Transversely isotropic parametrizations

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See also

Theory: The Walpole basis

Manual: Structured tensors · Parametrizations and conversions

Tutorials: The Walpole basis end to end · Acoustic and Hill tensors

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A transversely isotropic tensor with major symmetry has five degrees of freedom. Which five numbers one uses is a matter of discipline, not of mathematics, and TensND supports three conventions. All of them build a TensTI{4,T,5} and all are interconvertible through the Walpole coefficients of The Walpole basis.

Every matrix below is the Kelvin–Mandel matrix in the frame where  , with the ordering and factors of Kelvin–Mandel representation.

Summary

Constructor	Extractor	Parameters	Builds a
ti_from_constants	ti_constants		stiffness or compliance
ti_from_constants	ti_constants		compliance
ti_from_constants	ti_constants		compliance

To obtain a stiffness from either compliance form, invert: inv(ti_from_constants(…; form = :engineering)). Inversion is exact and stays in the class, by the synthetic rule of The Walpole basis.

Component form

The five independent components of a minor- and major-symmetric TI tensor:

The entry is not independent: in-plane isotropy forces    , which is what reduces nine orthotropic constants to five.

Relation to the Walpole coefficients

Inverted:

These are exactly _build_TI_KM and its inverse, exported as mandel_from_ti_params and ti_params_from_mandel.

Engineering form

The convention of composite mechanics, for the compliance:

• — transverse Young's modulus, in the isotropy plane;

• — axial Young's modulus, along ;

• — in-plane Poisson's ratio;

• — axial–transverse Poisson's ratio, with the reciprocity  ;

• — axial shear modulus.

The entry encodes the in-plane shear modulus   , which is determined by and — the isotropy of the transverse plane again.

Kelvin–Mandel, not Voigt

The shear entries read and  , where an engineering (Voigt) compliance matrix would show and  . The factor two is the Kelvin–Mandel convention, and it is what keeps   an ordinary matrix inverse.

Hoenig form

A dimensionless parametrization introduced for crack problems in an anisotropic medium (Hoenig, 1978), convenient when anisotropy ratios matter independently of the overall stiffness scale:

Parameter	Definition
	transverse Young's modulus,
	in-plane Poisson's ratio,
	axial–transverse Poisson's ratio,
	axial-to-transverse modulus ratio,
	shear anisotropy parameter,

so that the compliance components are

Isotropy is the point    with  : measures the departure of the axial stiffness from the transverse one, and that of the axial shear modulus from the in-plane one. The companion paper Hoenig (1979) applies the parametrization to the effective moduli of a non-randomly cracked body.

Choosing between them

Use	Form
a stiffness known by its components, or symbolic work	ti_from_constants
material data sheets, composites	ti_from_constants
anisotropy ratios, crack and inclusion problems	ti_from_constants

Conversion between any two is a round trip through the tensor itself: build with one constructor, extract with another extractor. Since all three produce the same TensTI{4,T,5}, the conversion is exact rather than fitted — unlike the projection of a general tensor onto the class, which is the subject of Projection onto a symmetry class.