This example deals with two-dimensional analysis of a more complicated
concrete specimen--a single-edge-notched beam loaded such that the central
portion around the notch is under high shear (Fig. 4a).
This so-called
Iosipescu geometry was adapted for concrete by
Arrea and Ingraffea [1].
For quasibrittle materials such as concrete, the fracture is still
locally driven by the maximal tensile stresses. The crack starts
propagating from the notch tip in an inclined direction and when it leaves
the region of high shear, it approaches the vertical direction and reaches
the surface of the specimen to the right of the applied force . Some
limited cracking also appears on the opposite side of the loading platen
in the region of high tensile stresses due to bending. The final crack
trajectory is curved and certain simulations from the literature,
especially on coarse meshes,
failed to reproduce it correctly. Better results are usually obtained
with anisotropic models (such as smeared crack models or anisotropic
damage models), but since the present study focuses mainly on the
numerical aspects, we will keep using the simple isotropic damage model
with a Rankine-type equivalent strain. On the fine mesh with
2135 nodes and 4132 constant-strain elements, shown in Fig. 4b,
the essential failure mechanism is captured properly.
The constitutive properties are set to: Young's modulus
GPa, Poisson's ratio
, tensile strength
MPa,
strain controling the softening
, and nonlocal interaction radius
mm.
The formation of the process zone and failure of the specimen are simulated in 40 incremental steps, controled by the increments of the crack mouth sliding displacement. Four solution strategies are compared:
For this model, the assembly of a local stiffness matrix
(ESM or SSM) takes
about 0.11 seconds, while the assembly of the nonlocal stiffness
matrix (TSM) takes between 0.11 and 0.28 seconds for the
skyline storage scheme (used with the direct solver)
or between 0.11 and 0.42 seconds for the compressed row storage
(used with the iterative solver). The assembly times vary depending
on the stage of analysis,
because the number of nonzero entries increases as the process zone evolves.
The factorization time for ESM and SSM is 2.19 seconds, while for TSM it varies
between 2.2 and 5.9 seconds.
The total times needed for the analysis using different solution strategies
are given in Table 4. Same as in the previous example, the
most efficient technique is SSM for a low accuracy and TSM with a direct solver for
a high accuracy. The ESM requires an extremely large number of iterations and is
prohibitively expensive. The TSM combined with an iterative solver (GMRES with
preconditioning by incomplete decomposition) is slightly slower than the
TSM with a direct solver. Additional details about the numbers of iterations per step
and user times per iteration are provided in Tables 5 and
6.
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