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.c om ng co an th cu u du o ng Chapter 4: Material Behavior – Linear elastic solid TDT University -‐ 2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn 4.1 Material characterization co ng 4.2 Linear elastic material – Hooke’s law an 4.3 Physical meaning of elastic module cu u du o ng th 4.4 Thermo-elastic constitutive relations CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn 4.1 Material characterization co an 4.3 Physical meaning of elastic module ng 4.2 Linear elastic material – Hooke’s law cu u du o ng th 4.4 Thermo-elastic constitutive relations CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn ⎧ ∂σ x ∂τ yx ∂τ zx ⎪ ∂x + ∂y + ∂z + Fx = ⎪ ⎪∂τ xy ∂σ y ∂τ zy + + + Fy = ⎨ ∂ x ∂ y ∂ z ⎪ ⎪ ∂τ ∂τ yz ∂σ z xz + + + Fz = ⎪ ∂y ∂z ⎩ ∂x ng co an th So far, we studied ∂u ∂v ∂w ⎧ ε = , ε = , ε = y z ⎪ x ∂x ∂y ∂z ⎪ ∂u ∂v ∂v ∂w ⎪ γ = + , γ = + , ⎨ xy yz ∂ y ∂ x ∂ z ∂ y ⎪ ⎪ ∂w ∂u γ = + ⎪ zx ∂ x ∂z ⎩ c om 4.1 Material characterization + equilibrium equations du o ng + strain-displacement equations cu u ⎧ ∂ ex ∂ e y ∂ 2exy ∂ ex ∂ ⎛ ∂eyz ∂ezx ∂exy ⎞ = + + + = ; ⎪ ⎜− ⎟ ∂ y ∂ z ∂ x ∂ x ∂ y ∂ z ∂ y ∂ x ∂ x ∂ y ⎝ ⎠ ⎪ 2 2 ∂ eyz ∂ ey ∂ ⎛ ∂ezx ∂exy ∂e yz ⎞ ⎪ ∂ e y ∂ ez + = ; = + + ⎨ ⎜− ⎟ ∂ z ∂ y ∂ y ∂ z ∂ z ∂ x ∂ y ∂ y ∂ z ∂ x ⎝ ⎠ ⎪ 2 2 ∂ ⎛ ∂exy ∂e yz ∂ezx ⎞ ⎪ ∂ ez + ∂ ex = ∂ ezx ; ∂ ez = + + ⎜− ⎟ ⎪ ∂x ∂z ∂z∂x ∂ x ∂ y ∂ z ∂ z ∂ x ∂ y ⎝ ⎠ ⎩ + compatibility equation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om 4.1 Material characterization ng In which, compatibility equations represent only independent relations, and these equations are needed only ensure that a given strain field will produce singlevalued continuous displacements => No need for the general problems th an co Excluding the compatibility relations, it is found that we have field equations The unknowns in these equations include displacement components, components of strain, and stress components => total 15 unknowns cu u du o ng So far, equations are not sufficient to solve for 15 unknowns • We need additional field equations • The material response => the relationship between the strains and stresses CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om 4.1 Material characterization u du o ng Solid Recovers Original Configuration When Loads Are Removed Linear Relation Between Stress and Strain Neglect Rate and History Dependent Behavior Include Only Mechanical Loadings Thermal, Electrical, Pore-Pressure, and Other Loadings Can Also Be Included As Special Cases cu • • • • • th an co ng Mechanical behavior of solids is normally defined by constitutive stress-strain relations Commonly, these relations express the stress as a function of the strain, strain rate, strain history, temperature, and material properties Here, we use the Linear Elastic Constitutive Solid Model in which the Stress-Strain Relations are under the Assumptions: CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om σ co ng 4.1 Material characterization an Typical One-Dimensional Tensile Sample Cast Iron Aluminum u du o ng th Stress-Strain Behavior Steel cu ε Applicable Region for Linear Elastic Behavior σ = Eε CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn 4.1 Material characterization an 4.3 Physical meaning of elastic module co ng 4.2 Linear elastic material – Hooke’s law cu u du o ng th 4.4 Thermo-elastic constitutive relations CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn σ ij = Cijkmε km C1113 C1211 C1212 C1213 C1311 C1312 C1122 C1123 C1221 C1222 C1223 C1321 C1322 C1132 C1133 C1231 C1232 C1233 C1331 C1332 co ng C1112 an C3123 C3221 C3222 C3133 C3231 C3232 du o C3122 C3132 ng th C2132 C2133 C2231 C2232 C2233 C2331 C2332 C3112 C3113 C3211 C3212 C3213 C3311 C3312 C3223 C3321 C3322 C3233 C3331 C3332 C1313 ⎤ C1323 ⎥⎥ C1333 ⎥ ⎥ C2313 ⎥ ⎡ε11 ε12 ε13 ⎤ C2323 ⎥ : ⎢⎢ε 21 ε 22 ε 23 ⎥⎥ ⎥ C2333 ⎥ ⎢⎣ε 31 ε 32 ε 33 ⎥⎦ C3313 ⎥ ⎥ C3323 ⎥ C3333 ⎥⎦ C1122 C1133 C1112 C1113 C1123 ⎤ ⎡ ε11 ⎤ C2222 C2233 C2212 C2213 C2223 ⎥⎥ ⎢⎢ ε 22 ⎥⎥ C3322 C3333 C3312 C3313 C3323 ⎥ ⎢ ε 33 ⎥ ⎥⎢ ⎥ C1222 C1233 C1212 C1213 C1223 ⎥ ⎢ 2ε12 ⎥ C1322 C1333 C1312 C1313 C1323 ⎥ ⎢ 2ε13 ⎥ ⎥⎢ ⎥ C2322 C2333 C2312 C2313 C2323 ⎥⎦ ⎢⎣ 2ε 23 ⎥⎦ u ⎡σ 11 ⎤ ⎡ C1111 ⎢σ ⎥ ⎢C ⎢ 22 ⎥ ⎢ 2211 ⎢σ 33 ⎥ ⎢C3311 ⎢ ⎥=⎢ ⎢σ 12 ⎥ ⎢ C1211 ⎢σ 13 ⎥ ⎢ C1311 ⎢ ⎥ ⎢ ⎢⎣σ 23 ⎥⎦ ⎢⎣C2311 ( K , M = 1, 2,3, 4,5, 6) C2112 C2113 C2211 C2212 C2213 C2311 C2312 C2122 C2123 C2221 C2222 C2223 C2321 C2322 cu ⎡σ 11 σ 12 ⎢σ ⎢ 21 σ 22 ⎢⎣σ 31 σ 32 ⎡ C1111 ⎢C ⎢ 1121 ⎢ C1131 ⎢ σ 13 ⎤ ⎢C2111 σ 23 ⎥⎥ = ⎢C2121 ⎢ σ 33 ⎥⎦ ⎢C2131 ⎢C ⎢ 3111 ⎢C3121 ⎢C ⎣ 3131 σ K = CKM ε M c om 4.2 Linear elastic material – Hooke’s law CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.2 Linear elastic material – Hooke’s law with Cijkl = C jikl ; Cijkl = Cijlk du o ( Due to the symmetry of stress and strain tensors) ng th an co σ ij = Cijkl ekl ng c om ⎧σ x = C11ex + C12e y + C13ez + 2C14exy + 2C15e yz + 2C16ezx ⎪ ⎪σ y = C21ex + C22e y + C23ez + 2C24exy + 2C25e yz + 2C26ezx ⎪σ = C e + C e + C e + 2C e + 2C e + 2C e 31 x 32 y 33 z 34 xy 35 yz 36 zx ⎪ z ⎨ ⎪τ xy = C41ex + C42e y + C43ez + 2C44exy + 2C45e yz + 2C46ezx ⎪τ = C e + C e + C e + 2C e + 2C e + 2C e 51 x 52 y 53 z 54 xy 55 yz 56 zx ⎪ yz ⎪⎩τ zx = C61ex + C62e y + C63ez + 2C64exy + 2C65e yz + 2C66ezx cu u ⎡σ x ⎤ ⎡ C11 C12 ⎢σ ⎥ ⎢C ⋅ ⎢ y ⎥ ⎢ 21 ⎢σ ⎥ ⎢ ⋅ ⋅ or ⎢ z ⎥ = ⎢ ⋅ ⎢τ xy ⎥ ⎢ ⋅ ⎢τ yz ⎥ ⎢ ⋅ ⋅ ⎢ ⎥ ⎢ ⎣τ zx ⎦ ⎣C61 ⋅ CuuDuongThanCong.com ⋅ ⋅ ⋅ C16 ⎤ ⎡ ex ⎤ ⋅ ⋅ ⋅ ⋅ ⎥ ⎢ ey ⎥ ⎥⎢ ⎥ ⋅ ⋅ ⋅ ⋅ ⎥ ⎢ ez ⎥ ⎥⎢ ⎥ ⋅ ⋅ ⋅ ⋅ ⎥ ⎢ 2exy ⎥ ⋅ ⋅ ⋅ ⋅ ⎥ ⎢ 2eyz ⎥ ⎥⎢ ⎥ ⋅ ⋅ ⋅ C66 ⎦ ⎣ 2ezx ⎦ https://fb.com/tailieudientucntt 36 Independent Elastic Constants 10 h"p://incos.tdt.edu.vn 4.2 Linear elastic material – Hooke’s law c om Prove Cijkl = QimQ jn Qkp Qlq Cmnpq co ng Cijkl = αδ ijδ kl + βδ ik δ jl + γδ ilδ jk an α ′δ ij′δ kl′ + β ′δ ik′ δ ′jl + γ ′δ il′δ ′jk = QimQ jnQkpQlq (αδ mnδ pq + βδ mpδ nq + γδ mqδ np ) th = α QimQ jmQkpQlp + β QimQ jnQkmQln + γ QimQ jnQknQlm du o ng = αδ ijδ kl + βδ ikδ jl + γδ ilδ jk u ′ = α ′δ ij′δ kl′ + β ′δ ik′ δ ′jl + γ ′δ il′δ ′jk Cijkl cu Cijkl = αδ ijδ kl + βδ ik δ jl + γδ ilδ jk 13 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.2 Linear elastic material – Hooke’s law c om Isotropic Materials ⎡σ x −ν (σ y + σ z ) ⎤⎦ Inverted Form - Strain in Terms of Stress E⎣ ⎡σ y −ν (σ z + σ x ) ⎤⎦ e = y σ kk = ( 3λ + 2µ ) ekk σ ij = λ ekk δ ij + 2µ eij E⎣ ez = ⎡⎣σ z −ν (σ x + σ y ) ⎤⎦ E ⎞ ⎛ λ eij = σ ij − σ kkδ ij ⎟ +ν e = τ = τ xy µ ⎜⎝ 3λ + µ ⎠ xy xy E 2µ +ν e = τ = τ yz yz yz +ν ν E µ eij = σ ij − σ kkδ ij E E +ν ezx = τ zx = τ zx E 2µ µ (3λ + 2µ ) E= Young’s modulus or modulus of elasticity λ+µ λ ν= Poisson’s ratio 2(λ + µ ) cu u du o ng th an co ng ex = 14 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn 4.1 Material characterization co ng 4.2 Linear elastic material – Hooke’s law an 4.3 Physical meaning of elastic module cu u du o ng th 4.4 Thermo-elastic constitutive relations 15 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.3 Physical meaning of elastic module c om Simple Tension an co ng Consider the simple tension with a sample subjected to tension in the x-direction The state of stress is represented by the one-dimensional field ⎡σ ⎤ 0 ⎢E ⎥ ⎢ ⎥ ⎡σ 0⎤ ν eij = ⎢ − σ ⎥ σ ij = ⎢⎢ 0 0⎥⎥ ⎢ ⎥ E ⎢⎣ 0 0⎥⎦ ⎢ ν ⎥ ⎢0 − σ⎥ E ⎦ ⎣ Slope of the stress-strain curve or E = σ / ex Elastic module in the x-direction du o cu u σ ng th σ ν = − e y / ex = − ez / ex Ratio of the transverse strain to the axial strain Standard measurement systems can easily collect axial stress and transverse and axial strain data, and thus through this, one type of test both elastic constants E 16 and ν can be determined for material of interest CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.3 Physical meaning of elastic module c om Pure Shear an th τ τ / 2µ ⎡ eij = ⎢τ / µ ⎢ ⎢⎣ 0 0⎤ 0⎥ ⎥ ⎥⎦ ng τ τ ⎡0 τ ⎤ σ ij = ⎢⎢τ 0 ⎥⎥ ⎢⎣0 0 ⎥⎦ co ng If a thin-walled cylinder is subjected to torsion loading, the state of stress on the surface of the cylindrical sample is given by du o τ = τ / γ xy Shear modulus which is the slope of the shear stress-shear strain curve cu u µ = τ / 2exy 17 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.3 Physical meaning of elastic module c om Hydrostatic Compression The final example is associated with the uniform compression (or tension) loading of a cubical specimen This type of test can be realizable if the sample was placed in a high-pressure compression chamber The state of stress for this case is given by ⎡ − 2ν ⎤ − p 0 p ⎢ ⎥ E ⎤ ⎡− p ⎢ ⎥ − ν ⎥ eij = ⎢ − p σ ij = ⎢⎢ − p ⎥⎥ ⎢ ⎥ E ⎢⎣ 0 − p ⎥⎦ ⎢ − 2ν ⎥ ⎢ 0 − p⎥ E ⎣ ⎦ p = − kekk = − kϑ Elastic constant k represents the ratio of pressure E to the dilatation (which represents the change in k= Bulk Modulus material volume) 3(1 − 2ν ) th cu u du o ng p an co ng p Note that when Poisson’s ratio approaches 0.5, the