces Stokes theorem for a scalar function f integrated on a volume Ω, n being the normal to Ω Z Z f,j dv = f n j ds Ω Ω Work of internal forces (real stress field, kinematically admissible displacement field) Z Z W i = σ ij ε ij dv = σ ij u i,j dv Ω Ω Z ( = (σij u i ),j σ ij,j u ) Z Z i dv = σ ij u i n jds σ ij,j u i dv Ω Ω Ω Work of external forces Z Z W e = fi d u i dv + Fi d u i ds Ω Ω F Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
14 Internal/external forces Application of virtual work theorem Total work (internal+external) should be zero for an isolated system with W i + W e = 0, it comes : Z Z Z σ ij u i n jds + σ ij,j u i dv + Ω Ω Ω equilibrium equation in Ω boundary condition on Ω F f d i u i dv + Z σ ij,j + f d i = 0 divσ + f = 0 Ω F F d i u i ds = 0 σ ij n j = F d i σ.n = F d These relations do not depend on the material The constitutive equations are the relations between stresses and strains Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
15 Elastic potential, linear elasticity Elastic potential The behaviour, eventually non linear, is fully defined by a potential, given by its volumetric density. Its shape depends on the representative variable Evolution between two equilibrium states, with σ = σ, and ε = d ε, the elastic potential W(ε ) writes, in linear elasticity : W(ε ) = 1 2 ε : C : ε σ = W ε = C : ε Evolution between two equilibrium states, with ε = ε, and σ = dσ, the complementary elastic potential W (σ ) writes, in linear elasticity : W (σ ) = 1 2 σ : S : σ ε = W σ = S : σ W and W convex and dw + dw = d(σ ij ε ij ) And : 2 W = σ ij = C ijkl = σ kl = C klij ε ij ε kl ε kl ε ij Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
16 Elastic potential, linear elasticity Linear elasticity Linear elasticity (stiffness and compliance matrices) : σ = C : ε ε = S : σ σ ij = C ijkl ε kl ε ij = S ijkl σ kl Symmetry relations : C ijkl = C ijlk = C jikl S ijkl = S ijlk = S jikl Energy related relations : C ijkl = C klij S ijkl = S klij General anisotropy = 21 coefficients ; orthotropy = 9 coeff ; cubic symmetry = 3 coeff ; isotropic material = 2 coefficients Isotropic material : s = 2µε dev σ ll = 3κε ll Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
17 Elastic potential, linear elasticity Isotropic elasticity Shear modulus µ such as τ = µγ Compressibility modulus κ such as p = 1 3 σ ll = κ V Young s modulus E such as σ = E ε in tension Poisson s ratio ν such as ε T = νε L in tension (ε T, transverse strain, ε L, longitudinal strain) Stress versus strain V σ = λtr ε I + 2µε σ ij = λε ll δ ij + 2µε ij Strain versus stress ε = 1 + ν E σ ν E Trσ I ε ij = 1 + ν E σ ij ν E ε llδ ij Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
18 Elastic potential, linear elasticity Relations between the elastic coefficients Expressions of λ, µ et κ λ = Eν (1 + ν)(1 2ν) Expressions of E et ν 2µ = E 1 + ν 3κ = E = 3λ + 2µ 1 2ν E = µ(3λ + 2µ) λ + µ ν = λ 2(λ + µ) Typical values : Rubber : ν 1/3 2µ 3E/4 κ E ν 1/2 µ E/3 κ E Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
19 A few particular stress states Pure tension Onedimensional stress state σ = σ 0 n n ; for instance in a prism of axis x 1, x 1 being the tensile direction, and the lateral faces being free : σ := σ For a section S 0, the force in direction x 1 is : F = σ 0 S 0 In the frame x 1 x 2 x 3, the strain tensor writes : ε := σ 0/E νσ 0 /E νσ 0 /E If the length is L 0, the elongation in direction x 1 is : L = εl 0 The stiffness of the prism is R = F/ L = ES 0 /L 0 Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
20 A few particular stress states Pure shear τ τ τ τ τ = σ 12 = 2µε 12 Deviatoric loading Example of pure shear in the plane x 1 x 2 σ := 0 τ 0 τ Rotation of π/4 σ := τ τ Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
