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Computational modeling of the mechanical response of lightweight foamed concrete over a wide range of temperatures and strain rates
Construction and Building Materials 96 (2015) 622–631
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Computational modeling of the mechanical response of lightweight foamed concrete over a wide range of temperatures and strain rates Hui Guo a, Weiguo Guo a,⇑, Yajie Shi b a b
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China China Academy of Civil Aviation Science and Technology, Beijing 100028, China
h i g h l i g h t s Uniaxial compression tests are conducted at various temperatures and strain rates. The characteristics of the stress-strain curve are analyzed and a model is developed. The relationships between model parameters and material density are discussed. The temperature effect of the material is illustrated based on the calculation model.
a r t i c l e
i n f o
Article history: Received 9 April 2015 Received in revised form 31 July 2015 Accepted 9 August 2015
Keywords: Lightweight foamed concrete Density effect Temperature effect Damage evolution Calculation model
a b s t r a c t Theoretical and experimental studies on the nonlinear mechanical properties of lightweight foamed concrete under uniaxial compression are conducted over the temperature range of 223–343 K and the strain rate range of 0.001–118/s in this paper. The experimental results show that the mechanical properties of the materials under uniaxial compression are strongly dependent on the density and temperature, but are weaker on strain rate. Based on the experimental results, the characteristics of density effect and temperature effect are analyzed and a calculation model is developed to describe the nonlinear mechanical behavior of lightweight foamed concrete. The model takes into account the effect of material damage, density and temperature. The experimental verification and the error analysis show that the model is shown to be able to predict the nonlinear deformation behavior of lightweight foamed concrete over a wide range of temperatures and densities. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Due to its special microstructure morphology and mechanical properties, lightweight foamed concrete is widely used as energy absorbers in various industries, for example, engineered material arresting system (EMAS). There are many ingredients influencing lightweight foamed concrete ability of energy absorption, such as density and nonlinear mechanical properties of the materials. Therefore, it is very necessary to study the nonlinear mechanical behavior of lightweight foamed concrete. The nonlinear mechanical properties of lightweight foamed concrete share some characteristics with the properties of metal foams and polymeric foams. Under the uniaxial compression, the stress–strain curves of these foams can be divided into three stages: the linear elastic stage, the plateau stage and the densification stage [1–3]. However, the components and the structure of lightweight foamed concrete ⇑ Corresponding author. E-mail address: [email protected] (W. Guo). http://dx.doi.org/10.1016/j.conbuildmat.2015.08.064 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.
are more complex compared with metal foams and polymeric foams, and there are less research literatures on its mechanical properties. Therefore, with the extensive use of lightweight foamed concrete, the mechanical properties of the materials have drawn great attention of researchers. Some experimental studies have been conducted to investigate the nonlinear response of lightweight foamed concrete under uniaxial compression [3–5]. Valore [4] studied varies factors that affect the strength of porous concrete, including the size and shape of specimen, the properties of matrix material, the porosity formation and the curing methods, loading direction, as well as the moisture content. Hengst and Tressler insisted that at constant density, the major factor affect the strength of porous concrete is the crack size, which is correlated with the size of holes [6]. Park studied the effect of the components of foamed concrete on the mechanical properties of the materials, and held that the content increasing of silica fume, fly ash and glass fiber can improve the mechanical properties of foamed concrete [7]. Visagie and Kearsely found the compressive strength of foamed concrete would decrease with the increase
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of the diameter of the aperture over the dry density ranges of 0.5– 1 g/cm3, while the porosity of the materials has little effect on the compressive strength when the dry density is over 1 g/cm3 [8]. Guo et al. studied systemically the compression behavior of one kind of new foamed concrete composite at the room temperature and strain rate ranges from 0.001/s to 0.1/s, and found that this material is insensitive to strain rates during the strain rate ranges, while the densities of this material have a great influence on the compression behavior [3]. Besides, few theoretical and numerical models had also been developed to describe the mechanical properties of foamed concrete recently. Hoff studied the relationship between the porosity and the compression strength of foamed concrete, and insisted that there be a certain calculating formula of strength and porosity for given cement content [9]. The prediction model proposed by Kearsely and Wainwright showed that the model from Hoff can efficiently predict the compression strength of foamed concrete at different time and densities [5]. Zhou et al. studied the characteristics of the compressive stress–strain curve of foamed concrete, and found that there are four steps in foamed concrete compression process, namely plateau step, compacting step, yield step and decline step, the compression mechanical property is affected by matrix material, volume weight, morphology and distribution of pore, etc. [10]. Gibson and Ashby derived the equation of compression strength and relative density of brittle foams, by which could get the parameter to describe the volume fraction of hole edge and the fracture strength of hole wall [1]. Recently, Guo et al. developed a compression phenomenological constitutive model of foamed concrete by analyzing platform crushing stress, compacting strain and density [3]. The theoretical study on the compressive mechanical properties of foam concrete in the above literature is almost all to set up the mathematics model through the relationship between the strength and porosity, matrix material and so on. However, there is not a more comprehensive study on the failure mechanism of the materials as well as on the macroscopic factors of affecting the compressive strength, such as strain rate, temperature, and relative density. In this paper, the uniaxial compression tests on the mechanical properties of one kind of new lightweight foamed concrete at the temperature range of 223–343 K and the strain rate range of 0.001–118/s are carried out. The deformation mechanism and the mechanical characteristics of the materials are analyzed and discussed. Based on the experimental results, a calculation model describing the nonlinear compressive behavior of lightweight foamed concrete is established in this paper, which includes the effect of material damage, density and temperature. 2. Experimental investigation 2.1. Materials The lightweight foamed concrete studied in this paper, mainly used in engineered material arresting system (EMAS). This new lightweight foamed concrete adds some polyester fiber materials into the generic foamed concrete using traditional manufacturing process [11–13]. The material of the polyester fiber is PP, the length is 6 mm, the melting temperature is 165–175 °C, and the fiber diameter is 25–45 lm. The mass of PP accounts for about 0.85% of the mass of the cement. The foaming agent used in this paper is 35% hydrogen peroxide. The main components of lightweight foamed concrete are cement, water, heavy calcium and fly ash. The particle size of the cement is about 10 lm. The particle size of the heavy calcium is about 18 lm. The calcium–cement ratio is 0.4, and the water–cement ratio is 0.8. The final optimization design scheme for the chemical constituents and mix proportion of the raw materials is determined through trial and error. The detailed preparation process of standard sample is as follows: (a) Mix the foaming agent and purified water thoroughly using agitator at room temperature, and whisk until the mixture is smooth. Then, add the mixture of clay, cement, sand, fly ash, polystyrene fiber and water into the mixed froth, and using the water reducer and the coagulation accelerator to make the mixture uniform flow slurry.
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(b) Fill the U = 80 mm, h = 60 mm mold with the uniform flow slurry, and vibrate it on the shaking table and smooth the upper surface with a spatula. Then, after letting it stand at room temperature for 3 days, demold and place it in constant humidity cabinet with the relative humidity of 90%, this stage will continue for 30 more days. Finally, paste numbering in the specimen for the subsequent experiments. The microstructure of the experimental materials was scrutinized using field emission scanning microscopy, and the scanning microscopic photos are shown in Fig. 1. It can be observed from Fig. 1a that the distribution and orientation of the pore wall materials are uniform approximately, and the thickness of hole wall is about 40 lm. As can be seen from Fig. 1b, the hole wall of the materials is accumulated by varieties of the salt crystals arranged without order at higher magnification. The crystal is a slender cylinder, and the maximum diameter is about 1 lm. In addition, the arrangement of the grain is loose, so it can be inferred that the weak crystal boundary is the essential reason of the low compression strength of the lightweight foamed concrete.
2.2. Testing equipment In order to cover a wide range of loading speeds, material tests are conducted on two types of testing machines: (1) a CSS-4410 electric universal testing machine equipped with a temperature chamber for low strain rate tests; (2) a DYN-9250 drop weight impact testing machine equipped with a temperature chamber for high-rate impact tests. During the loading process, the CSS4410 electric universal testing machine measures the displacements by automatic recording the movement distance of the crossbeam, and acquires the compressive load by high precision pulling and pushing dual-purpose load transducer installed on the crossbeam. The measuring range of compressive load is the sensor capacity of 2–100%, and the precision of the sensor is 0.5%. Based on the collected data from testing machine and the geometric size of specimen, the stress– strain curves of test materials are obtained by mathematical calculation method at low strain rates. The DYN-9250 drop weight impact testing machine adjusts the level of strain rate and its stability by controlling the weight of the hammer head and the initial height of free falling with the assistant of the Impulse control software. The testing machine releases the drop hammer to make it free falling in order to impact specimen, and collects a series of experimental data by using the Impulse data acquisition software in the process of impact. Finally, the curves of calculation results are generated by using the Impulse software at high strain rates.
