2.3.3. Solving for G and θ<sup>M</sup>

An advantage of "NV + T0/SW" is that there are no assumptions of the constitutive equation of materials, directly giving the stress–strain numerical relation. However, now there is a next step to take when the dynamic properties of materials are known. So, its description with a known standard linear solid constitutive model is provided in future work, and the dynamic shear modulus *G* and one Maxwell element material parameters *θ<sup>M</sup>* can be determined by the following method (6).

$$G = \frac{\sigma\_r - \sigma\_\theta}{2(\varepsilon\_r - \varepsilon\_\theta)}\tag{6}$$

$$\theta\_M = \frac{(\sigma\_r - \sigma\_\theta) - 2G\_d(\varepsilon\_r - \varepsilon\_\theta)}{2G\left(\frac{\partial \varepsilon\_r}{\partial t} - \frac{\partial \varepsilon\_\theta}{\partial t}\right) - \left(\frac{\partial \sigma\_r}{\partial t} - \frac{\partial \sigma\_\theta}{\partial t}\right)}\tag{7}$$

#### **3. Results**

#### *3.1. The Experimental Results*

Based on the experimental method as described in the previous section, the particle velocity profiles (Figure 3) in the low stress elastic region at the radii of 100.6 mm, 120.6 mm, 140.6 mm, 160 mm, and 180.6 mm are measured accurately in the semi-infinite space of concrete by using the mini-explosive ball and electromagnetic velocity measurement technology. Here, the radius of the mini-explosive ball is 5 mm, with an explosive equivalent of 1.00 g TNT. As shown in Figure 3b, the maximum particle velocity is lower than 4 m/s, and the experimental model is a one-dimensional spherical symmetry problem. At same time, the static mechanical property parameters of concrete can be easily measured, as shown in Table 1.

**Table 1.** Concrete static parameters for 'NV+T0/SW'.


## *3.2. The Inverse Numerical Results*

In the series of measured particle velocity histories shown in the Figure 4, the path-line is constructed from the initial zero value line. The path-line is divided into regions by the peak value. The analysis value of the path-line is interpolated at equal time intervals in each region to serve as the basis of the inversion analysis framework. With these path-line values covering the particle velocity field, the physical quantities of the spherical wave can easily be solved by the aforementioned method, i.e., "NV + T0/SW". Since the constitutive relation of materials is often described by volume deformation and shape deformation with multidimensional stress state, it is convenient to reflect the stress characteristics under 3-D stress. The results are expressed as spherical profiles of volumetric part and distortional part, such as stress histories σ<sup>r</sup> + 2σθ, strain histories ε<sup>r</sup> + 2εθ, stress histories σ<sup>r</sup> − σθ, and strain histories ε<sup>r</sup> − ε<sup>θ</sup> .The numerical results are shown in Figures 5 and 6, and the numerical constitutive relation, expressed in the form of volume and distortion, is shown in Figure 7. The volumetric constitutive relation satisfies linear elastic law with linear bulk modulus K, but the distortional constitutive relation does not. It is not difficult to find that the latter relation has an obvious rate effect.

**Figure 4.** The schemes of inversion analysis with path-line.

**Figure 5.** A comparison of positive and inverse results: (**a**) the volumetric part histories of stress σ<sup>r</sup> + 2σθ; (**b**) the volumetric part histories of strain ε<sup>r</sup> + 2εθ.

**Figure 6.** A comparison of positive and inverse results: (**a**) the distortional part histories of stress σ<sup>r</sup> − σθ; (**b**) the distortional part histories of strain ε<sup>r</sup> − εθ.

**Figure 7.** The numerical rate-dependent constitutive relation: (**a**) the volumetric relation of stress (σ<sup>r</sup> + 2σθ)/3 and strain ε<sup>r</sup> + 2ε<sup>θ</sup> stress σ<sup>r</sup> − σθ; (**b**) the distortional relation of stress σ<sup>r</sup> − σ<sup>θ</sup> and strain ε<sup>r</sup> − εθ.

