CONSTITUTIVE LAW
Robert S. Pritchard
IceCasting, Inc.
20 Wilson Court
San Rafael, CA 94901-1230
e-mail: pritchardr@asme.org
http://www.icecasting.com
Tel/fax: (415) 454-9899
The anisotropic plasticity constitutive law is actively being developed now [Coon, et al., 1992; 1998a,b; Pritchard, 1998a]. This law explicitly describes the formation and evolution of lead systems.
The new anisotropic constitutive law has not yet been tested. Its formulation is only now being completed. Two oriented thickness distribution models have been developed to describe ice conditions, including leads and ridges.
The constitutive law has been integrated in a 0d code, where the deformation history is specified and the stress and ice condition histories are calculated. There is much to do before this constitutive law can be declared finished. It must be implemented in finite element or finite difference codes, that will need to be coupled to ocean models. The detailed parameterizations of the model properties still need to be formulated and tested. Only then can the model performance be known.
What is the algorithm?
The anisotropic yield surface is a composite of surfaces (m=1,2,Š,M) that describe the ice conditions in all orientations.
Here km describes the material parameters that define yield surface size and shape.
A normal flow rule is assumed for the anisotropic plasticity constitutive law. This law is made complicated by the fact that the yield surface is (potentially) composed of many surfaces. When the stress lies on an intersection of multiple surfaces, the flow rule must be composed of a linear combination of normals to each of the contributing surfaces
where fl represents each branch of the composite yield surface on which flow is occurring.
The model is assumed to be elastic plastic, with a stiff linear elastic response
where M is the elastic modulus tensor and e is elastic strain. The rate of change of elastic strain
is related to the total and plastic stretching by
here D is total stretching, the symmetric part of the velocity gradient tensor , and W is spin the antisymmetric part of L.
The algorithm for integrating this anisotropic elastic
plastic constitutive law will be presented by Pritchard [1998b] at
the IAHR98 Conference later this summer.
What data are needed?
Model simulations and forecasts require ice condition fields to be specified, and data must be assimilated
when available to improve performance. The presence of leads and their directions are needed as part of the ice
conditions.
Is it practical?
The new constitutive law can be implemented in codes that use an explicit formulation with few serious
changes. the oriented thickness distribution, or lead size and orientation, must be added as primary variables. In
finite element codes that I work with, the constitutive law is integrated in each element by one subroutine call (many
subroutines are required, but only one call appears externally). I expect few differences in the stability criteria.
All ice model calculations could be performed on a modern PC, and data transferred back-and-forth to the ocean model via a network.
An implicit integration scheme will require more development. The implicit scheme will require that the elastic plastic modulus tensor be estimated. This step is quite complicated theoretically, but empirical approximations should be satisfactory in the integration step.
Data on lead orientations are available from SAR imagery. An excellent sample of SHEBA data can be found at the Internet URL: www-radar.jpl.nasa.gov/rgps/radarsat.html.
Will it make a difference?
Yes, and the differences are self-evident. Sea ice is oriented. When a lead or ridge forms, the ice strength
across the feature differs from its strength along the feature. Therefore an anisotropic constitutive law is essential if
we are to describe the physical behavior correctly.
If we ask only, 'Can the different behavior describe some aspect of the behavior better than an older isotropic model?' Then we do not yet know the answer. However, the cost of implementing and using the new anisotropic constitutive law is modest, and we should begin using it simply because it is more correct. As I said, 'Its benefits are self-evident.'
Less philosophically, the anisotropic will make a difference because lead orientation and width will appear as primary variables in the model.
References
Coon, M. D., Echert, D. C., and Knoke, G. S. (1992) "Pack Ice Anisotropic Constitutive Law," In IAHR 92,
Proceedings of 11th Int'l Symposium on Ice, Banff, Alberta, pp. 1188-1196.
Coon, M. D., Knoke, G. S., Echert, D. C., and Pritchard, R. S. (1998a) "The Architecture of an anisotropic elastic-plastic sea ice mechanics constitutive law," To appear in J. Geophys. Res.
Coon, M. D., Knoke, G. S., Echert, D. C., and Pritchard, R. S. (1998b) "An Oriented Thickness Distribution for Sea Ice," Submitted to J. Geophys. Res.
Pritchard, R. S. (1998a) "Ice Conditions in an Anisotropic Sea Ice Dynamics Model," Int=l. J. Offshore and Polar Engineering, vol. 8, no. 1 pp. 9-15.
Pritchard, R. S. (1998b) "Integrating an anisotropic plasticity law for sea ice," to appear in IAHR 98, Potsdam, NY, 26-31 July 1998.