Finite Element Modeling and Experimental Study of Foam Concrete and Polystyrene Concrete

Abstract

Predicting the physical and mechanical properties of polystyrene concrete is an important tool for determining its performance under various conditions. This article presents an experimental study and numerical modeling of polystyrene concrete under various types of loads: thermal and mechanical. The numerical model was developed in ANSYS in several stages. First, a foam concrete model was constructed in Materials Designer, and strength and thermal calculations were performed. The obtained data were entered into the polystyrene concrete model as input, polystyrene granules were added, and strength and thermal calculations were repeated. Using the Menetrey–Willam structural model, the numerical modeling sufficiently captured key mechanical properties of concrete. The parameters of the Menetrey–Willam model were adjusted based on experimental results from compression tests of foam concrete and polystyrene concrete. The results of numerical modeling, represented by stress and strain fields, allowed us to identify the dependence of thermal conductivity and compressive strength of polystyrene concrete on varying polystyrene granule contents. A comparison of the numerical analysis and experimental results showed good agreement. Errors in the obtained results were 6% for thermal conductivity and 7% for compressive strength. The resulting models revealed the characteristics of fracture sites, the relationship between structural changes, and the thermal and physical properties of polystyrene concrete, which can be used in the design of engineering structures.

1. Introduction

Lightweight concrete has become increasingly popular because of its unique properties [1,2,3] in recent years. Although its share is incomparable to the production volume of structural concrete, the polystyrene concrete market is growing and is part of broader building materials segments such as lightweight concrete and expanded polystyrene (EPS)-based thermal insulation materials.

As of early 2026, the key indicators of related markets driving the growth of polystyrene concrete are as follows [4,5]. The volume and growth rate of expanded polystyrene (EPS), the main component of polystyrene concrete, exceeded USD 17.98 billion in 2025. This figure is expected to reach USD 18.9 billion by 2026. The EPS segment is projected to grow at a CAGR of 5.7% to 6.2% through 2033–2035. In the construction sector, polystyrene concrete is the fastest-growing polystyrene application segment, with an expected growth rate of approximately 6.98%. This growth is driven by the global trend toward energy-efficient and green construction.

Higher building codes require the use of materials with high thermal insulation properties. That is why the “gray” (graphite) polystyrene (Grey EPS) segment is developing, which keeps heat 20% more effectively than standard compounds, making it possible to improve the formulation and manufacturing technology [6,7].

The high demand for polystyrene concrete in construction necessitates improving its properties. In [8,9], the authors used alkali activation to improve performance properties such as compressive strength, flexural strength, mass loss, impact resistance, and thermal insulation performance. The study included strengthening the matrix by adding granulated blast furnace slag (GBFS) and then developing lightweight, alkali-activated high-strength concrete by substituting 50% of the river sand with polystyrene (EPS). This combination of additives improved the strength properties of polystyrene concrete by up to 18%. However, the authors noted poor bonding between the exfoliated polystyrene particles and the cementitious material, which reduces the effectiveness of the EPS lightweight aggregate. The authors also noted increased shrinkage deformation when using EPS aggregate, which increases the likelihood of deformation and shrinkage cracks [10,11].

Global trends in sustainable construction and energy conservation are leading to the emergence of new structural elements incorporating polystyrene of various sizes and shapes [12]. This also ensures low construction and operating costs [13,14]. Spherical void shapes in reinforced concrete beams were studied in [15]. The spherical shape is attractive because it does not create stress concentrations. However, the authors note that voids introduce complex heat transfer dependencies that require empirical and theoretical analysis. The heat transfer mechanism is based on the fact that solid concrete zones around the voids act as thermal bridges, increasing thermal conductivity compared to polystyrene spheres. Thus, the next step suggests itself: increasing the voids to reduce thermal conductivity and strengthening the cement matrix to keep the strength properties of the composite.

In [16,17,18,19,20], the effect of nanosilica on a concrete composite using expanded polystyrene (EPS) was investigated. The authors showed that the combined effect of nanosilica and EPS can provide good thermal insulation, enhance water permeability, and keep structural integrity. Nanosilica (NS) was added in amounts ranging from 0.75% to 1.25% of the cement weight, and EPS replaced fine aggregates at 25% to 100%. Authors [20] found a synergistic effect with NS and EPS combined. Thermal conductivity was reduced by 39–86%, compressive strength increased to 36.5%, and water permeability was reduced by 40–52%. However, issues related to technological factors, polystyrene sizes, and their influence on the thermos-physical and strength properties of the final composite remain unexplored. The positive effects of nanomodifiers have been noted in many studies [21,22,23,24,25], which can contribute to strengthening the matrix in polystyrene concrete.

Understanding obstacles is significantly aided by analyzing composite materials’ mechanical properties without physical tests. Several promising methods have been developed for this purpose, including theoretical approaches, the discrete element method, and the finite element method (FEM). The effectiveness of these methods is demonstrated by the relatively high accuracy of predicting the mechanical properties of composite materials [26,27,28,29], as well as the visualization of stress and strain fields. One of the easiest ways to determine mechanical properties is by using the finite element method to generate a representative volume element (RVE). This involves elastic moduli, such as Young’s, shear, and Poisson’s ratio, alongside heat fluxes in foam concrete [30,31,32]. The FEM model made it possible to determine the porosity characteristics of lightweight cellular concrete and its density with an accuracy of up to 90%. However, existing approaches modeled polystyrene concrete as a two-phase composite and failed to take into account that the cement matrix is a porous medium. The presence of pores in the matrix creates nonlinear effects that manifest themselves both in the active loading phase (the ascending branch of the stress–strain curve) and in the softening phase (the descending branch).

