Statistical damage model with strain softening for lime-stabilized rammed earth after elevated temperature

Abstract

Many historical earthen buildings are damaged due to fire exposure in the past. It is important to understand the strength degradation of rammed earth after elevated temperature for guiding the strategy of building protection or rehabilitation. A total of 24 unconfined compression tests are conducted on lime-stabilized rammed earth specimens after elevated temperature up to 700°C. A quasi-linear reduction in strength and stiffness is found for rammed earth with the increase of temperature. At high temperature, the ductility of rammed earth is enhanced, e.g., strain at peak strength of 2.5% and 1.5% at 700°C and 20°C, respectively. Microstructural analyses demonstrate that with the increase of temperature, the specimen becomes more porous with reduced calcium carbonate precipitation, explaining the strength reduction. A new thermal damage model is proposed to describe the behavior of rammed earth after elevated temperature, in which the closure of pores is captured to show unrecoverable deformation, and the skeleton part is simulated using a thermal damage variable in a statistical manner to present the damage evolution (strain softening). By comparing with the measured stress-strain curves, one can confirm that the proposed method can provide effective prediction for the response of rammed earth after elevated temperature.

Keywords

Rammed earth elevated temperature thermal damage statistical model strain softening

Introduction

The use of rammed earth as construction materials can be traced back for over thousand years in the history of human civilization, through which shelters, ramparts or forts can be constructed (Recavarren et al., 2013). Essentially, native clays with a plasticity index of above 5 are mixed with coarse aggregates (e.g., silt, sand, or gravel), water and binding materials (e.g., lime, sticky rice soup (Liu et al., 2016) or sol-gel (Yang et al., 2016), brown sugar (Luo et al., 2022), tun oil and pig blood (Zhao et al., 2015)) to form the raw earth mixture, which is compacted in layers in formwork for gaining strength over time. Earthen buildings often employ rammed earth to form vertical support systems (walls) and adopt timber to manufacture horizontal structural components (floors and roofs) (Luo et al., 2021a, 2021b). Hence, rammed earth buildings can be extremely vulnerable subjected to fire damage. Due to the excellent nature of thermal insulation and strength (Adam and Jones 1995), rammed earth walls can often survive in fire, although timbers have been completely burned (see Figure 1). From the perspectives of historical building protection or rehabilitation, it is imperative to study the mechanical degradation of rammed earth after fire exposure.

At present, researchers put great efforts to study the performance of clay after elevated temperature, with the focus on the physical and chemical changes (Milheiro et al., 2005; Wang et al., 2021), strength, and thermal properties (Geng and Sun 2018; Han et al., 2017). To be specific, Milheiro et al. (2005) claimed that complex reactions could occur in clay upon a temperature up to 1150°C, including hydroxide dehydration, dehydroxylation of clay minerals, phase transformations and partial melting with formation of glassy viscous phase. Abuel-Naga et al. (2006) found that normally consolidated clay could become overconsolidated after heating/cooling cycles, and the elastic zone decreased with temperature. The strength reduction of clay after high-temperature heating was also observed by many researchers (Althoey et al., 2022; Beckett et al., 2019; Geng and Sun 2018; Han et al., 2017; Sun et al., 2016; Wang et al., 2021). Different types of microstructural analysis methods have been used to understand the changes in mineralogy, morphology and elemental composition, such as the use of X-ray diffraction, field emission scanning electron microscopy (FESEM) and energy-dispersive X-ray spectrometry (Goodman and Vahedifard, 2019).

It should be noted that rammed earth can behave differently compared to clay, with respect to strength change, damage evolution, and constitutive response. Rammed earth (or cemented soil) often shows a hardening regime due to cementation between earth and binders, and strain softening occurs after the peak strength is exceeded (Lee et al., 2004). Therefore, one can consider that there are some similarities between rammed earth and rock materials after elevated heating, showing typical brittle to semi-brittle deformation behavior (Schuster et al., 2021). Thermal response of rocks has been extensively studied. For example, Tian et al. (2014) stated that micro-crack closure in rocks occurred as a result of thermal expansion below 200°C, at which the strength increase was observed; baking effects contributed to the strength gain up to 800°C; but serious fracturing could cause a sharp reduction of strength for rocks at 1000°C. Ye et al. (2015) also found that at low temperature, closure of pores could alter the strength performance of rock specimens, showing a slight increase. There is no consensus on the effect of elevated temperature on rock strength. Liu and Xu (2015b) identified the detrimental effect of temperature on granite; but the impact of temperature on sandstone was insignificant. Interestingly, the strength behavior of rocks after elevated temperature is significantly affected by the rock type.

