**1. Introduction**

Concrete is widely used in engineering, and these concrete engineering facilities are often subjected to various effects, such as earthquakes, weapon strike explosions, and engineering blasting. There are usually spherical wave problems such as point explosion and point impact. Then, it is necessary to deal with the propagation of spherical waves in concrete [1–3]. The dynamic response or spherical wave propagation in concrete under spherical impact completely depends on the dynamic properties of the concrete. Therefore, it is important to study the dynamic properties of concrete under a high strain rate, which has attracted the attention of many researchers [4–6]. Bischoff et al. [4] review experimental techniques commonly used for high strain rate testing of concrete in compression and characteristics of the dynamic compressive strength and deformation behavior. Malvar and Ross [5] undertake a literature review to characterize the effects of strain rate on the tensile strength of concrete. Cusatis [6] presents a previously developed meso-scale model for concrete, including the effect of loading rate, and the rate dependence of concrete behavior is assumed to be caused by two different physical mechanisms. Some studies [7–10] indicated that the different strain-rate sensitivity is determined in concrete under different

**Citation:** Lai, H.; Wang, Z.; Yang, L.; Wang, L.; Zhou, F. Determining Dynamic Mechanical Properties for Elastic Concrete Material Based on the Inversion of Spherical Wave. *Materials* **2022**, *15*, 8181. https:// doi.org/10.3390/ma15228181

Academic Editor: Sérgio Manuel Rodrigues Lopes

Received: 13 September 2022 Accepted: 25 October 2022 Published: 17 November 2022

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strain rates. Al-Salloum et al. [7] studied the dynamic behavior of concrete experimentally by testing annular and solid concrete specimens using a split Hopkinson pressure bar (SHPB). Wang et al. [8] designed a large-diameter SHPB with a diameter of 100 mm used to carry out impact tests at different speeds. The results show that the increase in the strain rate has a hindering effect on the increase in damage variables and the increase rate (impact speeds of 5 m/s, 10 m/s, and 15 m/s). Wang et al. [9] provided guidance for selecting pulse shapers for concrete SHPB experiments. Grote et al. [10] applied SHPB and plate impact to achieve a range of loading rates and hydrostatic pressures.

Meanwhile, researchers have carried out many studies on rate-dependent materials of spherical waves [11–16]. Luk et al. [11] developed models for the dynamic expansion of spherical cavities from zero initial radii for elastic–plastic rate-independent materials with power-law strain hardening. Wegner et al. [12] presented a new formulation of the governing equations of spherical waves, in which the resulting system of five equations is treated as a strictly hyperbolic system of first-order hyperbolic partial differential equations, and the method of characteristics is adapted to obtain numerical solutions. Forrestal et al. [13] developed a spherical cavity-expansion penetration model for concrete targets, and predictions from the compressible penetration model are in good agreement with depth of penetration data. Lai et al. [14,15] used the ZWT linear and nonlinear visco-elastic constitutive model to set up the governing equations for linear and nonlinear visco-elastic spherical waves, and published numerical results using the characteristics method. Lu et al. [16] established the linear visco-elastic ZWT constitutive equation under a three-dimensional stress state by ignoring the relaxation effect of the low-frequency Maxwell element and the nonlinear spring element. The absorption and dispersion phenomena of the spherical wave propagation in the visco-elastic solid were analyzed. At present, with the development of experimental technology, researchers are interested in wave propagation technology (WPT) [17–20]. Zhu et al. [17] set up the error in the determination of dynamic stress–strain curve of rate-dependent brittle materials with the traditional SHPB techniques with either a three-wave method or a two-wave method, which is not accepted. Wang et al. [18] developed an experimental apparatus for spherical divergent wave propagation in solids. Liu et al. and Sollier et al. [19,20] completed a series of experiments to measure the shock initiation behavior using eleven embedded electromagnetic particle velocity gauges. The dynamic performance experiment of concrete is different from the quasi-static test. The behavior of materials under spherical impact cannot be separated from the analysis of spherical wave propagation (wave propagation effect). The core problem in carrying out this research is that the effects of wave propagation and strain rate are often coupled. When studying the dynamic constitutive relation of materials with high strain rates, the wave propagation effects in the experimental process, especially in the specimen, should not be ignored.

