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Generalized Hooke's CH.6. LINEAR ELASTICITY Multimedia Course on Continuum Mechanics Overview Hypothesis of the Linear Elasticity Theory Lecture 1 Linear Elastic Constitutive Equation Lecture 2 Generalized Hooke’s Law Elastic Potential Lecture 3 Isotropic Linear Elasticity Isotropic Constitutive Elastic Constants Tensor Lecture 4 Lamé Parameters Isotropic Linear Elastic Constitutive Equation Young’s Modulus and Poisson’s Ratio Lecture 5 Inverse Isotropic Linear Elastic Constitutive Equation Spherical and Deviator Parts of Hooke’s Law Lecture 6 Limits in Elastic Properties Lecture 7 2 Overview (cont’d) The Linear Elastic Problem Governing Equations Lecture 8 Boundary Conditions The Quasi-Static Problem Lecture 9 Solution Lecture 10 Displacement Formulation Stress Formulation Lecture 11 Saint-Venant’s Principle Lecture 12 Uniqueness of the solution Linear Thermoelasticity Hypothesis of the Linear Elasticity Theory Lecture 13 Linear Thermoelastic Constitutive Equation Inverse Constitutive Equation Lecture 14 Thermal Stress and Strain 3 Overview (cont’d) Thermal Analogies Solution to the linear thermoelastic problem Lecture 15 1st Thermal Analogy 2nd Thermal Analogy Lecture 16 Lecture 17 Superposition Principle in Linear Thermoelasticity Lecture 18 Hooke’s Law in Voigt Notation Lecture 19 4 6.1 Hypothesis of the Linear Elasticity Theory Ch.6. Linear Elasticity 5 Hypothesis of the Linear Elastic Model The simplifying hypothesis of the Theory of Linear Elasticity are: 1. ‘Infinitesimal strains and deformation’ framework 2. Existence of an unstrained and unstressed reference state 3. Isothermal, isentropic and adiabatic processes 8 Hypothesis of the Linear Elastic Model 1. ‘Infinitesimal strains and deformation’ framework the displacements are infinitesimal: material and spatial configurations or coordinates are the same ≈ 0 x= Xu + xX≈ material and spatial descriptions of a property & material and spatial differential operators are the same: xX= γγ()()()()xXXx,t= , t =Γ=Γ ,, tt ∂•() ∂•() = ∇∇()•=() • ∂∂Xx ∂x the deformation gradientF1 = ≈ F1 ≈ , so the current spatial ∂X density is approximated by the density at the reference configuration. ρρ0 =ttF ≈ ρ Thus, density is not an unknown variable in linear elastic problems. 9 Hypothesis of the Linear Elastic Model 1. ‘Infinitesimal strains and deformation’ framework the displacement gradients are infinitesimal: The strain tensors in material and spatial configurations collapse into the infinitesimal strain tensor. EX()()(),,,ttt≈= exε x 10 Hypothesis of the Linear Elastic Model 2. Existence of an unstrained and unstressed reference state It is assumed that there exists a reference unstrained and unstressed neutral state, such that, εε00()()xx0=,t = σσ00()()x= x0,t = The reference state is usually assumed to correspond to the reference configuration. 11 Hypothesis of the Linear Elastic Model 3. Isothermal and adiabatic (=isentropic) processes In an isothermal process the temperature remains constant. θθ()()()xx,,tt≡≡00 θ x ∀ x In an isentropic process the entropy of the system remains constant. ds sts(XX ,)= ( ) = = 0 s = 0 dt In an adiabatic process the net heat transfer entering into the body is zero. = ρ − ⋅ = ∀∆ ⊂ Qe0 ∫∫ r dV qn dS0 V V VV∂ heat conduction internal REMARK from the exterior sources An isentropic process is an ρ0 rt−⋅=∇ qx0 ∀∀ idealized thermodynamic process that is adiabatic, isothermal and reversible. 12 6.2 Linear Elastic Constitutive Equation Ch.6. Linear Elasticity 13 Hooke’s Law R. Hooke observed in 1660 that, for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. σ F= kl ∆ Fl∆ = E E Al F ε σ=E ε Hooke’s Law (for 1D problems) states that in an elastic material strain is directly proportional to stress through the elasticity modulus. 14 Generalized Hooke’s Law This proportionality is generalized for the multi-dimensional case in the Theory of Linear Elasticity. = σε(x,tt) () x:( x ,) C Generalized σεij= C ijkl kl ij,∈{ 1, 2, 3} Hooke’s Law It constitutes the constitutive equation of a linear elastic material. th The 4 order tensor is the constitutive elastic constants tensor: Has 34=81 components. Has the following symmetries, reducing the tensor to 21 independent components: REMARK minor CCijkl= jikl symmetries The current stress at a point CCijkl= ijlk depends only on the current strain at the point, and not on the past major = symmetries CCijkl klij history of strain states at the point. 