2.3. Experimental program and results Lightweight foamed concrete is a heterogeneous and anisotropic multi-phase composite material. Because the components of the material are complex and be many influence factors, the mechanical properties data obtained from the tests have the certain characteristics of discreteness. In order to carry on the system analysis of the experimental data, three major factors which influence the mechanical properties of lightweight foamed concrete are emphatically considered in this paper, and so as to provide valuable reference for the engineering design, namely material density, experimental temperature and strain rate. Test conditions are designed as three schemes as follows: (1) the effects of different density on the compressive strength of lightweight foamed concrete are studied when the temperature and strain rate are constant; (2) the effects of different temperature on the compressive strength of the materials are investigated when the density and strain rate are constant; (3) the effects of different strain rate on the compressive strength of the materials are discussed when the temperature and density are constant. The typical stress–strain curves of four different densities of lightweight foamed concrete specimens at the temperature of 293 K and strain rate of 0.1/s are shown in Fig. 2. The typical stress–strain curves of lightweight foamed concrete specimens with the density of 0.23 g/cm3 and strain rate of 0.1/s under four different temperatures are shown in Fig. 3. The typical stress– strain curves of lightweight foamed concrete specimens with the density of 0.18 g/cm3 and temperature of 293 K under six different strain rates are shown in Fig. 4. The typical stress–strain curves of lightweight foamed concrete specimens with the density of 0.37 g/cm3 and temperature of 293 K under six different strain rates are shown in Fig. 5. The results showed that the engineering stress–strain curves of low-density lightweight foamed concrete in low strain rate have the obvious characteristics of three stages, namely the linear elastic stage, the crushing plateau stage and the densification stage. The curves in low strain rate are smoother by comparison with high strain rate. The curves show the zigzag fluctuation at high strain rate and especially the larger fluctuation in the transition region of elastic segment and crushing plateau segment. In addition, as the density of the materials increases the ranges of fluctuation increases correspondingly. According to the analysis of the stress–strain curves of different density (0.18 g/cm3, 0.20 g/cm3, 0.27 g/cm3 and 0.37 g/cm3) lightweight foamed concrete in Fig. 2, it can be found that for a certain temperature and strain rate, the stress value at the same deformation amount increases as the density of the materials increases, and the crushing plateau segment of the curves
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Fig. 1. Scanning micrograph of lightweight foamed concrete: (a) extracellular pore size distribution microgram; (b) structure of cell wall microgram.
Fig. 2. Eng. stress–strain curves of materials of various densities at a temperature of 293 K and strain rate of 0.1/s.
Fig. 3. Eng. stress–strain curves of materials of a strain rate of 0.1/s and a density of 0.23 g/cm3 at various temperatures.
becomes shorter. According to the analysis of the stress–strain curves of lightweight foamed concrete under different temperature (223 K, 293 K, 313 K and 343 K) in Fig. 3, it can be found that for a certain density and strain rate, the stress value corresponding to the same deformation amount in the crushing plateau
Fig. 4. Eng. stress–strain curves of materials of a temperature of 293 K and a density of 0.18 g/cm3 at various strain rates.
Fig. 5. Eng. stress–strain curves of materials of a temperature of 293 K and a density of 0.37 g/cm3 at various strain rates.
segment of the curves decreases as the temperature increases, but the stress value in the elastic segment and the densification segment of the curves has little change. According to the analysis of the stress–strain curves of lightweight foamed concrete under different strain rate (0.001/s, 0.01/s, 0.1/s, 48.2/s, 68.2/s, 83.5/s, 96.4/s, 109.1/ s and 118/s) in Figs. 4 and 5, it can be found that the mechanical properties of the materials are less sensitive to strain rate effect at a given density and temperature.
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3. Theoretical model In developing a generalized computational model applicable to lightweight foamed concrete under finite deformation, the following relevant observations are noted from the available experimental data in this paper and related literature. (1) The shape of the compression stress–strain curve is obtained by the theoretical model is basically the same for almost all crushable foams [1–3], and can be characterized by three different regimes as follows: (a) the linear elastic regime where the stress keep near-linearly increasing with the increasing strain; (b) the crushing plateau regime extending over a large range strains where the material start to crack or crush progressively. In addition, the stress drop may be found before the stress reaches a plateau, which may be attributed to cracking of foam cell walls; (c) the densification regime where the stress rises sharply as the material starts to compact. (2) According to the experimental results, the theoretical model of lightweight foamed concrete should be able to describe the effect of material density and temperature, but does not consider the effect of strain rate. (3) The deformation mechanism of lightweight foamed concrete is different from that of metal foams and polymeric foams [14,15], and is usually guided by the generation and evolution of micro crack and damage of cell walls, because of which the material leans to brittle fracture. In addition, lightweight foamed concrete is a heterogeneous mixture of fiber and other aggregates by cementing materials, so the internal damage is an important factor affecting the nonlinear mechanical behavior of the material [16,17]. The goal of this paper is to capture all the above features of the stress–strain behavior with a continuous function defined in the whole real domain. Based on the above analysis of the failure mechanism and mechanical characteristics, the general form of the computational equation of lightweight foamed concrete under high temperature can be expressed as:
r ¼ rðe; q; T; DÞ
ð1Þ
where r is the stress, e is the strain, q is the material density, T is the temperature, and D is the damage factor which value range from 0 to 1. When D = 0, the material is perfect without damage; when D = 1, the material is damaged completely. According to strain equivalence principle proposed by Lemaitre [18,19], the damage computational relation of lightweight foamed concrete is obtained as follows:
rd ¼ ð1 DÞri ðe; q; TÞ
ð2Þ
where rd is the apparent stress, namely material stress considering damage, and ri is the material stress referring to no damage. 3.1. Damage evolution The progressive, partial interfacial debonding may occur under increasing deformations and affect the overall stress–strain behavior of lightweight foamed concrete [16,17]. After the interfacial debonding, the debonded fibers lose the load-carrying capacity along the debonded direction and are regarded as partially debonded fibers, namely material damage. The damage failure of lightweight foamed concrete is the time-correlated rheological process, which is also the time process of the different forms of micro crack with the finite rate evolution. The damage evolution law is the key to discover the functional relationship how the damage evolution is dependent on the constitutive state variables. Because the study on the behavior of single micro crack can not reflect global damage evolution on the micro-level, it needs to study the global evolution behavior of material damage from the viewpoint of probability. Following Zhao et al. and Ju et al. [20,21], the probability of material damage was modeled as a two-parameter Weibull process.
In addition, Lee and Liang also made use of the Weibull distribution function to characterize the damage evolution law of foamed concrete by statistical microscopic damage mechanics, and good effect was obtained [17]. Assuming that the Weibull statistics govern, the cumulative probability distribution function of material damage can be expressed as:
PðeÞ ¼
m
a
em1 exp
em a
ð3Þ
where a and m are scale parameter and shape parameter, respectively. Shape parameter m is the most important parameter, which determines the basic shape of the distribution density curve. Scale parameter a is a function of amplifying or scaling a curve, but does not affect the shape of the distribution density curve. In addition, the failure condition of the different stages can be expressed by changing the shape parameter. If the damage factor D is defined as the ratio of the damaged quantity of micro unit and its total quantity, then D and P(e) have the following relationship:
Z
D¼ 0
e
PðeÞde ¼
m
a
Z
e
0
em1 exp
em em de ¼ 1 exp a a
ð4Þ
3.2. Quasi-static damage model At present, the mechanical properties of lightweight foamed materials were studied to a degree in the literature, and some theoretical models for describing mechanical behavior of the materials were proposed. The work done by Gibson, Liu and Avalle, etc. has already been generally accepted in the study of theoretical modeling [1,2,22]. Based on the experimental results and mechanical characteristics of lightweight foamed concrete, this paper selects the three representative theoretical models to fit the experimental data, namely Gibson correction model, Liu-Subhash model and Avalle model. Finally, the quasi-static damage computational mode suitable for lightweight foamed concrete is established by the contrastive analysis. Because the stress–strain curves of foam materials, in general, exhibit three regions: the linear elastic region, the plateau region and the densification region, the Gibson model is formulated by three equations describing each region [1]. The observation of the experimental curves in Figs. 2 and 3 put in evidence that the crushing plateau region is not flat but has a negative slope. In order to take into account the described slope, the formula for the plateau region of the Gibson model is modified to improve its fitting capability. A sloped linear region is used instead of the constant stress value in Gibson correction model.