In order to solve the above-mentioned difficulties and deal with the coupling problem, people have developed WPT to study the dynamic properties of materials subjected to dynamic loads [21]. In various wave propagation analysis techniques, Lagrangian analysis has attracted the attention of many researchers [22–26], because there are no other preassumptions about the constitutive relation of the materials under study. In the case of spherical waves, the constitutive equation of spherical waves consists of two parts: the volumetric part and the distortional part [27]. The traditional Lagrange analysis of wave propagation is based on the conservation equations without any pre-assumption of material constitutive relation. However, when the radial particle velocity profiles are measured by velocity gauges at the Lagrangian coordinates ri (i = 1,2, ... ), it is still difficult to solve the other two unknowns from the two constitutive equations with unknown dynamic parameters (Equations (1a), (1b), and (2)), which is different from the rate-independent elastic problem for parameters of constitutive equations, which are constant. In the work outlined in this paper, a series of particle velocity wave profiles of concrete in the farfield or low-pressure region under spherical impact loading is measured. Then, based on the universal spherical wave conservation equation and the fact that the volumetric

part of constitutive relation satisfies linear elastic law, the Lagrangian inverse analysis of spherical wave problems and particle velocity history measurements (the inverse analysis) are carried out to obtain the numerical constitutive relation, expressed in the form of distortion. Furthermore, it is found that the rate-dependent characteristics of spherical wave distortion is different from the rate-independent case and therefore an appropriate rate-dependent constitutive model is chosen to describe this problem. Finally, the dynamic parameters in constitutive relation of concrete with high strain rates are obtained by the standard linear solid model.

#### **2. Materials and Methods**

#### *2.1. Theoretical Concepts of Spherical Waves in Concrete*

Many materials have significant rate correlation characteristics under the loading of short-duration explosion and impact [28–30]. Concrete materials also have relevant characteristics under short-history loading [31–33]. The fracture strain of concrete under a high strain rate is as low as a magnitude of 10−3, and the behavior of concrete under one-dimensional and multidimensional stress under static load also shows great differences. Therefore, the concrete can be regarded as a linear viscose-elastic material, not just a linear elastic material.

First, the description system of spherical wave propagation is established in the spherical coordinate system (Figure 1a). The governing equation system of a linear viscoseelasticity (Figure 1b) spherical wave is composed of two parts: the conservation Equations (1a) and (1b) and the constitutive Equations (2a) and (2b) (the volumetric part 2a and the distortional part 2b), representing the physical properties [34]. The linear viscose-elasticity is reflected in the distortion relation of the constitutive Equation (2b):

$$\frac{\partial \varepsilon\_r}{\partial t} = \frac{\partial v}{\partial r} \,\tag{1a}$$

$$\frac{\partial \varepsilon\_{\theta}}{\partial t} = \frac{v}{r'} \tag{1b}$$

$$\frac{\partial \sigma\_r}{\partial r} + \frac{2(\sigma\_r - \sigma\_\theta)}{r} = \rho\_0 \frac{\partial v}{\partial t'} \tag{1c}$$

**Figure 1.** Schemes of governing equations: (**a**) micro-element in spherical coordinate system; (**b**) the standard linear solid constitutive model.

The linear viscose-elastic constitutive equation in differential form based on the standard linear solid model can be effectively used to describe the dynamic constitutive properties of concrete (3a) [35], and Figure 1b shows how the model works.

$$2\frac{\partial \sigma\_r}{\partial t} + 2\frac{\partial \sigma\_\theta}{\partial t} - 3K\left(\frac{\partial \varepsilon\_r}{\partial t} + 2\frac{\partial \varepsilon\_\theta}{\partial t}\right) = 0,\tag{2a}$$

$$\frac{\partial \varepsilon\_r}{\partial t} - \frac{\partial \varepsilon\_\theta}{\partial t} = \frac{1}{2G} \left( \frac{\partial \sigma\_r}{\partial t} - \frac{\partial \sigma\_\theta}{\partial t} \right) + \frac{(\sigma\_r - \sigma\_\theta) - 2G\_a(\varepsilon\_r - \varepsilon\_\theta)}{2G\theta\_M} \tag{2b}$$

