Steinberg-Cochran-Guinan Hardening - Revision history

Nairnj: /* Plasticity Theory */

2019-10-30T06:54:35Z

Plasticity Theory

← Older revision		Revision as of 06:54, 30 October 2019
Line 15:		Line 15:
	The [[Steinberg-Lund Hardening]] law is similar to this law but adds a a strain rate- and temperature-dependent term.		The [[Steinberg-Lund Hardening]] law is similar to this law but adds a a strain rate- and temperature-dependent term.

	== Plasticity Theory ==		=== Plasticity Theory ===

	This law (and the [[Steinberg-Lund Hardening]] law) are implemented use J<sub>2</sub> plasticity theory. In associative, J<sub>2</sub> plasticity theory, the yield function has to depend only on deviatoric stress and cannot depend on pressure, but these laws have yield strengths that depend on pressure. They can, however be used in J<sub>2</sub> plasticity as non-associative laws. The plastic flow is determined by the failure surface defined by:		This law (and the [[Steinberg-Lund Hardening]] law) are implemented use J<sub>2</sub> plasticity theory. In associative, J<sub>2</sub> plasticity theory, the yield function has to depend only on deviatoric stress and cannot depend on pressure, but these laws have yield strengths that depend on pressure. They can, however be used in J<sub>2</sub> plasticity as non-associative laws. The plastic flow is determined by the failure surface defined by:

Nairnj: /* Plasticity Theory */

2019-10-30T06:54:12Z

Plasticity Theory

← Older revision		Revision as of 06:54, 30 October 2019
Line 22:		Line 22:
	<math>\Phi = \\|{\bf s}\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>		<math>\Phi = \\|{\bf s}\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>

	where <b>s<\b> is the deviatoric stress. The plastic strain and forces, however, are determined by a different plastic potential of:		where <b>s</b> is the deviatoric stress. The plastic strain and forces, however, are determined by a different plastic potential of:

Nairnj: /* Plasticity Theory */

2019-10-30T06:53:58Z

Plasticity Theory

← Older revision		Revision as of 06:53, 30 October 2019
Line 21:		Line 21:

	<math>\Phi = \\|{\bf s}\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>		<math>\Phi = \\|{\bf s}\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>

			where <b>s<\b> is the deviatoric stress. The plastic strain and forces, however, are determined by a different plastic potential of:


			<math>\Psi = \\|{\bf s}\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}</math>

			Because these two functions different, the implementation is formally non-associative plasticity.

	== Hardening Law Properties ==		== Hardening Law Properties ==

Nairnj: /* Plasticity Theory */

2019-10-30T06:51:47Z

Plasticity Theory

← Older revision		Revision as of 06:51, 30 October 2019
Line 20:		Line 20:


	<math>\Phi = \\|s\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>		<math>\Phi = \\|{\bf s}\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>

	== Hardening Law Properties ==		== Hardening Law Properties ==

Nairnj: /* Plasticity Theory */

2019-10-30T06:51:08Z

Plasticity Theory

← Older revision		Revision as of 06:51, 30 October 2019
Line 20:		Line 20:


	<math>\Phi = \min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>		<math>\Phi = \\|s\\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>

	== Hardening Law Properties ==		== Hardening Law Properties ==

Nairnj at 06:45, 30 October 2019

2019-10-30T06:45:32Z

← Older revision		Revision as of 06:45, 30 October 2019
Line 1:		Line 1:
			== Yield Stress ==

	In the Steinberg-Cochran-Guinan [[Hardening Laws\|hardening law]], the yield stress is given by		In the Steinberg-Cochran-Guinan [[Hardening Laws\|hardening law]], the yield stress is given by

Nairnj at 06:43, 30 October 2019

2019-10-30T06:43:24Z

← Older revision		Revision as of 06:43, 30 October 2019
Line 9:		Line 9:
	<math>{G(T,P)\over G_0} = 1 + {G_P'\over G_0} P J^{1/3} + {G_T'\over G_0}(T-T_0)</math>		<math>{G(T,P)\over G_0} = 1 + {G_P'\over G_0} P J^{1/3} + {G_T'\over G_0}(T-T_0)</math>

	where J is the relative volume change (V/V<sub>0</sub>), G<sub>P</sub>' and G<sub>T</sub>' are coefficients for pressure and temperature affects, T is current temperature, and T<sub>0</sub> is a reference temperature. For more details, see paper by Steinberg, Cochran, and Guinan.<ref>D. J. Steinberg S. G. Cochran, and M. W. Guinan, "A constitutive model for metals applicable at high strain rates," J. Appl. Phys., 51, 1498-1504 (~~1989~~).</ref>		where J is the relative volume change (V/V<sub>0</sub>), G<sub>P</sub>' and G<sub>T</sub>' are coefficients for pressure and temperature affects, T is current temperature, and T<sub>0</sub> is a reference temperature. For more details, see paper by Steinberg, Cochran, and Guinan.<ref>D. J. Steinberg S. G. Cochran, and M. W. Guinan, "A constitutive model for metals applicable at high strain rates," J. Appl. Phys., 51, 1498-1504 (1979).</ref>

	The [[Steinberg-Lund Hardening]] law is similar to this law but adds a a strain rate- and temperature-dependent term.		The [[Steinberg-Lund Hardening]] law is similar to this law but adds a a strain rate- and temperature-dependent term.