It should be noted that the application of numerical analysis methods integrates well with artificial intelligence methods, such as convolutional neural networks [33] and deep learning neural networks [34,35]. However, work related to computer modeling is limited to the study of linear elastic properties, such as elastic moduli. A comprehensive analysis of highly nonlinear porous materials requires an understanding of the stress–strain state generated by the inclusion of polystyrene spheres and the associated strength characteristics.

In [36], the mechanical response of structural lightweight aggregate concrete (LWAC) cubes to uniaxial, cyclic, and biaxial compression was analyzed. Using experiments, the authors found a biaxial failure stress surface. This helped them calibrate the Kupffer–Gerstle biaxial failure criterion for LWAC so that it could be used in numerical models. A mesoscale model using ABAQUS was developed in [37]. The authors presented LWAC-CB as a three-phase composite comprising cold-bonded (CB) fly ash aggregate, mortar, and an interfacial transition zone (ITZ). This model is designed to study the way stress develops, as well as the start of damage, in line with real-world physical events. The model correctly simulated stress concentration and the start of cracking in CB aggregates at normal temperatures. However, to date, there is no clear understanding of the relationship between the composition and size of polystyrene granules and the physical and mechanical properties of polystyrene concrete. The identified gaps in existing research allowed us to formulate the objective of this article: to develop a highly accurate computer model of polystyrene concrete with improved strength based on physical experimental data. This study mainly comprises a step-by-step modeling of a three-phase composite and an examination of its properties at each stage.

Research objectives:

1. To study the relationship between the composition, structure, and properties of polystyrene concrete.

2. To develop a computer model of polystyrene concrete with improved strength using formulation and technological methods.

3. To identify rational formulation and technological factors influencing the strength of polystyrene concrete based on literature analysis and physical experiments.

4. To identify patterns in the stress–strain state of polystyrene concrete under compressive loads and determine the relationship between the thermal and physical properties of foam concrete and polystyrene concrete.

The novelty of this study lies in the development of a mathematical model of polystyrene concrete as a multiphase material, including a porous matrix and polystyrene particles, and the use of this model to study the thermal, physical, and strength properties of the resulting composite. Mathematical modeling of the composite occurs in two stages. In the first stage, the effective properties of foam concrete as a composite of concrete and air pores are modeled. In the second stage, the obtained effective properties of foam concrete are used to model polystyrene concrete directly as a composite of foam concrete and inclusions of polystyrene particles. This is due to the different scales of air pores and polystyrene, ensuring the greatest reliability of the model and eliminating the need for many experiments, predicting the properties of polystyrene concrete (or other multiphase materials) with engineering accuracy.

2. Materials and Methods

Materials and Research Sequence

The following raw materials were used to produce the composites:

1. Portland cement CEM I 42.5 N (CEMROS, Stary Oskol, Russia). Properties: initial setting time—180 min; final setting time—220 min; standard consistency—30%; compressive strength at 28 days—49.2 MPa.

2. Polystyrene granules (Polistorg, St. Petersburg, Russia), 1–2 mm in size. Properties: density—25 kg/m³; thermal conductivity—0.038 W/(m × °C); compressive strength under 10% linear strain—0.025 MPa;

3. Polystyrene granules (Polistorg, St. Petersburg, Russia), 3–5 mm in size. Properties: density—12 kg/m³; thermal conductivity—0.041 W/(m × °C); Compressive strength under 10% linear strain—0.08 MPa;

4. Air-entraining admixture—SDO-MK saponified wood resin (NPF AKRIL, Omsk, Russia). Properties: solids content—52%; density of a 10% aqueous solution—1017 kg/m³.

Polystyrene concrete mix formulation: Portland cement CEM I 42.5 H—410 kg/m³; polystyrene granule volume—0.5 m³/m³; saponified wood resin SDO-MK—2 kg/m³; water—200 L/m³.

Experimental concrete samples were prepared as follows. The raw materials were dosed in the required quantities and poured into a laboratory concrete mixer. First, polystyrene granules were poured in and mixed for 30 s with 1/3 of the mixing water containing the SDO-MK additive dissolved in it. Treating the polystyrene granules with water containing the SDO-MK additive is necessary to improve their adhesion to the cement paste. Next, the cement was poured in, and the remaining mixing water with the additive and the entire mixture were mixed until a uniform consistency was achieved. The finished mixture was poured into metal molds, which were compacted on a laboratory vibrating platform, and the surface of the samples was leveled. The molds were removed after two days, and then the samples were cured in laboratory conditions at a temperature of 20 ± 2 °C and a relative humidity of 60 ± 10%.

The research diagram is presented in Figure 1

Figure 1 shows the study sequence, which included the following stages.

Stage 1. Laboratory testing of the cement matrix without a foaming agent. Properties of hardened cement paste: density—2200 kg/m³; coefficient of thermal expansion—1.4 × 10⁻⁵ 1/°C; young modulus—30,000 MPa; compressive strength—41.7 MPa; thermal conductivity—1.88 W/(m × °C).

