Modelagem do concreto usando modelos micromecânicos de múltiplas fases e análise multiescala

Silva, Rodrigo Mero Sarmento da; Barboza, Aline da Silva Ramos

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ORIGINAL ARTICLE • Rev. IBRACON Estrut. Mater. 16 (5) • 2023 • https://doi.org/10.1590/S1983-41952023000500001 copiar

Modelagem do concreto usando modelos micromecânicos de múltiplas fases e análise multiescala

Autoria SCIMAGO INSTITUTIONS RANKINGS

resumo:

O concreto em sua macroestrutura é um material compósito cimentício multifásico, contudo ao se reduzir sua escala consegue-se identificar as fases quem compõe o mesmo, dentre as fases estão as embutidas na microescala: os silicatos hidratados, na mesoescala: a pasta de cimento, zonas de transições e agregados e na fase macro: o próprio compósito. Modelagens desse tipo de material com modelos micromecânicos bifásicos são comuns na literatura, porém existe limitações já comprovadas de que modelos de duas fases podem aferir erros altos a modelagem não sendo recomendado para esse tipo de estudo. Diante dessa problemática, uma alternativa seria empregar modelos de múltiplas fases, aliado a uma perspectiva multiescala na tentativa de minimizar o erro na modelagem desse material. O presente trabalho modela o concreto em duas construções distinta: sem zona de transição interfacial e com a inclusão da zona de transição interfacial, verificando o erro de modelagem à negligência essa importante fase. Todo processo de homogeneização é realizado utilizando a técnica multiescala desacoplada obtendo resultados que descartam a utilização de modelos bifásicos e metodologias que não avaliam a zona de transição interfacial em concretos convencionais. Os resultados obtidos com a utilização de modelos de múltiplas fases reduziram o erro relativo a praticamente zero (em comparação com ensaios experimentais), demonstrando que a micromecânica pode ser uma ferramenta de modelagem do concreto desde que o processo multiescala leve em consideração o maior número possíveis de fases e modelos robustos que levem em conta essa natureza.

