A direct FEM approach to model mesoscale concrete and connect non-matching meshes in multiscale analysis

Vieira, Welington Hilário; Coda, Humberto Breves; Paccola, Rodrigo Ribeiro

doi:10.1590/S1983-41952024000100008

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Revista IBRACON de Estruturas e Materiais

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ORIGINAL ARTICLE • Rev. IBRACON Estrut. Mater. 17 (1) • 2024 • https://doi.org/10.1590/S1983-41952024000100008 copy

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10.1590/S1983-41952024000100011

A direct FEM approach to model mesoscale concrete and connect non-matching meshes in multiscale analysis

Uma abordagem direta pelo MEF para modelar concreto em mesoescala e conectar malhas não conformes em análises multiescala

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The mechanical degradation of concrete structures is a phenomenon dependent on the material heterogeneity observed at mesoscale. As the mechanical degradation is a localized phenomenon, structural members and structures may be simulated using the concurrent multiscale analysis technique. Thus, only the most critical regions are modeled in mesoscale, reducing the computational cost compared to the simulation of the entire structure at this scale. This work presents two contributions in concurrent multiscale analysis. The first contribution introduces an alternative representation of the mesoscale interfacial transition zone (ITZ) of the concrete together with a strategy that allows modeling particles (coarse aggregates) without degrees of freedom. The resulting ITZ representation allows the simulation of more realistic discrete cracks in concrete modeling. The second contribution uses particle-like elements without degrees of freedom as coupling elements to model non-matching meshes between different media. The proposed coupling technique does not add degrees of freedom and does not use penalty or Lagrange Multipliers methods. Experimental and numerical results are used in order to validate the proposed multiscale formulation regarding concrete specimen simulations.

Keywords:
mesoscale concrete; interfacial transition zone; coupling element; multiscale; non-matching meshes

INPUT: $E_{(n + 1)}, {\bar{σ}}_{n n_{(n)}}, r_{(n)}, Δ r_{(n)}$
(i) Compute the effective second Piola-Kirchhoff stress tensor
${\bar{S}}_{(n + 1)} = C : E_{(n + 1)}$
(ii) Compute the Cauchy effective stress tensor
${\bar{σ}}_{(n + 1)} = \frac{1}{J} A \cdot {\bar{S}}_{(n + 1)} \cdot A^{t}$
(iii) Compute the effective stress normal to the base of the element
${\bar{σ}}_{n n_{(n + 1)}} = \vec{n^{t}} \cdot {\bar{σ}}_{(n + 1)} \cdot \vec{n}$
(iv) Compute loading-unloading conditions
If ${\bar{σ}}_{n n_{(n + 1)}} \leq r_{(n)} ⟶ r_{(n + 1)} = r_{(n)}$
If ${\bar{σ}}_{n n_{(n + 1)}} > r_{(n)} ⟶ r_{(n + 1)} = {\bar{σ}}_{n n_{(n + 1)}}$
(v) Compute the strain-like internal variable increment
$Δ r_{(n + 1)} = r_{(n + 1)} - r_{(n)}$
(vi) Compute the explicit linear extrapolation of the strain-like internal variable
${\tilde{r}}_{(n + 1)} = r_{(n)} + \frac{Δ r_{(n)}}{Δ t_{(n)}} Δ t_{(n + 1)}$ ,
where $Δ t_{(n + 1)} = t_{(n + 1)} - t_{(n)} a n d Δ t_{(n)} = t_{(n)} - t_{(n - 1)}$
(vii) Update the damage variable
${\tilde{d}}_{(n + 1)} = 1 - \frac{q ({\tilde{r}}_{(n + 1)})}{{\tilde{r}}_{(n + 1)}}$
(viii) Compute the approximate nominal Cauchy effective stress tensor
${\tilde{S}}_{(n + 1)} = \{\begin{matrix} {(1 - {\tilde{d}}_{(n + 1)}) {\bar{S}}_{(n + 1)} i f \bar{σ}}_{n n_{(n)}} > 0 \\ {\bar{S}}_{(n + 1)} i f {\bar{σ}}_{n n_{(n)}} \leq 0 \end{matrix}$
(ix) Compute the approximate constitutive tensor
${\tilde{C}}_{(n + 1)} = \{\begin{matrix} (1 - {\tilde{d}}_{(n + 1)}) C i f {\bar{σ}}_{n n_{(n)}} > 0 \\ C i f {\bar{σ}}_{n n_{(n)}} \leq 0 \end{matrix}$
OUTPUT: ${\tilde{C}}_{(n + 1)}, {\tilde{σ}}_{(n + 1)}, {\bar{σ}}_{n n_{(n + 1)}}, r_{(n + 1)}, Δ r_{(n + 1)}$

Material	Young modulus (GPa)	$ν$	$G_{f}$ (N/mm)	$F_{t}$ (MPa)
Concrete	37	0.2	-	-
Coarse aggregate	50	0.2	-	-
Mortar	30.2	0.2	-	-
Coupling element	30.2	0.2	-	-
ITZ	30.2	0	0.05	2.6
Int. matrix-matrix	30.2	0	0.1	5.2

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E-mail: arlene@ibracon.org.br

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[1] Corresponding author: Rodrigo Ribeiro Paccola. E-mail: rpaccola@sc.usp.br