P-wave modulus

G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4

More information 3D Formulae, Knowns ...

Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two quantities among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part). 3D Formulae Knowns Bulk modulus (K) Young's modulus (E) Lamé's first parameter (λ) Shear modulus (G) Poisson's ratio (ν) P-wave modulus (M) Notes (K, E) 3K(1 + .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠6K/E − 9K⁠) ⁠E/3 − ⁠E/3K⁠⁠ ⁠1/2⁠ − ⁠E/6K⁠ ⁠3K + E/3 − ⁠E/3K⁠⁠ (K, λ) ⁠9K(K − λ)/3K − λ⁠ ⁠3(K − λ)/2⁠ ⁠λ/3K − λ⁠ 3K − 2λ (K, G) ⁠9KG/3K + G⁠ K − ⁠2G/3⁠ ⁠3K − 2G/6K + 2G⁠ K + ⁠4G/3⁠ (K, ν) 3K(1 − 2ν) ⁠3Kν/1 + ν⁠ ⁠3K(1 − 2ν)/2(1 + ν)⁠ ⁠3K(1 − ν)/1 + ν⁠ (K, M) ⁠9K(M − K)/3K + M⁠ ⁠3K − M/2⁠ ⁠3(M − K)/4⁠ ⁠3K − M/3K + M⁠ (E, λ) ⁠E + 3λ + R/6⁠ ⁠E − 3λ + R/4⁠ − ⁠E + R/4λ⁠ − ⁠1/4⁠ ⁠E − λ + R/2⁠ R = ±(E2 + 9λ2 + 2Eλ)⁠1/2⁠ (E, G) ⁠EG/3(3G − E)⁠ ⁠G(E − 2G)/3G − E⁠ ⁠E/2G⁠ − 1 ⁠G(4G − E)/3G − E⁠ (E, ν) ⁠E/3 − 6ν⁠ ⁠Eν/(1 + ν)(1 − 2ν)⁠ ⁠E/2(1 + ν)⁠ ⁠E(1 − ν)/(1 + ν)(1 − 2ν)⁠ (E, M) ⁠3M − E + S/6⁠ ⁠M − E + S/4⁠ ⁠3M + E − S/8⁠ ⁠E + S/4M⁠ − ⁠1/4⁠ S = ±(E2 + 9M2 − 10EM)⁠1/2⁠ (λ, G) λ + ⁠2G/3⁠ ⁠G(3λ + 2G)/λ + G⁠ ⁠λ/2(λ + G)⁠ λ + 2G (λ, ν) ⁠λ/3⁠(1 + ⁠1/ν⁠) λ(⁠1/ν⁠ − 2ν − 1) λ(⁠1/2ν⁠ − 1) λ(⁠1/ν⁠ − 1) (λ, M) ⁠M + 2λ/3⁠ ⁠(M − λ)(M+2λ)/M + λ⁠ ⁠M − λ/2⁠ ⁠λ/M + λ⁠ (G, ν) ⁠2G(1 + ν)/3 − 6ν⁠ 2G(1 + ν) ⁠2 G ν/1 − 2ν⁠ ⁠2G(1 − ν)/1 − 2ν⁠ (G, M) M − ⁠4G/3⁠ ⁠G(3M − 4G)/M − G⁠ M − 2G ⁠M − 2G/2M − 2G⁠ (ν, M) ⁠M(1 + ν)/3(1 − ν)⁠ ⁠M(1 + ν)(1 − 2ν)/1 − ν⁠ ⁠M ν/1 − ν⁠ ⁠M(1 − 2ν)/2(1 − ν)⁠ 2D Formulae Knowns (K) (E) (λ) (G) (ν) (M) Notes (K2D, E2D) ⁠2K2D(2K2D − E2D)/4K2D − E2D⁠ ⁠K2DE2D/4K2D − E2D⁠ ⁠2K2D − E2D/2K2D⁠ ⁠4K2D^2/4K2D − E2D⁠ (K2D, λ2D) ⁠4K2D(K2D − λ2D)/2K2D − λ2D⁠ K2D − λ2D ⁠λ2D/2K2D − λ2D⁠ 2K2D − λ2D (K2D, G2D) ⁠4K2DG2D/K2D + G2D⁠ K2D − G2D ⁠K2D − G2D/K2D + G2D⁠ K2D + G2D (K2D, ν2D) 2K2D(1 − ν2D) ⁠2K2Dν2D/1 + ν2D⁠ ⁠K2D(1 − ν2D)/1 + ν2D⁠ ⁠2K2D/1 + ν2D⁠ (E2D, G2D) ⁠E2DG2D/4G2D − E2D⁠ ⁠2G2D(E2D − 2G2D)/4G2D − E2D⁠ ⁠E2D/2G2D⁠ − 1 ⁠4G2D^2/4G2D − E2D⁠ (E2D, ν2D) ⁠E2D/2(1 − ν2D)⁠ ⁠E2Dν2D/(1 + ν2D)(1 − ν2D)⁠ ⁠E2D/2(1 + ν2D)⁠ ⁠E2D/(1 + ν2D)(1 − ν2D)⁠ (λ2D, G2D) λ2D + G2D ⁠4G2D(λ2D + G2D)/λ2D + 2G2D⁠ ⁠λ2D/λ2D + 2G2D⁠ λ2D + 2G2D (λ2D, ν2D) ⁠λ2D(1 + ν2D)/2ν2D⁠ ⁠λ2D(1 + ν2D)(1 − ν2D)/ν2D⁠ ⁠λ2D(1 − ν2D)/2ν2D⁠ ⁠λ2D/ν2D⁠ (G2D, ν2D) ⁠G2D(1 + ν2D)/1 − ν2D⁠ 2G2D(1 + ν2D) ⁠2 G2D ν2D/1 − ν2D⁠ ⁠2G2D/1 − ν2D⁠ (G2D, M2D) M2D − G2D ⁠4G2D(M2D − G2D)/M2D⁠ M2D − 2G2D ⁠M2D − 2G2D/M2D⁠

Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two quantities among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part).
3D Formulae
Knowns	Bulk modulus $(K)$	Young's modulus $(E)$	Lamé's first parameter $(λ)$	Shear modulus $(G)$	Poisson's ratio $(ν)$	P-wave modulus $(M)$	Notes
$(K, E)$			$3 K$ $($ $1 + .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠6K/E − 9K⁠$ $)$	$⁠ E / 3 - ⁠ E / 3 K ⁠ ⁠$	$⁠ 1 / 2 ⁠ - ⁠ E / 6 K ⁠$	$⁠ 3 K + E / 3 - ⁠ E / 3 K ⁠ ⁠$
$(K, λ)$		$⁠ 9 K (K - λ) / 3 K - λ ⁠$		$⁠ 3(K - λ) / 2 ⁠$	$⁠ λ / 3 K - λ ⁠$	$3 K - 2λ$
$(K, G)$		$⁠ 9 KG / 3 K + G ⁠$	$K - ⁠ 2 G / 3 ⁠$		$⁠ 3 K - 2 G / 6 K + 2 G ⁠$	$K + ⁠ 4 G / 3 ⁠$
$(K, ν)$		$3 K (1 - 2 ν)$	$⁠ 3 Kν / 1 + ν ⁠$	$⁠ 3 K (1 - 2 ν) / 2(1 + ν) ⁠$		$⁠ 3 K (1 - ν) / 1 + ν ⁠$
$(K, M)$		$⁠ 9 K (M - K) / 3 K + M ⁠$	$⁠ 3 K - M / 2 ⁠$	$⁠ 3(M - K) / 4 ⁠$	$⁠ 3 K - M / 3 K + M ⁠$
$(E, λ)$	$⁠ E + 3λ + R / 6 ⁠$			$⁠ E - 3λ + R / 4 ⁠$	$- ⁠ E + R / 4λ ⁠ - ⁠ 1 / 4 ⁠$	$⁠ E - λ + R / 2 ⁠$	$R = \pm$ $($ $E 2 + 9λ 2 + 2 E λ$ $)$ $⁠ 1 / 2 ⁠$
$(E, G)$	$⁠ EG / 3(3 G - E) ⁠$		$⁠ G (E - 2 G) / 3 G - E ⁠$		$⁠ E / 2 G ⁠ - 1$	$⁠ G (4 G - E) / 3 G - E ⁠$
$(E, ν)$	$⁠ E / 3 - 6 ν ⁠$		$⁠ Eν / (1 + ν)(1 - 2 ν) ⁠$	$⁠ E / 2(1 + ν) ⁠$		$⁠ E (1 - ν) / (1 + ν)(1 - 2 ν) ⁠$
$(E, M)$	$⁠ 3 M - E + S / 6 ⁠$		$⁠ M - E + S / 4 ⁠$	$⁠ 3 M + E - S / 8 ⁠$	$⁠ E + S / 4 M ⁠ - ⁠ 1 / 4 ⁠$		$S = \pm$ $($ $E 2 + 9M 2 - 10 E M$ $)$ $⁠ 1 / 2 ⁠$
$(λ, G)$	$λ + ⁠ 2 G / 3 ⁠$	$⁠ G (3λ + 2 G) / λ + G ⁠$			$⁠ λ / 2(λ + G) ⁠$	$λ + 2 G$
$(λ, ν)$	$⁠ λ / 3 ⁠$ $($ $1 + ⁠ 1 / ν ⁠$ $)$	$λ$ $($ $⁠ 1 / ν ⁠ - 2 ν - 1$ $)$		$λ$ $($ $⁠ 1 / 2 ν ⁠ - 1$ $)$		$λ$ $($ $⁠ 1 / ν ⁠ - 1$ $)$
$(λ, M)$	$⁠ M + 2λ / 3 ⁠$	$⁠ (M - λ)(M +2λ) / M + λ ⁠$		$⁠ M - λ / 2 ⁠$	$⁠ λ / M + λ ⁠$