8 ri ðeÞ ¼ Ee > > < ri ðeÞ ¼ ry þ he n > > : ri ðeÞ ¼ ry 1 eF F eF e
e 6 ey ey 6 e 6 ez
ð5Þ
e > ez
where ri and e are engineering stress and engineering strain, respectively, considered positive in compression. The model has six parameters, namely E the slope of the elastic part of the curve, ey the yield strain, eF the densification strain, and three constants h, F and n. The strain value ez of the intersection between the plateau region and the densification region is given by:
F þ r1 y Fhe ¼
eF
n
eF e
ð6Þ
To fit the ‘‘three-stage” characteristics of experimental stress– strain relation of foam materials better, Liu and Subhash also proposed a five-parameter theoretical model [2]:
ri ðeÞ ¼ p1
ep2 e 1 þ ep4 ðep5 e 1Þ 1 þ ep3 e
ð7Þ
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where ri and e are engineering stress and engineering strain, respectively, considered positive in compression. P1 is used to control the change of yield stress, P2 and P3 are used to describe the hardening performance in platform segment, P4 and P5 are used to describe the starting point of densification segment and the densification rate. Recently, a new model describing the mechanical properties of foam materials was presented by Avalle et al., the theoretical equation of Avalle model is shown as follows [22]:
ri ðeÞ ¼ Að1 eðE=AÞeð1eÞ Þ þ B k
e l 1e
ð8Þ
where ri and e are engineering stress and engineering strain, respectively, considered positive in compression. This model also has five parameters. The parameters E, A and B are density dependent, while k and l are not. The first term of theoretical equation fits the elastic and plateau region, the second one fits the densification region. The appropriate choice of the parameter k gives a relevant improvement in the fit of the curve knee at the connection of the elastic region with the plateau region [22]. The parameter l mainly adjusts the curve change trend of densification process of foam structure. Because the above three kinds of theoretical models of foam materials were obtained at room temperature, and the effect of material density has been included in the model parameters, the formula (2) can be simplified as follows:
rd ¼ ð1 DÞri ðeÞ
ð9Þ
The damage theoretical equation of foam materials can be obtained by substituting Eqs. (5), (7) and (8) into Eq. (9). The experimental verification and error analysis of the above three kinds of damage computational models can be conducted by using the stress–strain curves of lightweight foamed concrete. In this paper, the least squares method is used to calculate the deviation of the model prediction from the experimental data. Therefore, the model prediction error is defined as the difference of the experimental stress and the model stress at the same strain value. The three figures from Figs. 6–8 show examples of the fit results for the same density material, with exemplification purpose. Each figure is obtained using one of the three considered models. The right diagram of each figure shows the associated model prediction error as a function of strain. The fitting curves from Figs. 6–8 show that the quality of the fit is essentially well comparable between the three cases. In addition, it can be seen from model prediction error that the fluctuation of the fitting error curve of Gibson correction damage model is the least, and the fluctuation of the fitting error curve of Liu-Subhash damage model is the most. Nevertheless, Gibson correction damage model has two limitations in the curve-fitting process. First, the fitting curve of the model is not smooth at the boundaries of two regions (as shown in Fig. 6). Secondly, the strain value ez of the intersection between the plateau region and the densification region cannot be expressed explicitly, and it must be found numerically (as shown in Eq. (6)). Based on the above reasons, this paper chooses Avalle damage model to study the mechanical properties of lightweight foamed concrete. The quasi-static damage theoretical model is obtained by inserting Eq. (8) into (9).
m e l k e rd ðeÞ ¼ exp Að1 eðE=AÞeð1eÞ Þ þ B a 1e
ð10Þ
The experimental data of different density lightweight foamed concrete are fitted by using the model. The applicability of the damage theoretical model is examined by analyzing model prediction error. The comparison of the fitting curve and the
experimental curve is shown in Fig. 9, and the error analysis is shown in Fig. 10. The optimum parameter values determined by least square method are given in Table 1. As shown in Fig. 9, the theoretically predicted curves by the quasi-static damage theoretical model are in good agreement with the experimental data. In addition, it can be seen from the error analysis chart that the fitting effect of the model is remarkable in the crushing plateau region and decreases slightly in the elastic and densification region. The residual stress error is approximately in the range of 0.1 MPa < D < 0.1 MPa, and can be approximately negligible compared with the corresponding engineering stress amplitude. A direct comparison of the global fitting capability of the model can be performed by means of the total sums of the squared errors for each kind of density (as shown in Fig. 10). It can be seen from the histogram that the maximum accumulated error value is 0.2231 MPa in different densities, and is far lower than the allowable error range in engineering application. Based on the above analysis, it can be drawn that the new proposed damage model is suitable to describe the nonlinear mechanical response of lightweight foamed concrete. 3.3. Density effect characterization The parameter values in Table 1 show that the parameters E, A and B are density dependent, while the parameters m, a, k and l are not. The density dependence of the parameters tallies with the analysis conclusion obtained by Avalle and Jeong [22,23]. In order to illustrate the density effect of the quasi-static damage theoretical model, this paper further analyzes the relationship between model parameters and material density, and develops mathematical formulations suitable to describe this dependence. Eq. (10) has the following mathematical relationship:
jrd ðeÞje¼0 ¼ 0
ð11Þ
1 ðA 0 þ B 1Þ lim½rd ðeÞ ¼ exp e!1
@½rd ðeÞ ¼E @e e!0
a
ð12Þ
ð13Þ
The above mathematical relationships show that near the origin of the stress–strain curve the tangent modulus is equal to the parameter E, which can be considered the initial elastic modulus of lightweight foamed concrete. Eq. (12) shows that as the strain is close to the limit value, the stress is mainly controlled by the parameter B. Therefore, the parameter B can be regarded as the controlling factor of the stress–strain curve in the densification region. In addition, the result of Avalle et al. showed that the stress in the crushing plateau segment is mainly controlled by the parameter A [22]. Based on the above analysis, the parameter E is assumed to be the initial elastic modulus of the material. The variation of elastic modulus with material density mainly depends on the topology configuration and microstructure of the material. For open-cell foams, the variation of the elastic modulus with density is modeled with the following relation [1,24–26]:
2 E q ¼ C1 qs Es
ð14Þ
where C1 is the geometric proportional factor, E is the elastic modulus of the foam, ES is the elastic modulus of the matrix cell wall material, q is the density of the foam, and qS is the density of the matrix material. For closed-cell foams, the variation of the elastic modulus with density can be expressed by the following equation [1,27,28]:
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Fig. 6. Comparison between the curve predicted by the modified Gibson damage model and the experimental curve (q = 0.27 g/cm3), and model prediction error.
Fig. 7. Comparison between the curve predicted by the Liu-Subhash damage model and the experimental curve (q = 0.27 g/cm3), and model prediction error.
Fig. 8. Comparison between the curve predicted by the Avalle damage model and the experimental curve (q = 0.27 g/cm3), and model prediction error.
2 E q q ¼ C 01 þ C 02 qs qs Es
ð15Þ
where C01 and C02 are the geometric proportional factors. Eqs. (14) and (15) indicate that the elastic modulus of the foam depends on the density of the material. It can be seen from Fig. 1 that the cell structure of lightweight foamed concrete have both open-cell
morphology and closed-cell morphology. Therefore, the calculation formula of the material elastic modulus should have the characteristics of the above two equations. Because the density qS and elastic modulus ES of its matrix material are constant for given lightweight foamed concrete, the density qS and elastic modulus ES can be merged into the geometric proportional factors to form new independent parameters. Based on the above discussion, the general
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Fig. 9. Comparison between the curve predicted by the quasi-static damage model and the experimental curve of materials of various densities, and model prediction error.
rcr q ¼ C 03 qS rfs
3=2
þ C 04
q qS
ð18Þ
where C03 and C04 are the geometric proportional factors. Because the yield stress rfs of its matrix material is constant for given lightweight foamed concrete, as made for the elastic modulus the parameter A can be expressed as a linear combination of q and q3/2:
A ¼ C A q3=2 þ C 0A q
ð19Þ
The parameter B regulates the magnitude of the densification stress. Because the densification stress value approximately obeys the change law of power exponential function, it is convenient to use two parameters for the formulation of the power law of B, which could be of the form [2,33,34]: Fig. 10. Comparison of the total sums of the squared errors for each kind of density.
relationship of the elastic modulus as a function of density can be expressed as a linear combination of q and q2:
E ¼ C E q2 þ C 0E q
ð16Þ CE0
where the parameters CE and can be obtained with the least squares method on the basis of the previously identified values of parameter E in Table 1. The parameter A represents the stress in the crushing plateau segment and for its density dependence similar considerations can be made. For open-cell foams with brittle crushing behavior, the variation of the brittle crushing stress with density is modeled with the following relation [1,29,30]:
rcr q ¼ C3 qS rfs
3=2 ð17Þ
where C3 is the geometric proportional factor, rcr is the brittle crushing stress, and rfs is the yield stress of the matrix material. For closed-cell foams with brittle crushing behavior, the variation of the brittle crushing stress with density can be expressed by the following equation [1,31,32]:
0
B ¼ C B qC B
ð20Þ
These density dependence laws for the quasi-static damage theoretical model are identified for lightweight foamed concrete on the basis of the values in Table 1. The values calculated for the parameters of the density dependence laws are summarized in Table 2, while the corresponding curves are shown together with the identified model parameters points and the power regression curves in Figs. 11–13. Substituting the parameter values in Table 2 into Eqs. (16), (19) and (20) then yields:
8 2 > < E ¼ 968:2q þ 31:7q A ¼ 3:3 1013 q3=2 3:5 1013 q > : B ¼ 432:1q3:8
ð21Þ
The quasi-static damage theoretical model of lightweight foamed concrete with the density effect is finally obtained by substituting Eq. (21) into Eq. (10). 3.4. Temperature effect characterization The experimental results show that temperature has a certain effect on the mechanical behavior of lightweight foamed concrete.