Thirty-six samples were experimentally tested, divided into 12 series with three samples in each series. For samples with 1–2 mm and 3–5 mm polystyrene granules, six identical series of samples were produced for each filler type. The number of experimental samples allows for the most accurate determination of the properties of the developed composites and the calculation of standard deviations. The properties of polystyrene concrete at 28 days of age are presented in

Table 1. Properties of polystyrene concrete at 28 days of age.

Granule Diameter, mm	Density, kg/m3	Compressive Strength, MPa
1…2	780	3.6
	783	3.7
	784	3.2
	789	3.5
	782	3.4
	781	3.6
3…5	875	4.1
	870	4
	880	3.8
	876	4.2
	879	4.3
	882	4

.

The standard deviation of the density of a series of polystyrene concrete samples with a filler size of 1–2 mm was 3.2 kg/m³. The standard deviation of the compressive strength was 0.2 MPa.

The standard deviation of the density of a series of polystyrene concrete samples with a filler size of 3–5 mm was 4.3 kg/m³. The standard deviation of the compressive strength was 0.2 MPa.

Stage 2. In the second stage, modeling was performed using the Materials Designer module (ANSYS 2024 R2). This module allows for modeling the functional properties of composite materials and enables calculation of physical and mechanical properties (modulus of elasticity for each axis, thermal conductivity, and density). In the FEM simulation, spherical pores filled with air were considered. Pore size varied between 0.45 and 0.55 µm. The pore diameter distribution is described by a lognormal distribution with a probability density function.

$$f_X(x)={\displaystyle \frac{1}{x\sqrt{2\pi }\sigma }}{e}^{-{\displaystyle \frac{{\left(\ln x-\mu \right)}^2}{2{\sigma }^2}}},$$

(1)

here $\mu ,\hspace{0.17em}\sigma$—parameters of the log-normal distribution.

When modeling a representative volume of the porous material, the values of $\mu$ = 500 µm and $\sigma$ = 40 µm were chosen. The generated representative volume and finite element model are shown in Figure 2.

Since Materials Designer uses a random element generation algorithm, the model shown in Figure 2 contained 62 air-pore spheres with a pore volume fraction of 0.405 and an average diameter of 456 µm. The modeling results, in the form of predicted porous concrete properties and a comparison with experimental data, are presented in Section 3.

Stage 3. Modeling the strength properties of the resulting porous concrete model in ANSYS Static Structural (2024 R2). Figure 3 shows the geometric model of the porous concrete and its FEM discretization.

Stage 4. At this stage, laboratory tests of foam concrete were carried out for thermal conductivity, density and compressive strength. At this stage, laboratory tests of foam concrete for thermal conductivity and strength were conducted. The thermal conductivity of aerated concrete samples was experimentally measured using an ITP-MG4 meter. See Figure 4 for an overview of the ITP-MG4 meter.

Thermal conductivity measurements were conducted in accordance with GOST 7076 [38] Building Materials and Products. Test Method for Steady-State Thermal Conductivity and Thermal Resistance and ASTM C518 [39] Standard Test Method for Steady-State Heat-Transfer Properties. The samples were rectangular parallelepipeds with the largest faces square with sides measuring 100 × 100 × 21 mm. Before testing, the edges of the specimens in contact with the working surfaces of the instrument plates were ground. Deviations from parallelism did not exceed ±0.5 mm. The specimens were then dried to constant weight. Thickness was measured with calipers with an error of no more than ±0.1 mm at four corners at a distance of (50 ± 5) mm from the corner apex to the center of each side. Samples were put in the device; its heat was monitored and automatically controlled. The thermal conductivity values are automatically computed when the timer on the display’s bottom line reaches zero, marking the end of the observation time. The thermal conductivity coefficient of a series of samples was determined as the arithmetic mean of the test results for five samples.

Density was determined on 28-day-old cube samples measuring 100 × 100 × 100 mm, dried to constant weight in accordance with the requirements of [40]. The average density was calculated as the arithmetic mean of the test results for three samples, with an accuracy of 1 kg/m³, using Formula (2).

$$\rho ={\displaystyle \frac{m}{V}}\hspace{0.17em}\times \hspace{0.17em}1000$$

(2)

where m is the sample mass (g);

V is the sample volume (cm³).

Compressive strength was determined on 28-day-old 100 × 100 × 100 mm cube specimens dried to constant weight in accordance with standardized methodologies [41,42,43,44]. Composite specimens were centered in the testing machine and loaded to failure at a constant rate of (0.6 ± 0.2) MPa/s. The compressive strength of a series of specimens was calculated as the arithmetic mean of three specimens, the two with the highest strength. The calculation was performed using Formula (2) with an accuracy of 0.1 MPa.

$$R=\alpha {\displaystyle \frac{F}{A}}{K}_w$$

(3)

where F is the breaking load (N);

A is the cross-sectional area of the specimen (mm²);

α is a coefficient accounting for the specimen dimensions (for specimens with a side length of 100 mm, α = 0.95).

K_w is a correction factor accounting for the moisture content of the experimental specimens at the time of testing, which was 0%, K_w = 0.8.

Stage 5. The production of polystyrene concrete and a series of tests for thermal conductivity, density and compressive strength were carried out in accordance with the methods described in stage 4. Figure 5 shows photographs of the polystyrene concrete tests.