$(G, ν)$	$⁠ 2 G (1 + ν) / 3 - 6 ν ⁠$	$2 G (1 + ν)$	$⁠ 2 G ν / 1 - 2 ν ⁠$			$⁠ 2 G (1 - ν) / 1 - 2 ν ⁠$
$(G, M)$	$M - ⁠ 4 G / 3 ⁠$	$⁠ G (3 M - 4 G) / M - G ⁠$	$M - 2 G$		$⁠ M - 2 G / 2 M - 2 G ⁠$
$(ν, M)$	$⁠ M (1 + ν) / 3(1 - ν) ⁠$	$⁠ M (1 + ν)(1 - 2 ν) / 1 - ν ⁠$	$⁠ M ν / 1 - ν ⁠$	$⁠ M (1 - 2 ν) / 2(1 - ν) ⁠$
2D Formulae
Knowns	$(K)$	$(E)$	$(λ)$	$(G)$	$(ν)$	$(M)$	Notes
$(K 2D, E 2D)$			$⁠ 2 K 2D (2 K 2D - E 2D) / 4 K 2D - E 2D ⁠$	$⁠ K 2D E 2D / 4 K 2D - E 2D ⁠$	$⁠ 2 K 2D - E 2D / 2 K 2D ⁠$	$⁠ 4 K 2D^2 / 4 K 2D - E 2D ⁠$
$(K 2D, λ 2D)$		$⁠ 4 K 2D (K 2D - λ 2D) / 2 K 2D - λ 2D ⁠$		$K 2D - λ 2D$	$⁠ λ 2D / 2 K 2D - λ 2D ⁠$	$2 K 2D - λ 2D$
$(K 2D, G 2D)$		$⁠ 4 K 2D G 2D / K 2D + G 2D ⁠$	$K 2D - G 2D$		$⁠ K 2D - G 2D / K 2D + G 2D ⁠$	$K 2D + G 2D$
$(K 2D, ν 2D)$		$2 K 2D (1 - ν 2D)$	$⁠ 2 K 2D ν 2D / 1 + ν 2D ⁠$	$⁠ K 2D (1 - ν 2D) / 1 + ν 2D ⁠$		$⁠ 2 K 2D / 1 + ν 2D ⁠$
$(E 2D, G 2D)$	$⁠ E 2D G 2D / 4 G 2D - E 2D ⁠$		$⁠ 2 G 2D (E 2D - 2 G 2D) / 4 G 2D - E 2D ⁠$		$⁠ E 2D / 2 G 2D ⁠ - 1$	$⁠ 4 G 2D^2 / 4 G 2D - E 2D ⁠$
$(E 2D, ν 2D)$	$⁠ E 2D / 2(1 - ν 2D) ⁠$		$⁠ E 2D ν 2D / (1 + ν 2D)(1 - ν 2D) ⁠$	$⁠ E 2D / 2(1 + ν 2D) ⁠$		$⁠ E 2D / (1 + ν 2D)(1 - ν 2D) ⁠$
$(λ 2D, G 2D)$	$λ 2D + G 2D$	$⁠ 4 G 2D (λ 2D + G 2D) / λ 2D + 2 G 2D ⁠$			$⁠ λ 2D / λ 2D + 2 G 2D ⁠$	$λ 2D + 2 G 2D$
$(λ 2D, ν 2D)$	$⁠ λ 2D (1 + ν 2D) / 2 ν 2D ⁠$	$⁠ λ 2D (1 + ν 2D)(1 - ν 2D) / ν 2D ⁠$		$⁠ λ 2D (1 - ν 2D) / 2 ν 2D ⁠$		$⁠ λ 2D / ν 2D ⁠$
$(G 2D, ν 2D)$	$⁠ G 2D (1 + ν 2D) / 1 - ν 2D ⁠$	$2 G 2D (1 + ν 2D)$	$⁠ 2 G 2D ν 2D / 1 - ν 2D ⁠$			$⁠ 2 G 2D / 1 - ν 2D ⁠$
$(G 2D, M 2D)$	$M 2D - G 2D$	$⁠ 4 G 2D (M 2D - G 2D) / M 2D ⁠$	$M 2D - 2 G 2D$		$⁠ M 2D - 2 G 2D / M 2D ⁠$

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