Elastic moduli

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See also

Theory: Isotropic tensors

Manual: Structured tensors · Symbolic and numeric · Parametrizations and conversions

Tutorials: Elastic Green's functions · The elastic sphere, solved symbolically · Differentiation and interoperability · Performance of the structured types · Differential operators, symbolically

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An isotropic linear elastic material has two independent constants, and six of them are in common use: the bulk modulus , Young's modulus , Lamé's first parameter , the shear modulus , Poisson's ratio and the P-wave (oedometric) modulus . Any two of them fix the others. TensND makes the tensor the meeting point: the stiffness, or the compliance, is built from any pair, and any modulus is read from either, so that a conversion is one function inside another.

to …	write
build a stiffness from two moduli	iso_stiffness_<pair>(a, b), e.g. iso_stiffness_young_poisson(E, ν) or iso_stiffness_E_ν(E, ν)
build a compliance from two moduli	iso_compliance_<pair>(a, b), e.g. iso_compliance_young_poisson(E, ν)
read one modulus	young_modulus(ℂ), poisson_ratio(ℂ), …, or young_modulus(𝕊; role = :compliance)
read two moduli	young_poisson(ℂ) (E_ν(ℂ)), lame_shear(ℂ) (λ_μ(ℂ)), …, with the same role
work in 2-D (plane strain)	the same builders with Val(2) first
switch between plane strain and plane stress	plane_strain_stiffness, plane_stress_stiffness, plane_stress_to_plane_strain, plane_strain_to_plane_stress

The stiffness and its moduli

In dimension 3 the stiffness is          , a TensISO{4,3}. The other moduli follow from :

Each has its reader, bulk_modulus, young_modulus, lame_lambda, shear_modulus, poisson_ratio and pwave_modulus, and each pair too, named after its two moduli in the order : bulk_shear, young_poisson, lame_shear, young_pwave, …

using TensND

ℂ = iso_stiffness_young_poisson(70.0, 0.33)   # an aluminum, GPa
(k = bulk_modulus(ℂ), E = young_modulus(ℂ), λ = lame_lambda(ℂ),
 μ = shear_modulus(ℂ), ν = poisson_ratio(ℂ), M = pwave_modulus(ℂ))

(k = 68.62745098039215, E = 70.0, λ = 51.08359133126935, μ = 26.31578947368421, ν = 0.33, M = 103.71517027863777)

Stiffness or compliance

The compliance is       with   and  . Every reader takes the keyword role, which says which of the two it is given: :stiffness, the default, or :compliance. Each modulus then has a formula of its own on each, written directly on the two coefficients, never through an inverse:

modulus	on	on

𝕊 = iso_compliance_young_poisson(70.0, 0.33)   # the same aluminum
young_poisson(𝕊; role = :compliance)

(70.0, 0.33)

The compliance builders write directly as well: from ,       . Nothing in a TensISO tells a stiffness from a compliance: the role is the caller's to give, and a compliance read as a stiffness gives the moduli of another material.

Building from any pair

The stiffness of a pair is iso_stiffness_<pair>, and its compliance iso_compliance_<pair> (iso_compliance for ), with the same names. The fifteen pairs, and how each gives (the formulas are those of the classical conversion table, in this notation):

pair	builder
	iso_stiffness_bulk_young
	iso_stiffness_bulk_lame
	iso_stiffness
	iso_stiffness_bulk_poisson
	iso_stiffness_bulk_pwave
	iso_stiffness_young_lame
	iso_stiffness_young_shear
	iso_stiffness_young_poisson
	iso_stiffness_young_pwave
	iso_stiffness_lame_shear
	iso_stiffness_lame_poisson
	iso_stiffness_lame_pwave
	iso_stiffness_shear_poisson
	iso_stiffness_shear_pwave
	iso_stiffness_poisson_pwave

with   and  . These two pairs solve a quadratic equation. From the root kept is the only one with   . From both roots are materials, one with   and one with  , and and cannot tell them apart; the one kept is  .

