Cijkl

Cijkl is a symbol used in mathematics, physics, and engineering to denote the components of a fourth-order tensor with four indices i, j, k, and l. It is not a single universal object but a generic placeholder for a multilinear map or its component array, whose precise meaning depends on the context and the choice of basis.

Definition and notation: In a fixed basis, the tensor C is represented by the array of numbers Cijkl, where i, j, k, and l run over the dimension of the space (for example, 1 to 3 in three-dimensional space). As a tensor, C transforms according to the usual tensor transformation laws when the basis is changed, ensuring that its geometric meaning is basis-independent.

Symmetry and special cases: Depending on the problem, Cijkl may possess symmetries. Minor symmetries involve index pairs, such as Cijkl = Cjikl or Cijkl = Cijlk, and a major symmetry Cijkl = Cklij may occur if the tensor derives from a scalar potential. These symmetries reduce the number of independent components and often reflect underlying physical or geometric principles.

Typical contexts and applications: One common usage is in continuum mechanics, where Cijkl relates stress to strain through a constitutive law, for example sigma_ij = Cijkl epsilon_kl in linear elasticity. Fourth-order tensors also appear in differential geometry and material science to describe curvature-like properties or anisotropic responses in complex media.

Examples: In isotropic linear elasticity, a standard form is Cijkl = lambda delta_ij delta_kl + mu (delta_ik delta_jl + delta_il delta_jk), where delta is the Kronecker delta and lambda, mu are material constants.

See also: Tensor, Fourth-order tensor, Elasticity tensor, Isotropy.