#### *3.3. The Determination of Dynamic Parameters G and* θM

According to the above theory, the dynamic parameters G and θ<sup>M</sup> can be determined from Equations (6) and (7), and the concrete static parameters and the numerical distortion relations of stress σ<sup>r</sup> − σ<sup>θ</sup> and strain ε<sup>r</sup> − ε<sup>θ</sup> are taken as the known conditions using the inverse method. The concrete static parameters used in the inverse analysis are the results of our experimental research on concrete under one-dimensional stress, and ρ and *ν* are measured from concrete samples, as shown in Table 1.

Note that Equation (6) is suitable for the series numerical distortion relations with different strain rates at each radius, so the average value of dynamic parameters G can be calculated easily with Equation (6), as shown in Table 2. Similarly, the dynamic parameters θ<sup>M</sup> can be obtained through Equation (7), and the values of θ<sup>M</sup> are also listed in Table 2. The results show that the dynamic shear modulus G is larger than the static modulus Ga and decreases with the reducing of strain rate (Figure 7b). At the same time, the dynamic relaxation time θ<sup>M</sup> increases with a reducing strain rate and is in the magnitude range of 10−<sup>6</sup> s.


**Table 2.** Dynamic parameters by 'NV+T0/SW'.

#### **4. Discussion**

Firstly, a series of particle velocity histories of spherical waves in concrete is measured by magneto-electric velocimeters at each radius, which provides a basis for an experimental study on the dynamic properties of concrete in 3-D stress state under high strain rates. The particle velocimeter is a very thin ring coil, which is very suitable for measuring the physical quantities in spherical waves that change with the spherical radius. It is a good choice for measuring the signals of spherical waves for non-perspective materials, except for magnetic materials. Secondly, by analyzing the experimental data v(ri,t) of the spherical particle velocity wave of concrete, the Lagrangian "NV + T0/ SW" inverse analysis is carried out using the path-line method, and the wave propagation information of each physical quantity of the spherical wave is obtained. The numerical constitutive relation is expressed in the form of distortion and has an obvious rate effect. The results shown in Figure 7 demonstrate the obvious different behaviors of concrete between dynamic loading and static loading normally, and the strain rate effect of concrete cannot be ignored with the strain rate range of 102 1/s. The numerical constitutive relation is deduced directly from the measurements and analyses of wave propagation signals, which should be more appropriate for the coupled effects between wave propagation and rate dependency, are considered. Next, the rate-dependent dynamic parameters in concrete are determined by the standard linear solid model, which is a typical and useful model for analyzing stress relaxation and creep behaviors of viscoelastic solids. The results of dynamic parameters show that the dynamic shear modulus G is larger than the static modulus Ga and decreases with the reducing of strain rate (Figure 7b). Furthermore, the dynamic relaxation time θ<sup>M</sup> increases with reducing strain rate and is in the magnitude range of 10−<sup>6</sup> s.

#### **5. Conclusions**

The goal of this research was to expand the knowledge about the possibilities of studying rate-dependent constitutive relation and the determination of dynamic parameters based on spherical waves in concrete. According to the former, the main conclusions drawn from the above results are as follows:


It should be emphasized that, if more experimental data in the strain rate range and more continuous particle velocity profiles are measured through the improvement and development of experimental loading and data acquisition technology, the results obtained by this method will be enriched into a series. This method has good applicability, especially in the study of the dynamic behavior of multicomponent composite materials with largesize particle filler for the characteristic size of specimens in spherical wave experiments could be in the order of meters.

**Author Contributions:** Conceptualization, L.W. and F.Z.; Data curation, H.L.; Investigation, Z.W.; Methodology, H.L.; Project administration, Supervision L.Y.; Resources, Z.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (NSFC 11390361, 11172244), by K. C. Wong Magna Fund in Ningbo University.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