bulk modulus becomes unbounded and the material does not undergo any volumetric deformation and hence is incompressible 18 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om 4.3 Physical meaning of elastic module - Our discussion of elastic modulus for isotropic materials has led to the definition ng of five constants λ, µ, E, ν and k However, keep in mind that only two of these are co needed to characterize the material an - In can be shown that all five elastic constants are interrelated, and if any two are th given, the remaining three can be determined by using simple formulae Results of du o ng these relations are conveniently summarized in Table 4.1 - In addition, nominal values of elastic constants for particular engineering cu u materials are given in Table 4.2 19 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.3 Physical meaning of elastic module ν E,k E E,µ E E,λ E ν,k 3k (1 − 2ν) ν ,µ 2µ(1 + ν ) k,µ k,λ µ,λ k ng du o ν u λ(1 + ν )(1 − 2ν ) ν 9kµ 6k + µ 9k (k − λ ) 3k − λ µ(3λ + 2µ ) λ+µ cu ν ,λ 3k − E 6k E − 2µ 2µ 2λ E+λ+R ν ν 3k − 2µ 6k + 2µ λ 3k − λ λ 2(λ + µ) µE 3(3µ − E ) E + 3λ + R k 2µ(1 + ν ) 3(1 − 2ν ) λ (1 + ν ) 3ν µ E 2(1 + ν ) 3kE 9k − E ng E co E,ν k E 3(1 − 2ν ) an ν th E c om Table 4.1: Relations Among Elastic Constants µ E − 3λ + R 3k (1 − 2ν ) 2(1 + ν ) µ λ (1 − 2ν ) 2ν λ Eν (1 + ν )(1 − 2ν ) 3k (3k − E ) 9k − E µ(E − 2µ ) 3µ − E λ 3kν 1+ ν 2µν − 2ν λ k µ k− µ k (k − λ) λ 3λ + 2µ µ λ R = E + 9λ2 + Eλ CuuDuongThanCong.com 20 https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om 4.3 Physical meaning of elastic module Table 4.2: Typical Values of Elastic Moduli for Common Engineering Materials k(GPa) α(10-6/oC) 54.6 71.8 25.5 11.5 7.7 15.3 11 33.4 71 93.3 18 27.6 27.6 45.9 8.8 0.40 10.1 4.04 47.2 102 0.499 0.654x10-3 0.326 0.326 200 0.29 80.2 111 164 13.5 µ(GPa) Aluminum 68.9 0.34 25.7 Concrete 27.6 0.20 Cooper 89.6 0.34 Glass 68.9 0.25 Nylon 28.3 Rubber 0.0019 Steel 207 cu u du o ng th an co ν ng λ(GPa) E (GPa) 21 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.3 Physical meaning of elastic module c om Hooke’s Law in Cylindrical Coordinates ng x3 co z θ dθ dr cu x1 ng σr u r τrθ du o σθ τrz τ rθ τ rz ⎤ σ θ τ θ z ⎥⎥ τ θ z σ z ⎥⎦ th an σz τθz ⎡σ r σ = ⎢τ rθ ⎢ ⎢⎣τ rz x2 σ r = λ (er + eθ + ez ) + µ er σ θ = λ (er + eθ + ez ) + µ eθ σ z = λ (er + eθ + ez ) + µ ez τ rθ = µ erθ τ θ z = µ eθ z τ zr = µ ezr 22 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn 4.3 Physical meaning of elastic module c om Hooke’s Law in Spherical Coordinates τRθ du o u cu x1 σφ ng R τφθ θ an th σθ co σR τRφ φ ⎡ σ R τ Rϕ τ Rθ ⎤ ⎢ ⎥ σ = ⎢τ Rϕ σ ϕ τ ϕθ ⎥ ⎢⎣τ Rθ τ ϕθ σ θ ⎥⎦ ng x3 σ R = λ (eR + eϕ + eθ ) + µ eR x2 σ ϕ = λ (eR + eϕ + eθ ) + µ eϕ σ θ = λ (eR + eϕ + eθ ) + µ eθ τ Rϕ = µ eRϕ τ ϕθ = µ eϕθ τ θ R = µ eθ R 23 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn 4.1 Material characterization co an 4.3 Physical meaning of elastic module ng 4.2 