I want to find out the elasticity matrix for my composite film, I need 36 values to fill my matrix , how can I measure it practically,
Please help me.
Dear Chandrasekhar; Because of symmetry the cubic system has only three independent elastic constant. C1, C2, C44. The anisotropy constant, in the honor of the scientist who introduced it, the ZENER anisotropy ratio is defined by the following connection: A= 2C44((C1-C2), where A=0 corresponds to isotropic elastic media. The maxiumu value of A= 1. Aluminum and tungsten, have value very clsoe to 1. The stiffness trigonal upper matric of an isotropic system is:
C11 C12 C12 0 0 0
. C11 C12 0 0 0
. . C11 0 0 0
. . . C44 0 0
. . . . C44 0
. . . . . C44
Where Lame' constants are given by: u= C44= 1/2(C11-C12) , which is also known as G rigidity or shear modulus, and Lamda =C12.
C11 C12 C44
Aluminum 10.62 6.13 2.85 10-10 Pa
Copper 16.84 12.14 7.54
Iron 23.70 14.10 11.60
Tungsten 50.10 19.80 15.14
While the matrix is indeed 6 by six in size and thus has 36 elements, there are not 36 independent elements. The matrix does have some symmetry, which reduces the number of independent elements to no more than 21, in the worst case (atomic lattice of lowest symmetry).
Lots of simplifications usually come in through knowledge of structure. Knowing the (crystal) structure is therefore pivotal to assessing which elements of the matrix still need being addressed (resp. can be addressed) by experiments.
If your film is not single crystalline then any experiment shall average (fully or to some extent) over spatial directions. This usually simplifies the resultant (effective) elasticity matrix but also means that not all the quantities intrinsic to the crystalline material can be obtained from experiments.
Concerning practical experiments on thin films, I am not an expert, though. The speed of sound and its dependence on spatial direction and polarisation is a very useful quantity in crystals. I could also imagine that determining structural changes (XRD) under external constraints (stress/strain) should be useful as well. Others might have more comprehensive recommendations.
Dear Chandrasekhar; Because of symmetry the cubic system has only three independent elastic constant. C1, C2, C44. The anisotropy constant, in the honor of the scientist who introduced it, the ZENER anisotropy ratio is defined by the following connection: A= 2C44((C1-C2), where A=0 corresponds to isotropic elastic media. The maxiumu value of A= 1. Aluminum and tungsten, have value very clsoe to 1. The stiffness trigonal upper matric of an isotropic system is:
C11 C12 C12 0 0 0
. C11 C12 0 0 0
. . C11 0 0 0
. . . C44 0 0
. . . . C44 0
. . . . . C44
Where Lame' constants are given by: u= C44= 1/2(C11-C12) , which is also known as G rigidity or shear modulus, and Lamda =C12.
C11 C12 C44
Aluminum 10.62 6.13 2.85 10-10 Pa
Copper 16.84 12.14 7.54
Iron 23.70 14.10 11.60
Tungsten 50.10 19.80 15.14
Dear Chandrasekhar,
attached you will find two files: Elasticity Matrix and Anisotropic Elasticity. These are parts of my textbook on structural mechanics by McGraw-Hill. If the elasticity is one of that anisotropic ones treated in the second file, you will perform selected experimental tests to detect the few material constants necessary to construct the elasticity Matrix of your material. In case you need more suggestions, please send me more information about your type of laminate to [email protected].
regards,
Luciano Nunziante.
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