Anisotropic, Elastic-Plastic Material - Revision history

Nairnj: /* Hardening Laws */

2024-07-24T16:41:17Z

Hardening Laws

← Older revision		Revision as of 16:41, 24 July 2024
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	<math>\alpha_{max} = -\frac{\ln s_{rel}}{\beta} \qquad{\rm where}\qquad s_{rel} = \frac{K^*}{K}</math>		<math>\alpha_{max} = -\frac{\ln s_{rel}}{\beta} \qquad{\rm where}\qquad s_{rel} = \frac{K^*}{K}</math>

			Note that these are slopes of hardening law and not slope of a uniaxial stress strain curves. In 2D, the value of <math>\alpha_{max}</math> needed to match experimental final slopes will depend on if the analysis is plane stress or plane strain.

	== Why Two Hill Criteria ==		== Why Two Hill Criteria ==

Nairnj: /* Constitutive Law */

2024-06-21T00:16:34Z

Constitutive Law

← Older revision		Revision as of 00:16, 21 June 2024
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	<math>{\sigma_{Y,ii}\over {1+R}} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over {\|1-R\|}}</math>		<math>{\sigma_{Y,ii}\over {1+R}} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over {\|1-R\|}}</math>

	When using this material, the code automatically verifies the normal yield stresses define a physically acceptable yield surface. The shear yield stresses ~~can~~ have any values.		When using this material, the code automatically verifies the normal yield stresses define a physically acceptable yield surface. The shear yield stresses are allowed to have any positive values (although some might not make sense due to symmetry considerations<ref name="Hill"/>).

	=== Example Yield Strengths ===		=== Example Yield Strengths ===

Nairnj: /* Constitutive Law */

2024-06-20T21:39:56Z

Constitutive Law

← Older revision		Revision as of 21:39, 20 June 2024
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	<math>{\sigma_{Y,ii}\over {1+R}} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over {\|1-R\|}}</math>		<math>{\sigma_{Y,ii}\over {1+R}} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over {\|1-R\|}}</math>

	When using this material, the code automatically verifies the normal yield stresses define a physically acceptable yield surface.		When using this material, the code automatically verifies the normal yield stresses define a physically acceptable yield surface. The shear yield stresses can have any values.

	=== Example Yield Strengths ===		=== Example Yield Strengths ===

Nairnj: /* Constitutive Law */

2024-06-20T21:39:02Z

Constitutive Law

← Older revision		Revision as of 21:39, 20 June 2024
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	<math>{\sigma_{Y,ii}\over {1+R}} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over {\|1-R\|}}</math>		<math>{\sigma_{Y,ii}\over {1+R}} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over {\|1-R\|}}</math>

			When using this material, the code automatically verifies the normal yield stresses define a physically acceptable yield surface.

	=== Example Yield Strengths ===		=== Example Yield Strengths ===

Nairnj: /* Why Two Hill Criteria */

2024-06-20T21:11:01Z

Why Two Hill Criteria

← Older revision		Revision as of 21:11, 20 June 2024
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	where elements of <math>\mathbf{A}</math> are derived from material properties ''F'' through ''N''. A discussion with implementation details for <tt>HillStyle=1</tt> (the first one with a square root) can be found in Simo and Hughes.<ref name="SH"/> Unfortunately, that derivation has a step that multiplies by <math>\mathbf{A}^{-1}\mathbf{A}</math> but the tensor for Hill plasticity is singular. Despite this inconsistency, the overall implementation seems to work well.		where elements of <math>\mathbf{A}</math> are derived from material properties ''F'' through ''N''. A discussion with implementation details for <tt>HillStyle=1</tt> (the first one with a square root) can be found in Simo and Hughes.<ref name="SH"/> Unfortunately, that derivation has a step that multiplies by <math>\mathbf{A}^{-1}\mathbf{A}</math> but the tensor for Hill plasticity is singular. Despite this inconsistency, the overall implementation seems to work well.

	To avoid this problem, the criteria can use <tt>HillStyle=2</tt> instead (with squared terms). Implementation by this approach<ref name="dB/> does not need to use <math>\mathbf{A}^{-1}</math>. It is now the default style for this material type.		To avoid this problem, the criteria can use <tt>HillStyle=2</tt> instead (with squared terms). Implementation by this approach<ref name="dB/> does not need to use <math>\mathbf{A}^{-1}</math>. It is now the default style for this material type. Furthermore, <tt>HillStyle=2</tt> can be used in both 2D plane stress and plane strain analysis while <tt>HillStyle=1</tt> can only be used in plane strain.

	~~If you want to use~~ 2D plane stress analysis~~, the modeling must use~~ <tt>HillStyle=2</tt>.