Table 1 The optimal model parameters determined by the least square method.
q (g/cm3)
m
a
A (MPa)
E (MPa)
k
B (MPa)
l
0.18 0.20 0.27 0.37
0.33 0.33 0.33 0.33
0.17 0.17 0.17 0.17
3.687E12 4.120E12 5.012E12 5.540E12
40.83147 47.21908 71.68527 146.73597
1.1 1.1 1.1 1.1
1.02906 1.25912 2.84793 10.38101
3.9 3.9 3.9 3.9
Table 2 The parameter values for the proposed density dependence laws. CE
CE0
CA
CA0
CB
CB0
968.2
31.7
3.3E13
3.5E13
432.1
3.8
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It can be seen from Fig. 3 that the stress of lightweight foamed concrete exhibits the softening effect as temperature rises. This implies that the energy-absorbing capability is reduced as temperature rises. According to the conclusion of Sherwood and Zhang et al. [35,36], Eq. (1) can be decoupled into the following form:
-3.00E+012
r ¼ HðTÞrd ðe; q; DÞ
-4.00E+012
where H(T) is the function of temperature, and rd(e, q, D) is the stress of the material at room temperature. For the temperature softening factor H(T), Williams et al. proposed a simple equation to describe the time–temperature effect. This equation allows the modulus in very short-time or extremely long-time tests at one temperature to be obtained from data at more reasonable times at other temperatures [1,36]. The temperature softening factor H (T) is defined as follows [37]:
G1 ðT T r Þ HðTÞ ¼ exp G2 þ T T r
ð23Þ
A parameter
ð22Þ
Identified parameters Theoretical curve
-5.00E+012
-6.00E+012
0.1
0.2
0.3
0.4
3
Density (g/cm )
4. Conclusions
Fig. 12. A parameters of the proposed density dependence laws for the foamed concrete.
12
Identified parameters Theoretical curve 8
B parameter
where G1 and G2 are material constants to be determined directly from the experimental data for each specific material, and Tr is the reference temperature (Tr = 293 K). When T equals to Tr, H(T) becomes unity. In order to examine the validity of Eq. (22), the lightweight foamed concrete (0.23 g/cm3) is used as an example. At room temperature, the material parameters of the quasi-static damage theoretical model rd (e, q, D) can be obtained by Table 1 and Eq. (21), the material parameters G1 and G2 are determined by the stress–strain curves at other temperatures. The comparison between model predictions and experimental results is shown in Fig. 14. The calculated values are: G1 = 0.261 and G2 = 50.958 K. As shown in Fig. 14, the theoretical prediction is in good agreement with the available experimental data. Therefore, the new proposed model is suitable to describe the nonlinear mechanical behavior of lightweight foamed concrete in a certain temperature range.
4
The systematic experiment research on the mechanical properties of lightweight foamed concrete at the temperature range of 223–343 K and the strain rate range of 0.001–118/s are carried out in this paper. The deformation mechanism and mechanical characteristics of the materials observed in the experiment are analyzed and discussed. Based on the experimental results and mathematical derivation methods, the calculation model
0 0.1
0.2
0.3
0.4
3
Density (g/cm ) Fig. 13. B parameters of the proposed density dependence laws for the foamed concrete.
150
E parameter
Identified parameters Theoretical curve
describing the nonlinear compressive mechanical behavior of lightweight foamed concrete is proposed in this paper, which includes the effect of material damage, density and temperature. Findings of this study can be summarized as follows:
100
50
0
0.1
0.2
0.3
0.4
3
Density (g/cm ) Fig. 11. E parameters of the proposed density dependence laws for the foamed concrete.
(1) The mechanical behavior of the lightweight foamed concrete under uniaxial compression is dependent on the density and temperature, while weaker on strain rate by comparison. For a certain temperature and strain rate, the stress value at the same deformation amount increases as the density of the materials increases, the crushing plateau segment of the curves becomes shorter. For a certain density and strain rate, the stress value corresponding to the same deformation amount in the crushing plateau segment of the curves decreases as the temperature increases, but the stress value in the elastic segment and the densification segment of the curves has little change.
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Fig. 14. Comparison of model prediction and experimental result of foamed concrete at various temperatures (q = 0.23 g/cm3).
(2) The shape of the stress–strain curves of lightweight foamed concrete under uniaxial compression is basically the same for almost all crushable foams, which can be characterized by three distinct regions: the linear elastic region, the crushing plateau region and the densification region. Due to the special topology configuration and deformation mechanism of lightweight foamed concrete, the stress–strain curves in the crushing plateau region are not flat but have a negative slope. (3) The quasi-static damage calculation model is established to describe the nonlinear compressive behavior of lightweight foamed concrete in this paper. The experimental verification and the error analysis show that the model can effectively predict the nonlinear deformation behavior of lightweight foamed concrete over a wide range of densities. (4) Considering the density effect of the quasi-static damage calculation model, this paper further analyzes the relationship between model parameters and material density, and develops mathematical formulations suitable to describe this dependence. (5) In order to illustrate the stress softening effect of lightweight foamed concrete over a wide range of temperatures, the paper further analyzes the temperature effect of the material on the basis of the quasi-static damage calculation model. Finally, the applicability of the theoretical equation including the temperature softening factor is verified by using the experimental data at different temperatures.
Acknowledgements The authors acknowledge with thanks the technical support of the work by China Academy of Civil Aviation Science and Technology. The authors wish to thank W.J. Hu and J.L. Tao for fruitful discussions on developing the constitutive model framework employed in this paper. References [1] L.J. Gibson, M.F. Ashby, Cellular Solids Structure and Properties, Cambridge University Press, 1987. [2] Q. Liu, G. Subhash, Polym. Eng. Sci. 44 (2004) 463–473. [3] G.L. Li, W.G. Guo, R. Zhao, Y.J. Shi, J. Mater. Sci. Eng. 30 (2012) 428–431. [4] R.C. Valore, ACI J. 50 (1954) 773–796. [5] E.P. Kearsley, P.J. Wainwright, Cem. Concr. Res. 32 (2002) 233–239. [6] R.R. Hengst, R.E. Tressler, Cem. Concr. Res. 13 (1983) 127–134. [7] S.B. Park, Cem. Concr. Res. 29 (1999) 193–200. [8] M. Visagie, E.P. Kearsely, Concrete/Beton 101 (2002) 8–14. [9] G.C. Hoff, Cem. Concr. Res. 2 (1972) 91–100. [10] S.E. Zhou, Z.Y. Lu, L. Jiao, S.X. Li, Mater. Sci. Ed. 32 (2010) 9–13. [11] C. Tuman, Aircraft Arrestment System, United States Patent, No. US4393996, 1983. [12] R. Zhao, W.G. Guo, J.J. Wang, Y.J. Shi, L. Zeng, J. Mater. Sci. Eng. 27 (2012) 354– 360. [13] S.C. Marisetty, E.D. Bailey, W.M. Hale, E.P. Heymsfield, MBTC-2089, OMB No. 0704-0188. [14] R. Brezny, D.J. Green, J. Am. Ceram. Soc. 76 (1993) 2185–2192. [15] M.R. Jones, A. McCarthy, Mag. Concr. Res. 57 (2005) 21–31. [16] Y.H. Zhao, G.J. Weng, Int. J. Solids Struct. 34 (1997) 493–507. [17] H.K. Lee, Z. Liang, J. Comput. Struct. 82 (2004) 581–592. [18] J. Lemaitre, J. Eng. Mater. Tech. 107 (1985) 83–89. [19] J. Lemaitre, R. Desmorat, M. Sauzay, J. Mech.-A/Solids 19 (2000) 187–203.
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Y.H. Zhao, G.J. Weng, Int. J. Plast. 12 (1996) 781–804. J.W. Ju, H.K. Lee, Comput. Meth. Appl. Mech. Eng. 183 (2000) 201–222. M. Avalle, G. Belingardi, A. Ibba, Int. J. Impact Eng. 34 (2007) 3–27. K.Y. Jeong, S.S. Cheon, M.B. Munshi, J. Mech. Sci. Tech. 34 (2012) 2033–2038. M.F. Ashby, R.F. Mehl Medalist, Metall. Trans. A 14 (1983) 1755–1769. J.M. Williams, J.J. Bartosi, M.H. Wilkerson, J. Mater. Sci. 25 (1990) 5134–5141. J.G.F. Wismans, J.A.W. van Dommelen, Characterization of Polymeric Foams, Eindhoven University of Technology, 2009. [27] A.E. Simone, L.J. Gibson, Acta Mater. 46 (1998) 3929–3935. [28] A.P. Roberts, E.J. Garboczi, J. Mech. Phys. Solids 50 (2002) 33–55. [29] A.M. Hodge, J. Biener, J.R. Hayes, P.M. Bythrow, C.A. Volkert, A.V. Hamza, Acta Mater. 55 (2007) 1343–1349.
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[30] E. Amsterdam, J.Th.M. De Hosson, P.R. Onck, Scripta Mater. 59 (2008) 653–656. [31] Y. Sugimura, J. Meyer, M.Y. He, H. Bart-Smith, J. Grenstedt, A.G. Evans, Acta Mater. 45 (1997) 5245–5259. [32] N.J. Mills, H.X. Zhu, J. Mech. Phys. Solids 47 (1999) 669–695. [33] K.C. Rusch, J. Appl. Polym. Sci. 13 (1969) 2297–2311. [34] K.C. Rusch, J. Appl. Polym. Sci. 14 (1970) 1263–1273. [35] J.A. Sherwood, C.C. Frost, Polym. Eng. Sci. 32 (1992) 1138–1146. [36] J. Zhang, N. Kikuchi, V. Li, A. Yee, G. Nusholtz, Polym. Eng. Sci. 21 (1998) 369– 386. [37] M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701–3707.
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Computational modeling of the mechanical response of lightweight foamed concrete over a wide range of temperatures and strain rates Hui Guo a, Weiguo Guo a,⇑, Yajie Shi b a b
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China China Academy of Civil Aviation Science and Technology, Beijing 100028, China
h i g h l i g h t s Uniaxial compression tests are conducted at various temperatures and strain rates. The characteristics of the stress-strain curve are analyzed and a model is developed. The relationships between model parameters and material density are discussed. The temperature effect of the material is illustrated based on the calculation model.
a r t i c l e
i n f o
Article history: Received 9 April 2015 Received in revised form 31 July 2015 Accepted 9 August 2015
Keywords: Lightweight foamed concrete Density effect Temperature effect Damage evolution Calculation model
a b s t r a c t Theoretical and experimental studies on the nonlinear mechanical properties of lightweight foamed concrete under uniaxial compression are conducted over the temperature range of 223–343 K and the strain rate range of 0.001–118/s in this paper. The experimental results show that the mechanical properties of the materials under uniaxial compression are strongly dependent on the density and temperature, but are weaker on strain rate. Based on the experimental results, the characteristics of density effect and temperature effect are analyzed and a calculation model is developed to describe the nonlinear mechanical behavior of lightweight foamed concrete. The model takes into account the effect of material damage, density and temperature. The experimental verification and the error analysis show that the model is shown to be able to predict the nonlinear deformation behavior of lightweight foamed concrete over a wide range of temperatures and densities. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Due to its special microstructure morphology and mechanical properties, lightweight foamed concrete is widely used as energy absorbers in various industries, for example, engineered material arresting system (EMAS). There are many ingredients influencing lightweight foamed concrete ability of energy absorption, such as density and nonlinear mechanical properties of the materials. Therefore, it is very necessary to study the nonlinear mechanical behavior of lightweight foamed concrete. The nonlinear mechanical properties of lightweight foamed concrete share some characteristics with the properties of metal foams and polymeric foams. Under the uniaxial compression, the stress–strain curves of these foams can be divided into three stages: the linear elastic stage, the plateau stage and the densification stage [1–3]. However, the components and the structure of lightweight foamed concrete ⇑ Corresponding author. E-mail address: [email protected] (W. Guo). http://dx.doi.org/10.1016/j.conbuildmat.2015.08.064 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.
are more complex compared with metal foams and polymeric foams, and there are less research literatures on its mechanical properties. Therefore, with the extensive use of lightweight foamed concrete, the mechanical properties of the materials have drawn great attention of researchers. Some experimental studies have been conducted to investigate the nonlinear response of lightweight foamed concrete under uniaxial compression [3–5]. Valore [4] studied varies factors that affect the strength of porous concrete, including the size and shape of specimen, the properties of matrix material, the porosity formation and the curing methods, loading direction, as well as the moisture content. Hengst and Tressler insisted that at constant density, the major factor affect the strength of porous concrete is the crack size, which is correlated with the size of holes [6]. Park studied the effect of the components of foamed concrete on the mechanical properties of the materials, and held that the content increasing of silica fume, fly ash and glass fiber can improve the mechanical properties of foamed concrete [7]. Visagie and Kearsely found the compressive strength of foamed concrete would decrease with the increase
H. Guo et al. / Construction and Building Materials 96 (2015) 622–631
of the diameter of the aperture over the dry density ranges of 0.5– 1 g/cm3, while the porosity of the materials has little effect on the compressive strength when the dry density is over 1 g/cm3 [8]. Guo et al. studied systemically the compression behavior of one kind of new foamed concrete composite at the room temperature and strain rate ranges from 0.001/s to 0.1/s, and found that this material is insensitive to strain rates during the strain rate ranges, while the densities of this material have a great influence on the compression behavior [3]. Besides, few theoretical and numerical models had also been developed to describe the mechanical properties of foamed concrete recently. Hoff studied the relationship between the porosity and the compression strength of foamed concrete, and insisted that there be a certain calculating formula of strength and porosity for given cement content [9]. The prediction model proposed by Kearsely and Wainwright showed that the model from Hoff can efficiently predict the compression strength of foamed concrete at different time and densities [5]. Zhou et al. studied the characteristics of the compressive stress–strain curve of foamed concrete, and found that there are four steps in foamed concrete compression process, namely plateau step, compacting step, yield step and decline step, the compression mechanical property is affected by matrix material, volume weight, morphology and distribution of pore, etc. [10]. Gibson and Ashby derived the equation of compression strength and relative density of brittle foams, by which could get the parameter to describe the volume fraction of hole edge and the fracture strength of hole wall [1]. Recently, Guo et al. developed a compression phenomenological constitutive model of foamed concrete by analyzing platform crushing stress, compacting strain and density [3]. The theoretical study on the compressive mechanical properties of foam concrete in the above literature is almost all to set up the mathematics model through the relationship between the strength and porosity, matrix material and so on. However, there is not a more comprehensive study on the failure mechanism of the materials as well as on the macroscopic factors of affecting the compressive strength, such as strain rate, temperature, and relative density. In this paper, the uniaxial compression tests on the mechanical properties of one kind of new lightweight foamed concrete at the temperature range of 223–343 K and the strain rate range of 0.001–118/s are carried out. The deformation mechanism and the mechanical characteristics of the materials are analyzed and discussed. Based on the experimental results, a calculation model describing the nonlinear compressive behavior of lightweight foamed concrete is established in this paper, which includes the effect of material damage, density and temperature. 2. Experimental investigation 2.1. Materials The lightweight foamed concrete studied in this paper, mainly used in engineered material arresting system (EMAS). This new lightweight foamed concrete adds some polyester fiber materials into the generic foamed concrete using traditional manufacturing process [11–13]. The material of the polyester fiber is PP, the length is 6 mm, the melting temperature is 165–175 °C, and the fiber diameter is 25–45 lm. The mass of PP accounts for about 0.85% of the mass of the cement. The foaming agent used in this paper is 35% hydrogen peroxide. The main components of lightweight foamed concrete are cement, water, heavy calcium and fly ash. The particle size of the cement is about 10 lm. The particle size of the heavy calcium is about 18 lm. The calcium–cement ratio is 0.4, and the water–cement ratio is 0.8. The final optimization design scheme for the chemical constituents and mix proportion of the raw materials is determined through trial and error. The detailed preparation process of standard sample is as follows: (a) Mix the foaming agent and purified water thoroughly using agitator at room temperature, and whisk until the mixture is smooth. Then, add the mixture of clay, cement, sand, fly ash, polystyrene fiber and water into the mixed froth, and using the water reducer and the coagulation accelerator to make the mixture uniform flow slurry.
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(b) Fill the U = 80 mm, h = 60 mm mold with the uniform flow slurry, and vibrate it on the shaking table and smooth the upper surface with a spatula. Then, after letting it stand at room temperature for 3 days, demold and place it in constant humidity cabinet with the relative humidity of 90%, this stage will continue for 30 more days. Finally, paste numbering in the specimen for the subsequent experiments. The microstructure of the experimental materials was scrutinized using field emission scanning microscopy, and the scanning microscopic photos are shown in Fig. 1. It can be observed from Fig. 1a that the distribution and orientation of the pore wall materials are uniform approximately, and the thickness of hole wall is about 40 lm. As can be seen from Fig. 1b, the hole wall of the materials is accumulated by varieties of the salt crystals arranged without order at higher magnification. The crystal is a slender cylinder, and the maximum diameter is about 1 lm. In addition, the arrangement of the grain is loose, so it can be inferred that the weak crystal boundary is the essential reason of the low compression strength of the lightweight foamed concrete.
2.2. Testing equipment In order to cover a wide range of loading speeds, material tests are conducted on two types of testing machines: (1) a CSS-4410 electric universal testing machine equipped with a temperature chamber for low strain rate tests; (2) a DYN-9250 drop weight impact testing machine equipped with a temperature chamber for high-rate impact tests. During the loading process, the CSS4410 electric universal testing machine measures the displacements by automatic recording the movement distance of the crossbeam, and acquires the compressive load by high precision pulling and pushing dual-purpose load transducer installed on the crossbeam. The measuring range of compressive load is the sensor capacity of 2–100%, and the precision of the sensor is 0.5%. Based on the collected data from testing machine and the geometric size of specimen, the stress– strain curves of test materials are obtained by mathematical calculation method at low strain rates. The DYN-9250 drop weight impact testing machine adjusts the level of strain rate and its stability by controlling the weight of the hammer head and the initial height of free falling with the assistant of the Impulse control software. The testing machine releases the drop hammer to make it free falling in order to impact specimen, and collects a series of experimental data by using the Impulse data acquisition software in the process of impact. Finally, the curves of calculation results are generated by using the Impulse software at high strain rates.
2.3. Experimental program and results Lightweight foamed concrete is a heterogeneous and anisotropic multi-phase composite material. Because the components of the material are complex and be many influence factors, the mechanical properties data obtained from the tests have the certain characteristics of discreteness. In order to carry on the system analysis of the experimental data, three major factors which influence the mechanical properties of lightweight foamed concrete are emphatically considered in this paper, and so as to provide valuable reference for the engineering design, namely material density, experimental temperature and strain rate. Test conditions are designed as three schemes as follows: (1) the effects of different density on the compressive strength of lightweight foamed concrete are studied when the temperature and strain rate are constant; (2) the effects of different temperature on the compressive strength of the materials are investigated when the density and strain rate are constant; (3) the effects of different strain rate on the compressive strength of the materials are discussed when the temperature and density are constant. The typical stress–strain curves of four different densities of lightweight foamed concrete specimens at the temperature of 293 K and strain rate of 0.1/s are shown in Fig. 2. The typical stress–strain curves of lightweight foamed concrete specimens with the density of 0.23 g/cm3 and strain rate of 0.1/s under four different temperatures are shown in Fig. 3. The typical stress– strain curves of lightweight foamed concrete specimens with the density of 0.18 g/cm3 and temperature of 293 K under six different strain rates are shown in Fig. 4. The typical stress–strain curves of lightweight foamed concrete specimens with the density of 0.37 g/cm3 and temperature of 293 K under six different strain rates are shown in Fig. 5. The results showed that the engineering stress–strain curves of low-density lightweight foamed concrete in low strain rate have the obvious characteristics of three stages, namely the linear elastic stage, the crushing plateau stage and the densification stage. The curves in low strain rate are smoother by comparison with high strain rate. The curves show the zigzag fluctuation at high strain rate and especially the larger fluctuation in the transition region of elastic segment and crushing plateau segment. In addition, as the density of the materials increases the ranges of fluctuation increases correspondingly. According to the analysis of the stress–strain curves of different density (0.18 g/cm3, 0.20 g/cm3, 0.27 g/cm3 and 0.37 g/cm3) lightweight foamed concrete in Fig. 2, it can be found that for a certain temperature and strain rate, the stress value at the same deformation amount increases as the density of the materials increases, and the crushing plateau segment of the curves
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Fig. 1. Scanning micrograph of lightweight foamed concrete: (a) extracellular pore size distribution microgram; (b) structure of cell wall microgram.
Fig. 2. Eng. stress–strain curves of materials of various densities at a temperature of 293 K and strain rate of 0.1/s.
Fig. 3. Eng. stress–strain curves of materials of a strain rate of 0.1/s and a density of 0.23 g/cm3 at various temperatures.
becomes shorter. According to the analysis of the stress–strain curves of lightweight foamed concrete under different temperature (223 K, 293 K, 313 K and 343 K) in Fig. 3, it can be found that for a certain density and strain rate, the stress value corresponding to the same deformation amount in the crushing plateau
Fig. 4. Eng. stress–strain curves of materials of a temperature of 293 K and a density of 0.18 g/cm3 at various strain rates.
Fig. 5. Eng. stress–strain curves of materials of a temperature of 293 K and a density of 0.37 g/cm3 at various strain rates.
segment of the curves decreases as the temperature increases, but the stress value in the elastic segment and the densification segment of the curves has little change. According to the analysis of the stress–strain curves of lightweight foamed concrete under different strain rate (0.001/s, 0.01/s, 0.1/s, 48.2/s, 68.2/s, 83.5/s, 96.4/s, 109.1/ s and 118/s) in Figs. 4 and 5, it can be found that the mechanical properties of the materials are less sensitive to strain rate effect at a given density and temperature.
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3. Theoretical model In developing a generalized computational model applicable to lightweight foamed concrete under finite deformation, the following relevant observations are noted from the available experimental data in this paper and related literature. (1) The shape of the compression stress–strain curve is obtained by the theoretical model is basically the same for almost all crushable foams [1–3], and can be characterized by three different regimes as follows: (a) the linear elastic regime where the stress keep near-linearly increasing with the increasing strain; (b) the crushing plateau regime extending over a large range strains where the material start to crack or crush progressively. In addition, the stress drop may be found before the stress reaches a plateau, which may be attributed to cracking of foam cell walls; (c) the densification regime where the stress rises sharply as the material starts to compact. (2) According to the experimental results, the theoretical model of lightweight foamed concrete should be able to describe the effect of material density and temperature, but does not consider the effect of strain rate. (3) The deformation mechanism of lightweight foamed concrete is different from that of metal foams and polymeric foams [14,15], and is usually guided by the generation and evolution of micro crack and damage of cell walls, because of which the material leans to brittle fracture. In addition, lightweight foamed concrete is a heterogeneous mixture of fiber and other aggregates by cementing materials, so the internal damage is an important factor affecting the nonlinear mechanical behavior of the material [16,17]. The goal of this paper is to capture all the above features of the stress–strain behavior with a continuous function defined in the whole real domain. Based on the above analysis of the failure mechanism and mechanical characteristics, the general form of the computational equation of lightweight foamed concrete under high temperature can be expressed as:
r ¼ rðe; q; T; DÞ
ð1Þ
where r is the stress, e is the strain, q is the material density, T is the temperature, and D is the damage factor which value range from 0 to 1. When D = 0, the material is perfect without damage; when D = 1, the material is damaged completely. According to strain equivalence principle proposed by Lemaitre [18,19], the damage computational relation of lightweight foamed concrete is obtained as follows:
rd ¼ ð1 DÞri ðe; q; TÞ
ð2Þ
where rd is the apparent stress, namely material stress considering damage, and ri is the material stress referring to no damage. 3.1. Damage evolution The progressive, partial interfacial debonding may occur under increasing deformations and affect the overall stress–strain behavior of lightweight foamed concrete [16,17]. After the interfacial debonding, the debonded fibers lose the load-carrying capacity along the debonded direction and are regarded as partially debonded fibers, namely material damage. The damage failure of lightweight foamed concrete is the time-correlated rheological process, which is also the time process of the different forms of micro crack with the finite rate evolution. The damage evolution law is the key to discover the functional relationship how the damage evolution is dependent on the constitutive state variables. Because the study on the behavior of single micro crack can not reflect global damage evolution on the micro-level, it needs to study the global evolution behavior of material damage from the viewpoint of probability. Following Zhao et al. and Ju et al. [20,21], the probability of material damage was modeled as a two-parameter Weibull process.
