Invariants and Structural Invariants of the Anisotropic Elasticity Tensor

Ahmad, Faiz

doi:10.1093/qjmam/55.4.597

Cited by 16 publications

(22 citation statements)

References 3 publications

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“…Now we turn to the decomposition of the tensor S ijkl . We denote the two subtensors (1) S ijkl := αSg (ij g kl) ,…”

Section: Irreducible Decomposition Under the Rotation Groupmentioning

confidence: 99%

Quadratic Invariants of the Elasticity Tensor

Itin

2016

J Elast

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We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.

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“…Now we turn to the decomposition of the tensor S ijkl . We denote the two subtensors (1) S ijkl := αSg (ij g kl) ,…”

Section: Irreducible Decomposition Under the Rotation Groupmentioning

confidence: 99%

Quadratic Invariants of the Elasticity Tensor

Itin

2016

J Elast

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“…An important thing to note here is that, in order to define characters and orthogonality theorems, we require the density function. We do not have a direct expression for the density function of SO (4). But we can make use of the isomorphism…”

Section: T Iijj T Ij Ij T Ijj I Ij Kl T Ij Klmentioning

confidence: 99%

“…Due to (4.1), we can use the density function of SO (3) to obtain the density function of SO (4). The matrix representing an orthogonal coordinate transformation, in three dimensions, for a rotation through an angle ϕ, about x 3 -axis, is given by…”

Section: So(4) ∼ = So(3) ⊗ So(3)mentioning

confidence: 99%

“…Ting [3] considered quadratic invariants of the elasticity tensor and found two invariants for arbitrary rotations and 15 invariants when the transformation is confined to rotations about a fixed axis. Ahmad [4] enlarged these numbers respectively to four and 17. Norris [5] showed that the set of four invariants of Ahmad, combined with C 2 1 , C 2 2 and C 1 C 2 (which are indeed quadratic invariants) is complete, also the complete set of quadratic invariants, with one axis fixed, contains 35 members.…”

Section: Introductionmentioning

confidence: 99%

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Linear Invariants of a Cartesian Tensor Under SO(2), SO(3) and SO(4)

2010

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“…This tensor was used by Ahmad (2002) to find a quadratic invariant of the elasticity tensor. Its off-diagonal components are the same as those of V À U.…”

Section: An Axis Of Symmetrymentioning

confidence: 99%

The Cowin–Mehrabadi theorem for an axis of symmetry

Ahmad

2010

International Journal of Solids and Structures

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Keywords:Plane of symmetry Axis of symmetry Normal to a plane of symmetry Common eigenvectors a b s t r a c tThe well-known Cowin-Mehrabadi Theorem deals with necessary and sufficient conditions for a normal n to a symmetry plane. Necessary conditions require that n be a common eigenvector of C ijkk , C ikjk and C ijkl n j n l . It is shown that a vector parallel to an axis of symmetry must also satisfy these conditions. An axis of rotational symmetry is also a normal to a plane of symmetry except in the case of a trigonal material. Being a common eigenvector of C ijkk and C ikjk belonging to a nondegenerate eigenvalue guarantees it to be an axis of symmetry.

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Invariants and Structural Invariants of the Anisotropic Elasticity Tensor

Cited by 16 publications

References 3 publications

Quadratic Invariants of the Elasticity Tensor

Quadratic Invariants of the Elasticity Tensor

Linear Invariants of a Cartesian Tensor Under SO(2), SO(3) and SO(4)

The Cowin–Mehrabadi theorem for an axis of symmetry

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