The polystyrene concrete specimen containing 3–5 mm granules, shown in Figure 5a, exhibits the following failure pattern. The specimen exhibits delamination on the side faces and partial fracture of the top face. A crack extends through the central portion of the specimen. The failure pattern of the polystyrene concrete specimen containing 1–2 mm granules, shown in Figure 5b, is different. A significant fracture is observed, with the specimen disintegrating into five large fragments and numerous smaller fragments.

Stage 6. Modeling polystyrene concrete in Materials Designer to determine elastic moduli, density, and thermal conductivity. In this stage, similar to Stage 2, an REV element of a representative volume was generated with the properties obtained numerically in Stage 3 for foam concrete.

Stage 7. At this stage, computer modeling of polystyrene concrete for compressive strength was performed. The data obtained in Stage 6 were used as matrix properties. Figure 6 shows the geometric model of polystyrene concrete and its FEM discretization.

The problem shown in Figure 6 was numerically calculated under the following boundary conditions. On the upper face of the cube, the displacement of all nodes belonging to this plane, $UY=-6.0\times {10}^{-5}$ m, $UX=UZ=0$ m was applied. On the lower face, the displacement of all nodes $UY=0,\hspace{0.17em}\hspace{0.17em}UX\hspace{0.17em}and\hspace{0.17em}UZ\hspace{0.17em}free.\hspace{0.17em}$ The choice of boundary conditions is determined by the conditions of compression tests on cubic specimens. Along the lower boundary, horizontal deformations are limited by frictional forces, while vertical deformations are limited by a fixed support. A displacement corresponding to the press platen displacement is applied to the upper boundary of the cubic specimen.

3. Results

3.1. Calibration of the Parameters of the Menetrey–Willam Model

This paper examines the Menetrey–Willam model, in which the yield surface represents a nonlinear stress–strain relationship with increasing load and exponential softening after the plastic potential function $\Omega$ reaches a critical value. According to this model, the increment in total strain can be decomposed into elastic and plastic components.

$$d\epsilon =d{\epsilon }^{el}+d{\epsilon }^{pl}$$

(4)

Elastic deformations are related to stresses based on the generalized Hooke’s law

$$d{\epsilon }^{el}={\left[D\right]}^{-1}d\sigma$$

(5)

The increment of plastic deformations obeys the law of non-associated flow, in which the increment of plastic deformations is associated with the plastic potential $\Omega$

$$d{\epsilon }^{pl}=d\zeta {\displaystyle \frac{\partial \Omega }{\partial \sigma }}$$

(6)

here $\sigma \hspace{0.17em}$ is the stress tensor, $\zeta \hspace{0.17em}$ is a plastic parameter, and $\Omega$ is a plastic potential function.

The compression hardening and exponential softening functions are determined by the relations [36,37]

$${\Omega }_c={\Omega }_{ci}+\left(1-{\Omega }_{ci}\right)\hspace{0.17em}\sqrt{2{\displaystyle \frac{{\epsilon }_c}{{\epsilon }_{c1}^{pl}}}-{\displaystyle \frac{{\epsilon }_c^2}{{\epsilon }_{c1}^{pl\hspace{0.17em}2}}}}$$

(7)

here ${\Omega }_c\hspace{0.17em}$ is the compression hardening/softening function, ${\Omega }_{ci}\hspace{0.17em}$ is the stress corresponding to the transition from elastic to plastic deformations, ${\epsilon }_c\hspace{0.17em}$ is the strain, ${\epsilon }_{c1}^{pl}\hspace{0.17em}$ is the plastic strain corresponding to the peak stress, ${\Omega }_c={\sigma }_c/f_c\hspace{0.17em}$, where $f_c\hspace{0.17em}$ is the compressive strength under uniaxial compression. The branch that represents material softening has two segments:

$$\left\{\begin{array}{l}{\Omega }_c=1-\left(1-{\Omega }_{cu}\right)\hspace{0.17em}{\left({\displaystyle \frac{{\epsilon }_c-{\epsilon }_{c1}^{pl}}{{\epsilon }_{c,\lim }^{pl}-{\epsilon }_{c1}^{pl}}}\right)}^2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_{c1}^{pl}<{\epsilon }_c<{\epsilon }_{c,\lim }^{pl}\\ {\Omega }_c={\Omega }_{cr}+\left({\Omega }_{cu}-{\Omega }_{cr}\right)\hspace{0.17em}\exp \left(2{\displaystyle \frac{{\Omega }_{cu}-1}{{\epsilon }_{c,\lim }^{pl}-{\epsilon }_{c,1}^{pl}}}\hspace{0.17em}{\displaystyle \frac{{\epsilon }_c-{\epsilon }_{c,\lim }^{pl}}{{\Omega }_{cu}-{\Omega }_{cr}}}\right),\hspace{0.17em}\hspace{0.17em}{\epsilon }_c>{\epsilon }_{c,\lim }^{pl}\end{array}\right.$$

(8)

here ${\epsilon }_{c,\lim }^{pl}\hspace{0.17em}$ is the plastic strain at the transition point from power-law to exponential softening; ${\Omega }_{cu}\hspace{0.17em}$ is the residual stress at the transition point to material softening.