15 Elastic Potential The internal energy balance equation for the (adiabatic) linear elastic model is global form stress power heat transfer rate d duρ ρρ u dV = 0 dV=σ :d dV +() r −∇⋅q dV dt ∫00 ∫dt ∫∫ V V VV= ε internal energy infinitesimal strains REMARK ()Vt ≡∀ Vt local form The rate of strain tensor is related to the d material derivative of the material strain T ()ρρ0 ur=σ : ε + −∇⋅q = ⋅⋅ dt tensor through: E F dF In this case, E = ε and F = 1 . Where: u is the specific internal energy (energy per unit mass). r is the specific heat generated by the internal sources. q is the heat conduction flux vector per unit surface. 16 Elastic Potential The stress power per unit of volume is an exact differential of the internal energy density, uˆ , or internal energy per unit of volume: d duˆ(,)x t ()ρ uu= =ˆ = σε: REMARK 0 dt dt The symmetries of the uˆ constitutive elastic Operating in indicial notation: constants tensor are used: duˆ 1 = =σε: =εσ=εε=εε+εε=C() CC minor CCijkl jikl ij ij ij ijkl kl ij ijkl kl ij ijkl kl dt C:ε 2 symmetries = Cijklε kl ik↔ CCijkl ijlk jl↔ major 11 CCijkl= klij =εε+ε=εε+εε=()ijC ijkl klε kl C klij ij ( ij CC ijkl kl ij ijkl kl ) symmetries 22 C d ijkl (εεC ) dt ij ijkl kl 1 d duˆ d 1 =εεijC ijkl kl = ()εε::C 2 dt dt dt 2 εε::C 17 Elastic Potential duˆ 1 d =σε: = () ε ::C ε dt2 dt Consequences: 1. Consider the time derivative of the internal energy in the whole volume: dd d ∫∫uˆˆ()xx,, t dV= u() t dV=Uˆ() t = ∫σε : dV stress VVdt dt dt Vpower In elastic materials we talk about deformation energy because the stress power is an exact differential. REMARK The stress power, in elastic materials is an exact differential of the internal energy U ˆ . Then, in elastic processes, we can talk of the elastic energy U ˆ () t . 18 Elastic Potential duˆ 1 d =σε: = () ε ::C ε dt2 dt Consequences: 2. Integrating the time derivative of the internal energy density, 1 utˆ ()x,,= εε()()() x t ::xC , ta+ x 2 and assuming that the density of the internal energy vanishes at the neutral reference state, ut ˆ () xx ,0 0 = ∀ : 1 = 0 11 εε+==∀ ˆ ε= ε ε = σε ε ()()()x,t00 ::xC , ta x a () x 0 x u () :C : ( ): 2 22σ Due to thermodynamic reasons the internal energy is assumed always positive 1 uˆ ()ε= εε::C > 0∀≠ ε0 2 19 Elastic Potential σε()()x,tt= C () x: x , The internal energy density defines a potential for the stress tensor, and is thus, named elastic potential. The stress tensor can be computed as =σσT = ∂∂utˆ(ε (x , )) 1 1 1 1 ∂uˆ ()ε =(ε::CC ε) = : ε + ε : C = ( σ += σ ) σ σ = ∂∂εε ∂ε 2 222= σ The constitutive elastic constants tensor can be obtained as the second derivative of the internal energy density with respect to the strain tensor field, 2 ∂σε() ∂∂2uˆ ()εε()C : ∂ uˆ ()ε = = = C Cijkl = ∂ε ∂ εε ⊗∂ ∂ ε ∂εij ∂ε kl 20 6.3 Isotropic Linear Elasticity Ch.6. Linear Elasticity 21 Isotropic Constitutive Elastic Constants Tensor An isotropic elastic material must have the same elastic properties (contained in C ) in all directions. All the components of C must be independent of the orientation of the chosen (Cartesian) system C must be a (mathematically) isotropic tensor. C =λµ11 ⊗+2 I =++λδ δ µ δ δ δ δ ∈ Cijkl ij kl( ik jl il jk ) i, jkl , ,{} 1, 2, 3 Where: 1 is the 4th order unit tensor defined as []I =δδ + δδ I ijkl 2 ik jl il jk λ and µ are scalar constants known as Lamé‘s parameters or coefficients. REMARK The isotropy condition reduces the number of independent elastic constants from 21 to 2. 22 Isotropic Linear Elastic Constitutive Equation Introducing the isotropic constitutive elastic constants tensor C = λµ 11 ⊗+ 2 I into the generalized Hooke’s Law σε = C : , in index notation: σij=C ijkl ε kl =( λδ ij δ kl + µ( δ ik δ jl + δ il δ jk) ) ε= kl = ε 11ij =λδδij kl ε+ kl 2 µδδik jl ε+ kl δδ il jk ε kl = λTr()ε δµij + 2 ε ij 22 =εε = =εll = Tr(ε) ji ij 1 1 = ε+ εε= 2 ij 2 ij ij And the resulting constitutive equation is, Isotropic linear elastic σεε=λµTr ()1 + 2 constitutive equation. σ=+∈ λδ ε µ ε ij ij ll2 ij ij ,{} 1, 2, 3 Hooke’s Law 23 Elastic Potential If the constitutive equation is, σεε=λµTr ()1 + 2 Isotropic linear elastic constitutive equation. σij=+∈ λδ ij ε ll2 µ ε ij ij ,{} 1, 2, 3 Hooke’s Law Then, the internal energy density can be reduced to: 11 REMARK uˆ ()ε= σ : ε=(λµTr () ε1 + 2 ε) : ε= 22 The internal energy density is an =σ elastic potential of the stress tensor 11 as: =λµ ε1 ε+ ε:ε= Tr () :2 22ε ∂uˆ ()ε Tr( ) =σε()() =λµTr εε1 + 2 ∂ε 1 =λµTr2 ()ε + ε:ε 2 24 Inversion of the Constitutive Equation 1. ε is isolated from the expression derived for Hooke’s Law 1 σεε=λµTr ()1 + 2 ε=( σ−λ Tr () ε 1) 2µ 2. The trace of σ is obtained: Tr ()()σ= Tr(λ Tr ε11 +2 µ ε=) λ Tr()()()()() ε Tr+=+2 µ Tr ε 3 λµ 2 Tr ε =3 3.
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