Elastic Properties of Solids, Part III Topics Discussed in Kittel, Ch. 3, pages Another Lecture Found on the Internet!

Elastic and Complimentary Energy Density

 = U o + C o

U o = U o (  xx,  yy,  zz,  xy,  yz,  zx, x, y, z, T)

Expressed in compliance matrix form  = S· 

Expressed in stiffness matrix form  = C· 

constitutive relations In general, stress-strain relationships such as these are known as constitutive relations Note that the stiffness matrix is traditionally represented by the symbol C, while S is reserved for the compliance matrix!

Internal Energy

Strains  xy = 2  xy  yz = 2  yz  zx = 2  zx

Hooke’s Law (Anisotropic) The 36 coefficients C 11 to C 66 are called elastic coefficients

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

anisotropic 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 anisotropic case. Isotropic 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

lower bound restriction on the range for Poisson's ratio, 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 restriction 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 material

The 9 elastic constants in orthotropic constitutive equations are comprised of 3 Young's modulii E x, E y, E z, 3 Poisson's ratios yz, zx, xy, 3 shear modulii G yz, G zx, G xy. Note that, in orthotropic materials, there are no interaction between the normal stresses  x,  y,  z and the shear strains  yz,  zx,  xy

Hooke’s law in compliance matrix form

Hooke’s law in stiffness matrix form End of session 2