Constitutive equations

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Constitutive Equations
David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 October 4, 2000

Introduction
The modules on kinematics (Module 8), equilibrium (Module 9), and tensor transformations (Module 10) contain concepts vital to Mechanics of Materials, but they do not provide insight on the role of the material itself.The kinematic equations relate strains to displacement gradients, and the equilibrium equations relate stress to the applied tractions on loaded boundaries and also govern the relations among stress gradients within the material. In three dimensions there are six kinematic equations and three equilibrum equations, for a total of nine. However, there are ﬁfteen variables: three displacements, sixstrains, and six stresses. We need six more equations, and these are provided by the material’s consitutive relations: six expressions relating the stresses to the strains. These are a sort of mechanical equation of state, and describe how the material is constituted mechanically. With these constitutive relations, the vital role of the material is reasserted: The elastic constants that appear inthis module are material properties, subject to control by processing and microstructural modiﬁcation as outlined in Module 2. This is an important tool for the engineer, and points up the necessity of considering design of the material as well as with the material.

Isotropic elastic materials
In the general case of a linear relation between components of the strain and stress tensors, we mightpropose a statement of the form
ij

= Sijkl σkl

where the Sijkl is a fourth-rank tensor. This constitutes a sequence of nine equations, since each component of ij is a linear combination of all the components of σij . For instance:
23

= S2311 σ11 + S2312 σ12 + · · · + S2333

33

Based on each of the indices of Sijkl taking on values from 1 to 3, we might expect a total of 81independent components in S. However, both ij and σij are symmetric, with six rather than nine independent components each. This reduces the number of S components to 36, as can be seen from a linear relation between the pseudovector forms of the strain and stress:

1

        

x y z

        

 γyz     γxz   

γxy

  S21 = .   .   .     S61  S11 S12 · · · S16 S22 · · · S26 . . .. . . . . . S26 · · · S66

                 

σx σy σz τyz τ xz τxy

                

(1)

It can be shown1 that the S matrix in this form is also symmetric. It therefore it contains only 21 independent elements, as can be seen by counting the elements in the upper right triangle of the matrix, including thediagonal elements (1 + 2 + 3 + 4 + 5 + 6 = 21). If the material exhibits symmetry in its elastic response, the number of independent elements in the S matrix can be reduced still further. In the simplest case of an isotropic material, whose stiﬀnesses are the same in all directions, only two elements are independent. We have earlier shown that in two dimensions the relations between strains andstresses in isotropic materials can be written as
x y 1 = E (σx − νσy ) 1 = E (σy − νσx ) 1 γxy = G τxy

(2)

along with the relation G= E 2(1 + ν)

Extending this to three dimensions, the pseudovector-matrix form of Eqn. 1 for isotropic materials is
        
x y z

        

 γyz     γxz   

γxy

    =   0       0   

1 E −ν E −ν E−ν E 1 E −ν E

0

0 0 0

−ν E −ν E 1 E

0 0 0

1 G

0 0 0

0 0

1 G

0 0 0 0

0

1 G

0 0 0 0 0

                  

σx σy σz τyz τ xz τxy

                

(3)

The quantity in brackets is called the compliance matrix of the material, denoted S or Sij . It is important to grasp the physical signiﬁcance of its various...