In addition, Lee and Liang also made use of the Weibull distribution function to characterize the damage evolution law of foamed concrete by statistical microscopic damage mechanics, and good effect was obtained [17]. Assuming that the Weibull statistics govern, the cumulative probability distribution function of material damage can be expressed as:
PðeÞ ¼
m
a
em1 exp
em a
ð3Þ
where a and m are scale parameter and shape parameter, respectively. Shape parameter m is the most important parameter, which determines the basic shape of the distribution density curve. Scale parameter a is a function of amplifying or scaling a curve, but does not affect the shape of the distribution density curve. In addition, the failure condition of the different stages can be expressed by changing the shape parameter. If the damage factor D is defined as the ratio of the damaged quantity of micro unit and its total quantity, then D and P(e) have the following relationship:
Z
D¼ 0
e
PðeÞde ¼
m
a
Z
e
0
em1 exp
em em de ¼ 1 exp a a
ð4Þ
3.2. Quasi-static damage model At present, the mechanical properties of lightweight foamed materials were studied to a degree in the literature, and some theoretical models for describing mechanical behavior of the materials were proposed. The work done by Gibson, Liu and Avalle, etc. has already been generally accepted in the study of theoretical modeling [1,2,22]. Based on the experimental results and mechanical characteristics of lightweight foamed concrete, this paper selects the three representative theoretical models to fit the experimental data, namely Gibson correction model, Liu-Subhash model and Avalle model. Finally, the quasi-static damage computational mode suitable for lightweight foamed concrete is established by the contrastive analysis. Because the stress–strain curves of foam materials, in general, exhibit three regions: the linear elastic region, the plateau region and the densification region, the Gibson model is formulated by three equations describing each region [1]. The observation of the experimental curves in Figs. 2 and 3 put in evidence that the crushing plateau region is not flat but has a negative slope. In order to take into account the described slope, the formula for the plateau region of the Gibson model is modified to improve its fitting capability. A sloped linear region is used instead of the constant stress value in Gibson correction model.
8 ri ðeÞ ¼ Ee > > < ri ðeÞ ¼ ry þ he n > > : ri ðeÞ ¼ ry 1 eF F eF e
e 6 ey ey 6 e 6 ez
ð5Þ
e > ez
where ri and e are engineering stress and engineering strain, respectively, considered positive in compression. The model has six parameters, namely E the slope of the elastic part of the curve, ey the yield strain, eF the densification strain, and three constants h, F and n. The strain value ez of the intersection between the plateau region and the densification region is given by:
F þ r1 y Fhe ¼
eF
n
eF e
ð6Þ
To fit the ‘‘three-stage” characteristics of experimental stress– strain relation of foam materials better, Liu and Subhash also proposed a five-parameter theoretical model [2]:
ri ðeÞ ¼ p1
ep2 e 1 þ ep4 ðep5 e 1Þ 1 þ ep3 e
ð7Þ
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where ri and e are engineering stress and engineering strain, respectively, considered positive in compression. P1 is used to control the change of yield stress, P2 and P3 are used to describe the hardening performance in platform segment, P4 and P5 are used to describe the starting point of densification segment and the densification rate. Recently, a new model describing the mechanical properties of foam materials was presented by Avalle et al., the theoretical equation of Avalle model is shown as follows [22]:
ri ðeÞ ¼ Að1 eðE=AÞeð1eÞ Þ þ B k
e l 1e
ð8Þ
where ri and e are engineering stress and engineering strain, respectively, considered positive in compression. This model also has five parameters. The parameters E, A and B are density dependent, while k and l are not. The first term of theoretical equation fits the elastic and plateau region, the second one fits the densification region. The appropriate choice of the parameter k gives a relevant improvement in the fit of the curve knee at the connection of the elastic region with the plateau region [22]. The parameter l mainly adjusts the curve change trend of densification process of foam structure. Because the above three kinds of theoretical models of foam materials were obtained at room temperature, and the effect of material density has been included in the model parameters, the formula (2) can be simplified as follows:
rd ¼ ð1 DÞri ðeÞ
ð9Þ
The damage theoretical equation of foam materials can be obtained by substituting Eqs. (5), (7) and (8) into Eq. (9). The experimental verification and error analysis of the above three kinds of damage computational models can be conducted by using the stress–strain curves of lightweight foamed concrete. In this paper, the least squares method is used to calculate the deviation of the model prediction from the experimental data. Therefore, the model prediction error is defined as the difference of the experimental stress and the model stress at the same strain value. The three figures from Figs. 6–8 show examples of the fit results for the same density material, with exemplification purpose. Each figure is obtained using one of the three considered models. The right diagram of each figure shows the associated model prediction error as a function of strain. The fitting curves from Figs. 6–8 show that the quality of the fit is essentially well comparable between the three cases. In addition, it can be seen from model prediction error that the fluctuation of the fitting error curve of Gibson correction damage model is the least, and the fluctuation of the fitting error curve of Liu-Subhash damage model is the most. Nevertheless, Gibson correction damage model has two limitations in the curve-fitting process. First, the fitting curve of the model is not smooth at the boundaries of two regions (as shown in Fig. 6). Secondly, the strain value ez of the intersection between the plateau region and the densification region cannot be expressed explicitly, and it must be found numerically (as shown in Eq. (6)). Based on the above reasons, this paper chooses Avalle damage model to study the mechanical properties of lightweight foamed concrete. The quasi-static damage theoretical model is obtained by inserting Eq. (8) into (9).
m e l k e rd ðeÞ ¼ exp Að1 eðE=AÞeð1eÞ Þ þ B a 1e
ð10Þ
The experimental data of different density lightweight foamed concrete are fitted by using the model. The applicability of the damage theoretical model is examined by analyzing model prediction error. The comparison of the fitting curve and the
experimental curve is shown in Fig. 9, and the error analysis is shown in Fig. 10. The optimum parameter values determined by least square method are given in Table 1. As shown in Fig. 9, the theoretically predicted curves by the quasi-static damage theoretical model are in good agreement with the experimental data. In addition, it can be seen from the error analysis chart that the fitting effect of the model is remarkable in the crushing plateau region and decreases slightly in the elastic and densification region. The residual stress error is approximately in the range of 0.1 MPa < D < 0.1 MPa, and can be approximately negligible compared with the corresponding engineering stress amplitude. A direct comparison of the global fitting capability of the model can be performed by means of the total sums of the squared errors for each kind of density (as shown in Fig. 10). It can be seen from the histogram that the maximum accumulated error value is 0.2231 MPa in different densities, and is far lower than the allowable error range in engineering application. Based on the above analysis, it can be drawn that the new proposed damage model is suitable to describe the nonlinear mechanical response of lightweight foamed concrete. 3.3. Density effect characterization The parameter values in Table 1 show that the parameters E, A and B are density dependent, while the parameters m, a, k and l are not. The density dependence of the parameters tallies with the analysis conclusion obtained by Avalle and Jeong [22,23]. In order to illustrate the density effect of the quasi-static damage theoretical model, this paper further analyzes the relationship between model parameters and material density, and develops mathematical formulations suitable to describe this dependence. Eq. (10) has the following mathematical relationship:
jrd ðeÞje¼0 ¼ 0
ð11Þ
1 ðA 0 þ B 1Þ lim½rd ðeÞ ¼ exp e!1
@½rd ðeÞ ¼E @e e!0
a
ð12Þ
ð13Þ
The above mathematical relationships show that near the origin of the stress–strain curve the tangent modulus is equal to the parameter E, which can be considered the initial elastic modulus of lightweight foamed concrete. Eq. (12) shows that as the strain is close to the limit value, the stress is mainly controlled by the parameter B. Therefore, the parameter B can be regarded as the controlling factor of the stress–strain curve in the densification region. In addition, the result of Avalle et al. showed that the stress in the crushing plateau segment is mainly controlled by the parameter A [22]. Based on the above analysis, the parameter E is assumed to be the initial elastic modulus of the material. The variation of elastic modulus with material density mainly depends on the topology configuration and microstructure of the material. For open-cell foams, the variation of the elastic modulus with density is modeled with the following relation [1,24–26]:
2 E q ¼ C1 qs Es
ð14Þ
where C1 is the geometric proportional factor, E is the elastic modulus of the foam, ES is the elastic modulus of the matrix cell wall material, q is the density of the foam, and qS is the density of the matrix material. For closed-cell foams, the variation of the elastic modulus with density can be expressed by the following equation [1,27,28]:
H. Guo et al. / Construction and Building Materials 96 (2015) 622–631
627
Fig. 6. Comparison between the curve predicted by the modified Gibson damage model and the experimental curve (q = 0.27 g/cm3), and model prediction error.
