Elastic Properties of Solids, Part IIITopics Discussed in Kittel, Ch. 3, pages 73-85

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Elastic and Complimentary Energy Density

εdσC

σdεU

0

0

,0

0,

,

0U

0C = Uo + Co

Uo= Uo(xx , yy , zz , xy , yz , zx , x, y, z, T)

xx 1 xx yy zz xy xz yz

yy 2 xx yy zz xy xz yz

yz 6 xx yy zz xy xz yz

f , , , , ,

f , , , , ,

f , , , , ,

M-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Expressed in compliance matrix form

= S·

Expressed in stiffness matrix form

= C·

In general, stress-strain relationships such as these are known as constitutive relationsconstitutive relations

Note that the stiffness matrix is traditionally represented by the symbol C, while S is

reserved for the compliance matrix!

Internal Energy

0 0 xx xx yy yy zz zz

xy xy xz xz yz yz

C U σ ε σ ε σ ε

2σ ε 2σ ε 2σ ε

yy0 0 xx 0 0 zz

xx xx xx yy xx zz xx

xy yz0 0 xz 0

xy xx xz xx yz zz

U U U U

U U U

Strains

0 0 0xx yy zz

xx yy zz

0 0 0xy xz yz

xy xz yz

C C Cε ε ε

σ σ σ

1 C 1 C 1 Cε ε ε

2 σ 2 σ 2 σ

xyxy = 2 = 2xyxy yzyz = 2 = 2yzyz zxzx = 2 = 2zxzx

Hooke’s Law (Anisotropic)

xx 11 xx 12 yy 13 zz 14 xy 15 xz 16 yz

yy 21 xx 22 yy 23 zz 24 xy 25 xz 26 yz

zz 31 xx 32 yy 33 zz 34 xy 35 xz 36 yz

xy 41 xx 42 yy 43 zz 44 xy 45 xz 46 yz

xz 51 xx 52 yy 5

σ C ε C ε C ε C γ C γ C γ

σ C ε C ε C ε C γ C γ C γ

σ C ε C ε C ε C γ C γ C γ

σ C ε C ε C ε C γ C γ C γ

σ C ε C ε C

3 zz 54 xy 55 xz 56 yz

yz 61 xx 62 yy 63 zz 64 xy 65 xz 66 yz

ε C γ C γ C γ

σ C ε C ε C ε C γ C γ C γ

The 36 coefficients C11 to C66 are called elastic coefficients

The 36 coefficients C11 to C66 are called elastic coefficients

ij jiC C

21 independent constants

Hooke’s Law

Hooke’s Law

The generalized Hooke’s law is an assumption, which is reasonably accurate

for many material subjected to small strain, for a given temperature, time and location

Strain Energy Density

21 1 1 1 1 10 11 xx 12 xx yy 13 xx zz 14 xx xy 15 xx xz 16 xx yz2 2 2 2 2 2

21 1 1 1 1 121 yy xx 22 yy 23 yy zz 24 yy xy 25 yy xz 26 yy yz2 2 2 2 2 2

21 1 1 1 131 zz xx 32 zz yy 33 zz 34 zz xy 35 zz x2 2 2 2 2

U C ε C ε ε C ε ε C ε γ C ε γ C ε γ

C ε ε C ε C ε ε C ε γ C ε γ C ε γ

C ε ε C ε ε C ε C ε γ C ε γ

1z 36 zz yz2

21 1 1 1 1 141 xy xx 42 xy yy 43 xy zz 44 xy 45 xy xz 46 xy yz2 2 2 2 2 2

21 1 1 1 1 151 xz xx 52 xz yy 53 xz zz 54 xz xy 55 xz 56 xz yz2 2 2 2 2 2

1 1 1 161 yz xx 62 yz yy 63 yz zz 64 yz x2 2 2 2

C ε γ

C γ ε C γ ε C γ ε C γ C γ γ C γ γ

C γ ε C γ ε C γ ε C γ γ C γ C γ γ

C γ ε C γ ε C γ ε C γ γ

21 1y 65 yz xz 66 yz2 2C γ γ C γ

Isotropic materials have only 2 independent variables (i.e. elastic constants) in their stiffness and compliance matrices, as opposed to the 21 elastic constants in the general anisotropicanisotropic case.

Isotropic materialIsotropic material

Eg: Metallic alloys and thermo-set polymers

The two elastic constants are usually expressed as the Young's modulus E and the Poisson's ratio .

Alternatively, elastic constants K (bulk modulus) and/or G (shear modulus) can also be used. For isotropic materials

G and K can be found from E and by a set of equations, and vice-versa.

Hooke's Law in Compliance Form

Hooke's Law in Stiffness Form

An isotropic material subjected to uniaxial tension in x direction, xx is the only non-zero stress. The strains in

the specimen are

Youngs Modulus from Uniaxial Tension

The modulus of elasticity in tension, Young's modulus E, is the ratio of stress to strain on the loading plane along the loading direction.

2nd Law of Thermodynamics and understanding that under uniaxial tension, material must elongate in length implies:

E > 0

Shear Modulus for Pure Shear

Isotropic material subjected to pure shear, for instance, a cylindrical bar under torsion in the xy sense, xy is the

only non-zero stress. The strains in the specimen are

Shear modulus G:Ratio of shear stress to engineering shear strain on the loading plane

2nd Law of Thermodynamics and understanding that a positive shear stress leads to a positive shear strain implies

G > 0

Since both G and E are required to be positive, the quantity in the denominator of G must also be positive. This requirement places a lower bound lower bound restriction on the range for Poisson's ratiorestriction on the range for Poisson's ratio, ,

> -1

G=E/2(1+)

Bulk Modulus for Hydrostatic stress

For an isotropic material subjected to hydrostatic pressure , all shear stress will be zero and the normal stress will be uniform

Also note: K > 0

Under hydrostatic load, material will change its volume. Its resistance to do so is termed as bulk modulus K, or modulus of compression.

hydrostatic pressure

K =

relative volume change

The fact that both bulk modulus K and the elastic modulus E are required to be positive, it sets an upper bound of Poisson's ratio

< 1/2

K=E/ 3(1-2)

Orthotropic material has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane.

Eg: Certain engineering materials, 2-ply fiber-reinforced composites, piezoelectric materials (e.g.Rochelle salt)

Orthotropic material require 9 independent variables (i.e. elastic constants) in their constitutive matrices.

Orthotropic materialOrthotropic material

The 9 elastic constants in orthotropic constitutive equations are comprised of

3  Young's modulii Ex, Ey, Ez,

3 Poisson's ratios yz, zx, xy,

3 shear modulii Gyz, Gzx, Gxy.

Note that, in orthotropic materials, there are no Note that, in orthotropic materials, there are no interaction between the normal stresses interaction between the normal stresses xx, , yy, , zz

and the shear strains and the shear strains yzyz, , zxzx, , xyxy

Hooke’s law in compliance matrix form

Hooke’s law in stiffness matrix form

End of session 2