As a series of experiments has shown, foam concrete and polystyrene concrete behave linearly at approximately 33% of the ultimate load. Therefore, the stress corresponding to the transition from elastic to plastic deformations is calculated as ${\Omega }_{ci}\hspace{0.17em}$ = 0.33. The remaining parameters are determined in accordance with recommendations [45,46] and are listed in

Table 2. Parameters of the Menetrey–Willam model.

Title	Units	Value
Uniaxial compressive strength	MPa	From 9 to 47.1
Uniaxial tensile strength	MPa	From 1 to 4.2
Biaxial compressive strength	MPa	From 16 to 54
Dilatancy angle	degree	9
Plastic strength at uniaxial compressive strength	-	0.0012667
Plastic strain at transition from power law to exponential softening	-	0.0025067
Relative stress at the start of nonlinear softening	-	0.33
Residual relative stress at transition from power law to exponential softening	-	0.85
Residual compressive relative stress	-	0.2
Mode 1 area specific fracture energy	N/m	100
Residual tensile relative stress	-	0.1

.

In general, it should be noted that the Menetrey–Willam model behaves quite stable when implemented in numerical algorithms and gives results close to the experiment.

3.2. Thermal Conductivity Properties of Polystyrene Concrete Analysis

The material properties are largely determined by the porosity of foam concrete and polystyrene concrete. The porous structure of foam concrete is the result of adding ready-to-use technical foam to the mixture, which creates pores—cells. The addition of polystyrene creates a composite two-phase structure, which further reduces thermal conductivity. The porosity of foam concrete and polystyrene concrete is determined by the spatial arrangement of the pores (polystyrene spheres), the pore size distribution, the maximum and average pore size, their shape, and the thickness of the inter-pore partitions. Foam concrete’s characteristics, like how it conducts heat, its strength, and how it absorbs moisture, are influenced by its porosity. The heterogeneous properties of polystyrene concrete preclude an analytical solution, so numerical methods are more accurate [47].

The proportion and size of pores in the concrete matrix, and therefore the average density, primarily determine the thermal conductivity of foam concrete. This is explained by the fact that the material of the walls forming the pores consists of cement stone or a similar hydrosilicate framework. Modeling the structure of polystyrene concrete to determine its thermal properties is of interest. Resistance to moisture absorption is improved, since the closed-cell configuration promotes the resistance of the foam concrete mass to moisture absorption.

However, porosity can also have negative consequences: for example, the porosity of the structure makes polystyrene concrete brittle, especially at the edges of structures, and the strength of the material is unstable. Therefore, it is important to evaluate these characteristics using computer modeling.

Table 3. Physical Properties of Polystyrene Concrete.

Initial Density of Foam Concrete, kg/m3	Polystyrene Particle Volume Fraction	Average Particle Diameter, µm	Thermal Conductivity of Polystyrene Concrete, W/(m × °C)	Polystyrene Concrete Density, kg/m3
1187	0.3	5000	0.208	822
1187	0.329	5000	0.2	800
1187	0.406	4495	0.177	710
1187	0.415	3600	0.174	695
1187	0.469	4063	0.161	635

presents the findings from the ANSYS Materials Designer module, specifically regarding the density and thermal conductivity calculations for polystyrene concrete.

According to Table 3 (Figure 7), the density and thermal conductivity of foam concrete have a nearly linear relationship with the proportion of polystyrene.

The dependence presented in Figure 7 shows that the density and thermal conductivity of polystyrene concrete decrease as the volume fraction of polystyrene increases. As the volume fraction of polystyrene increases in the cement matrix structure, the number of macropores increases and the thickness of the interpore walls decreases. These structural changes are significant and affect the physical and strength properties of the resulting composite. The diameter of the polystyrene granules also affects the structure and properties of polystyrene concrete. Polystyrene granules with a diameter of 1–2 mm are distributed more evenly within the composite structure than granules with a diameter of 3–5 mm. This is primarily due to the specifics of the polystyrene concrete mixture preparation process. Granules with a smaller diameter are uniformly coated with cement paste, and the mixture undergoes less stratification during preparation.

These relationships are well approximated by a linear function with a determination coefficient of R².

$$\lambda =0.327-0.467\hspace{0.17em}x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.99$$

(9)

$$d_{pps}=1071.5-\hspace{0.17em}621.6\hspace{0.17em}x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.997$$

(10)

here $\lambda \hspace{0.17em}$ is the thermal conductivity of polystyrene concrete, W/(m × °C),

$d_{pps}\hspace{0.17em}$ is the density of polystyrene concrete, kg/m³

Figure 8 presents the relationship between the density and thermal conductivity of polystyrene concrete.

An analysis of Figure 8 shows that the thermal conductivity coefficient increases with increasing polystyrene concrete density. Polystyrene granules have a low density and act as a lightweight, porous filler in the composite. Therefore, decreasing the volumetric content of polystyrene granules reduces the number of pores that slow heat transfer, resulting in faster heat flow through the composite and an increased thermal conductivity coefficient.

A second-degree polynomial effectively models the relationship in Figure 8.

$$\lambda =0.170-0.000217\hspace{0.17em}{d}_{pps}+3.191\hspace{0.17em}{d}_{pps}^2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.999$$

(11)

here $\lambda$ is the thermal conductivity of polystyrene concrete, W/(m × °C),

$d_{pps}\hspace{0.17em}$ is the density of polystyrene concrete, kg/m³

3.3. Analysis of Strength Properties of Polystyrene Concrete

The strength of polystyrene concrete decreases; the more pores and the thinner the interpore walls of the cement stone, the lower the average density and, consequently, the compressive strength of the polystyrene concrete. To analyze the strength properties of polystyrene concrete, the RVE-generated volume was subjected to a load in the form of displacement of the upper face of the cube. Analysis of the stress–strain state revealed that the volume fraction of polystyrene particles plays a crucial role in forming the strength properties.