Different constitutive models have been proposed to describe the response of geomaterial after heating, showing the damage nature. Lee et al. (2004) developed a plasticity based constitutive model for cement treated clay, where a bonding stress ratio was defined to characterize the increase in initial stiffness and shear strength and the reduction in stiffness and strength after the breaking in bonding between cementitious compounds. There are many statistical damage-based thermal constitutive models for simulating the brittle feature of rocks upon elevated temperature. For example, Cao et al. (2010) derived a statistical damage model with strain hardening and softening for rocks, in which the rock mass was considered to have a part of voids, a damaged part, and an undamaged part. At the initial loading, void compaction occurred to cause a hardening behavior; damage could evolve to change the proportion between damaged and undamaged parts due to the thermal effect. For simplicity, Cao et al. (2016) refined the damage model by introducing unrecoverable deformation characteristics for the void part and statistical damage characteristics for the skeleton part. Similarly, nonequilibrium statistical methods were adopted by Liu and Xu (2015a) to describe the occurrence of thermal damage in marble as a function of elastic modulus. Xu et al. (2018) formulated a thermo-mechanical coupling damage constitutive model with the Hoek-Brown strength criterion for rocks, based on the Weibull distribution and the continuous damage theory, which could account for the damage evolution from non-damage of loading, damage stability expansion, damage intensification expansion, to saturation. Based on the Lemaitre’s strain-equivalent principle (Lemaitre and Desmorat, 2005), a thermal damage model for granite was derived through a nonlinear coupled total damage parameter (Xu and Karakus, 2018). Zhao et al. (2019) considered the microelement failure using the Drucker-Prager criterion to calibrate a statistical thermal damage model for oil shale. Zymnis et al. (2019) calculated the progressive accumulation of strains with heating and cooling cycles. It should be emphasized that all these models were originally derived for rocks or cemented clays, but their applicability to rammed earth after fire exposure requires further investigation.

The strength of rammed earth can be lower than that of rock or concrete, and as such the stage of void compaction (closure of pores) cannot be neglected. Hence, a statistical thermal damage model is proposed for rammed earth after elevated temperature, in which the closure of pores is simulated to describe the initial hardening phase (unrecoverable deformation), and the damage evolution is captured for the skeleton part statistically (strain softening phase). In this study, a total of 24 rammed earth specimens are prepared to form the testing matrix for unconfined compression tests subjected to elevated temperature (three repetitive specimens, and eight temperatures of 20°C, 100°C, 200°C, 300°C, 400°C, 500°C, 600°C, and 700°C). Changes in microstructure of rammed earth at different temperatures are identified with scanning electron microscopy (SEM) and X-ray diffraction (XRD). In the end, model predictions are compared with experimental data for evaluation.

Thermal damage model

Definition of thermal damage

Rammed earth has been used worldwide to construct buildings, and Fujian Tulou is a typical type of earthen structures. Chinese Hakkas people built fort-like residences, having giant rammed earth walls for the outside appearance. Throughout the history of more than 600 years, rammed earth walls show significant deterioration due to various reasons, such as typhoon, heavy rainfall, temperature and moisture variation, sunshine, and salts attack. Most existing studies focused on the degradation of rammed earth due to wetting and drying cycles and salts attack (Beckett et al., 2020; Beckett and Ciancio, 2016; Luo et al., 2020, 2021c), considering the easiness in testing scheme/equipment and the associated costs. However, from historical statistics, fire exposure can also result in catastrophic damages to Fujian Tulou buildings, as demonstrated in Figure 1. The current study is done to understand the impact of elevated temperature on the behavior of rammed earth only, and other influencing factors (e.g., wetting and drying cycles and salts attack) are eliminated from the analysis.