	== Material Properties ==		== Material Properties ==

Nairnj: /* Why Two Hill Criteria */

2024-06-20T21:09:41Z

Why Two Hill Criteria

← Older revision		Revision as of 21:09, 20 June 2024
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	== Why Two Hill Criteria ==		== Why Two Hill Criteria ==

	The two Hill criteria described [[#Constitutive Law\|above]] ~~can be written with a symmetric, fourth-rank tensor as~~		The two Hill criteria described [[#Constitutive Law\|above]] are:

Nairnj: /* Example Yield Strengths */

2024-06-20T21:08:33Z

Example Yield Strengths

← Older revision		Revision as of 21:08, 20 June 2024
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	</math>		</math>

	If that limit does not include experimental results for thickness direction yielding, Hill plasticity may need be a good model. Overall, Hill plasticity was developed for modeling sheet metals where the three anisotropic yields stresses ~~were~~ expected to be a similar magnitude.		If that limit does not include experimental results for thickness direction yielding, Hill plasticity may need be a good model. Overall, Hill plasticity was developed for modeling sheet metals where the three anisotropic yields stresses are expected to be a similar magnitude.

	=== Hardening Laws ===		=== Hardening Laws ===

Nairnj: /* Example Yield Strengths */

2024-06-20T21:08:19Z

Example Yield Strengths

← Older revision		Revision as of 21:08, 20 June 2024
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	</math>		</math>

	If that limit does not include experimental results for thickness direction, Hill plasticity may need be a good model. Overall, Hill plasticity was developed for modeling sheet metals where the three anisotropic yields stresses were expected to be a similar magnitude.		If that limit does not include experimental results for thickness direction yielding, Hill plasticity may need be a good model. Overall, Hill plasticity was developed for modeling sheet metals where the three anisotropic yields stresses were expected to be a similar magnitude.

	=== Hardening Laws ===		=== Hardening Laws ===

Nairnj: /* Example Yield Strengths */

2024-06-20T21:07:47Z

Example Yield Strengths

← Older revision		Revision as of 21:07, 20 June 2024
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	Similarly for <math>0<R<1</math>, the limits in <math>\sigma_{Y,kk}</math> define two limits that degenerate into parallel planes while yeild stresses within those limits are ellipsoidal surfaces. Picking values outside the limits gives opposing hyperbolic surfaces that allow a non-physical expanding elastic domain.		Similarly for <math>0<R<1</math>, the limits in <math>\sigma_{Y,kk}</math> define two limits that degenerate into parallel planes while yeild stresses within those limits are ellipsoidal surfaces. Picking values outside the limits gives opposing hyperbolic surfaces that allow a non-physical expanding elastic domain.

	This limitation of yield stress limits ability to model materials that might have high yield stresses in one plane, but a low yield stress in the thickness directions. Some examples are in-plane, random fiber composites such as medium density fiber board, particle board, fiberglass composites, or paperboard. After picking reasonable in-plane ~~yields stress, which are likely similar~~, the thickness direction yield stress must be:		This limitation of yield stress limits ability to model materials that might have high yield stresses in one plane, but a low yield stress in the thickness directions. Some examples are in-plane, random fiber composites such as medium density fiber board, particle board, fiberglass composites, or paperboard. After picking reasonable, and similar, in-plane yield stresses, the thickness direction yield stress must be:

Nairnj: /* Example Yield Strengths */

2024-06-20T21:07:10Z

Example Yield Strengths

← Older revision		Revision as of 21:07, 20 June 2024
Line 77:		Line 77:
	Similarly for <math>0<R<1</math>, the limits in <math>\sigma_{Y,kk}</math> define two limits that degenerate into parallel planes while yeild stresses within those limits are ellipsoidal surfaces. Picking values outside the limits gives opposing hyperbolic surfaces that allow a non-physical expanding elastic domain.		Similarly for <math>0<R<1</math>, the limits in <math>\sigma_{Y,kk}</math> define two limits that degenerate into parallel planes while yeild stresses within those limits are ellipsoidal surfaces. Picking values outside the limits gives opposing hyperbolic surfaces that allow a non-physical expanding elastic domain.

	This limitation of yield stress limits ability to model materials that might have high yield stresses in one plane, but a low yield stress in the thickness directions. Some examples are in-plane, random fiber composites, such as medium density fiber board, particle board, fiberglass composites, or paperboard. After picking reasonable in-plane yields stress, which are likely similar, the thickness direction yield stress must be:		This limitation of yield stress limits ability to model materials that might have high yield stresses in one plane, but a low yield stress in the thickness directions. Some examples are in-plane, random fiber composites such as medium density fiber board, particle board, fiberglass composites, or paperboard. After picking reasonable in-plane yields stress, which are likely similar, the thickness direction yield stress must be: