Concrete modeling using micromechanical multiphase models and multiscale analysis

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

doi:10.1590/S1983-41952023000500001

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

Concrete modeling using micromechanical multiphase models and multiscale analysis

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

Authorship SCIMAGO INSTITUTIONS RANKINGS

abstract:

Concrete in its macrostructure is a multiphase cementitious composite material, however, by reducing its scale, it is possible to identify the phases that compose it, among the phases are those embedded in the microscale: the hydrated silicates, in the mesoscale: the cement paste, transition zones and aggregates and in the macro phase: the composite itself. Modeling this type of material with two-phase micromechanical models is common in the literature, but there are already proven limitations that two-phase models can provide high modeling errors and are not recommended for this type of study. Faced with this problem, an alternative would be to use multiple-phase models, combined with a multiscale perspective in an attempt to minimize the error in modeling this material. The present paper models the concrete in two different constructions: without an interfacial transition zone and with the inclusion of the interfacial transition zone, verifying the modeling error when neglecting this important phase. The entire homogenization process is performed using the decoupled multiscale technique, obtaining results that rule out the use of two-phase models and methodologies that do not evaluate the interfacial transition zone in conventional concrete. The results obtained with the use of multiple-phase models reduced the relative error to practically zero (compared to experimental tests), demonstrating that micromechanics can be a concrete modeling tool provided that the multiscale process considers as many as possible phases and robust models that take this nature into account.

Keywords:
multiscale modeling; micromechanics; concrete

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