From the perspective of chemical reaction mechanism, the compounds of SiO₂ and Al₂O₃ in rammed earth can react with calcium hydroxide (Ca(OH)₂) in binders (e.g., lime) to form calcium silicate hydrate (CSH) for enhancing the strength gain (El-Mahllawy and Kandeel, 2014). Figure 2 depicts the formation of different compounds, including CSH gel, CSH crystal, CSH precipitation, and restricted reaction due to loss of water. It should be understood that partial reaction causes different mechanical response for rammed earth. Figure 3 illustrates the typical stress-strain curves for uncemented and cemented soils subjected to normal and/or elevated temperatures. One can quite anticipate the behavior of uncemented soil (i.e., normally consolidated clay), which presents the strain hardening behavior until a peak strength is mobilized. For cemented soil, Lee et al. (2004) claimed that the strength can increase due to the formation of cementation, following by a strain softening behavior (due to breaking of bond) once the peak strength was exceeded. The response of rammed earth after elevated temperature could be similar to the behavior of rocks, showing a sudden change in volume initially, before the occurrence of strain hardening (Cao et al., 2010, 2016). This is because the pores can change from the open state to the closure state at the initial loading. Then, the typical strain hardening and softening behavior can be observed.

Essentially, the variation of temperature causes the formation of microcracks within the rammed earth. These microcracks propagate and spread with the increase of temperature, leading to a reduction of elastic modulus. The thermal damage is unrecoverable in nature. In the following, a thermal damage variable is defined to describe the detrimental effect of temperature on mechanical integrity of rammed earth.

D_{T} = 1 - \frac{E_{T}}{E_{0}}

(1)

where D_T represents the thermal damage variable for rammed earth, E₀ is the elastic modulus of rammed earth under ambient temperature, and E_T denotes the degraded elastic modulus of rammed earth after elevated temperature.

Deformation analysis of rammed earth

Compression dominates in many cases for rammed earth, although the walls could have some degrees of confinement. Hence, the development of thermal damage model for rammed earth is considerer as a one-dimensional problem for unconfined compression only, but tension and shear are not examined. The behavior of rammed earth is quite similar to that of rocks. Cao et al. (2016) established a deformation analysis model for rocks, containing a void part and a skeleton part. In their model, void compaction could occur at the initial loading stage, in which voids within the rock mass changed from the open to the closure state, along with some unrecoverable deformations. At a later stage, continued loading could cause damage to the skeleton part, which was described using a statistical method to capture the strain softening behavior. For rammed earth, the influence of thermal damage can be captured in a similar manner.

Figure 4 demonstrates the development of thermal damage model for lime-stabilized rammed earth. For clayey soils, researchers often use a Maxwell rheology model to characterize the time-dependent stress-strain behavior, containing a linear elastic spring element and a linear viscous dashpot element (Yin and Graham, 1989, 1996). Essentially, linear spring can deform upon loading, but the dashpot can dissipate the energy to incorporate the time effect. For rammed earth, one can adopt unconfined compression testing to derive its strength characters. For any type of soil, there are three phases, namely solid, air and water. Then, following the work of Cao et al. (2016), a thermal damage model is proposed for rammed earth by considering two parts, i.e., a part of pores (air and water phases) and a skeleton part (solid). It is anticipated that the part of pores can deform upon loading easily, showing the closure of pores (before point A in Figure 3). The skeleton part can present the strain hardening response up to a certain strength due to cementation (up to point C in Figure 3), and then show thermal damage due to breaking of bond, which is the main reason for the strain softening behavior (after point C in Figure 3).

A rammed earth cylinder is assumed to have an initial length of l₀, which consists of the length of pores in $l_{0}^{v}$ and the length of skeleton in $l_{0}^{r}$ . One can define the ratio of pores in the cylinder as $γ_{0} = l_{0}^{v} / l_{0}$ . Hence, the strain compatibility equation for different components can be established as follows:

ε = (1 - γ_{0}) ε^{r} + γ_{0} ε^{v}

(2)

where

ε

is the total strain,

ε^{r}

is the strain of skeleton, and

ε^{v}

is the strain of pores. It should be emphasized that the associated strain of pores

ε^{v}

in the dashpot diminishes instantaneously as long as it is loaded (i.e., no time-dependent load-deformation behavior upon the closure of pores in the dashpot).