The pair says nothing of at  , where   as well. A conversion reads the result on the stiffness:

E, ν = young_poisson(iso_stiffness_lame_shear(40.0, 26.0))

(67.75757575757575, 0.30303030303030304)

young_shear(iso_stiffness_bulk_poisson(70.0, 0.25))

(105.0, 42.0)

Shorter names

The pairs without the P-wave modulus also have the names of their symbols, in ASCII and with Greek letters, as exact aliases: k_mu and k_μ are bulk_shear, E_nu and E_ν are young_poisson, λ_μ and lambda_mu are lame_shear, and the builders likewise, iso_stiffness_E_nu, iso_stiffness_E_ν, iso_stiffness_λ_μ, … and iso_compliance_E_nu, iso_compliance_E_ν, iso_compliance_λ_μ, …. The pairs with have none: k_M or μ_M name the moduli of a matrix in MeanFieldHomogenization, and the two meanings would be read one for the other.

k, μ = k_μ(iso_stiffness_E_ν(70.0, 0.33))

(68.62745098039215, 26.31578947368421)

Symbolic moduli

No formula depends on the sign of a value, so the moduli may be symbols. With SymPy, the closed forms come out exactly:

using SymPyPythonCall
λ, μ = symbols("λ μ", positive = true)
ℂ = iso_stiffness_lame_shear(λ, μ)
simplify(young_modulus(ℂ)), simplify(poisson_ratio(ℂ))

(μ*(3*λ + 2*μ)/(λ + μ), λ/(2*(λ + μ)))

They go through ForwardDiff as well: a derivative of a modulus with respect to another pair is that of the composition.

using ForwardDiff
ForwardDiff.derivative(ν -> shear_modulus(iso_stiffness_young_poisson(70.0, ν)), 0.33)

-19.78630787495053

Two dimensions

A TensISO{4,2} is the in-plane stiffness of a 3-D material under plane strain,      . Its , , , and are those of the 3-D material, while its bulk modulus is the in-plane one,   , the factor of over the dimension, as for bulk_shear. Every builder takes the dimension as a first argument Val(2) (Val(3) is the default):

ℂ₂ = iso_stiffness_young_poisson(Val(2), 70.0, 0.33)
bulk_shear(ℂ₂), young_poisson(ℂ₂)

((77.39938080495357, 26.31578947368421), (70.0, 0.33000000000000007))

On a 2-D compliance     , with    and  , the readers use

and the compliance builders take Val(2) as the stiffness builders do:

𝕊₂ = iso_compliance_young_poisson(Val(2), 70.0, 0.33)
young_poisson(𝕊₂; role = :compliance)

(70.0, 0.32999999999999996)

The pairs with then use the in-plane bulk modulus:   ,   ,   , and, from ,   . That last quadratic has two roots of physical materials, of Poisson's ratios and  ; the one kept is   .

Plane strain and plane stress

A 2-D computation on a 3-D material assumes either plane strain,  , or plane stress,  . plane_strain_stiffness and plane_stress_stiffness give the in-plane stiffness of each. Plane strain keeps the in-plane block of ; plane stress eliminates from  , which keeps and replaces by

ℂ = iso_stiffness_young_poisson(70.0, 0.33)
lame_shear(plane_strain_stiffness(ℂ)), lame_shear(plane_stress_stiffness(ℂ))

((51.08359133126936, 26.31578947368421), (25.92301649646505, 26.31578947368421))

Both are TensISO{4,2}, so a plane-stress state is computed with the plane-strain formulas of another material. plane_stress_to_plane_strain gives its moduli and plane_strain_to_plane_stress goes back:

E′, ν′ = plane_stress_to_plane_strain(70.0, 0.33)
young_poisson(plane_stress_stiffness(ℂ)) .≈ (E′, ν′)

(true, true)