Introduction to materials modelling Lecture 6 - Transversely isotropic elasticity Reijo Kouhia Tampere University, Structural Mechanics October 4, 2019 R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 1 / 8
Constitutive models - symmetries Eight possible linear elastic symmetries Figure from Chadwick, Vianello, Cowin, JMPS , 2001. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 2 / 8
Transverse isotropy Definition: For transversely isotropic material there exists a material direction defined by a unit vector m m such m that the constitutive relations are unchanged for arbitrary rotations of the coordinate system about that axis. Examples: unidirectionally reinforced materials, stratified soils and rocks. Materials with hexagonal close packed crystal structure. Figures: unidirectional fibres from mscsoftware.com and Grand Canyon by Luca Galuzzi R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 3 / 8
Representation theorem for transversely isotropic elasticity The spesific strain energy (or alternatively the spesific complementary energy) can depend on five invarints W = W ( I 1 , I 2 , I 3 , I 4 , I 5 ) , where the invariants can be defined as I 2 = 1 2 tr( ε 2 ) , I 3 = 1 3 tr( ε 3 ) , I 1 = tr ε , I 5 = tr( ε 2 M ) , I 4 = tr( ε M ) , where M = mm T is the structural tensor of transverse isotropy. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 4 / 8
Linear transversely isotropic elasticity Five material parameters , engineering constants E T Modulus of elasticity in the transverse isotropy plane , E L Modulus of elasticity in the longitudinal direction m m m, G L Shear modulus in the plane containing the symmetry axis , ν T Poisson’s ratio in the isotropy plane , ν L Poisson’s ratio in the isotropy plane when load in the longitudinal direction . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 5 / 8
Linear transverse isotropy, restrictions to the elastic constants Thermodynamic restrictions to the material parameters E T > 0 , E L > 0 , G L > 0 , − 1 < ν T < 1 , � � E L /E T < ν L < E L /E T , − � � E L (1 − ν T ) E L (1 − ν T ) < ν L < − . − 2 E T 2 E T In addition the monotonicity of the modulus of elasticity in an arbitrary direction requires E L G L ≤ 2(1 + ν L ) . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 6 / 8
Linear transverse isotropy, determination of material constants Stress in the longitudinal direction 1 , i.e. σ 11 , measure ε 11 , ε 22 = ε 33 , then E 1 = E L = σ 11 /ε 11 1 and ν L = ν 12 = ν 13 = − ε 22 /ε 11 . Stress in the transverse direction, i.e. σ 22 , measure strain in the three perpendicular direction 2 ε 11 , ε 22 and ε 33 , then E 2 = E T = σ 22 /ε 22 , ν 23 = − ε 33 /ε 22 = ν T Shear in the 1-2 plane, then G 12 = G L = τ 12 /γ 12 . Note G 12 = G 13 . 3 This test is not necessary. Shear in the isotropy plane, i.e. in the 2-3 plane. G 23 = τ 23 /γ 23 . Could 4 also be obtained from G 23 = E 2 / (1 + ν 23 ) . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 7 / 8
Linear transverse isotropy - determination of material constants (cont’d) Figure from http://nptel.ac.in/courses/101104010/lecture12/12 4.htm R. Kouhia (Tampere University, Structural Mechanics) Introduction to materials modelling October 4, 2019 8 / 8
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