Deformation analysis of pores

The applied stress is divided into n levels, and each increment (s = 1, 2, …, n) is imposed to the part of pores. Hence, the strain of pores can be considered as the sum of deformation in the pores due to stress increment $Δ σ^{s}$ . It is assumed that the instantaneous length of pores after loading is $l_{s}^{v}$ . For preventing the void space from falling below zero, the logarithmic form of instantaneous strain of pores, $ε^{v s}$ , can then be calculated as:

ε^{v s} = - \int_{l_{s - 1}^{v}}^{l_{s}^{v}} \frac{d l}{l} = - l n \frac{l_{s}^{v}}{l_{s - 1}^{v}}

(3)

Following the Hooke’s law, the relationship between $Δ σ^{s}$ and $ε^{v s}$ can be written as:

ε^{v s} = Δ σ^{s} / E_{T}^{v}

(4)

in which,

E_{T}^{v}

is the compression modulus of pores in rammed earth. Both the air and water phases exist in the pores, but compression is assumed to occur only in the air phase. This is because the water phase in rammed earth becomes negligible especially after elevated temperature. Compression in the water phase is neglected in the compression modulus of pores.

One can further derive the following expressions:

\sum_{s = 1}^{n} ε^{v s} = (\sum_{s = 1}^{n} Δ σ^{s}) / E_{T}^{v}

(5)

- \ln [l_{n}^{v} / l_{0}^{v}] = σ / E_{0}^{v}

(6)

Hence, the integral form of strain of pores can be determined by:

ε^{v} = \frac{Δ l^{v}}{l_{0}^{v}} = \frac{l_{0}^{v} - l_{n}^{v}}{l_{0}^{v}} = 1 - \exp [- \frac{σ}{E_{0}^{v}}]

(7)

Deformation analysis of skeleton

Due to the inhomogeneity and/or the nonuniform distribution of strength for rammed earth materials, one can assume that the strength of rammed earth follows the Weibull distribution (Cao et al., 2010). Damage in mesoscopic elements can govern the macroscopic behavior of rammed earth.

P (ε) = \frac{m}{ε_{0}} {(\frac{ε}{ε_{0}})}^{m - 1} e xp [- {(\frac{ε}{ε_{0}})}^{m}]

(8)

where

P (ε)

is a shape parameter for the density function, representing the degree of damage in rammed earth.

ε

is the strain for the element, and

ε_{0}

and m are the scale parameters as a function of strength of rammed earth.

If the number of damaged mesoscopic elements is n, and the total number of mesoscopic elements is N, the damage variable (Cao et al., 2010) is then expressed as:

D_{ε} = \frac{n (ε)}{N} = \frac{\int_{0}^{ε} N P (ε) d ε}{N} = 1 - \exp [- {(\frac{ε}{ε_{0}})}^{m}]

(9)

Following the Lemaitre’s strain-equivalent principle (Lemaitre and Desmorat, 2005), the constitutive relationship between stress and strain for rammed earth can be obtained. Damaged mesoscopic elements can transfer stress due to the existence of friction. Hence, a damage modification factor, $μ$ (0 $< μ <$ 1), is introduced to consider the effect of friction mobilization. At a temperature of 300°C, the water phase in rammed earth diminishes almost completely. Microcracking then occurs in the solid phase of rammed earth, changing the contact state between solid compounds to enhance the friction character. Beyond a temperature of 300°C, macrocracking continues to occur in rammed earth, which causes the reduction of friction, and the strength is dominated by cracking only (thermal damage evolves).

σ = E_{0} (1 - D_{T}) (1 - {μ D}_{ε}) ε

(10)

It is reasonable to assume that the deformation due to the closure of pores ( $ε^{v}$ ) should have already reached the maximum, when damage is initiated in the skeleton part. One could argue that porous media like rammed earth will always have some voids as the particles do not fit exactly, depending on whether particle crushing is occurring or not. Hence, once the $ε^{v}$ term reaches the maximum, the compression of pores is considered to be complete, leading to $σ \geq σ_{ε}$ . The strain in the skeleton part is then written as:

ε^{r} = \{\begin{array}{l} \frac{σ}{E_{T} (1 - {μ D}_{ε})} σ \geq σ_{ε} \\ σ / E_{T} σ < σ_{ε} \end{array}

(11)

Statistical constitutive model

Substituting equations (7) and (12) into equation (2), a new statistical thermal damage constitutive model is derived for rammed earth to capture the whole deformation pattern.

ε = \{\begin{array}{l} γ_{0} [1 - \exp (- \frac{σ}{E_{0}^{v}})] + \frac{(1 - γ_{0}) σ}{E_{T} (1 - {μ D}_{ε})} σ \geq σ_{ε} \\ γ_{0} [1 - \exp (- \frac{σ}{E_{0}^{v}})] + \frac{(1 - γ_{0}) σ}{E_{T}} σ < σ_{ε} \end{array}

(12)

The deformation characteristics of different components for rammed earth are illustrated in Figure 5. When the curve enters the linear deformation stage (AB), the deformation due to the closure of pores is considered as zero. Therefore, the first term in equation (12) reaches the maximum value of $γ_{0}$ . The vertical interception of line AB is defined as $γ_{0}$ , which corresponds to the stress of $σ_{ε}$ . Before point A, a total of z points (k = 1, 2, …, z) can be selected on the stress-strain curve, which are analyzed by fitting with the least square method to derive the parameters for the skeleton part ( $γ_{0}$ and $E_{0}^{v}$ ).

For the stress-strain curve, the peak point is written as $σ_{c}$ and $ε_{c}$ , leading to a first derivative of zero as follows:

{\frac{d σ}{d ε}|}_{\begin{matrix} {σ = σ}_{c} \\ ε = ε_{c} \end{matrix}} = 0

(13)

One can combine equations (12) and (13) to calculate the parameters $m$ and $ε_{0}$ .

Experiments

Materials and specimen preparation

To simulate the behavior of rammed earth used in Fujian Tulou, local soil materials are collected from Longyan City, Fujian Province, China. The protocols defined in GB/T 50123-2019 (2019) are strictly followed to determine the soil properties. Sieve analysis is conducted to obtain the grain size distribution for both sand and earth, as shown in Figure 6. The mass ratio between sand and earth is taken as 4:3, which follows the normal practice of Chinese Hakkas people (Luo et al., 2022). Hydrated lime is purchased from a local company, and the content of Ca(OH)₂ is found to be greater than 95%. Normally, the binder content varies from 5% to 10% for soil stabilization (Li et al., 2019b), which can achieve a good balance between treatment efficiency and cost. In this work, the content of lime (against the combined weight of solids) is fixed at 10%. The initial water content is controlled at 20%. For each specimen compacted at a bulk density of 1.5 g/cm³, the mass of earth, sand, lime and water is 77.55, 58.16, 15.08, and 31.67 g, respectively.

As suggested in GB/T 50123-2019 (2019), a rammed earth specimen should be prepared into a cylinder, with a diameter of 3.5–4.0 cm and a height-to-diameter ratio of 2.0–2.5. In the current work, the mold is manufactured to have a diameter of 40 mm and a height of 80 mm. Before mixing, all soil materials are dried fully in an oven at a temperature of 105 ± 5°C. After weighting, different materials are poured in a mixer to achieve the uniform mixture. The mixture is then put into a plastic bag for 24 hours to enable the occurrence of reaction between different components. A house-made hydraulic compactor is then used to conduct compaction for rammed earth. All specimens are cured in a curing room for 60 days. After curing, a thermocouple (diameter of 1 mm and length of 40 mm) is installed into the specimen through a drilled hole with a diameter of 1.5 mm and a depth of 40 mm (see Figure 7). Thermal insulation materials are employed to cover the thermocouple. Installation of thermocouple could potentially damage the specimen, changing its mechanical behavior. However, the use of three specimens for each testing condition is to check the repeatability of the testing program and eliminate the influencing factor induced by thermocouple installation. A specimen without thermocouple could also demonstrate the negligible impact of thermocouple installation.

A resistance testing chamber (see Figure 8) is used to simulate the behavior of rammed earth exposed to elevated temperature. The chamber has an inside dimension of 300 mm × 200 mm ×120 mm. The highest temperature that can be set is 1000°C, with an accuracy of 1°C. The temperature raising rate is set at 5°C/min. The chamber is equipped with an auto controller, allowing a constant temperature once the target value is reached. Different specimens are put in the chamber at a target temperature for 3 hours to enable a uniform temperature within the specimen.

Testing program

In total, 24 rammed earth specimens are tested, with a matrix of three repetitive specimens and eight temperatures of 20°C (ambient temperature), 100°C, 200°C, 300°C, 400°C, 500°C, 600°C, and 700°C. The highest temperature is selected as 700°C. This can effectively avoid that rammed earth could be burned into brick (Wang et al., 2021). It should be noted that after the removal from the resistance testing chamber, all specimens are cooled naturally for 7 days before they are tested under the unconfined compression condition.

Unconfined compression tests are conducted at a loading rate of 0.8 mm/min. The unconfined compression strength (UCS) can be determined as the average value from three repetitive tests. The change in color due to elevated temperature is monitored, and the failure patterns of different specimens after unconfined compression tests are closely examined.

Microstructural analyses

After UCS tests, crushed soils are collected for use in microstructural analyses. The Talos F200X 200 kV field emission transmission electron microscopy is adopted to carry out scanning electron microscopy (SEM) analyses. Freeze-drying is conducted for collected soils, which can minimize the chance for accelerating or retarding continued chemical reaction during the drying process. For each scan, the soil sample is peeled off with epoxy resin through Electrodag silver glue, and sputtered with a layer of gold. The electric conductivity on the sample surface is improved with a gold layer, and at the same time, it helps to prevent spalling of particles from the sample during testing.

The Bruker D8 Advance X-ray diffractometer is used to conduct X-ray diffraction (XRD) analyses on collected samples. The continuous scan mode with a scan rate of 5°/min is set for the X-ray tube. In total, the scan range from 5° to 70° is covered. The “MDI jade 9.0” software is then adopted to analyze the obtained XRD patterns to reveal qualitative descriptions.

Results

Stress-strain response

Figure 9 shows the photos of rammed earth specimens subjected to elevated temperature before UCS testing. It is interesting that with the increase of temperature, the color of rammed earth gradually becomes darker. With the increase of temperature, the iron (Fe) ion reacts to form the precipitation of ferric oxide (Fe₂O₃), showing dark red color. Due to the loss of water at high temperature, the specimen becomes a more porous structure. For the specimen subjected to an elevated temperature of 700°C, the top part of the specimen turns into yellowish. This color change could be a result of chemical compound decomposition (from calcium carbonate (CaCO₃) or Ca(OH)₂ to calcium oxide (CaO)). All specimens are initially cooled in the chamber. The air in the top portion of the chamber has a higher flow, leading to a faster cooling process. Hence, the top portion of the specimen becomes yellowish, and the color changes into dark red at the bottom. Overall, there is no apparent cracking on the surface of the specimen.

The variations of stress-strain curves with temperature are plotted in Figure 10. It should be emphasized that only one stress-strain curve is presented in the figure for a single temperature, since the measured curves for all three repetitive tests do not change much. For all cases, one can clearly see four stages: closure of pores (void compaction), linear elastic, strain hardening, and strain softening. The patterns of all curves are very similar, but showing different peak strength and elastic/plastic regime. In general, the slope of the linear elastic regime becomes gentler, when the temperature is higher. As expected, the strength is lowered with temperature. One can directly detect that there is a detrimental effect of temperature on strength of rammed earth (lime-stabilized soils). This is consistent with the observations of many previous studies (Althoey et al., 2022; Beckett et al., 2019; Geng and Sun, 2018; Han et al., 2017; Sun et al., 2016; Wang et al., 2021). However, it should also be noted that the ductility of rammed earth subjected to high temperature is improved. Taking the specimen under a temperature of 700°C as an example, the peak strength occurs at a strain level of 2.5%, which is much higher than the value of 1.5% for the specimen under ambient temperature of 20°C.

The UCS values of all specimens are calculated as shown in Figure 11. Error bars are plotted to characterize the variance of data from repetitive tests. Compared to the UCS of approximately 2.4 MPa for rammed earth at ambient temperature, the cases subjected to elevated temperature show clear strength reduction by 26%, 30%, 22%, and 30% at 100°C, 200°C, 300°C, and 400°C, respectively. It is interesting that the strength of rammed earth after exposure to 300°C shows a slightly higher value compared to other high temperatures. This is because thermal expansion can cause a certain extent of closure of micro-cracking, which improves the strength performance slightly due to the enhanced effect of friction between minerals or chemical compounds (Liu and Xu, 2015b). When the temperature is beyond 300°C, temperature can result in cracking in rammed earth continuously, and deterioration within the specimen becomes significant. Hence, one can see apparent strength reduction with temperature (quasi-linear reduction), i.e., 35%, 43%, and 52% strength reduction at 500°C, 600°C, and 700°C, respectively, compared to the case under ambient temperature.

