# Tag Archives: algorithms

# Publication: A model for statistical variation of fracture properties in a continuum mechanics code

NEWS FLASH: The print version of the Meyer-Brannon paper on statistical variation of fracture patterns in a continuum code (CTH) is now available at http://dx.doi.org/10.1016/j.ijimpeng.2010.09.007.

# Publication: On the thermodynamic requirement of elastic stiffness anisotropy in isotropic materials

T. Fuller and R.M. Brannon

In general, thermodynamic admissibility requires isotropic materials develop reversible deformation induced anisotropy (RDIA) in their elastic stiffnesses. Taking the elastic potential for an isotropic material to be a function of the strain invariants, isotropy of the elastic stiffness is possible under distortional loading if and only if the bulk modulus is independent of the strain deviator and the shear modulus is constant. Previous investigations of RDIA have been limited to applications in geomechanics where material non-linearityand large deformations are commonly observed. In the current paper, the degree of RDIA in other materials is investigated. It is found that the resultant anisotropy in materials whose strength does not vary appreciably with pressure, such as metals, is negligible, but in materials whose strength does vary with pressure, the degree of RDIA can be significant. Algorithms for incorporating RDIA in a classical elastic–plastic model are provided.

Available Online:

http://www.mech.utah.edu/~brannon/pubs/7-2011FullerBrannonInducedElasticAnisotropy.pdf

http://www.sciencedirect.com/science/article/pii/S0020722511000024

# Publication: A multi-stage return algorithm for solving the classical damage component of constitutive models for rocks, ceramics, and other rock-like media

R. M. Brannon and S. Leelavanichkul

Classical plasticity and damage models for porous quasi-brittle media usually suffer from mathematical defects such as non-convergence and nonuniqueness.Yield or damage functions for porous quasi-brittle media often have yield functions with contours so distorted that following those contours to the yield surface in a return algorithm can take the solution to a false elastic domain. A steepest-descent return algorithm must include iterative corrections; otherwise,the solution is non-unique because contours of any yield function are non-unique. A multi-stage algorithm has been developed to address both spurious convergence and non-uniqueness, as well as to improve efficiency. The region of pathological isosurfaces is masked by first returning the stress state to the Drucker–Prager surface circumscribing the actual yield surface. From there, steepest-descent is used to locate a point on the yield surface. This first-stage solution,which is extremely efficient because it is applied in a 2D subspace, is generally not the correct solution,but it is used to estimate the correct return direction.The first-stage solution is projected onto the estimated correct return direction in 6D stress space. Third invariant dependence and anisotropy are accommodated in this second-stage correction. The projection operation introduces errors associated with yield surface curvature,so the two-stage iteration is applied repeatedly to converge. Regions of extremely high curvature are detected and handled separately using an approximation to vertex theory. The multi-stage return is applied holding internal variables constant to produce a non-hardening solution. To account for hardening from pore collapse (or softening from damage), geometrical arguments are used to clearly illustrate the appropriate scaling of the non-hardening solution needed to obtain the hardening (or softening) solution.

For errata (transcription errors in two of the verification solutions), please see:

https://csmbrannon.net/2015/07/12/errata-for-two-verification-publications/

Available Online:

http://www.mech.utah.edu/~brannon/pubs/7-2009BrannonLeelavanichkul-IJF.pdf

http://dx.doi.org/10.1007/s10704-009-9398-4

# Tutorial: Radial Return

A tutorial on the underlying theory of projecting a stress state back to the plastic yield surface.

You may download the document here, or view just view the graph.

# Tutorial: Classical constitutive material models for engineering materials

An overview book chapter of classical constitutive material models by Kaspar Willam.

You may download the rest of the document here.