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Ch.6. Linear Elasticity

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Ch.6. Linear Elasticity CH.6. LINEAR ELASTICITY Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Hypothesis of the Linear Elasticity Theory Linear Elastic Constitutive Equation Generalized Hooke’s Law Elastic Potential Isotropic Linear Elasticity Isotropic Constitutive Elastic Constants Tensor Lamé Parameters Isotropic Linear Elastic Constitutive Equation Young’s Modulus and Poisson’s Ratio Inverse Isotropic Linear Elastic Constitutive Equation Spherical and Deviator Parts of Hooke’s Law Limits in Elastic Properties 2 Overview (cont’d) The Linear Elastic Problem Governing Equations Boundary Conditions The Quasi-Static Problem Solution Displacement Formulation Stress Formulation Uniqueness of the solution Saint-Venant’s Principle Linear Thermoelasticity Hypothesis of the Linear Elasticity Theory Linear Thermoelastic Constitutive Equation Inverse Constitutive Equation Thermal Stress and Strain 3 Overview (cont’d) Thermal Analogies Solution to the linear thermoelastic problem 1st Thermal Analogy 2nd Thermal Analogy Superposition Principle in Linear Thermoelasticity Hooke’s Law in Voigt Notation 4 6.1 Hypothesis of the Linear Elasticity Theory Ch.6. Linear Elasticity 5 Introduction Elasticity: Property of solid materials to deform under the application of an external force and to regain their original shape after the force is removed. Stress: external force applied on a specified area. Strain: amount of deformation. The (general) Theory of Elasticity links the strain experienced in any volume element to the forces acting on the macroscopic body. Linear Elasticity is a simplification of the more general (nonlinear) Theory of Elasticity. 6 Hypothesis of the Linear Elastic Model The simplifying hypothesis of the Theory of Linear Elasticity are: 1. ‘Infinitesimal strains and deformation’ framework Both the displacements and their gradients are infinitesimal. 2. Existence of an unstrained and unstressed reference state The reference state is usually assumed to correspond to the reference configuration. 00xx0 ,t 00xx0 ,t 3. Isothermal, isentropic and adiabatic processes Isothermal - temperature remains constant Isentropic - entropy of the system remains constant Adiabatic - occurs without heat transfer 7 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 xXu xX material and spatial descriptions of a property & material and spatial differential operators are the same: xX xXXx,,,,tttt Xx F1 x the deformation gradient X F1 , so the current spatial density is approximated by the density at the reference configuration. 0 ttF 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 ex 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, 00xx0 ,t 00xx0 ,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. xx,,tt 00 x x 0 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. Q rdV qn dS0 V V e0 VV internal heat conduction fonts from the exterior 0 rt qx0 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. Fkl Fl E E l A l F E 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. xx:x,(),tt C Generalized ijC ijkl kl ij,1,2,3Hooke’s Law It constitutes the constitutive equation of a linear elastic material. The 4th 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 variation ddu udV 0 dV :d dV r q dV dt0 dt 0 VVVV internal energy infinitesimal strains REMARK VVtt local form The deformation rate tensor is related to d the material derivative of the material T 0 ur : q dt strain tensor through: EFdF In this case, E 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: ˆ ddut(,)x REMARK ()0 uuˆ : dt dt uˆ The symmetries of the constitutive elastic Operating in indicial notation: constants tensor are used: duˆ 1 : ()minor CCijkl jikl ij ij ijCCC ijkl kl ij ijkl kl ij ijkl kl dt : 2 symmetries C Cijkl kl ik CCijkl ijlk jl major 11 symmetries CCijkl klij ()ijCC ijkl kl kl klij ij ij CC ijkl kl ij ijkl kl 22 d Cijkl ) dt ijCijkl kl 1 d duˆ d 1 ijC ijkl kl ::C 2 dt dt dt 2 ::C 17 Elastic Potential duˆ 1 d ::: C dt2 dt Consequences: 1. Consider the time derivative of the internal energy in the whole volume: dd dˆ utdVutdVˆˆxxx,,,: U t dVstress VVdt dt dt Vpower In elastic materials we talk about deformation energy because the stress power is an exact differential. REMARK Internal energy in elastic materials is an exact differential. 18 Elastic Potential duˆ 1 d ::: C dt2 dt Consequences: 2. Integrating the time derivative of the internal energy density, 1 utˆ xx::xx,, tC , ta 2 and assuming that the density of the internal energy vanishes at the neutral reference state, ut ˆ xx ,0 0 : 1 0 11 x::x,,ttaa00C x ()0 x x uˆ ::C (): 2 22 Due to thermodynamic reasons the elastic energy is assumed always positive 1 uˆ ::C 0 0 2 19 Elastic Potential The internal energy density defines a potential for the stress tensor, and is thus, named elastic potential. The stress tensor can be computed as T utˆ((,)) x 1 1 1 1 uˆ (:: CC : : C ( ) 2222 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 2 () uˆ C : uˆ C Cijkl ij 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 112 I ijkl,,, 1,2,3 Cijkl ij kl ik jl il jk Where: 1 is the 4th order unit tensor defined as I I ijkl 2 ik jl il jk and are scalar constants known as Lamé 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 112 I into the generalized Hooke’s Law C : , (in indicial notation) ijC ijkl kl ij kl ik jl il jk kl 11ij ij kl kl2()2 ik jl kl il jk kl Tr ij ij 22 ll Tr() jii j 1 1 2 ij 2 ij ij The resulting constitutive equation is, Isotropic linear elastic Tr 1 2 constitutive equation. ij ij ll2,1,2,3 ij ij Hooke’s Law 23 Elastic Potential If the constitutive equation is, Tr 1 2 Isotropic linear elastic constitutive equation. ij ij ll2,1,2,3 ij ij Hooke’s Law Then, the internal energy density can be reduced to: 11 REMARK uTrˆ :2:1 22 The internal energy density is an σ elastic potential of the stress tensor 11 as: Tr 1:2 22Tr uˆ Tr 1 2 1 Tr 2 2 24 Inversion of the Constitutive Equation 1. is isolated from the expression derived for Hooke’s Law 1 Tr 1 2 Tr 1 2 2. The trace of is obtained: Tr Tr Tr 112 Tr Tr 2 Tr 32 Tr =3 3. The trace of is easily isolated: 1 Tr Tr 32 4. The expression in 3. is introduced into the one obtained in 1. 11 1 Tr 1 Tr 1 232 23 2 2 25 Inverse Isotropic Linear Elastic Constitutive Equation The Lamé parameters in terms of E and : 32 E E 112 E G 2 21 So the inverse const. eq. is re-written: 1 Tr 1 Inverse isotropic linear EE elastic constitutive equation. 1 ij,1,2,3 Inverse Hooke’s Law. ijEE ll ij ij 11 x xyz xyxy EG 11 y y x z xz xz In engineering notation: EG 11 z z x y yz yz E G 26 Young’s Modulus and Poisson’s Ratio Young's modulus E is a measure of the stiffness of an elastic material.
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