Abstracts - Conference Planning and Management - Iowa State ...

Nonlocal Boundary Layer Method for Overcoming Boundary Condition Ambiguity
in Nonlocal and Statistical Models of Quasibrittle Structures
Christian Hoover
Northwestern University, 2145 Sheridan Road, Technological Institute Suite A123, Evanston, 60208, US
Phone: 1-847-491-4025, Email: ChristianHoover2010@u.northwestern.edu
Jialiang Le
Northwestern University, Evanston, IL
Zdenek P. Bazant
Northwestern University, Evanston, IL
Abstract:
The nonlocal models used in computational analysis for regularization of the boundary value problems
of softening damage are still hampered by unresolved issues in the treatment of boundary conditions.
The existing models are non-physical and treat the protrusion of the domain of the nonlocal weighting
function beyond the structure boundary differently. For example, the protruding part is deleted while
the interior part is either rescaled, or enhanced by a Dirac delta peak at the structure boundary or at the
centroid of the domain. The boundary conditions are also unclear for gradient models. Various models
can yield very different results.
A more physical approach to the numerical treatment of the boundaries, inspired by the
weakest-link statistical size effect on structural strength, is proposed. Its modeling requires the structure
(of positive geometry) to be subdivided into elements roughly equal to the representative volume
element (RVE) of material. The RVE is defined as the smallest material volume whose failure causes
the entire structure to fail. In the proposed nonlocal boundary layer model (NBL), the structure is
divided into two parts: an interior domain, and a boundary layer of a thickness equal to the RVE size,
which approximately equals the fracture process zone width. In the boundary layer, the stress depends
solely on the average strain over its thickness, which is approximately given by the continuum strain at
the middle surface of the layer. In the interior domain, nonlocal averaging may be applied without
modification because the nonlocal integral domain cannot protrude outside the structure boundary.
Since statistically the structure is equivalent to the weakest link model, its survival probability 1
- Pf is the joint probability of survival of all the RVEs. The subdivision into RVEs is generally nonunique
and for irregular geometries, exactly equal RVE sizes are impossible. This gives inconsistent
estimates of failure probability Pf. One way to overcome the problem is to calculate the nonlocal stress
in the joint probability expression by averaging the strains over the zone of influence using a weighting
function. Five different methods of boundary treatment were used for comparing the size effect on
structure strength: 1) calculating Pf based on the weakest-link model, 2) redistributing the protruding
volume by placing it into a Dirac delta function at either the structure boundary or 3) at the center
nonlocal domain, 4) uniformly rescaling the weighting function, 5) and the present NBL method. The
last three methods were also used in and integral type formulation of deterministic nonlocal softening
damage. These models were compared by simulating the size effect on the modulus of rupture due to
crack initiation, which is a problem experimentally studied by Rocco (1995), and the size effect in the
notched specimens of Bazant and Pfeiffer (1987). The NBL model and the implicit gradient model of
Peerlings et al. were also compared by simulating the size effect tests of Rocco (1995).
373 ABSTRACTS