21 A few particular stress states Circular bending Only one component in the stress tensor, but non uniform in space : σ := σ 11(x 3 ) For instance : σ 11 = Mx 3, where M is the bending moment around x 2, and Z I I = x3 2 ds is the inertia of the section with respect to x 2 S A prism of axis x 1 submitted to such a loading type presents a relative rotation of its sections characterized by an angle θ such as θ, 1 = M EI The strain can be expressed as a function of the curvature u 3,11 by : ε 11 = x 3 u 3,11 For a rectangular section of height h along x 3 and width b along x 2 : I = bh3 12 Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
22 A few particular stress states Torsion x 3 β Displacements u 1 = αx 3 x 2 u 2 = αx 3 x 1 u 3 = αφ(x 1,x 2 ) Stress γ σ 13 = µα(φ,1 x 2 ) =µαθ,2 (1) σ 23 = µα(φ,2 + x 1 )= µαθ,1 (2) avec φ = 0 θ + 2 = 0 θ = 0 sur Γ Γ contour of the section A line parallel to the prism axis becomes helicoidal Torsion moment : Z M = (x 1 σ 23 x 2 σ 13 )ds S Torsion stiffness modulus : Z D = 2µ θdx 1 dx 2 = M/α S Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
23 A few particular stress states Torsion, circular section For a circular prism of length L, and internal radius R e : β = αl At the external surface γ = 2ε θz = αr Shear stress τ = µαr θ can be expressed as θ = 1 2 (R2 x 2 1 x 2 2 ) A section perpendicular to x 3 remains plane : φ = 0 Tube with an internal radius R i and external R e : D = µ π(r4 e Ri 4 ) 2 Thin tube, with a radius R and a width e : D = 2µπeR 3 = M/α thus α = τ µr = M 2πµeR 3, and τ = M 2eπR 2 Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
24 A few particular stress states Cylindrical coordinates Expressions limited to the classical examples where : - the displacement is colinear to e r, u = u r e r - the only non zero volumetric force is f r e r. Equilibrium σ rr,r + σ rr σ θθ + f r = 0 r Strain that is : rε θθ, r = ε rr ε θθ ε rr = u r,r ε θθ = u r r Assuming zero volume forces, internal radius a, external radius b σ rr = A B r 2 σ θθ = A + B r 2 u r = Cr + D/r Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
25 A few particular stress states Cylindre under pressure Tube under pressure, internal pressure p i, external pressure p e Solid cylinder (p i = 0, a = 0, p e = p), A = p ia 2 p e b 2 b 2 a 2 B = (p i p e )a 2 b 2 b 2 a 2 σ rr = σ θθ = p Internal pressure (p i = p, a, b), ) σ rr = (1 a2 )p b2 σ b 2 a 2 r 2 θθ = (1 a2 + b2 b 2 a 2 r 2 p Thin tube under internal pressure p, radius R, thickness e, σ rr negligeable σ θθ = pr/e Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
26 A few particular stress states Spherical coordinates Expressions limited to the classical examples where : - the displacement is colinear to e r, u = u r e r - the only non zero volumetric force is f r e r. Equilibrium σ rr,r + 2 σ rr σ θθ + f r = 0 r Strain that is : rε θθ, r = ε rr ε θθ ε rr = u r,r ε θθ = u r r Zero volume force, internal radius a, external radius b σ rr = A 2B r 3 σ θθ = σ φφ = A + B r 3 u r = Cr + D/r 2 Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
27 A few particular stress states Sphere under pressure Sphere under pressure, internal pressure p i, external pressure p e Solid sphere (p i = 0, a = 0, p e = p), A = p ia 3 p e b 3 b 3 a 3 B = (p i p e )a 3 b 3 2(b 3 a 3 ) σ rr = σ θθ = σ φφ = p Internal pressure (p i = p, a, b), ) σ rr = (1 a3 )p b3 σ b 3 a 3 r 3 θθ = (1 a3 + b3 b 3 a 3 2r 3 p Thin tube mince under internal pressure p, radius R, thickness e, σ rr négligeable σ θθ = pr/2e Georges Cailletaud (Centre des Matériaux/UMR 7633 ) Basic concepts for MoM October / 24
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