The variations of elastic modulus with temperature are illustrated in Figure 12, along with error bars showing the variance of data. As an interpretation of Figure 10, one can infer that the elastic modulus should decline with temperature, since all stress-strain curves shift towards the right. Again, the elastic modulus calculated for rammed earth exposed to 300°C is slightly higher, showing a local peak. However, a general linear decreasing pattern of elastic modulus with temperature can be seen.

Based on equation (1), the variations of thermal damage with temperature can be computed as depicted in Figure 13. Overall, an increasing pattern of thermal damage as a function of temperature is derived. Interestingly, there is a sudden change in gradient in the thermal damage versus temperature curve at 300°C. When the temperature is applied at the range of 100–200°C, water inside a rammed earth specimen is evaporated along the pores, leading to an increase in pore size. At 300°C, the magnitude of thermal damage is reduced, since water is almost completely lost. Rammed earth particles or chemical compounds can then fill the pores to improve the strength performance. After 400°C, macro-cracking further propagates, causing increased deterioration or thermal damage.

Failure pattern and mechanism

The failure patterns of rammed earth exposed to different temperatures after UCS testing are demonstrated in Figure 14. One can see two typical types of failure: necking-type failure and hourglass-type failure. At a temperature of 20°C, 200°C, 400°C or 500°C, one can see clearly the presence of necking-type failure. In the initial loading stage, there is no apparent cracking on the surface of the specimen. With continued loading, some vertical micro-cracks can be observed from both ends of the specimen. Then, plastic deformation occurs in the weakest position, leading to a reduction of the cross-sectional area (necking-type failure). For the condition of high temperature (i.e., 600°C and 700°C), the initiation of micro-cracking also starts from the two ends of the specimen. As the load increases, vertical cracks from the two ends become thorough cracks, and the mode of hourglass-type failure can be observed.

Figure 15 shows the SEM images of lime-stabilized rammed earth after elevated temperature. Researchers identified that the morphology of stabilized rammed earth materials often show a densely compacted microstructure under ambient temperature as a result of chemical compounds produced within the soil, while it becomes an irregular pattern when experiencing a high temperature exposure (Li et al., 2019a; Luo et al., 2022; Ming et al., 2020). One can see that different forms of CaCO₃ have precipitated in the soil. For example, flake crystals of CaCO₃ are formed to bond soil particles, leading to the formation of large clusters, primarily for those cases subjected to low temperature (below 400°C). With the increase of temperature, the flake mode of CaCO₃ turns into the mode of being particle-like with a smaller size, which is similar to the observations of Li et al. (2019a). Furthermore, the intact microstructure with small pore size at ambient temperature is deteriorated by the effect of elevated temperature as well. The loose porous structure can be seen, or the pore size is increased for the case subjected to high temperature (above 300°C). More dissolved gels can be observed in the microstructure, which is the sign of chemical compound decomposition (from CaCO₃ to CaO). The occurrence of chemical compound decomposition is also demonstrated from the color change from dark red to yellowish, especially after a temperature of 700°C (see Figure 9). Hence, the combined actions of morphology change (from flake crystals to particle-like), microstructure change (from dense to loose) and chemical compound decomposition all contribute to the reduction in strength for rammed earth experiencing elevated temperature.

Figure 16 presents the XRD diffractograms of lime-stabilized rammed earth subjected to different temperatures. For rammed earth at ambient temperature, the primary mineralogical constituents are identified as quartz (SiO₂), kaolinite (Al₂(SiO₂)(OH)₄) and CaCO₃. Essentially, Ca(OH)₂ in lime can interact with SiO₂ and Fe₂O₃ to form calcium silicate hydrate (CSH) through pozzolanic reaction. Partial calcium ion can enable cation exchange with sodium ion in water to increase the double layer thickness, enhancing the bonding effect between particles. These chemical or physical actions promote the strength gain in stabilized rammed earth. With the increase of temperature, the constituent of kaolinite reduces heavily, and the relative intensity for the CaCO₃ peak decreases slightly. It should be noted that the decomposition of CaCO₃ into CaO and carbon dioxide (CO₂) often occurs at a temperature of 530°C. This can explain the strength reduction of rammed earth with temperature.

Comparison between model predictions and experimental data

To demonstrate the effectiveness of the proposed thermal damage model, the measured stress-strain curves of rammed earth after elevated temperature are evaluated for calibration. Based on the observations of Figure 10, the parameters of $γ_{0}$ and $E_{0}^{v}$ do not change with temperature. Hence, in subsequent predictions, the average values of $γ_{0} = 0.0075$ and $E_{0}^{v} = 0.04$ MPa are adopted for comparison. All other parameters used for thermal damage model are listed in Table 1. Figure 17 shows the comparison between measured and calculated stress-strain curves for rammed earth. One should emphasize that the degree of thermal damage is simulated without modification for high temperatures (above 300°C, $μ =$ 1), whereas a damage modification factor is introduced for low temperatures (at ambient temperature of 20°C, $μ =$ 0.9; and at 100°C and 200°C, $μ =$ 0.7). It is interesting that the four stages can be captured well by the proposed approach, including the closure of pores (void compaction), elastic regime, strain hardening and strain softening. This demonstrates that the proposed method is effective to analyze the behavior of rammed earth after elevated temperature.

Table 1.

Parameters used for thermal damage model.

Temperature (°C)	$γ_{0}$	$E_{0}^{v}$ (MPa)	µ	$m$	$ε_{0}$	$D_{T}$
20	0.0075	0.04	0.9	0.95	0.009	0
100	0.0075	0.04	0.7	2.03	0.010	0.31
200	0.0075	0.04	0.7	1.95	0.016	0.53
300	0.0075	0.04	1	1.06	0.019	0.38
400	0.0075	0.04	1	1.57	0.021	0.53
500	0.0075	0.04	1	1.09	0.022	0.62
600	0.0075	0.04	1	1.28	0.026	0.74
700	0.0075	0.04	1	2.04	0.035	0.81

Conclusions

In this study, for revealing the mechanical degradation of rammed earth after fire exposure, a set of unconfined compression tests are conducted on lime-stabilized rammed earth specimens subjected to elevated temperature. Microstructural analyses are conducted to understand the mechanism of strength change. In addition, four stages of closure of pores, elastic regime, strain hardening and strain softening in the stress-strain curve are analyzed, based on which a thermal damage model is proposed for prediction. The main findings from this work are summarized below:

Temperature has apparent detrimental effects on the strength of lime-stabilized rammed earth. Compared with the strength of rammed earth at ambient temperature of 20°C, the strength reduction is determined as 26%, 30%, 22%, 30%, 35%, 43%, and 52% for the case with an elevated temperature of 100°C, 200°C, 300°C, 400°C, 500°C, 600°C, and 700°C, respectively.

With the increase of temperature, rammed earth shows an increased level of ductility. At a high temperature, the stress-strain curve shifts towards the right. At 700°C, the strain level of rammed earth corresponding to the peak strength is 2.5%, being much higher than the value of 1.5% at 20°C.

Upon increasing the temperature, a rammed earth specimen shows a more porous structure, with the evidence of reduced calcium carbonate precipitation from microstructural analyses. This can explain the strength reduction in rammed earth experiencing high temperature.

A new statistical thermal damage model is introduced for rammed earth after elevated temperature, where the closure of pores is modeled to show unrecoverable deformation, and the skeleton part is simulated with a thermal damage variable statistically to present the damage evolution (strain softening phase). The model is found to be effective in predicting the stress-strain response of rammed earth experiencing elevated temperature.

It should be recognized that rammed earth walls primarily withstand compression due to gravity. The derived statistical damage model can only consider the one-dimensional strain softening behavior of lime-stabilized rammed earth under unconfined compression conditions exposed to elevated temperature, which is a limitation in the applicability of the model in practice. Other failure modes in the wall are neglected, in terms of tension and shear. Future work should be carried out to understand the degradation mechanism of rammed earth materials experiencing other loading schemes.

Footnotes

Acknowledgements

We would like to extend our sincere gratitude to the Instrumental Analysis Center of Huaqiao University for their invaluable support and assistance.

Data availability statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (52078225, 52479101, and 52078506), the Fujian Province Foreign Cooperation Project (2023I0015), and the Guangdong Basic and Applied Basic Research Foundation (2023A1515012159).

ORCID iDs

Yi Luo

Pengpeng Ni

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