The sequential development of von Mises stresses is shown in Figure 9.

Figure 9 shows that the loss of strength in polystyrene concrete depends on the volume fraction of polystyrene (the spherical blue shapes are visible in the figures). The strength of the cement matrix, which includes spherical granules of expanded polystyrene, determines the strength characteristics of polystyrene concrete. Polystyrene granules are necessary to create a uniform cellular structure, which ensures strength. The polystyrene itself does not support loads; it only forms macro-voids. The failure pattern largely corresponds to experimental data (Figure 10).

Comparing the nature of the destruction of polystyrene concrete, presented by means of a model and a real photograph reflecting the process of destruction of a cube sample with dimensions of 100 × 100 × 100 mm, it can be noted that the nature of the destruction of the experimental sample with delamination of the side and upper face coincides with the image presented by means of numerical analysis.

The sequential development of equivalent plastic strains (cracks) at various stages of deformation and destruction of polystyrene foam is shown in Figure 11.

A comparison of the stress–strain curves obtained experimentally during compression testing of polystyrene concrete and as a result of numerical modeling is shown in Figure 12.

The applicability and adequacy of the Menetrey–William model are illustrated in Figure 12, which shows that during the active loading phase (the ascending branch of the stress–strain curve), the Menetrey–William model satisfactorily describes the nature of the stress–strain state. It should be noted that the Menetrey–William model is only applicable to the concrete matrix. Then, when porous concrete is modeled in stage 2, the curve changes, since porous concrete is a two-phase composite. In stage 6, polystyrene is introduced into this porous concrete, and the stress–strain curve changes again. This sequential modeling alters the characteristics of the stress–strain curve and the final curve. Figure 12 demonstrates that the modeling sequence satisfactorily reflects the experimental data.

As can be seen in Figure 11a,b, the development of plastic deformations and, consequently, cracks in the matrix occurs along the cross-sectional zones weakened by the polystyrene spheres. The partitions separating the polystyrene granules are destroyed. Subsequently, after the stress peak (Figure 12), the spheres collapse, and an avalanche-like failure of the matrix occurs (blue curve). At this stage (the descending branch), the numerical model qualitatively reflects the failure mode. Quantitatively, a significant deviation from the experiment is observed. However, for problems related to the modeling of polystyrene concrete, the ascending branch does not play a significant role. The descending branch is significant for reinforced concrete structures reinforced with bars made of steel or other composite materials, when the reinforcing bars bear the load after the concrete’s bearing capacity has been exhausted. In the case of polystyrene concrete, the softening curve is more brittle, and therefore, this stage should be avoided in the design.

Currently, the Menetrey–William, Drucker–Prager, and Willam–Warnke nonlinear concrete deformation models are frequently used. They differ in their description of material behavior and scope of application. The Menetrey–William model has been experimentally verified and certified for use in ANSIS. It is based on the theory of plastic flow and describes concrete properties such as different tensile and compressive strengths, nonlinear hardening, softening, and dilatancy. The constitutive relations depend on three stress tensor invariants, hardening and softening functions, and the plastic potential function. The Menetrey–William model is used for nonlinear calculations of reinforced concrete structures.

The Drucker–Prager model is based on parabolic strength surfaces constructed in the principal stress space. In the principal stress space, the surface is represented by a cone, open at the base.

The apex of the cone is located in the region of three tensile principal stresses. For concrete, the classical Drucker–Prager strength condition correlates poorly with experimental data. Specifically, under uniform triaxial tension, the material’s strength is overestimated, while under non-uniform triaxial compression, there are also regions where the strength is overestimated. The Drucker–Prager model is used for nonlinear analysis of bending and tension elements. However, to correctly describe the behavior of concrete, a surface dependent on the third invariant of the stress tensor deviator is required, as concrete is not circular in its deviatoric section.

The Willam–Warnke model is based on a five-parameter failure surface model. The Willam–Warnke model is used to analyze reinforced concrete structures under volumetric stress conditions, taking into account both brittle fracture and elastic-plastic deformation of concrete. Determining the parameters of the Willam–Warnke model requires experimental data, which presents technical difficulties. For example, to use the Willam–Warnke criterion, five types of tests are required: axial tension and compression, biaxial compression, and strength values at two different stress states corresponding to a higher level of hydrostatic compression.

Parameterized modeling and sensitivity analysis of the transition zone at the phase boundary (TZB) revealed that failure occurs along the partitions in close proximity to the phase boundary (Figure 13a,b).

It is evident that polystyrene, due to its significantly lower elastic modulus and strength, offers virtually no resistance, and the primary load-bearing capacity is determined by the concrete matrix (Figure 13a,b).

The dependence of the specimen load on the displacement of the upper face of the press is shown in Figure 14.

Polystyrene (granulated expanded polystyrene) influences the density of polystyrene concrete due to its low particle bulk density. This allows for a wide range of material density variations, which is important for the various applications of polystyrene concrete. Polystyrene particles create the porous structure of the cement stone. This ensures a relatively low mass and the flexibility to select a density range that meets the requirements of a specific application.