The relevant material parameters are characterized as a linear elastic response (3b), volume deformation (3c), linear bulk modulus (3d), linear Young's modulus (3e), and linear shear modulus (3f). According to conventional considerations, it is assumed that Poisson's ratio u is constant, and the elastic stage is independent of other strains and strain rates.

$$\frac{\partial \sigma}{\partial t} + \frac{\sigma}{\theta\_M} = (E\_a + E\_M) \frac{\partial \varepsilon}{\partial t} + \frac{E\_a \varepsilon}{\theta\_M} \tag{3a}$$

$$
\sigma = E\_d \varepsilon \tag{3b}
$$

$$
\Delta = \varepsilon\_r + \mathfrak{L}\varepsilon\_\theta \tag{3c}
$$

$$K = \frac{E}{\Im(1 - 2\nu)}\tag{3d}$$

$$E = E\_a + E\_M \tag{3e}$$

$$G = \frac{E}{2(1+\nu)},\tag{3f}$$

In this way, in order to describe the linear viscose-elastic spherical wave propagation problem, based on the standard linear solid constitutive relation, the governing equation reflecting the linear and high strain rate effect of materials is established.

## *2.2. Experimental Method*

In order to understand the propagation characteristics of spherical waves in concrete, an experimental method is developed to measure the particle velocity histories of spherical waves. The experiment adopts the electromagnetic method, and the sample is a cylinder with a diameter equal to the height. Because the arrangement of particle velocimeters have accurate representative characteristics, the method has strong advantages in studying the dynamic properties of multicomponent composites containing fillers, such as polymer–matrix composite materials, concrete, and rock in 3-D stress. In the spherical wave experiment, the characteristic size of the sample can be meters, which is more than ten times larger than the size of concrete coarse fillers, so that the information of wave histories can accurately reflect the wave propagation characteristics. A group of particle velocity waves v(ri,t) at different radii distance ri from the center of the sphere is measured by a series of embedded magneto-electric velocimeters.

In the experiment, a mini-charge is detonated in the center of a cylindrical concrete block with a diameter of 25 cm and a length of 25 cm, and a spherical impact is loaded by detonating an explosive with a weight of 0.1 g/0.8 g. The principle of the spherical particle velocity history device is shown in Figure 2 [36]. The experimental specimen consists of two equal-height cylinder parts. A series of concentric toroidal magneto-electric particle string gauges is arranged on the mating surface. Explosive charges are placed in the cavity at the center of the sample; the soft detonating cord for initiation is entered along the mini hole of the upper half of the sample, and the upper and lower parts are bonded with epoxy resin after the gauge and the explosive charge are placed. After initiation, the particle velocimeters move to cut the magnetic field to form voltage signals, and the particle velocity histories at a series of radii can be obtained from the calibration results.

**Figure 2.** Scheme for velocity history test device in concrete with mini-charge: (**a**) mini-charge with soft detonating cord and long cylindrical block of concrete; (**b**) experimental concept for spherical wave experiments.

#### *2.3. Inverse Method*

The particle velocities in spherical wave propagation are easy to measure, but other physical quantities are difficult to measure directly at the same time. In order to obtain accurate information about other physical quantities during spherical wave propagation, and then obtain the constitutive relation of materials, Lagrangian inverse analysis is a good alternative, which is based on conservation equations and does not make any assumptions [37–40]. Next, the "second type inverse problem" in mathematics is dealt with to determine the dynamic constitutive properties of concrete. In the study of spherical waves, when the particle velocities at a series of different Lagrangian coordinates ri are obtained, it is difficult to calculate other unknown quantities from the former (2a, 2b). So we developed a new spherical wave analysis method "NV+T0/SW" to deal with this problem [14].