Fig. 7. Comparison between the curve predicted by the Liu-Subhash damage model and the experimental curve (q = 0.27 g/cm3), and model prediction error.
Fig. 8. Comparison between the curve predicted by the Avalle damage model and the experimental curve (q = 0.27 g/cm3), and model prediction error.
2 E q q ¼ C 01 þ C 02 qs qs Es
ð15Þ
where C01 and C02 are the geometric proportional factors. Eqs. (14) and (15) indicate that the elastic modulus of the foam depends on the density of the material. It can be seen from Fig. 1 that the cell structure of lightweight foamed concrete have both open-cell
morphology and closed-cell morphology. Therefore, the calculation formula of the material elastic modulus should have the characteristics of the above two equations. Because the density qS and elastic modulus ES of its matrix material are constant for given lightweight foamed concrete, the density qS and elastic modulus ES can be merged into the geometric proportional factors to form new independent parameters. Based on the above discussion, the general
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Fig. 9. Comparison between the curve predicted by the quasi-static damage model and the experimental curve of materials of various densities, and model prediction error.
rcr q ¼ C 03 qS rfs
3=2
þ C 04
q qS
ð18Þ
where C03 and C04 are the geometric proportional factors. Because the yield stress rfs of its matrix material is constant for given lightweight foamed concrete, as made for the elastic modulus the parameter A can be expressed as a linear combination of q and q3/2:
A ¼ C A q3=2 þ C 0A q
ð19Þ
The parameter B regulates the magnitude of the densification stress. Because the densification stress value approximately obeys the change law of power exponential function, it is convenient to use two parameters for the formulation of the power law of B, which could be of the form [2,33,34]: Fig. 10. Comparison of the total sums of the squared errors for each kind of density.
relationship of the elastic modulus as a function of density can be expressed as a linear combination of q and q2:
E ¼ C E q2 þ C 0E q
ð16Þ CE0
where the parameters CE and can be obtained with the least squares method on the basis of the previously identified values of parameter E in Table 1. The parameter A represents the stress in the crushing plateau segment and for its density dependence similar considerations can be made. For open-cell foams with brittle crushing behavior, the variation of the brittle crushing stress with density is modeled with the following relation [1,29,30]:
rcr q ¼ C3 qS rfs
3=2 ð17Þ
where C3 is the geometric proportional factor, rcr is the brittle crushing stress, and rfs is the yield stress of the matrix material. For closed-cell foams with brittle crushing behavior, the variation of the brittle crushing stress with density can be expressed by the following equation [1,31,32]:
0
B ¼ C B qC B
ð20Þ
These density dependence laws for the quasi-static damage theoretical model are identified for lightweight foamed concrete on the basis of the values in Table 1. The values calculated for the parameters of the density dependence laws are summarized in Table 2, while the corresponding curves are shown together with the identified model parameters points and the power regression curves in Figs. 11–13. Substituting the parameter values in Table 2 into Eqs. (16), (19) and (20) then yields:
8 2 > < E ¼ 968:2q þ 31:7q A ¼ 3:3 1013 q3=2 3:5 1013 q > : B ¼ 432:1q3:8
ð21Þ
The quasi-static damage theoretical model of lightweight foamed concrete with the density effect is finally obtained by substituting Eq. (21) into Eq. (10). 3.4. Temperature effect characterization The experimental results show that temperature has a certain effect on the mechanical behavior of lightweight foamed concrete.
Table 1 The optimal model parameters determined by the least square method.
q (g/cm3)
m
a
A (MPa)
E (MPa)
k
B (MPa)
l
0.18 0.20 0.27 0.37
0.33 0.33 0.33 0.33
0.17 0.17 0.17 0.17
3.687E12 4.120E12 5.012E12 5.540E12
40.83147 47.21908 71.68527 146.73597
1.1 1.1 1.1 1.1
1.02906 1.25912 2.84793 10.38101
3.9 3.9 3.9 3.9
Table 2 The parameter values for the proposed density dependence laws. CE
CE0
CA
CA0
CB
CB0
968.2
31.7
3.3E13
3.5E13
432.1
3.8
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It can be seen from Fig. 3 that the stress of lightweight foamed concrete exhibits the softening effect as temperature rises. This implies that the energy-absorbing capability is reduced as temperature rises. According to the conclusion of Sherwood and Zhang et al. [35,36], Eq. (1) can be decoupled into the following form:
-3.00E+012
r ¼ HðTÞrd ðe; q; DÞ
-4.00E+012
where H(T) is the function of temperature, and rd(e, q, D) is the stress of the material at room temperature. For the temperature softening factor H(T), Williams et al. proposed a simple equation to describe the time–temperature effect. This equation allows the modulus in very short-time or extremely long-time tests at one temperature to be obtained from data at more reasonable times at other temperatures [1,36]. The temperature softening factor H (T) is defined as follows [37]:
G1 ðT T r Þ HðTÞ ¼ exp G2 þ T T r
ð23Þ
A parameter
ð22Þ
Identified parameters Theoretical curve
-5.00E+012
-6.00E+012
0.1
0.2
0.3
0.4
3
Density (g/cm )
4. Conclusions
Fig. 12. A parameters of the proposed density dependence laws for the foamed concrete.
12
Identified parameters Theoretical curve 8
B parameter
where G1 and G2 are material constants to be determined directly from the experimental data for each specific material, and Tr is the reference temperature (Tr = 293 K). When T equals to Tr, H(T) becomes unity. In order to examine the validity of Eq. (22), the lightweight foamed concrete (0.23 g/cm3) is used as an example. At room temperature, the material parameters of the quasi-static damage theoretical model rd (e, q, D) can be obtained by Table 1 and Eq. (21), the material parameters G1 and G2 are determined by the stress–strain curves at other temperatures. The comparison between model predictions and experimental results is shown in Fig. 14. The calculated values are: G1 = 0.261 and G2 = 50.958 K. As shown in Fig. 14, the theoretical prediction is in good agreement with the available experimental data. Therefore, the new proposed model is suitable to describe the nonlinear mechanical behavior of lightweight foamed concrete in a certain temperature range.
4
The systematic experiment research on the mechanical properties of lightweight foamed concrete at the temperature range of 223–343 K and the strain rate range of 0.001–118/s are carried out in this paper. The deformation mechanism and mechanical characteristics of the materials observed in the experiment are analyzed and discussed. Based on the experimental results and mathematical derivation methods, the calculation model
0 0.1
0.2
0.3
0.4
3
Density (g/cm ) Fig. 13. B parameters of the proposed density dependence laws for the foamed concrete.
150
E parameter
Identified parameters Theoretical curve
describing the nonlinear compressive mechanical behavior of lightweight foamed concrete is proposed in this paper, which includes the effect of material damage, density and temperature. Findings of this study can be summarized as follows:
100
50
0
0.1
0.2
0.3
0.4
3
Density (g/cm ) Fig. 11. E parameters of the proposed density dependence laws for the foamed concrete.
(1) The mechanical behavior of the lightweight foamed concrete under uniaxial compression is dependent on the density and temperature, while weaker on strain rate by comparison. For a certain temperature and strain rate, the stress value at the same deformation amount increases as the density of the materials increases, the crushing plateau segment of the curves becomes shorter. For a certain density and strain rate, the stress value corresponding to the same deformation amount in the crushing plateau segment of the curves decreases as the temperature increases, but the stress value in the elastic segment and the densification segment of the curves has little change.
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Fig. 14. Comparison of model prediction and experimental result of foamed concrete at various temperatures (q = 0.23 g/cm3).
(2) The shape of the stress–strain curves of lightweight foamed concrete under uniaxial compression is basically the same for almost all crushable foams, which can be characterized by three distinct regions: the linear elastic region, the crushing plateau region and the densification region. Due to the special topology configuration and deformation mechanism of lightweight foamed concrete, the stress–strain curves in the crushing plateau region are not flat but have a negative slope. (3) The quasi-static damage calculation model is established to describe the nonlinear compressive behavior of lightweight foamed concrete in this paper. The experimental verification and the error analysis show that the model can effectively predict the nonlinear deformation behavior of lightweight foamed concrete over a wide range of densities. (4) Considering the density effect of the quasi-static damage calculation model, this paper further analyzes the relationship between model parameters and material density, and develops mathematical formulations suitable to describe this dependence. (5) In order to illustrate the stress softening effect of lightweight foamed concrete over a wide range of temperatures, the paper further analyzes the temperature effect of the material on the basis of the quasi-static damage calculation model. Finally, the applicability of the theoretical equation including the temperature softening factor is verified by using the experimental data at different temperatures.
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