			== Plasticity Theory ==

			This law (and the [[Steinberg-Lund Hardening]] law) are implemented use J<sub>2</sub> plasticity theory. In associative, J<sub>2</sub> plasticity theory, the yield function has to depend only on deviatoric stress and cannot depend on pressure, but these laws have yield strengths that depend on pressure. They can, however be used in J<sub>2</sub> plasticity as non-associative laws. The plastic flow is determined by the failure surface defined by:


			<math>\Phi = \min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math>

	== Hardening Law Properties ==		== Hardening Law Properties ==

Nairnj at 22:59, 2 June 2015

2015-06-02T22:59:19Z

← Older revision		Revision as of 22:59, 2 June 2015
Line 21:		Line 21:
	! Property !! Description		! Property !! Description
	\|-		\|-
	\| yield \|\| Initial yield stress (σ<sub>0</sub> at zero pressure and the reference temperature). Enter in ~~units of~~ [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]].		\| yield \|\| Initial yield stress (σ<sub>0</sub> at zero pressure and the reference temperature). Enter in [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]].
	\|-		\|-
	\| betahard \|\| Yield stress hardening term β. It is dimensionless.		\| betahard \|\| Yield stress hardening term β. It is dimensionless.
Line 31:		Line 31:
	\| GTpG0 \|\| The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units K<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus.		\| GTpG0 \|\| The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units K<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus.
	\|-		\|-
	\| yieldMax \|\| Maximum yield stress. Enter in ~~units of~~ [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]].		\| yieldMax \|\| Maximum yield stress. Enter in [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]].
	\|}		\|}

Nairnj at 21:51, 2 June 2015

2015-06-02T21:51:08Z

← Older revision		Revision as of 21:51, 2 June 2015
Line 21:		Line 21:
	! Property !! Description		! Property !! Description
	\|-		\|-
	\| yield \|\| Initial yield stress (σ<sub>0</sub> at zero pressure and the reference temperature). Enter in units of ~~MPa~~.		\| yield \|\| Initial yield stress (σ<sub>0</sub> at zero pressure and the reference temperature). Enter in units of [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]].
	\|-		\|-
	\| betahard \|\| Yield stress hardening term β. It is dimensionless.		\| betahard \|\| Yield stress hardening term β. It is dimensionless.
Line 27:		Line 27:
	\| nhard \|\| Exponent on cumulative plastic strain in hardening term. It is dimensionless.		\| nhard \|\| Exponent on cumulative plastic strain in hardening term. It is dimensionless.
	\|-		\|-
	\| GPpG0 \|\| The (G<sub>p</sub>'/G<sub>0</sub>) ratio term for pressure dependence of shear modulus. Enter in units ~~MPa~~<sup>-1</sup>. Enter 0 to omit pressure dependence in shear modulus.		\| GPpG0 \|\| The (G<sub>p</sub>'/G<sub>0</sub>) ratio term for pressure dependence of shear modulus. Enter in units [[ConsistentUnits Command#Legacy and Consistent Units\|pressure<sup>-1</sup> units]]. Enter 0 to omit pressure dependence in shear modulus.
	\|-		\|-
	\| GTpG0 \|\| The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units K<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus.		\| GTpG0 \|\| The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units K<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus.
	\|-		\|-
	\| yieldMax \|\| Maximum yield stress. Enter in units of ~~MPa~~.		\| yieldMax \|\| Maximum yield stress. Enter in units of [[ConsistentUnits Command#Legacy and Consistent Units\|pressure units]].
	\|}		\|}

Nairnj: /* Hardening Law Properties */

2015-03-12T16:07:22Z

Hardening Law Properties

← Older revision		Revision as of 16:07, 12 March 2015
Line 29:		Line 29:
	\| GPpG0 \|\| The (G<sub>p</sub>'/G<sub>0</sub>) ratio term for pressure dependence of shear modulus. Enter in units MPa<sup>-1</sup>. Enter 0 to omit pressure dependence in shear modulus.		\| GPpG0 \|\| The (G<sub>p</sub>'/G<sub>0</sub>) ratio term for pressure dependence of shear modulus. Enter in units MPa<sup>-1</sup>. Enter 0 to omit pressure dependence in shear modulus.
	\|-		\|-
	\| GTpG0 \|\| The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units ~~MPa~~<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus.		\| GTpG0 \|\| The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units K<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus.
	\|-		\|-
	\| yieldMax \|\| Maximum yield stress. Enter in units of MPa.		\| yieldMax \|\| Maximum yield stress. Enter in units of MPa.