The numerical calculation allowed us to compare the strength of polystyrene concrete with its density and polystyrene content. Figure 15 shows the resulting dependence of the strength of polystyrene concrete on its density.

Figure 15 shows that as the composite density increases, so does its compressive strength. Reducing the number of polystyrene granules in the cement matrix structure leads to a decrease in the total volume of macropores formed by these granules, an increase in the thickness of the interpore walls, and, consequently, an increase in the composite density.

The dependence of the compressive strength of polystyrene concrete on its density, shown in Figure 15, is well approximated by a second-degree polynomial.

$$R_c=2.586-0.001135\hspace{0.17em}{d}_{pps}+2.3367\times {10}^{-6}\hspace{0.17em}{d}_{pps}^2,\hspace{0.17em}\hspace{0.17em}{R}^2=0.976$$

(12)

Here $d_{pps}\hspace{0.17em}$ is the density of polystyrene concrete, and R² is the coefficient of determination.

Figure 16 shows the calculated dependence of concrete strength on the volume fraction of polystyrene in the total concrete volume.

A statistical analysis of the regression equations was conducted. To test the significance of the regression equation, the calculated Fisher’s criterion F = 144.0 was compared with the tabulated value, taken for the number of degrees of freedom at the selected significance level of 0.05. The calculated Fisher’s exact test was significantly higher than the tabulated value. This indicates that the explained variance is significantly greater than the unexplained variance, and the model is significant. An analysis of the residuals revealed that the residuals are normally distributed with a mean of zero, indicating an unbiased regression estimate. The significance of the regression coefficients was assessed using Student’s t-test. The resulting t-statistic t = 42.4 significantly exceeds the tabulated value at the 95% confidence level, demonstrating the significance of the calculated regression coefficients.

An analysis of Figure 16 shows that the change in compressive strength of polystyrene concrete depends linearly on the volume fraction of polystyrene. As the volume fraction of polystyrene increases, the strength of the composite decreases. Polystyrene granules in the cement matrix act as a porous filler, creating a uniform cellular structure. Polystyrene granules do not absorb the forces and stresses generated in the composite and are more focused on stress relaxation. The interface between polystyrene granules and the cement matrix plays a crucial role in ensuring the strength properties of the composite. The hydrophobic surface of polystyrene granules is poorly wetted, resulting in poor adhesion to the cement paste, and the resulting adhesive contacts have low strength. In this study, the problem of poor wettability of polystyrene granules is addressed by mixing them in water with a dissolved SDO-MK additive. The presence of this additive allows for the formation of adsorbent layers in the contact zone between the hydrophobic surface of polystyrene granules and the surface of Portland cement particles and their hydration products, which improve the adhesion of granules to the cement paste [48].

A linear function (13) closely approximates the relationship between the polystyrene volume fraction and compressive strength of polystyrene concrete, as seen in Figure 15.

$$R_c=4.105-2.773\hspace{0.17em}x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.979$$

(13)

where x is the volume fraction of polystyrene particles.

4. Discussion

4.1. Dependence of Thermal Conductivity of Polystyrene Concrete on Its Physical Characteristics

Current literature notes the absence of a direct linear relationship between thermal conductivity and the volume fraction of polystyrene in polystyrene concrete. It is logical to assume that the higher the proportion of polystyrene granules, the lower the thermal conductivity. However, research shows that the thermal conductivity of polystyrene concrete depends on the mixture composition, manufacturing technology, and the uniformity of the polystyrene granule distribution. Structural features of polystyrene concrete can lead to deviations from this linear relationship. Polystyrene concrete contains a layer of gas bubbles in the matrix, which acts as a barrier to heat transfer, and its low density also reduces thermal conductivity. These factors produce a synergistic effect that prevents the establishment of a precise linear relationship between thermal conductivity and the proportion of polystyrene in polystyrene concrete. However, with minor air pores in the matrix, such a linear relationship exists (Figure 7, Equation (7)), as demonstrated by this study.

However, experimental data on the dependence of thermal conductivity on the composition of polystyrene concrete are contradictory, as researchers have used different methods. A clear mathematical mechanism for the thermal properties of polystyrene concrete calculation, which are largely dependent on the composition of the material, has not been developed.

The presence of a gas bubble layer in the material structure introduces certain nonlinear adjustments. Gas bubbles reduce thermal conductivity by acting as a barrier to heat transfer and further contribute to a decrease in thermal conductivity. This study showed that when inert polystyrene is mechanically mixed with a cement matrix, the density decrease is proportional to the volume fraction of polystyrene spheres. This proportionality is close to linear. This, in turn, is related to thermal conductivity. The nonlinear regression is clearly evident in the dependence of compressive strength. Since the cross-sectional area resisting external load decreases quadratically with the addition of polystyrene spheres, it is logical to assume that the dependence of strength on density will also be quadratic.

In [49], it is confirmed that thermal conductivity and porosity have a near-linear relationship, which suggests a closed pore structure that computed tomography showed.

