Page 113 - The Mechatronics Handbook

0066_Frame_C08 Page 4 Wednesday, January 9, 2002 3:48 PM

rotations do not cause deformation of the body, they do not affect the internal stress field either. In fact,
the latter is a function of the gradient of the displacement, called deformation gradient. When this
gradient is small, a linear relationship between the displacements and strains can be used

e x = --------, e y = --------, e z = --------, e xy = -------- + --------, e xz = -------- + --------, e zy = -------- + ∂u y (8.14)
∂u y
∂u y
∂u x
∂u z
∂u z
∂u x
∂u z
∂u x
--------
∂x ∂y ∂z ∂y ∂x ∂z ∂x ∂y ∂z
The conservation of momentum and kinematic relations does not contain any information about the
material. Constitutive laws provide this additional information. The most common such law describes a
linear elastic material and can be conveniently expressed using a symmetric matrix c ij , called stiffness matrix:
T xx c 11 c 12 c 13 c 14 c 15 c 16 e x
T yy c 22 c 23 c 24 c 25 c 26 e y
= ⋅ (8.15)
T zz c 33 c 34 c 35 c 36 e z
T yz c 44 c 45 c 46 e yz
symm.
T zx c 55 c 56 e zx
T xy c 66 e xy
In the most general case, the matrix c ij has 21 independent elements. When the material has a crystal
symmetry, the number of independent constants is reduced. For example, single crystal Si is a common
structural material in MEMS with a cubic symmetry. In this case there are only three independent
constants:

0 0 0
T xx c 11 c 12 c 12 e x
0 0 0
T yy c 11 c 12 e y
0 0 0
T zz c 11 ⋅ e z
= (8.16)
0 0
T yz c 44 e yz
symm. 0
T zx c 44 e zx
T xy c 44 e xy
If the material is isotropic (amorphous or polycrystalline), the number of independent elastic constants
is further reduced to two by the relation c 44 = (c 11 − c 12 )/2. The elastic constants of several most commonly
used materials are listed in Table 8.1 (from [Kittel 1996]).
Additional information on other symmetry classes can be found in [Nye 1960].

TABLE 8.1 Elastic Constants of Several Common Cubic
Crystals
11
Stiffness Constants at Room Temperature, 10 N/m 2
Crystal c 11 c 12 c 44
W 5.233 2.045 1.607
Ta 2.609 1.574 0.818
Cu 1.684 1.214 0.754
Ag 1.249 0.937 0.461
Au 1.923 1.631 0.420
Al 1.608 0.607 0.282
K 0.0370 0.0314 0.0188
Pb 0.495 0.423 0.149
Ni 2.508 1.500 1.235
Pd 2.271 1.761 0.17
Si 1.66 0.639 0.796

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