In the previous instalments of this series (see Part 1 and Part 2), I have shown that regardless of the dimensionality (3D or plane strain elasticity), the constitutive law of an isotropically elastic material reads
$$\sigma_{ij}=\kappa\varepsilon_{kk}\delta_{ij}+2\mu\left(\varepsilon_{ij}-\frac{\varepsilon_{kk}}d\delta_{ij}\right),\tag{1}$$
where $d$ is the dimension of the physical space ($d=3$ for 3D elasticity, $d=2$ for plane strain elasticity), $\mu$ is the shear modulus, and $\kappa$ is the bulk modulus, whose expression depends on $d$
$$\kappa=\frac23\frac{1+\nu}{1-2\nu}\mu\qquad(d=3),\tag{2}$$
$$\kappa=\frac\mu{1-2\nu}\qquad(d=2).\tag{3}$$
In this instalment, I am going to introduce some classical isotropic, fourth-rank tensors which will prove extremely useful and will allow us to cast Eq. (1) in an intrinsic (component-free) form. It should be noted that the following developments are restricted to fourth-rank tensors $\tens T$ with both minor symmetries
$$T_{ijkl}=T_{jikl}=T_{ijlk},$$
and major symmetry
$$T_{ijkl}=T_{klij}.$$
Spherical and deviatoric projection tensors
In the present section, I am going to introduce the fourth-rank identity, spherical and deviatoric projection tensors.
Fourth-rank identity tensor
The fourth-rank identity tensor $\tens{I}$ maps any second-rank, symmetric tensor $\tens u$ onto itself
$$\tens I\dbldot\tens u=\tens u.$$
It is straightforward to work out the components of $\tens I$ (accounting for minor symmetries)
$$I_{ijkl}=\frac12\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right).$$
Fourth-rank spherical projection tensor
The fourth-rank spherical projection tensor $\tens J$ extracts the spherical part of any symmetric, second-rank tensor $\tens u$
$$\tens J\dbldot\tens u=\frac1d\tr(\tens u)\tens\delta,$$
where $\tens\delta$ is the second-rank identity tensor (with components $\delta_{ij}$). In particular, $\tens J\dbldot\tens\delta=\tens\delta$. In intrinsic form, $\tens J$ reads
$$\tens J=\frac1d\tens\delta\otimes\tens\delta,$$
where $\otimes$ denotes the tensor product. The components of $\tens J$ are
$$J_{ijkl} = \frac1d\delta_{ij}\delta_{kl}.$$
Fourth-rank deviatoric projection tensor
The fourth-rank deviatoric projection tensor $\tens K$ extracts the deviatoric part of any symmetric, second-rank tensor $\tens u$
$$\tens K\dbldot\tens u=\tens u-\frac1d\tr(\tens u)\tens{\delta}=\tens u-\tens J\dbldot\tens u,$$
from which it results that
$$\tens K=\tens I-\tens J.$$
Stiffness tensor
Going back the constitutive law of the linearly elastic, isotropic material given by Eq. (1), it is found that
$$\tens\sigma=d\kappa\tens J\dbldot\tens\varepsilon+2\mu\tens K\dbldot\tens\varepsilon=\tens C\dbldot\tens\varepsilon,$$
where $\tens C$ is the fourth-rank stiffness tensor of the material
$$\tens C=d\kappa\tens J+2\mu\tens K.$$
In isotropic elasticity, the stiffness tensor is a linear combination of the two tensors $\tens J$ and $\tens K$. This is in fact a general result, which is heavily used in materials science: any isotropic fourth-rank tensor with minor and major symmetries is a linear combination of $\tens J$ and $\tens K$. In other words, $\tens J$ and $\tens K$ form a basis of the space of fourth-rank isotropic tensors with minor and major symmetries. Why this is useful will become obvious in the next section, where I will show that algebra in this basis is dead simple.
Algebra of the $\tens{J}$ and $\tens{K}$ tensors
It can readily be verified that
$$\tens J\dbldot\tens J=\tens J,\quad\tens K\dbldot\tens K=\tens K\quad\text{and}\quad\tens K\dbldot\tens J=\tens J\dbldot\tens K=\tens0.$$
Therefore, multiplication of two isotropic tensors $\tens T_i=a_i\tens{J}+b_i\tens{K}$ ($i=1,2$) is trivial
$$\tens T_1\dbldot\tens T_2=a_1a_2\tens J+b_1b_2\tens K.$$
Also, inversion of an isotropic tensor $\tens T=a\tens{J}+b\tens{K}$ is straightforward
$$\tens T^{-1}=\frac1a\tens J+\frac1b\tens K.$$
All these expressions will prove extremely useful in due time.
Conclusion
This is the end of this series on the elastic constants of isotropic materials. I have shown that (regardless of the dimension $d$ of the physical space), such materials are characterized by two constants. The shear modulus $\mu$, the Poisson ratio $\nu$ and Young's modulus $E$ do not depend on $d$, while the plane strains expression of the bulk modulus $\kappa$ differs from its 3D expression (see Eqs. (2) and (3)).
This series was also a good opportunity to introduce the tensors $\tens J$ and $\tens K$, which form a basis for the isotropic, fourth-rank tensors. Using this basis is very convenient, as algebra within this basis is extremely simple.