Palavras-chave:
modelagem multiescala; micromecânica; concreto

Models	Particulars and recurrence equation	Source
Reuss	Its limitation is the imposition of a constant state of tension, in addition to not evaluating the interaction between inclusions	[³⁶36 A. Reuss, “Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle,” J. Appl. Math. Mechanics, vol. 9, pp. 49-58, 1929.]
Reuss	$C^{H} = C_{m} {: C}_{i} : {[C_{i} (1 - f_{i}) + C_{m} f_{i}]}^{- 1}$
Voigt	Its limitation is a constant of strain state, in addition to not evaluating the integration between the inclusions	[³²32 A. Kaw, Mechanics of Composite Materials, 2nd ed. New York, NY, USA: Taylor & Francis, 2006.]
Voigt	$C^{H} = C_{m} (1 - f_{i}) + C_{i} f_{i}$
Hashin	It does not estimate constant stress and strain fields, instead, it estimates auxiliary fields representing a variation of the reference solution. When the formulation of the energy obtained is maximized, the upper limit is found and when it is minimized, the lower limit is found.	[³³33 Z. Hashin and S. Shtrikman, “On some variational principles in anisotropic and nonhomogeneous elasticity,” J. Mech. Phys. Solids, vol. 10, no. 4, pp. 335-342, Oct./Dec., 1962.]
	$K^{-} = K_{m} + [\frac{f_{i}}{[\frac{1}{K_{i} - K_{m}} + \frac{3 (1 - f_{i})}{3 K_{m} + 4 G_{m}}]}]$
	$K^{+} = K_{i} + [\frac{{1 - f}_{i}}{[\frac{1}{K_{m} - K_{i}} + \frac{3 f_{i}}{3 K_{i} + 4 G_{i}}]}]$
	$G^{-} = G_{m} + [\frac{f_{i}}{[\frac{1}{G_{i} - G_{m}} + \frac{6 (1 - f_{i}) (K_{m} + 2 G_{m})}{5 G_{m} (3 K_{m} + 4 G_{m})}]}]$
	$G^{+} = G_{i} + [\frac{{1 - f}_{i}}{[\frac{1}{G_{m} - G_{i}} + \frac{6 f_{i} (i + 2 G_{i})}{5 G_{i} (3 K_{i} + 4 G_{i})}]}]$
Mori-Tanaka	It is the most used model for homogenization of composites, it considers the interaction between the particles and can be used with larger volumetric fractions.	[³⁴34 T. Mori and K. Tanaka “Average stree in matrix and average energy of materials with mis-fitting inclusions,” Act. Metall., vol. 21, no. 5, pp. 571-574, May. 1973.]
Mori-Tanaka	$C^{H} = C_{m} : {\begin{matrix} \{I + f_{i} (S - I) : {[{(C_{m} - C_{i})}^{- 1} : C_{m} - S]}^{- 1}\} : \\ \{I + f_{i} S : {[{(C_{m} - C_{i})}^{- 1} : C_{m} - S]}^{- 1}\} \end{matrix}}^{- 1}$
Dilute Suspension	It is limited by the amount of volumetric fraction of inclusion in the homogenization process, and can only be applied to low inclusion rates	[³⁷37 H. Zhang, K. Anupam, A. Scarpas, and C. Kasbergen, “Comparison of different micromechanical models for predicting the effective properties of open graded mixes,” Transp. Res. Rec., vol. 2672, no. 28, pp. 404-415, Sep. 2018.]
Dilute Suspension	$C^{H} = C_{m} + f_{i} (C_{i} - C_{m}) : A_{i}$
Self-Consistent	$A_{i} = [I - S : C_{m} : (C_{m} - C_{i})]$
	${C^{H}}_{n + 1} = {C_{m} + f_{i} (C_{i} - C_{m}) : [I - {S^{H}}_{n} : {C^{H}}^{- 1}_{n} : ({C^{H}}_{n} - C_{i})]}^{- 1}$	[³⁵35 R. Hill, “A self-consistent mechanics of composite materials,” J. Mech. Phys. Solids, vol. 13, pp. 213-222, Aug. 1965.]
	Step 1:
	$S^{H} = S$ , $C^{H} = C_{m}$
	$\frac{‖{C^{H}}_{n} - {C^{H}}_{n - 1}‖}{‖{C^{H}}_{n - 1}‖} < δ$
Generalized Self-Consistent	Based on the theory of elasticity, it manages to evaluate the interaction between inclusions being developed as a three-phase model, essentially it only evaluates two phases. It considers the inclusion-matrix interaction and between inclusions	[²⁷27 R. Christensen and K. Lo, “Solutions for effective shear properties in three phase sphere and cylinder models,” J. Mech. Phys. Solids, vol. 27, no. 4, pp. 315-330, Aug. 1979.]
	$A {[\frac{G^{H}}{G_{m}}]}^{2} + B [\frac{G^{H}}{G_{m}}] + C$
	A, B, and C, are constants that can be obtained in Christensen and Lo [²⁷27 R. Christensen and K. Lo, “Solutions for effective shear properties in three phase sphere and cylinder models,” J. Mech. Phys. Solids, vol. 27, no. 4, pp. 315-330, Aug. 1979.]
	$K^{H} = [K_{m} + \frac{f_{i} (K_{i} - K_{m}) (3 K_{m} + {4 G}_{m})}{3 K_{m} + {4 G}_{m} + 3 (1 - f_{i}) (K_{i} - K_{m})}]$
Differential Scheme	Unlike other methods that assume an inclusion immersed in an infinite matrix, the differential scheme works with incremental doses of inclusions.	[³⁰30 Z. Hashin, “The differential scheme and its application to cracked materials,” J. Mech. Phys. Solids, vol. 36, no. 6, pp. 719-734, 1988.]
	${C^{H}}_{n + 1} = {C^{H}}_{n} + \frac{∆ f_{i}}{1 - f_{i}} (C_{i} - {C^{H}}_{n}) : {A_{i}}^{D}$
	${A_{i}}^{D} = {[I - {S^{H}}_{n} : {C^{H}}_{n}^{- 1} : ({C^{H}}_{n} - C_{i})]}^{- 1}$
	Step 1:
	$S^{H} = S, {C^{H}}_{n} = C_{m}$
Four-phases	which is based on the three-phase model (Generalized Self-Consistent), associated with an analytical model, namely: “Composite Sphere Assemblage (CSA)” [²²22 Z. Hashin, “Analysis of composite materials-a survey,” ASME J. Appl. Mech., vol. 50, pp. 481-505, Sep. 1983.] or “Composite Cylinder Assemblage (CCA)” [³³33 Z. Hashin and S. Shtrikman, “On some variational principles in anisotropic and nonhomogeneous elasticity,” J. Mech. Phys. Solids, vol. 10, no. 4, pp. 335-342, Oct./Dec., 1962.].	[²¹21 G. Li, Y. Zhao, and S. Pang, “Four-phase sphere modeling of effective bulk modulus of concrete,” Cement Concr. Res., vol. 29, no. 6, pp. 839-845, Feb. 1999.]
	$K^{H} = [K_{m} + \frac{f_{i} (K_{i} - K_{m}) (3 K_{m} + {4 G}_{m})}{3 K_{m} + {4 G}_{m} + 3 (1 - f_{i}) (K_{i} - K_{m})}]$
	$E_{L}^{H} = f_{i} E_{i} + f_{m} E_{m} + \frac{4 f_{i} f_{m} {(υ_{i} - υ_{m})}^{2} G_{m}}{1 + \frac{3 f_{i} G_{m}}{3 K_{m} + G_{m}} + \frac{3 f_{m} G_{m}}{3 K_{i} + G_{i}}}$
	$K_{T}^{H} = K_{m} + \frac{G_{m}}{3} + \frac{f_{i}}{[\frac{3}{3 K_{i} + G_{i} - 3 K_{m} - G_{m}}] + [\frac{3 f_{i}}{3 K_{m} + {4 G}_{m}}]}$
	$G_{L}^{H} = \frac{(f_{i} G_{i} + f_{m} G_{m} + G_{i}) G_{m}}{f_{m} G_{i} + f_{i} G_{m} + G_{m}}$
	$υ_{L T}^{H} = f_{i} E_{i} + f_{m} E_{m} + \frac{3 f_{i} f_{m} (υ_{i} - υ_{m}) G_{m} [\frac{f_{i} G_{m}}{3 K_{m} + G_{m}} + \frac{f_{m} G_{m}}{3 K_{i} + G_{i}}]}{[\frac{3}{3 K_{i} + G_{i} - 3 K_{m} - G_{m}}] + [\frac{3 f_{i}}{3 K_{m} + {4 G}_{m}}]}$
Multiphase	That bases its formulation on the famous study of double inclusion [³⁸38 M. Hori and S. Nemat-Nasser, “Double-inclusion model and overall moduli,” Mech. Mater., vol. 14, no. 3, pp. 189-206, Jan. 1993.]. The model proposed [³⁹39 C. Shi, H. Fan, and S. Li, “Interphase model for effective moduli of nanoparticle-reinforced composites,” J. Eng. Mechanic, vol. 141, no. 12, pp. 141-153, Dec. 2020.], takes into consideration the assemblage of the Eshelby [²⁸28 J. Eshelby “The determination of the elastic field of an ellipsoidal inclusion and related problems,” Proc. Royal Society London Ser. A, vol. 241, pp. 376-396, Mar. 1957.] tensor for all of the layers of existing materials in the modeling.	[³⁹39 C. Shi, H. Fan, and S. Li, “Interphase model for effective moduli of nanoparticle-reinforced composites,” J. Eng. Mechanic, vol. 141, no. 12, pp. 141-153, Dec. 2020.]
	$f_{m} + \sum_{1}^{n} {(f}_{i} + f_{r}) = 1$
	${A_{i}}^{D} = I + S_{i} : Φ_{i} + ∆ S : Φ_{r}$
	${A_{r}}^{D} = I + S_{r} : Φ_{r} + \frac{f_{i}}{f_{r}} ∆ S : (Φ_{i} - Φ_{r})$
	$∆ S = S_{i} - S_{r}$
	$Φ_{i} = - {[(S_{i} + A_{i}) + ∆ S : (S_{i} + A_{i} - \frac{f_{i}}{f_{r}} ∆ S) : {(S_{r} + A_{r} - \frac{f_{i}}{f_{i}} ∆ S)}^{- 1}]}^{- 1}$
	$Φ_{r} = - {[∆ S + (S_{i} + A_{i}) : {(S_{i} + A_{i} - \frac{f_{i}}{f_{r}} ∆ S)}^{- 1} : (S_{r} + A_{r} - \frac{f_{i}}{f_{i}} ∆ S)]}^{- 1}$
	$A_{i} = {(C_{i} - C_{m})}^{- 1} : C_{m}$
	$A_{r} = {(C_{r} - C_{m})}^{- 1} : C_{m}$
	$C^{H} = C_{m} + [f_{r} (C_{r} - C_{m}) : {A_{r}}^{D} + f_{i} (C_{i} - C_{m}) : {A_{i}}^{D}]$

