Introduction to materials modelling Lecture 6 - Transversely - - PowerPoint PPT Presentation

Introduction to materials modelling

Lecture 6 - Transversely isotropic elasticity Reijo Kouhia

Tampere University, Structural Mechanics

October 4, 2019

• R. Kouhia (Tampere University, Structural Mechanics)

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Constitutive models - symmetries

Eight possible linear elastic symmetries

Figure from Chadwick, Vianello, Cowin, JMPS, 2001.

• R. Kouhia (Tampere University, Structural Mechanics)

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Transverse isotropy

Definition: For transversely isotropic material there exists a material direction defined by a unit vector m m m such 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)

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Representation theorem for transversely isotropic elasticity

The spesific strain energy (or alternatively the spesific complementary energy) can depend on five invarints W = W(I1, I2, I3, I4, I5), where the invariants can be defined as I1 = trε, I2 = 1

2tr(ε2),

I3 = 1

3tr(ε3),

I4 = tr(εM ), I5 = tr(ε2M ), where M = mmT is the structural tensor of transverse isotropy.

• R. Kouhia (Tampere University, Structural Mechanics)

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Linear transversely isotropic elasticity

Five material parameters, engineering constants ET Modulus of elasticity in the transverse isotropy plane, EL Modulus of elasticity in the longitudinal direction m m m, GL Shear modulus in the plane containing the symmetry axis, νT Poisson’s ratio in the isotropy plane, νL Poisson’s ratio in the isotropy plane when load in the longitudinal direction.

• R. Kouhia (Tampere University, Structural Mechanics)

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Linear transverse isotropy, restrictions to the elastic constants

Thermodynamic restrictions to the material parameters ET > 0, EL > 0, GL > 0, −1 < νT < 1, −

• EL/ET < νL <
• EL/ET ,

−

• EL(1 − νT )

2ET < νL < −

• EL(1 − νT )

2ET . In addition the monotonicity of the modulus of elasticity in an arbitrary direction requires GL ≤ EL 2(1 + νL).

• R. Kouhia (Tampere University, Structural Mechanics)

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Linear transverse isotropy, determination of material constants

1

Stress in the longitudinal direction 1, i.e. σ11, measure ε11, ε22 = ε33, then E1 = EL = σ11/ε11 and νL = ν12 = ν13 = −ε22/ε11.

2

Stress in the transverse direction, i.e. σ22, measure strain in the three perpendicular direction ε11, ε22 and ε33, then E2 = ET = σ22/ε22, ν23 = −ε33/ε22 = νT

3

Shear in the 1-2 plane, then G12 = GL = τ12/γ12. Note G12 = G13.

4

This test is not necessary. Shear in the isotropy plane, i.e. in the 2-3 plane. G23 = τ23/γ23. Could also be obtained from G23 = E2/(1 + ν23).

• R. Kouhia (Tampere University, Structural Mechanics)

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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)

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