On the added value of multi-scale modeling of concrete

doi:10.1007/s11709-021-0790-0

Front. Struct. Civ. Eng.

2022, Vol. 16

Issue (1) : 1-23 https://doi.org/10.1007/s11709-021-0790-0

REVIEW

On the added value of multi-scale modeling of concrete

Jiaolong ZHANG¹, Eva BINDER², Hui WANG³, Mehdi AMINBAGHAI⁴, Bernhard LA PICHLER⁴, Yong YUAN¹, Herbert A MANG^1,⁴(

)

¹. College of Civil Engineering, Tongji University, Shanghai 200092, China
². Department of Building Technology, Linnaeus University, Växjö 35195, Sweden
³. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
⁴. Institute for Mechanics of Materials and Structures, TU Wien, Vienna 1040, Austria

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Abstract

This review of the added value of multi-scale modeling of concrete is based on three representative examples. The first one is concerned with the analysis of experimental data, taken from four high-dynamic tests. The structural nature of the high-dynamic strength increase can be explained by using a multi-scale model. It accounts for the microstructure of the specimens. The second example refers to multi-scale thermoelastic analysis of concrete pavements, subjected to solar heating. A sensitivity analysis with respect to the internal relative humidity (RH) of concrete has underlined the great importance of the RH for an assessment of the risk of microcracking of concrete. The third example deals with multi-scale structural analysis of a real-scale test of a segmental tunnel ring. It has turned out that multi-scale modeling of concrete enables more reliable predictions of crack opening displacements in tunnel segments than macroscopic models taken from codes of practice. Overall, it is concluded that multi-scale models have indeed a significant added value. However, its degree varies with these examples. In any case, it can be assessed by means of a comparison of the results from three sources, namely, multi-scale structural analysis, conventional structural analysis, and experiments.

Keywords experiments multi-scale analysis conventional structural analysis concrete reinforced concrete

Corresponding Author(s): Herbert A MANG

Just Accepted Date: 07 December 2021 Online First Date: 21 January 2022 Issue Date: 07 March 2022

Cite this article:

Jiaolong ZHANG,Eva BINDER,Hui WANG, et al. On the added value of multi-scale modeling of concrete[J]. Front. Struct. Civ. Eng., 2022, 16(1): 1-23.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-021-0790-0
https://academic.hep.com.cn/fsce/EN/Y2022/V16/I1/1

Fischer et al. [12] tested cement paste samples (6 months old)		Kühn et al. [15] tested concrete samples (5 months old)		Zhang et al. [16] tested mortar samples with two different sizes
				small specimens		large specimens
$f u l t$ (MPa)	$ε ˙$ (1/s)	$f u l t$ (MPa)	$ε ˙$ (1/s)	$f u l t$ (MPa)	$ε ˙$ (1/s)	$f u l t$ (MPa)	$ε ˙$ (1/s)
74.15	700	95.1	136.6	70.66	49.46	64.45	37.01
42.01	200	104.2	150.1	73.82	71.01	71.22	69.67
74.41	500	119.2	185.5	74.75	85.09	71.47	76.68
48.6	500	125.7	190	75.11	130.74	75.02	56.43
65.4	500	142.5	202.7	75.61	153.16	72.13	89.38
114.26	500	–	–	79.67	183.52	76.01	126.2
132.73	1900	–	–	81.66	219.90	77.00	152.86
164.39	2100	–	–	86.57	205.48	79.14	126.2
156.36	2100	–	–	90.43	295.02	82.03	165.04
133.73	2100	–	–	96.52	337.87	88.8	192.39
152.64	2100	–	–	98.16	345.6	93.34	165.04
143.78	1900	–	–	99.45	322.94	93.75	228.61
–	–	–	–	101.67	330.32	98.62	261.43
–	–	–	–	109.04	453.30	–	–

Tab.1 Selected experimental data from high-dynamic compression tests in the literature

reference	Tang et al. [26]		Gebbeken and Greulich [27]				Brara and Klepaczko [28]		Li et al. [29]			Fischer et al. [12]
reference	$C$ (MPa)	$D$ (MPa/s)	$F m$ (MPa)	$W x$	$W y$ (MPa)	$S$	$t C 0$ (s)	$ξ$	$B 3$	$B 4$	$B 5$	$F h$
Fischer et al. [12]	72.33	0.021	143.4	2.912	96.47	5.27	9.323e–6	1.95	–1.256	9.46	–14.78	0.59
Kühn et al. [15]	6.986	0.638	1463	1.876	30.65	0.17	2.193e–5	0.06	56.47	–241	261	0.49
Zhang et al. [16] (large specimens)	61.11	0.142	166.2	2.6	116.1	1.95	3.296e–5	3.73	0.5787	–1.60	2.373	0.74
Zhang et al. [16] (small specimens)	64.11	0.099	124.8	2.527	98.5	3.32	2.839e–5	3.69	0.684	–2.01	2.608	0.74

Tab.2 Identiﬁed ﬁtting parameters of various models for the investigated experimental test series