Material	Module of Elasticity (GPa)	Poisson ratio	Volumetric Fraction (%)
Cement Paste	21.34	0.25	15.40
Gravel	51.31	0.15	40.30
ITZ-1	10.17	0.30	2.30
Sand	77.60	0.15	26.90
ITZ-2	12.70	0.25	15.10
Concrete	34.50	-	-

Norm	Module of Elasticity estimate	Comments
NBR 6118 (2014)	$20 \leq f_{c k} \leq 50$ MPa $E_{c} = α ∙ 5600 \sqrt{f_{c k}}$ $55 \leq f_{c k} \leq 90$ MPa $E_{c} = 21.5 ∙ 10^{3} ∙ α \sqrt[3]{\frac{f_{c k}}{10} ∙ 1.25}$	$α = 1.2$ (Basalt, dense limestone aggregates)
		$α = 1.0$ (Quartzite aggregates)
		$α = 0.9$ (Limestone aggregates)
ACI 3018 (2014)	$E_{c} = 5170 \sqrt{f_{c k}}$	$α = 0.7$ (Sandstone aggregates)
Fib Model Code (2010)	$12 \leq f_{c k} \leq 80$ MPa $E_{c} = 21.5 ∙ 10^{3} ∙ α \sqrt[3]{\frac{f_{c k} + 8}{10}}$	$α = 1.2$ (Basalt, dense limestone aggregates)
		$α = 1.0$ (Quartzite aggregates)
		$α = 0.9$ (Limestone aggregates)
		$α = 0.7$ (Sandstone aggregates)
Eurocode 2 (2004)	$12 \leq f_{c k} \leq 90$ MPa $E_{c} = 23.1 ∙ 10^{3} ∙ α {(\frac{f_{c k} + 8}{10})}^{0.3}$	$α = 1.2$ (Basalt, dense limestone aggregates)
		$α = 1.0$ (Quartzite aggregates)
		$α = 0.9$ (Limestone aggregates)
		$α = 0.7$ (Sandstone aggregates)

Level 4		Volumetric Fraction in Composite	Volumetric Fraction in Homogenization Level
Mortar homogenized	Cement paste	15.40%	36.41%
	Sand	26.90%	63.59%
	Total	42.30%	100.00%
Level 5		Volumetric Fraction in Composite	Volumetric Fraction in Homogenization Level
Concrete homogenized	Mortar homogenized	42.30%	48.79%
	Gravel	40.30%	51.21%
	Total	42.30%	100.00%
Level 6		Volumetric Fraction in Composite	Volumetric Fraction in Homogenization Level
Homogenized concrete adjusted	Concrete homogenized	100.00%	97.22%
	Pores	2.42%	2.78%
	Total	102.48%	100.00%

Models	Modulus of elasticity (GPa)			Error %
Models	Level 4	Level 5	Level 6	Level 4	Level 5	Level 6
Reuss	39.596	44.559	-	14.770	26.156	-
Voigt	57.324	54.400	52.887	66.157	57.680	53.295
Hashin +	51.179	51.262	48.496	48.345	48.584	40.569
Hashin -	45.614	48.341	-	32.210	40.118	-
Concrete [⁴⁰40 Y. Li, Y. Li, and R. Wang, “Quantitative evaluation of elastic modulus of concrete with nanoidentation and homogenization method,” Constr. Build. Mater., vol. 212, pp. 295-303, Jul. 2019.]	34.50

Models	Modulus of elasticity (GPa)			Error %
Models	Level 4	Level 5	Level 6	Level 4	Level 5	Level 6
Mori-Tanaka	45.614	48.341	45.671	32.215	40.118	32.379
Self-Consistent	49.304	50.290	47.436	42.909	45.768	37.496
Generalized Self-Consistent	46.144	48.624	45.937	33.751	40.938	33.325
Differential Scheme	46.772	48.923	46.981	35.571	41.806	36.177
Concrete [⁴⁰40 Y. Li, Y. Li, and R. Wang, “Quantitative evaluation of elastic modulus of concrete with nanoidentation and homogenization method,” Constr. Build. Mater., vol. 212, pp. 295-303, Jul. 2019.]	34.50