In [50], the authors discovered a nearly linear correlation between polystyrene concrete’s thermal conductivity and density. The authors identified two ranges of polystyrene concrete density in which the relationship is close to linear. Nonlinear effects were detected in the range of 400–850 kg/m³, which is in good agreement with the data obtained in this study. The authors of [51,52] studied the influence of density, water-cement ratio, polystyrene content, glass powder, sand, and silica on the thermal characteristics of polystyrene granulated concrete. A model of the thermal conductivity of polystyrene granulated concrete under various external loads was also obtained using the principal component analysis method. A positive linear correlation between concrete density and granulated polystyrene and its thermal conductivity was shown by the authors. These data also agree well with the results obtained in this article. Based on Graph 8, the thermal conductivity’s relationship with density appears nearly linear.

4.2. Dependence of the Compressive Strength of Polystyrene Concrete on Its Physical Properties

In expanded polystyrene concrete, there is a positive correlation between compressive strength and dry density. With increasing density, compressive strength usually increases proportionally or exponentially. In [47], the authors showed that the dependence of strength on density has a power-law character. However, the authors examined the density range from 1170 to 2000 kg/m³, and the excessive dispersion of the results does not allow us to judge the high accuracy of the regression. In an earlier study [53], a power-law dependence with an exponent of 1.918 was proposed, which is very close to the results of this article (Figure 15, Equation (12)).

In [50], it is also evident that in the range of 400 … 1200 kg/m³, the dependence of strength on density has a parabolic shape, which is also consistent with the results of this article. The dynamics of the main relationships are determined by the fact that at higher density, a more compact cement matrix with fewer voids is formed, which leads to significantly greater strength.

The effect of expanded polystyrene (EPS) content is determined by the fact that expanded polystyrene (EPS) granules have insignificant strength; therefore, increasing the EPS volume to reduce density directly reduces the overall compressive strength of the material. In expanded polystyrene concrete, the relationship between compressive strength and the volume fraction of polystyrene is characterized by an inverse decrease [54]. Studies conducted in [55,56] show that compressive strength after 28 days follows a nearly linear decrease function relative to the volume of polystyrene granules. This is in good agreement with the results of the present study (Figure 16, Equation (13)). An exponential dependence of strength on the EPS volume fraction was observed at a low water-cement ratio (W/C = 0.45). The strength reduction mechanism is based on the development of stresses from internal structural defects in the form of spherical polystyrene inclusions. Polystyrene granules act as “weakness zones” or artificial voids, reducing the effective load-bearing cross-sectional area of the concrete. Furthermore, the hydrophobic nature of polystyrene leads to weak interfacial bonds between the granules and the cement matrix, further reducing strength. At very high-volume fractions (e.g., 80–100%), compressive strength can drop from the standard level (approximately 40 MPa) to 1.0 MPa.

The limitations of polystyrene concrete’s application are primarily related to its low compressive strength, high flammability, and low vapor permeability. Therefore, when designing buildings and structures, it is important to consider their actual properties, which will determine their suitability for specific applications. It should be noted that the polystyrene concretes developed in this study have low compressive strengths of 3.4 MPa to 4.3 MPa, which is typical for structural thermal insulation materials. Therefore, this type of composite can be used as a standalone building material in the construction of external and internal building walls, in combination with other finishing materials, for creating partitions, and as a thermal insulation material for façade cladding.

This study developed a mathematical model for the dependence of the thermos-physical, deformation, and strength properties of polystyrene concrete on the content of expanded polystyrene in the composite structure. This model enables numerical analysis to determine the optimal ratio of polystyrene in the composite structure. The description of the correlation laws between the polystyrene particle content and the properties of polystyrene concrete proposed in the manuscript is expanded.

Future research plans include testing this model on an expanded set of experimental data. New, improved polystyrene concrete compositions modified with various types of chemical and mineral additives and nanoadditives will be developed. Their physical, mechanical, and durability properties will be determined. Furthermore, new mechanisms for the influence of formulation solutions and aggressive operating conditions on the properties of polystyrene concrete will be identified.

5. Conclusions

This article presents a numerical simulation and experimental study of polystyrene concrete to develop a mathematical model for the dependence of its thermos-physical, deformation, and strength properties on the EPS content in the composite structure.

The following results were obtained:

(1) A FEM model of polystyrene concrete as a multiphase material, including a porous matrix and polystyrene particles, was developed, and the thermos-physical and strength characteristics of the resulting composite were studied based on this model. Mathematical modeling of the composite comprises two stages. In the first stage, the effective properties of foam concrete as a composite of concrete and air pores are modeled. In the second stage, the obtained effective properties of foam concrete are used to model polystyrene concrete directly, as a composite of foam concrete and inclusions of polystyrene particles.

(2) Verification of the model by comparison with experimental data showed that the model adequately reflects the characteristics of polystyrene concrete for assessing its thermo-physical properties. For modeling compressive strength properties, the model satisfactorily captures the nature of the stress–strain state up to the ultimate strength (ascending branch of the stress–strain curve). However, when modeling properties under compression (descending branch), the model qualitatively reflects stress decay but quantitatively introduces significant errors.

(3) Based on the mathematical model and experimental data, regularities linking thermal conductivity to the polystyrene granule content were identified.

(4) Models of the stress–strain state of polystyrene concrete as a three-phase material were obtained using the Menetrey–Willam model. The resulting models allowed us to identify the dependence of compressive strength on the polystyrene granule content and the initial porosity of the cement matrix.

(5) An analysis of the stress–strain state of polystyrene concrete under compressive stresses was conducted, and crack formation sites limiting the material’s strength were identified.