Linear elastic material – Hooke’s law cu u du o ng th 4.4 Thermo-elastic constitutive relations 24 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om 4.4 Thermo-elastic constitutive relations M) + eij( T) an eij = eij( co ng - It is well known that a temperature change in an unrestrained elastic solid produces deformation Thus a general strain field results from both mechanical and thermal effects Within the context of linear small deformation theory, the total strain can be decomposed into the sum of mechanical and thermal components as ng th - If T0 is taken as the reference temperature and T as an arbitrary temperature, the thermal strains in an unrestrained solid can be written in the linear form eij( ) = α ij (T − T0 ) du o T cu u where αij is the coefficient of thermal expansion tensor Notice that it is the temperature difference that creates thermal strain If the material is taken as isotropic, then eij must be an isotropic second-order tensor, and eij( ) = α (T − T0 )δ ij T where α is the coefficient of thermal expansion Table 4.2 provides typical values of this constant for some common materials 25 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn c om 4.4 Thermo-elastic constitutive relations - Notice that for isotropic materials, no shear strains are created by temperature change This result can be combined with the mechanical relation to give ng +ν ν σ ij − σ kkδ ij + α (T − T0 )δ ij E E (4.4.4) co eij = an - The corresponding results for the stress in terms of strain can be written as ng th σ ij = Cijkl ekl − βij (T − T0 ) u du o where βij is a second-order tensor containing thermo-elastic modulus This result is sometimes referred to as the Duhamel-Neumann thermo-elastic constitutive law The isotropic case can be found by simply inverting relation (4.4.4) to get cu σ ij = λ ekkδ ij + 2µ eij − ( 3λ + 2µ )α (T − T0 )δ ij - Having developed the necessary constitutive relations, the elasticity field equation system is now complete with 15 equations (6 strain-displacement, equilibrium, Hooke’s law) for 15 unknowns (3 displacements, strains and stresses) 26 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om ng co an th ng du o u cu CuuDuongThanCong.com https://fb.com/tailieudientucntt ... h"p://incos.tdt.edu.vn 4. 1 Material characterization co an 4. 3 Physical meaning of elastic module ng 4. 2 Linear elastic material – Hooke’s law cu u du o ng th 4. 4 Thermo -elastic constitutive relations 24 CuuDuongThanCong.com... h"p://incos.tdt.edu.vn 4. 1 Material characterization co an 4. 3 Physical meaning of elastic module ng 4. 2 Linear elastic material – Hooke’s law cu u du o ng th 4. 4 Thermo -elastic constitutive... h"p://incos.tdt.edu.vn 4. 1 Material characterization an 4. 3 Physical meaning of elastic module co ng 4. 2 Linear elastic material – Hooke’s law cu u du o ng th 4. 4 Thermo -elastic constitutive
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