Level 4		Volumetric Fraction in Composite	Volumetric Fraction in Homogenization Level
Mortar homogenized + Interfacial Transition Zone of Sand	Sand	26.90%	46.86%
	ITZ-s	15.10%	26.31%
	Cement Paste	15.40%	26.83%
	Total	57.40%	100.00%
Level 5		Volumetric Fraction in Composite	Volumetric Fraction in Homogenization Level
Concrete homogenized + Interfacial Transition Zone of Gravel	Gravel	40.30%	40.30%
	ITZ-g	2.30%	2.30%
	Mortar homogenized + Interfacial Transition Zone of Sand	57.40%	57.40%
	Total	100.00%	100.00%
Level 6		Volumetric Fraction in Composite	Volumetric Fraction in Homogenization Level
Homogenized concrete adjusted	Concrete homogenized + Interfacial Transition Zone of Gravel	100.00%	97.64%
	Pores	2.42%	2.36%
	Total	102.48%	100.00%

Model	Modulus of elasticity (GPa)			Error %
Model	Level 4	Level 5	Level 6	Level 4	Level 5	Level 6
Four-Phases	30.638	36.167	34.501	11.193	4.830	0.0031
Multiphase	32.101	37.774	36.302	6.952	9.488	4.441
Concrete [⁴⁰40 Y. Li, Y. Li, and R. Wang, “Quantitative evaluation of elastic modulus of concrete with nanoidentation and homogenization method,” Constr. Build. Mater., vol. 212, pp. 295-303, Jul. 2019.]	34.50

Models	Poisson ratio
Models	Level 4	Level 5	Level 6
Mori-Tanaka	0.1997	0.1751	0.1759
Self-Consistent	0.1829	0.1671	0.1683
Generalized Self-Consistent	0.1942	0.1732	0.1741
Differential Scheme	0.182	0.172	0.173
NBR 6118	0.11 - 0.22

Model	Poisson ratio
Model	Level 4	Level 5	Level 6
Four-Phases	0.183	0.173	0.174
Multiphase	0.201	0.182	0.182
NBR 6118	0.11 - 0.22

Models	$f_{c k}$ estimates (MPa)				Error %
Models	NBR 6118	ACI 3018	Fib Model Code	Eurocode 2	NBR 6118	ACI 3018	Fib Model Code	Eurocode 2
Mori-Tanaka	66.512	78.036	87,850	88.997	75.244	75.244	99.736	195.878
Self-Consistent	71.753	84.185	99.401	102.066	89.053	89.052	198.340	239.326
Generalized Self-Consistent	67.289	78.948	89.537	90.893	77.291	77.292	168.735	202.181
Differential Scheme	70.383	82.577	96.340	98.586	85.443	85.441	189.153	227.757
Concrete [⁴⁰40 Y. Li, Y. Li, and R. Wang, “Quantitative evaluation of elastic modulus of concrete with nanoidentation and homogenization method,” Constr. Build. Mater., vol. 212, pp. 295-303, Jul. 2019.]	37.954	44.530	33.318	30.079	-	-	-	-

Models	$f_{c k}$ estimates (MPa)				Error %
Models	NBR 6118	ACI 3018	Fib Model Code	Eurocode 2	NBR 6118	ACI 3018	Fib Model Code	Eurocode 2
Four-Phases	37.956	44.533	33.321	30.083	0.005	0.007	0.009	0.013
Multiphase	42.022	49.303	40.136	37.122	10.718	10.719	20.463	23.415
Concrete [⁴⁰40 Y. Li, Y. Li, and R. Wang, “Quantitative evaluation of elastic modulus of concrete with nanoidentation and homogenization method,” Constr. Build. Mater., vol. 212, pp. 295-303, Jul. 2019.]	37.954	44.530	33.318	30.079	-	-	-	-

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[1] Corresponding author: Rodrigo Mero Sarmento da Silva. E-mail: rodrigo.mero@gmail.com