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HGEOP 315
Fall 2003
Lab assignment #1
Elastic tensor
(Due date: October 5, 2003)
Objective
To understand the meaning of elastic constants and the symmetry relations that exist in the stress, strain, and elastic constants.
Introduction
Hookes law states that, at sufficiently small strains, the stress (s) is proportional to the strain (e). Mathematically:
s = C e, (1)
where C, the proportionality constant, is called the elastic constant. Equation (1) applies to scalar, vector, and matrix representations of stress and strain. If s and e are scalars, then C is also a scalar; if s and e are vectors, then C is a 2D matrix; if s and e are 2D matrices, then C is a tensor of rank 4. A tensor is a generalized mathematical quantity. A tensor of rank 0 is a scalar, a tensor of rank 1 is a vector, a tensor of rank 2 is a 2D matrix, a tensor of rank 3 is a 3D matrix, a tensor of rank 1 is a 4D matrix, and so on.
In order to handle tensors of rank 4, we have to change our representation of the indices of stress and strain. The following representation will be used throughout this exercise: x ( 1, y ( 2, z ( 3. For example, sxy will be written as s12, ezz will be e33, and so on. Since we have only 3 axis (x,y,z), then the stress and strain matrices will have 9 elements (3(3) and the elastic constant tensor will have 81 elements (3(3(3(3).
Exercises
(1) The 81 indices of the elastic tensor C can be represented by the following table. The first row is shown. Fill out the rest of the table.
111213212223313233111111111211131121112211231131113211331213212223313233
(2) Transform the indices according to the following rules:
11 ( 1, 22 ( 2, 33 ( 3, 23 ( 4, 13 ( 5, 12 ( 6, 32 ( 4, 31 ( 5, 21 ( 6. For example, the elements of the first row will become: 11, 16, 15, 16, 12, 14, 15, 14, 13. Write and fill out the rest of the table using this index representation.
(3) Omit any repeated indices. For example, in the first row, you should omit one of the 14, 15, and 16. You should end up with 36 elements. Arrange them in a 6(6 matrix.
(4) The symmetry in the stress, strain, and strain energy gives us the following symmetry in the elastic constant matrix: Cij = Cji. This means that not all of the 36 elements of C are independent. How many independent elements are there in C%4?NQ\
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(5) What is the lowest number of independent elastic constants that exists in a material? What do we call such a material? What are these constants called? What are their conventional symbols?
(6) Repeat steps (1) (4) for the stress matrix. You should end up with 6 independent elements. Write them out as a vector.
(7) Repeat steps (1) (4) for the strain matrix. You should end up with 6 independent elements. Write them out as a vector.
(8) Now, write out the vector form of Hooke s law using the stress and strain vectors and the elastic constant matrix.
(9) Write out each element sij in terms of Cij and eij. What are the strain elements that s1 depends on?
(10) What are the units of stress, strain, and elastic constants?
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