Fig.1 Dynamic increase factor of the ultimate stress depending on the strain rate: model of Li et al. [29], in orange, of Gebbeken and Greulich [27], in green, of Brara and Klepaczko [28], in red, of Tang et al. [26], in dark blue, and of Fischer et al. [12], in light blue; Markers represent the experimental results of (a) Fischer et al. [12], (b) Kühn et al. [15], (c) Zhang et al. [16], small specimens, (d) Zhang et al. [16], large specimens.

experimental data sets and content of comparison	sum of squared errors (SSE)
experimental data sets and content of comparison	M1	M2	M3	M4	M5	M6
Fischer et al. [12]	5.57	0.99	3.31	2.07	6.68	86.15
Kühn et al. [15]	0.13	0.16	0.13	0.07	0.19	15.59
Zhang et al. [16] (large specimens)	0.06	0.06	0.09	0.07	0.16	4.39
Zhang et al. [16] (small specimens)	0.04	0.03	0.11	0.11	0.24	10.19
$Σ S S E$	5.8	1.3	3.6	2.3	7.3	116.3
number of ﬁtting parameters	2	4	2	3	1	0
prediction without high-dynamic test	–	–	–	–	yes	yes

Tab.3 Differences between modeled values and experimental results: the sum of squared errors (SSE) is calculated individually for each model and each experimental data set; the sum of all errors (Σ SSE) according to Eq. (7) for the different experimental data enables a comparison of the different models

Fig.2 Dynamic increase factor of the ultimate stress depending on the strain rate: the black curve shows the ﬁb Model Code prediction, the gray-shaded area illustrates the model prediction according to Fischer et al. [12], and the markers represent the experimental results of (a) Fischer et al. [12], (b) Kühn et al. [15], (c) Zhang et al. [16], for small specimens, (d) Zhang et al. [16], for large specimens.

Fig.3 Pavement plate resting on an elastic Winkler foundation, loaded by a temperature change.

Fig.4 Thermal expansion coefficient of the cement paste, depending on the internal RH, see Ref. [43]; the curve refers to a fitting function used by Emanuel and Hulsey [40] and the markers represent the experimental results of Meyers [44], Mitchell [45], and Dettling [46].

Fig.5 (a) Distribution of the temperature change over the thickness of the pavement plate and decomposition of the resulting thermal eigenstrains into those related to the eigenstretch, the eigencurvature, and the eigendistortion for a pavement, with internal RH amounting to (b) 50%, (c) 65%, and (d) 100%, respectively.

material	volume fraction	bulk modulus (GPa)	shear modulus (GPa)	thermal expansion coefficient (10⁻⁶ °C^–1)
material	volume fraction	bulk modulus (GPa)	shear modulus (GPa)	$R H = 50 %$	$R H = 65 %$	$R H = 100 %$
concrete	–	17.8	13.2	11.5	12.1	9.8
mortar	–	17.4	14.0	14.2	15.1	11.1
cement paste	0.30	10.2	7.7	18.0	20.2	10.5
fine aggregate	0.28	33.8	30.8	–	11.5	–
coarse aggregate	0.42	18.5	12.2	–	8.0	–

Tab.4 Properties of concrete and its constituents: volume fractions, elastic constants, and thermal expansion coefficients; with output highlighted in bold face

Fig.6 Macroscopic thermal stresses resulting from (a) the constrained eigencurvature and (b) the prevented eigendistortion.

Fig.7 Total thermal stresses of the pavement plate, loaded by a daily temperature evolution.

Fig.8 Thermally loaded concrete: macroscopic stresses of concrete and microscopic stresses of its constituents (con = concrete, cagg = coarse aggregates, mor = mortar, fagg = fine aggregates, cp = cement paste), with internal relative humidity amounting to (a) and (b) 50%; (c) and (d) 65%; as well as (e) and (f) 100%, respectively.

Fig.9 Set-ups of real-scale tests of segmental tunnel rings at Tongji University: (a) single ring [55] and (b) three rings allowing consideration of ring-to-ring interaction [64].

Fig.10 Test of a single ring, reported in Ref. [55]: (a) geometric dimensions of the ring [67] and (b) layout of the hydraulic jacks, producing the loads P₁, P₂, P₃ [68].

Fig.11 Stress-strain relations of the concrete of tunnel segments of the tested ring in Ref. [1], obtained by means of formulae in (a) the fib Model Code [69]; (b) the JSCE Guidelines for Concrete [70]; (c) the GB50010-2010 [71]; (d) by the multi-scale model.

Tab.5 Mix design of the concrete of the segments (kg/m³)

Fig.12 Comparison of convergences obtained from experimental measurements with those from conventional structural analysis based on (a) the fib Model Code [69]; (b) the JSCE Guidelines for Concrete [70]; (c) the GB50010-2010 [71]; and from (d) multi-scale structural analysis.

Fig.13 Development of bending-induced crack openings at the segment surface, obtained from conventional structural analysis by means of (a) the fib Model Code [69]; (b) the JSCE Guidelines for Concrete [70]; (c) the GB50010-2010 [71], and from (d) multi-scale structural analysis; the green arrows point to

P 1 = 0.117 M N

, corresponding to the experimentally-observed onset of cracking; the line segments in green color represent the experimentally measured range of the maximum crack opening.

Tab.6 Input for different approaches for determination of the tensile strength and of softening of concrete, and prediction quality of the corresponding structural analyses

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